Research ArticleConditional Analysis for Mixed Covariates withApplication to Feed Intake of Lactating Sows

S Y Park 1 C Li 2 S M Mendoza Benavides3 E van Heugten3 and A M Staicu2

1Eli Lilly and Company Indianapolis IN 46285 USA2Department of Statistics North Carolina State University Raleigh NC 27695 USA3Department of Animal Science North Carolina State University Raleigh NC 27695 USA

Correspondence should be addressed to S Y Park parksoyoung3859gmailcom and C Li cli9ncsuedu

Received 5 December 2018 Revised 28 April 2019 Accepted 28 May 2019 Published 16 July 2019

Guest Editor Rahim Alhamzawi

Copyright copy 2019 S Y Park et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose a novel modeling framework to study the effect of covariates of various types on the conditional distribution of theresponse The methodology accommodates flexible model structure allows for joint estimation of the quantiles at all levels andprovides a computationally efficient estimation algorithm Extensive numerical investigation confirms good performance of theproposedmethodThemethodology ismotivated by and applied to a lactating sow studywhere the primary interest is to understandhow the dynamic change of minute-by-minute temperature in the farrowing rooms within a day (functional covariate) is associatedwith low quantiles of feed intake of lactating sows while accounting for other sow-specific information (vector covariate)

1 Introduction

Many modern applications routinely collect data on studyparticipants comprising scalar responses and covariates ofvarious types vector function and image and the mainquestion of interest is to examine how the covariates affectthe response For example in our motivating experimentalstudy the goal is to analyze how the minute-by-minute dailytemperature and humidity of the farrowing rooms wheresows are placed after giving birth for nursing affect theirfeed intake during a lactation period The covariates consistof temperature profile humidity and sow age where theresponse is a total amount of daily feed intake of sows Apopular approach in these cases is to use a nonparametricframework and assume that the mixed covariates solely affectthe mean response see Cardot et al [1] James [2] Ramsayand Silverman [3 4] Ferraty and Vieu [5 6] Goldsmithet al [7] McLean et al [8] and others However for ourapplication while it is important to study the mean feedintake animal scientists are often more concerned with theleft tail of the feed intake distribution This is because lowfeed intake of lactating sows could lead to many seriousissues including decrease in milk production and negative

impact on the sows reproductive system see for referenceQuiniou and Noblet [9] Renaudeau and Noblet [10] St-Pierre et al [11] among others In this paper we focus onregression models that study the effects of covariates onthe entire distribution of response Our contribution is thedevelopment of a modeling framework that accommodates acomprehensive study of various types (vector and functional)of covariates on a scalar response

Quantile regression models the effect of scalarvectorcovariates beyond the mean response it provides a morecomprehensive study of the covariates on the response andhas attracted great interest [12 13] For prespecified quantilelevels quantile regression models the conditional quantilesof the response as a function of the observed covariatesthis approach has been extended more recently to ensurenoncrossing of quantile functions [14] Quantile regressionhas been also extended to handle functional covariatesCardot et al [15] discussed quantile regression models byemploying a smoothing spline modeling based approachKato [16] considered the same problem and used a functionalprincipal component (fPC) based approach Both papersmainly discussed the case of having a single functionalcovariate and it is not clear how to extend them to the

HindawiJournal of Probability and StatisticsVolume 2019 Article ID 3743762 14 pageshttpsdoiorg10115520193743762

2 Journal of Probability and Statistics

case where there are multiple functional covariates or mixedcovariates (vector and functional)

More recently Tang and Cheng [17] Lu et al [18] andYu et al [19] studied quantile regression when the covariatesare of mixed types and introduced the partial functionallinear quantile regression modeling frameworkThe first twopublications used fPC basis while the last one consideredpartial quantile regression (PQR) basis These approachesall are suitable when the interest is studying the effect ofcovariate at a particular quantile level and do not handle thestudy of covariate effects at simultaneous quantile levels dueto the well-known crossing-issue

Ferraty et al [20] and Chen and Muller [21] con-sidered a different perspective and studied the effect ofa functional predictor on the quantiles of the responseby modeling the conditional distribution of the responsedirectly However their approach is limited to one functionalpredictor In this paper we fill this gap and propose aunifying modeling framework and estimation technique thatallows studying the effect of mixed type covariates (iescalar vector and functional) on the conditional distri-bution of a scalar response in a computationally efficientmanner

Let 119876119884|119883(sdot)(120591) denote the 120591th conditional quantile of 119884given a functional covariate 119883(sdot) and let 119865119884|119883(sdot)(119910) denotethe conditional distribution of 119884 given 119883(sdot) We model theconditional distribution using a generalized function-on-function regression framework ie 119865119884|119883(sdot)(119910) = 119864[1(119884 le119910) | 119883(sdot)] = 119892minus1int119883(119905)120573(119905 119910)119889119905 where 1(sdot) is anindicator function and 119892 is the logit link function and westudy the conditional quantiles by exploiting the relationshipbetween 119876119884|119883(sdot)(120591) and 119865119884|119883(sdot)(119910) through 119876119884|119883)(sdot)(120591) =inf119910 119865119884|119883(sdot)(119910) ge 120591 for 0 lt 120591 lt 1 The advantageand contribution of our proposed method mainly comefrom the following reasons (1) our modeling approach isspline-based and as a result it can easily accommodatesmooth effects of scalar variables as well as of functionalcovariates and (2) our estimation approach is based ona single step function-on-function (or function-on-scalar)penalized regression which enables efficient implementationby exploiting off-the-shelf software and leads to competitivecomputations

The remainder of the paper is structured as followsSection 2 discusses the details of the proposed method andSection 3 describes the estimation procedure and extensionsSection 4 performs a thorough simulation study evaluatingthe performance of the proposedmethod and its competitorsWe apply the proposed method to analyze the sow datain Section 5 We conclude the paper with a discussion inSection 6

2 Methodology

21 Statistical Framework Let 119894 be index subjects 119895 indexrepeated measurements 119899 the number of subjects and 119898119894the number of observations for subject 119894 Suppose we observe119884119894X1198941 1198831198942 119905119894119895 1198831198943(119905119894119895)1le119895le1198981198941le119894le119899 where119884119894 is the responsewhich is a scalar random variable X1198941 is a 119901-dimensional

vector of nuisance covariates 1198831198942 is a scalar covariate and1198831198943(sdot) is a functional covariate which is assumed to be square-integrable on a closed domainT

We propose the following model for the conditionaldistribution of 119884119894 given X11989411198831198942 and1198831198943(sdot)

119865119884119894|X1198941 11988311989421198831198943(sdot) (119910) = 119864 1 (119884119894 le 119910) | X1198941 1198831198942 1198831198943 (sdot)

= 119892minus1 1205730 (119910) + X11987911989411205731 + 11988311989421205732 (119910)

+ int1198831198943 (119905) 1205733 (119905 119910) 119889119905 (1)

where 119865(sdot) denotes the conditional distribution function asbefore 119892(sdot) is a known monotone link function namelythe logit link function defined as 119892(119909) = log119909(1 minus 119909)for arbitrary scalar 119909 isin [0 1] 1205730(sdot) is an unknown andsmooth functional intercept1205731 is a119901-dimensional parametercapturing the linear additive effect of the covariate vectorX11989411205732(sdot) is an unknown and smooth function and 1205733(sdot sdot) is anunknown and smooth bivariate function Here the effect ofthe nuisance covariates X1198941 is 1205731 it is assumed to be constantover 119910 while the smooth intercept 1205730(119910) is 119910-variant Theeffect of 1198831198942 is 1205732(119910) which varies smoothly over 119910 1205733(sdot 119910)quantifies 119910-variant linear effects of the covariate 1198831198943(sdot) Ifthe parameter function 1205732(sdot) is zero then the covariate 1198831198942has no effect on the distribution of the response 119884119894 which isequivalent to 1198831198942 having no effect on any quantile level of 119884119894Similarly it is easy to see that a null effect say 1205733(sdot sdot) equiv 0is equivalent to the case that the functional covariate 1198831198943(sdot)has no effect on any quantile level of the response Chen andMuller [21] (CM henceforth) considered a similar modelhowever their approach is restrictive to a single functionalcovariate We discuss the differences between their methodand ours in Section 22

To explain our ideas we consider the case that thefunctional covariates are observed without noise on a fineregular and common grid of sampling points ie 119905119894119895 = 119905119895with 119895 = 1 119898 for all 119894 Bear in mind this assumptionis made for illustration only and our framework can beextended to more general cases including settings where thefunctional covariate is observed with noise and at irregularsampling points see Section 33

22 Modeling of the Covariate Effects We model 1205730(119910) and1205732(119910) by using prespecified truncated univariate basis Let11986101198890(sdot) 1198890 = 1 1205810 and 11986111198891(sdot) 1198891 = 1 1205811be two bases of dimensions 1205810 and 1205811 respectively 1205730(119910) asympsum12058101198890=1

11986101198890(119910)12057901198890 and 1205732(119910) asymp sum12058111198891=1

11986111198891(119910)12057911198891 where12057901198890 rsquos and 12057911198891 rsquos are unknown basis coefficientsWe represent1205733(119905 119910) using the tensor product of two univariate basesfunctions 11986111990521198892(119905) 1198892 = 1 1205812119905 and 119861119910211988910158402(119910) 119889

10158402 =

1 1205812119910 where 1205812119905 and 1205812119910 are the bases dimensions1205733(119905 119910) asymp sum1205812119905

1198892=1sum120581211991011988910158402=1

11986111990521198892(119905)119861119910

211988910158402(119910)12057921198892 11988910158402 where 12057921198892 11988910158402 rsquos

are unknown basis coefficients In practice the integrationterm int1198831198943(119905)1205733(119905 119910)119889119905 is approximated by Riemann integra-tion int1198831198943(119905)1205733(119905 119910)119889119905 asymp sum119898119895=11198831198943(119905119895)1205733(119905119895 119910)(119905119895+1 minus 119905119895) butother numerical approximation scheme can be also used

Journal of Probability and Statistics 3

Define 119885119894(119910) = 1(119884119894 le 119910) for 119910 isin R In practicefor each 119910 in a fine grid we view 119885119894(119910) as a binary-valued random functional variable It follows that model (1)can be written equivalently as a generalized function-on-function regression model through relating the ldquoartificialrdquobinary functional response 119885119894(119910) and the mixed covariatesX1198941 1198831198942 1198831198943(sdot)Thismodel can be fitted by using for examplethe ideas of Scheipl et al [22] which we briefly summarizenext

Model (1) can be represented as the following generalizedadditive model

119864 [119885119894 (119910) | X1198941 1198831198942 1198831198943 (sdot)] = 119892minus1 120578119894 (119910)

120578119894 (119910) =1205810

sum1198890=1

11986101198890 (119910) 12057901198890 + X11987911989411205731

+ 11988311989421205811

sum1198891=1

11986111198891 (119910) 12057911198891 +119898

sum119895=1

(119905119895+1 minus 119905119895)1198831198943 (119905119895)

sdot1205812119905

sum1198892=1

1205812119910

sum11988910158402=1

11986111990521198892 (119905119895) 119861119910

211988910158402(119910) 12057921198892 11988910158402

(2)

For convenience we use the notation 1198611198831198942 1198891(119910) =119883119894211986111198891(119910) 1198831198943(119905119895) = (119905119895+1 minus 119905119895)1198831198943(119905119895) We let B0(119910) =11986101(119910) 11986101205810(119910)119879 B1198941(119910) = 1198611198831198942 1(119910) 1198611198831198942 1205811(119910)119879B2119905(119905) = 11986111990521(119905) 11986111990521205812119905(119905)119879 B2119910(119910) = 11986111991021(119910) 11986111991021205812119910(119910)119879 1205790 = 12057901 12057901205810119879 1205791 = 12057911 12057911205811119879 andΘ2 = [12057921198892 11988910158402]1le1198892le1205812119905 1le11988910158402le1205812119910 is a coefficient matrix Then1205733(119905 119910) = B2119905(119905) otimesB2119910(119910)1198791205792 where 1205792 is the vectorizationof Θ2 We let X1198943(t) = 1198831198943(1199051) 1198831198943(119905119898)119879 isin R119898B2(t 119910) = B2119905(1199051) otimes B2119910(119910) B2119905(119905119898) otimes B2119910(119910)119879 isinR119898times12058121199051205812119910 and B1198942(119910) = B2(t 119910)119879X1198943(t) Now model (2) canbe written as

119864 [119885119894 (119910) | X1198941 1198831198942 1198831198943 (sdot)] = 119892minus1 X11987911989411205731 + B0 (119910)119879 1205790+ B1198941 (119910)119879 1205791 + B1198942 (119910)119879 1205792

(3)

The general idea is to set the bases dimensions 1205810 1205811 1205812119905and 1205812119910 to be sufficiently large to capture the complexity ofthe coefficient functions and control the smoothness of theestimator through some roughness penalties This approachof using roughness penalties has been widely used see forexample Eilers and Marx [23] Ruppert [24] Wood [25 26]among many others

It is important to emphasize that even in the case ofa single functional covariate our methodology differs from[21] in two directions (1) our proposed method is based onmodeling the unknown smooth coefficient functions usingprespecified basis function expansion and using penaltiesto control their roughness In contrast CM uses data-driven basis and chooses the number of basis functionsthrough the percentage of explained variance (PVE) of thefunctional predictors This key difference allows our methodto accommodate covariates of different types as well as

nonlinear effects (2) Our estimation approach is based on asingle step penalized function-on-function regression whileCM uses pointwise estimation based on functional principalcomponent bases and thus requires fitting multiple general-ized regressions This nice feature leads to an computationaladvantage

3 Estimation

31 Estimation via Penalized Log-Likelihood Let 119910ℓ ℓ =1 119871 be a set of equally spaced points in the range ofthe response variable 119884119894rsquos Conditioning on X1198941 1198831198942 1198831198943(sdot)we model 119885119894(119910ℓ) as independently distributed Bernoullivariables with mean 120583119894(119910ℓ) where 119892120583119894(119910ℓ) = 120578119894(119910ℓ) Thecoefficients 1205731 1205790 1205791 and 1205792 are estimated by minimizingthe penalized log-likelihood criterion

minus 2L (1205731 1205790 1205791 1205792 | 119885119894 (119910ℓ) forall119894 ℓ) + 12058201198750 (1205790)+ 12058211198751 (1205791) + 12058221199051198752119905 (1205792) + 12058221199101198752119910 (1205792)

(4)

where L is the log-likelihood function of data 119885119894(119910ℓ) ℓ =1 1198711le119894le119899 1205820 1205821 1205822119905 and 1205822119910 are smoothing parameterswhich control the balance between the model fit and itscomplexity and1198750(sdot)1198751(sdot)1198752119905(sdot) and1198752119910(sdot) are all penalties

There are several choices to define the penalty matrixin nonparametric regression see Eilers and Marx [23]and Wood [26] We use quadratic penalties which penal-ize the size of the curvature of the estimated coefficientfunctions Let 1198750(1205790) = int12059721205730(119910)12059711991022119889119910 = 1205791198790D01205790where D0 is of dimension 1205810 times 1205810 with its (119904 1199041015840) ele-ment equal to int12059721198610119904(119910)1205971199102120597211986101199041015840(119910)1205971199102119889119910 Similarly1198751(1205791) = int12059721205732(119910)12059711991022119889119910 = 1205791198791D11205791 where D1 isof dimension 1205811 times 1205811 with its (119904 1199041015840) element equal toint12059721198611119904(119910)1205971199102120597211986111199041015840(119910)1205971199102119889119910 As 1205733(sdot sdot) is a bivariatefunction the choice of penalty implies penalizing the sizeof curvature in each direction respectively 1198752119905(1205792) =int int12059721205733(119910 119905)12059711990522119889119910119889119905 = 1205791198792D21199051205792 where D2119905 = P2119905 otimesI1205812119910 is of dimension 12058121199101205812119905 times 12058121199101205812119905 with the (119904 1199041015840)element of P2119905 equal to int12059721198611199052119904(119905)1205971199052120597211986111990521199041015840(119905)1205971199052119889119905for some orthonormal spline bases Similarly 1198752119910(1205792) =int int12059721205733(119910 119905)12059711991022119889119910119889119905 = 1205791198792D21199101205792 where D2119910 = I1205812119905 otimesP2119910 is of dimension 12058121199101205812119905 times 12058121199101205812119905 with the (119904 1199041015840) elementof P2119910 equal to int12059721198611199102119904(119910)1205971199102120597211986111991021199041015840(119910)1205971199102119889119910

Criterion (4) can be viewed as a penalized quasilikelihood(PQL) of the corresponding generalized linear mixed model

119885119894 (119910ℓ) | 1205731 1205790 1205791 1205792 sim Bernoulli (120583119894 (119910ℓ)) ℓ = 1 119871

1205790 sim 119873(0 120582minus10 Dminus0 ) 1205791 sim 119873(0 120582minus11 Dminus1 ) 1205792 sim 119873 (0Dminus2 )

(5)

where Dminus0 is the generalized inverse matrix of D0 Dminus1 andDminus2 = (1205822119905D2119905 + 1205822119910D2119910)minus are defined similarly Wood [26]

4 Journal of Probability and Statistics

discusses an alternative way to deal with the rank-deficientmatrices in the context of restricted maximum likelihood(REML) estimation Here we do not account for the depen-dence over 119910 see Scheipl et al [22] for a general formulationSee also Ivanescu et al [27] who use the mixed modelrepresentation of a similar regression model to (5) but with aGaussian functional responseThe smoothing parameters areestimated using REML

32 Extension to Nonlinear Model One advantage of theproposed framework is that it can be easily extended toallow for more flexible effects ie extending the ideasto accommodate multiple covariates scalar or functionaland varied types of effects In particular the smooth effect11988311989421205732(119910) can be replaced by ℎ1(1198831198942 119910) and int1198831198943(119905)1205733(119905 119910)119889119905byint ℎ21198831198943(119905) 119905 119910119889119905 where ℎ1(sdot sdot) and ℎ2(sdot sdot sdot) are unknownbivariate and trivariate smooth functions respectively seeScheipl et al [22] and Kim et al [28] These changes requirelittle additional computational effort The modeling andestimation follow roughly similar ideas as Scheipl et al [22]We consider the nonlinear model in the simulation studyfor the case of having a scalar covariate only ie ℎ1(119883119894 119910)and the corresponding results are presented in Section S11of the Supplementary Materials The results show excellentprediction performance compared to the competitive nonlin-ear quantile regressionmethod namely constrained B-splinesmoothing (COBS) [29]

33 Extension to Sparse and Noisy Functional CovariatesIn practice the functional covariates are often observed atirregular times across the units and the measurements arepossibly corrupted by noises In such case one needs tofirst smooth and denoise the trajectories before fitting Whenthe sampling design of the functional covariate is dense thecommon approach is to smooth each trajectory using splinesor local polynomial smoothing as proposed in Ramsay andSilverman [3] and Zhang and Chen [30] When the designis sparse the smoothing can be done by pooling all thesubjects and following the PACE method proposed in Yaoet al [31] As recovering the trajectories has been extensivelydiscussed in the literatures we do not review the procedureshere Instead we discuss some available computing resourcesthat can be used to fit these methods In our numericalstudy we used fpcasc function in the refund R package[32] for recovering the latent trajectories irrespective of asampling design (dense or sparse) Alternatively one can usefpcaface [33] in refund for regular dense design andfacespare [34] in the R package face [35] for irregularsparse design Once the latent trajectories are estimated theycan be used in the fitting criterion (4)

34 Estimation of Conditional Quantile Let 1 0 1 and2 be parameter estimates in (5) It follows that the estimateddistribution function 119865119884119894|X1198941 11988311989421198831198943(sdot)(119910) can be obtained byplugging in the estimated coefficients The 120591th conditionalquantile is estimated by inverting the estimated distributionie 119876119884119894|X1198941 1198831198942 1198831198943(sdot)(120591) = inf119910 119865119884119894|X1198941 1198831198942 1198831198943(sdot)(119910) ge 120591 The

estimated distribution function is not a monotonic functionyet In practice we suggest to first apply a monotonizationmethod as described in Section 35 and then estimate theconditional quantiles by inverting the resulting estimateddistribution

35 Monotonization and Implementation While a condi-tional quantile function is nondecreasing the resulting esti-mated quantiles may not be Two approaches are widely usedone is to monotonize the estimated conditional distributionfunction and the other is to monotonize the estimatedconditional quantile function We choose the former as119865119884119894|X1198941 11988311989421198831198943(sdot)(119910) is readily available at dense grid points 119910ℓrsquosWeuse an isotonic regressionmodel [36] formonotonizationwhich imposes an order restriction this is done by using the Rfunction isoreg Other monotonization approaches includeChernozhukov et al [37] which was employed in Kato [16]

Our approach is implemented by first creating an artificialbinary response and then fitting a penalized function-on-function regression model and using the logit link functionFitting models in (4) can be done by extending the ideasof Ivanescu et al [27] for Gaussian functional responsethe extension of the model to the non-Gaussian functionalresponse has recently been studied and implemented byScheipl et al [22] as the pffr function in refund package[32]

4 Simulation Study

41 Simulation Setting In this section we evaluate theempirical performance of the proposed method We presentresults for the case when we have both functional and scalarcovariates additional results when there is only a singlescalar or a single functional covariate are discussed in theSupplementary Materials Section S1

Suppose the observed data for the 119894th subject are[119884119894 1198831119894 (1198821198941 1199051198941) (119882119894119898119894 119905119894119898119894)] 119905119894119895 isin [0 10] where

1198831119894 119894119894119889sim 119880119899119894119891(minus16 16)119882119894119895 = 1198832119894(119905119894119895) + 120598119894119895 for 1 le 119894 le 119899 1 le119895 le 119898119894 Let1198832119894(119905119894119895) = 120583(119905119894119895)+sum4119896=1 120585119894119896120601119896(119905119894119895)+120598119894119895 where 120583(119905) =119905 + sin(119905) 120601119896(119905) = cos(119896 + 1)12058711990510radic5 for odd values of 119896120601119896(119905) = sin11989612058711990510radic5 for even values of 119896 120585119894119896 119894119894119889sim 119873(0 120582119896)(1205821 1205822 1205823 1205824) = 16 9 756 506 and 120598119894119895 119894119894119889sim 119873(0 1205902120598 ) Weassume three cases for generating response 119884119894

(i) Gaussian119884119894 | 1198831119894 1198832119894(sdot) sim 119873(2 int1198832119894(119905)120573(119905)119889119905+2119883111989452) this corresponds to the quantile regressionmodel119876119884|1198831 1198832(sdot)(120591) = 2 int1198832119894(119905)120573(119905)119889119905 + 21198831119894 + 5Φminus1(120591)whereΦ(sdot) is the distribution function of the standardnormal

(ii) Mixture ofGaussians119884119894 | 11988311198941198832119894(sdot) sim 05119873(int1198832119894(119905)120573(119905)119889119905+119883111989412) + 05119873(3 int1198832119894(119905)120573(119905)119889119905+ 31198831119894 42) where thetrue quantiles can be approximated numerically byusing qnorMix function in the R package norMix

(iii) Gaussian with heterogeneous error 119884119894 | 11988311198941198832119894(sdot) sim 119873(2 int1198832119894(119905)120573(119905)119889119905 + 21198831119894 52 int1198832119894(119905)2119889119905

Journal of Probability and Statistics 5

sum4119896=1 120582119896) the true quantiles are given by119876119884|1198831 1198832(sdot)(120591) = 2 int1198832119894(119905)120573(119905)119889119905 + 21198831119894 +

5radicint1198832119894(119905)2119889119905sum4119896=1 120582119896Φminus1(120591)

Let 120573(119905) = sum4119896=1 120573119896120601119896(119905) where 120573119896 = 1 for 119896 = 1 4For each case we use different combinations of signal to

noise ratio (SNR) sample size and sampling designs to gen-erate 500 simulated datasets We define SNR asradicsum4119896=1 120582119896120590120598and we consider five levels of noise SNR = 150 10 5 2 1Two levels of sample size are 119899 = 100 and 119899 = 1000 Twosampling designs are considered (i) sparse design where 119905119894119895 119895 = 1 119898 are119898 = 15 randomly selected points from a setof 30 equispaced grids in [0 10] and (ii) dense design wherethe sampling points 119905119894119895 = 119905119895 119895 = 1 119898 are 119898 = 30equispaced time points in [0 10]

The performance is evaluated on a test set of 100 subjectsfor whichwe have 1198831119894lowast (119882119894lowast119895 119905119894lowast119895) 119895 = 1 119898 available interms of mean absolute error (MAE) for quantile levels 120591 =005 01 025 and 05

MAE (120591) = 1100100

sum119894lowast=1

10038161003816100381610038161003816119876119894lowast (120591) minus 119876119894lowast (120591)10038161003816100381610038161003816 (6)

42 CompetingMethods Wedenote the proposedmethod byJoint QR to emphasize the single step estimation approachWe compare our method with two alternative approaches(1) a variant of our proposed approach using pointwiseestimation denoted by Pointwise QRThis approach consistsof fittingmultiple regressionmodels with binomial link func-tion as implemented by the penalized functional regressionpfr developed byGoldsmith et al [7] of therefundpackagefor generalized scalar responses (2) a modified version ofthe CM method denoted by Mod CM that we developedto account for additional scalar covariates and which fitsmultiple generalized linear models with scalar covariates andfPC scores as predictors (3) a linear quantile regressionapproach using the quantile loss function and the partialquantile regression bases for functional covariates proposedby Yu et al [19] and denoted by PQRNotice that although theformulation of the first twomethods implicitly accounts for avarying effect of the covariates on the response distributionthey do not ensure that this effect is smooth The thirdapproach can only estimate a specific quantile rather than theentire conditional distribution Note that all the competingmethods are monotonized for a fair comparison

The R function pfr can incorporate both scalarvectorand functional predictors by adopting a mixed effects modelframework The functional covariates are presmoothed byfPC analysis [31] Throughout the simulation study we fixPVE as 095 for fPC analysis to determine the number ofprincipal components and use REML to select the smoothingparameters for our proposed methods Other basis settingsare set to their default values We use 100 equally distancedpoints between the minimum andmaximum of the observed119884119894rsquos to set the grid 119910ℓ ℓ = 1 119871 for the conditionaldistribution function

43 Simulation Results Tables 1 and 2 show the accuracyof the quantile prediction for the two cases (normal andmixture) when the functional covariate is observed sparselyand the sample size is 119899 = 100 (Table 1) and 119899 = 1000(Table 2) Table 3 presents the prediction accuracy for thecase of heteroskedasticity with sparsely observed functionalcovariates The results based on dense sampling design showsimilar patterns and thus are relegated to the supplement seeSection S13 The comparison of running times is presentedin Table 4

For the case when the response is Gaussian Tables 1and 2 suggest that the Joint QR typically outperforms itscompetitors especially for lower quantile levels (120591 = 005and 120591 = 01) For very small noise level (SNR = 150) PQRperforms the best followed closely by the proposed Joint QRThe variant Pointwise QR which has a poorer performanceis generally better than the modified CM approach Asexpected as the sample size increases (119899 = 1000) all theaccuracy results improve the proposed Joint QR continuesto yield most accurate quantiles for the low quantile levelsFor mixture of Gaussians the results are somewhat similarThe accuracy of the quantile estimators with the PointwiseQR improves greatly in fact the Joint QR and PointwiseQR outperform the other approaches for quantile levels 120591 =005 01 025 irrespective of the SNR Finally Table 3 showsthat the results for Gaussian with heterogeneous error areclose to those for the case of Gaussian Again the proposedmethod has competitive performance in terms of predictionaccuracy

Table 4 compares the three methods that involve esti-mating the conditional distribution in terms of the runningtime required for fitting The times are reported based ona computer with a 23 GHz CPU and 8GB of RAM Notsurprisingly by fitting the model a single time Joint QR is thefastest in some cases being order ofmagnitude faster than therest Pointwise QR can be up to twice as fast as Mod CM

For completeness we also compare our proposedmethodto the appropriate competitivemethods for the cases (1) whenthere is a single scalar covariate and (2) when there is a singlefunctional covariate In the SupplementaryMaterials SectionS11 discusses the former case and compares Joint QR andPointwise QR with the linear quantile regression (LQR) [12]implemented by rq function in the R package quantreg andthe constrained B-splines nonparametric regression quantiles(COBS) implemented by the cobs function in the R pack-age COBS [29] in an extensive simulation experiment thatinvolves both linear quantile settings and nonlinear quantilesettings Overall the results show that the proposed methodshave similar behavior as LQR see Table S1 Furthermore weconsider the proposed method and its variant with nonlinearmodeling of the conditional distribution as discussed inSection 32 which we denote with Joint QR (NL) for jointfitting andPointwiseQR (NL) for pointwise fittingNonlinearversions of the proposed methods have an excellent MAEperformance which is comparable to or better than that ofthe COBS method

Finally Section S12 in the Supplementary Materialsdiscusses the simulation study for the case of having a singlefunctional covariate and compares the proposed methods

6 Journal of Probability and Statistics

Table 1 Average MAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 100Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 367 (003) 353 (003) 330 (002) 317 (002)Pointwise QR 496 (003) 461 (003) 422 (002) 418 (002)Mod CM 604 (003) 581 (003) 555 (003) 514 (003)PQR 317 (004) 271 (003) 231 (002) 216 (002)

Normal 10

Joint QR 632 (003) 600 (003) 576 (002) 573 (002)Pointwise QR 744 (004) 685 (003) 639 (003) 628 (003)Mod CM 820 (004) 810 (004) 804 (004) 801 (004)PQR 682 (005) 611 (004) 534 (003) 509 (002)

Normal 5

Joint QR 784 (004) 734 (003) 693 (003) 684 (003)Pointwise QR 891 (004) 812 (004) 745 (003) 726 (003)Mod CM 934 (004) 923 (004) 914 (005) 906 (005)PQR 868 (006) 781 (005) 674 (003) 634 (002)

Normal 2

Joint QR 1005 (005) 922 (004) 847 (003) 828 (003)Pointwise QR 1091 (006) 986 (004) 887 (004) 854 (003)Mod CM 1085 (005) 1055 (005) 1034 (006) 1021 (006)PQR 1121 (008) 1003 (006) 856 (004) 796 (003)

Normal 1

Joint QR 1150 (006) 1041 (005) 940 (004) 911 (003)Pointwise QR 1212 (006) 1088 (005) 970 (004) 930 (003)Mod CM 1195 (006) 1146 (006) 1107 (006) 1105 (007 )PQR 1282 (008) 1138 (006) 960 (004) 886 (003)

Mixture 150

Joint QR 692 (006) 623 (006) 616 (006) 481 (006)Pointwise QR 810 (008) 680 (006) 666 (006) 525 (006)Mod CM 918 (007) 899 (007) 893 (007) 790 (007)PQR 843 (006) 718 (004) 622 (004) 548 (014)

Mixture 10

Joint QR 902 (006) 795 (005) 785 (005) 619 (006)Pointwise QR 1011 (007) 852 (005) 795 (005) 642 (006)Mod CM 1133 (007) 1095 (007) 1079 (007) 980 (008)PQR 1072 (008) 899 (005) 763 (004) 533 (009)

Mixture 5

Joint QR 1018 (006) 891 (005) 853 (005) 682 (005)Pointwise QR 1118 (007) 938 (005) 858 (005) 698 (005)Mod CM 1219 (007) 1175 (007) 1152 (007) 1047 (008)PQR 1212 (009) 1012 (006) 840 (004) 573 (007)

Mixture 2

Joint QR 1193 (007) 1026 (005) 946 (005) 761 (005)Pointwise QR 1268 (008) 1063 (006) 951 (005) 770 (005)Mod CM 1333 (008) 1260 (008) 1227 (008) 1108 (010)PQR 1416 (010) 1172 (006) 955 (004) 641 (005)

Mixture 1

Joint QR 1317 (008) 1119 (005) 1006 (005) 804 (005)Pointwise QR 1371 (009) 1144 (006) 1010 (005) 813 (005)Mod CM 1416 (008) 1322 (008) 1271 (009) 1144 (011)PQR 1544 (011) 1284 (007) 1023 (004) 689 (004)

with CM in terms of MAE as well as computation timesee results displayed in Tables S2 and S3 The results showthat the proposed Joint QR is comparable to CM in termsof the prediction accuracy and has less computation timeIn our simulation study we also consider the joint fitting ofthe model by treating the binary response as normal and usepffr [27] with Gaussian link denoted by Joint QR (G)

5 Sow Data Application

Our motivating application is an experimental study carriedout at a commercial farm in Oklahoma from July 21 2013 toAugust 19 2013 [38] The study comprises 480 lactating sowsof different parities (ie number of previous pregnancieswhich serves as a surrogate for age and body weight) that

were observed during their first 21 lactation days theirfeed intake was recorded daily as the difference betweenthe feed offer and the feed refusal In addition the studycontains information on the temperature and humidity of thefarrowing rooms each recorded at five minute intervals Thefinal dataset we used for the analysis consists of 475 sows afterfive sows with unreliable measurements were removed by theexperimenters

The experiment was conducted to gain better insightsinto the way that the ambient temperature and humidity ofthe farrowing room affect the feed intake of lactating sowsPrevious studies seem to suggest a reduction in the sowrsquosfeed intake due to heat stress above 29∘C sows decrease feedintake by 05 kg per additional degree in temperature [9]Studying the effect of heat stress on lactating sows is a very

Journal of Probability and Statistics 7

Table 2 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 1000Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 168 (001) 165 (001) 162 (001) 160 (001)Pointwise QR 194 (001) 192 (001) 188 (001) 181 (001)Mod CM 193 (001) 188 (001) 187 (001) 187 (001)PQR 172 (001) 161 (001) 151 (001) 148 (001)

Normal 10

Joint QR 545 (002) 497 (002) 466 (002) 465 (002)Pointwise QR 564 (002) 513 (002) 478 (002) 475 (002)Mod CM 569 (002) 521 (002) 487 (002) 485 (002)PQR 585 (002) 537 (002) 481 (002) 460 (002)

Normal 5

Joint QR 734 (003) 654 (002) 594 (002) 585 (002)Pointwise QR 753 (003) 669 (002) 604 (002) 594 (002)Mod CM 753 (003) 677 (002) 618 (002) 606 (002)PQR 784 (003) 705 (002) 616 (002) 581 (002)

Normal 2

Joint QR 997 (003) 870 (003) 762 (002) 738 (002)Pointwise QR 1015 (003) 885 (003) 771 (002) 745 (002)Mod CM 1012 (003) 893 (003) 787 (003) 760 (003)PQR 1055 (004) 934 (003) 791 (003) 734 (002)

Normal 1

Joint QR 1169 (004) 1010 (003) 868 (003) 832 (003)Pointwise QR 1188 (004) 1025 (004) 877 (003) 839 (003)Mod CM 1185 (004) 1037 (004) 896 (003) 857 (003)PQR 1229 (004) 1077 (004) 902 (003) 829 (003)

Mixture 150

Joint QR 456 (002) 444 (002) 433 (003) 368 (003)Pointwise QR 434 (003) 424 (002) 420 (003) 366 (004)Mod CM 468 (003) 459 (002) 438 (003) 401 (003)PQR 784 (003) 645 (002) 529 (002) 319 (001)

Mixture 10

Joint QR 756 (003) 679 (002) 614 (003) 502 (003)Pointwise QR 749 (003) 661 (002) 588 (003) 502 (003)Mod CM 767 (004) 697 (003) 617 (003) 552 (004)PQR 1011 (004) 829 (003) 671 (002) 362 (002)

Mixture 5

Joint QR 920 (004) 713 (003) 694 (003) 56 (003)Pointwise QR 917 (004) 701 (003) 671 (003) 558 (003)Mod CM 932 (004) 737 (003) 708 (003) 618 (004)PQR 1146 (004) 936 (003) 749 (003) 432 (002)

Mixture 2

Joint QR 1157 (005) 902 (003) 805 (003) 635 (003)Pointwise QR 1158 (005) 898 (004) 790 (003) 631 (003)Mod CM 1169 (005) 938 (004) 838 (004) 701 (004)PQR 1349 (005) 1095 (004) 861 (003) 532 (002)

Mixture 1

Joint QR 1318 (005) 1031 (004) 879 (004) 679 (003)Pointwise QR 1321 (005) 1028 (004) 869 (004) 673 (003)Mod CM 1326 (005) 1067 (004) 921 (004) 747 (004)PQR 1493 (006) 1205 (004) 935 (004) 596 (002)

important scientific question because of a couple of reasonsFirst the reduction of feed intake of the lactating sows isassociated with a decrease in both their bodyweight (BW)and milk production as well as the weight gain of their litter[10 39 40] Sows with poor feed intake during lactationcontinue the subsequent reproductive period with negativeenergy balance [41] which leads to preventing the onset of anew reproductive cycle Second heat stress reduces farrowingrate (number of sows that deliver a new litter) and number

of piglets born [42] the reduction in reproduction due toseasonality is estimated to cost 300 million dollars per yearfor the swine industry [11] Economic losses are estimated toincrease [43] because high temperatures are likely to occurmore frequently due to global warming [44]

Our primary goal is to understand the thermal needsof the lactating sows for proper feeding behavior duringthe lactation time We are interested in how the interplaybetween the temperature and humidity of the farrowing room

8 Journal of Probability and Statistics

Table 3 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of heteroskedasticitywith a scalar covariateand a sparsely observed functional covariate

Sample size SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

100 150

Joint QR 441 (003) 399 (002) 348 (002) 325 (002)Pointwise QR 540 (003) 484 (003) 426 (002) 421 (003)Mod CM 600 (003) 571 (003) 542 (003) 539 (003)PQR 438 (005) 343 (003) 252 (002) 214 (002)

100 10

Joint QR 693 (003) 640 (003) 589 (002) 576 (002)Pointwise QR 794 (004) 720 (003) 648 (003) 630 (003)Mod CM 841 (004) 822 (004) 813 (004) 805 (004)PQR 776 (006) 657 (004) 547 (003) 508 (002)

100 5

Joint QR 844 (004) 772 (003) 705 (003) 686 (003)Pointwise QR 944 (005) 850 (004) 757 (003) 726 (003)Mod CM 965 (005) 940 (005) 936 (005) 926 (005)PQR 951 (007) 825 (005) 687 (003) 633 (002)

100 2

Joint QR 1068 (005) 963 (004) 860 (003) 830 (003)Pointwise QR 1155 (006) 1029 (005) 900 (004) 854 (003)Mod CM 1127 (005) 1080 (005) 1043 (005) 1040 (006)PQR 1207 (008) 1053 (006) 866 (004) 792 (003)

100 1

Joint QR 1223 (006) 1091 (005) 954 (004) 911 (003)Pointwise QR 1287 (007) 1139 (005) 985 (004) 929 (003)Mod CM 1243 (006) 1180 (006) 1119 (006) 1109 (007)PQR 1370 (009) 1191 (006) 970 (004) 884 (003)

1000 150

Joint QR 287 (001) 242 (001) 189 (001) 165 (001)Pointwise QR 306 (001) 260 (001) 207 (001) 186 (001)Mod CM 313 (001) 265 (001) 210 (001) 191 (001)PQR 310 (001) 247 (001) 178 (001) 148 (001)

1000 10

Joint QR 621 (002) 545 (002) 479 (002) 466 (002)Pointwise QR 638 (002) 559 (002) 490 (002) 477 (002)Mod CM 646 (002) 570 (002) 501 (002) 486 (002)PQR 668 (003) 582 (002) 492 (002) 460 (002)

1000 5

Joint QR 808 (003) 701 (002) 607 (002) 585 (002)Pointwise QR 827 (003) 715 (002) 616 (002) 594 (002)Mod CM 830 (003) 726 (002) 631 (002) 608 (002)PQR 864 (003) 750 (003) 627 (002) 581 (002)

1000 2

Joint QR 1076 (003) 921 (003) 776 (002) 738 (002)Pointwise QR 1095 (004) 934 (003) 785 (003) 745 (002)Mod CM 1092 (003) 945 (003) 801 (003) 760 (003)PQR 1138 (004) 979 (003) 803 (003) 734 (002)

1000 1

Joint QR 1254 (004) 1065 (003) 883 (003) 832 (003)Pointwise QR 1274 (004) 1081 (004) 892 (003) 839 (003)Mod CM 1270 (004) 1091 (004) 910 (003) 856 (003)PQR 1316 (005) 1126 (004) 913 (003) 829 (003)

Table 4 Average computing time (in seconds) of the three approaches that involve estimating the conditional distribution for the case ofhaving a scalar covariate and a densely observed functional covariate

Distribution Method 119899 = 100 119899 = 1000

NormalJoint QR 12 133

Pointwise QR 148 271Mod CM 278 511

MixtureJoint QR 13 154

Pointwise QR 151 296Mod CM 327 532

Journal of Probability and Statistics 9

affects the feed intake demeanor of lactating sows of differentparities We focus on three specific time points during thelactation periodmdashbeginning (lactation day 4) middle (day11) and end (day 18)mdashand the analyses are done separatelyfor each time point We consider two types of responses thatare meant to assess the feed intake behavior using the currentand the previous lactation days The first one quantifies theabsolute change in the feed intake over two consecutive daysand the second one quantifies the relative change and takesinto account the usual sowrsquos feed intake We define themformally after introducing some notation

Let 119865119868119894119895 be the 119895th measurement of the feed intakeobserved for the 119894th sow and denote by the lactation day 119871119863119894119895when 119865119868119894119895 is measured here 119895 = 1 119899119894 Most sows areobserved every day within the first 21 lactation days and thushave 119899119894 = 21 First define the absolute change in the feedintake between two consecutive days asΔ(1)

119894(119895+1)= 119865119868119894(119895+1)minus119865119868119894119895

for 119895 that satisfies 119871119863119894(119895+1)minus119871119863119894119895 = 1 For instanceΔ(1)119894(119895+1) = 0means there was no change in feed intake of sow 119894 betweenthe current day and the previous day while Δ(1)

119894(119895+1)lt 0means

that the feed intake consumed by the 119894th sow in the currentday is smaller than the feed intake consumed in the previousday However the same amount of change in the feed intakemay reflect some stress level for a sow who typically eats a lotand a more serious stress level for a sow that usually has alower appetite For this we define the relative change in thefeed intake by Δ(2)

119894(119895+1)= (119865119868119894(119895+1) minus 119865119868119894119895)(119871119863119894(119895+1) minus 119871119863119894119895) sdot

119879119860 119894 where 119879119860 119894 is the trimmed average of feed intake of 119894thsow calculated as the average feed intake after removing thelowest 20 and highest 20 of the feed intake measurements1198651198681198941 119865119868119894119899119894 taken for the corresponding sow Here 119879119860 119894 issurrogate for the usual amount of feed intake of the 119894th sowTrimmed average is used instead of the common average toremove outliers of very low feed intakes in first few lactatingdays For example consider the situation of two sows sow119894 that typically consumes 10lb food per day and sow 1198941015840 thatconsumes 5lb food per day A reduction of 5lb in the feedintake over two consecutive days corresponds to Δ(2)

119894(119895+1)=

minus50 for the 119894th sow and Δ(2)1198941015840(119895+1)

= minus100 for the 1198941015840th sowClearly both sows are stressed (negative value) but the secondsow is stressed more as its absolute relative change is largerin view of this we refer to the second response as the stressindex Due to the construction of the two types of responsesthe data size varies for lactation days 4 (119895 = 3) 11 (119895 = 10)and 18 (119895 = 17) for the first response Δ(1)

119894(119895+1) we have sample

sizes of 233 350 and 278 whereas for Δ(2)119894(119895+1)

the sample sizesare 362 373 and 336 for the respective lactation days

In this analysis we center the attention on the effect ofthe ambient temperature and humidity on the 1st quartile ofthe proxy stress measures and gain more understanding ofthe food consumption of sows that are most susceptible toheat stress While the association between the feed intake oflactating sows and the ambient conditions of the farrowingroom has been an active research area for some timeaccounting for the temperature daily profile has not beenconsidered yet hitherto Figure 1 displays the temperature

and humidity daily profiles recorded at a frequency of 5-minute window intervals for three different days Preliminaryinvestigation reveals that temperature is negatively correlatedwith humidity at each time this phenomenon is causedbecause the farm uses cool cell panels and fans to controlthe ambient temperature Furthermore it appears that thereis a strong pointwise correlation between temperature andhumidity In view of these observations in our analysis weconsider the daily average of humidity Exploratory analysisof the feed intake behavior of the sows suggests similaritiesfor the sows with parity greater than older sows (ones whoare at their third pregnancy or higher) thus we use a parityindicator instead of the actual parity of the sow The parityindicator 119875119894 is defined as one if the 119894th sow has parity oneand zero otherwise

For the analysis we smooth daily temperature measure-ments of each sow using univariate smoother with 15 cubicregression bases and quadratic penalty REML is used toestimate smoothing parameter The smoothed temperaturecurve for sow 119894rsquos 119895th repeatedmeasure is denoted by119879119894119895(119905) 119905 isin[0 24) and the corresponding daily average humidity isdenoted by119860119867119894119895 Both temperature and average humidity arecentered before being used in the analysis

For convenience we denote the response with Δ 119894119895 byremoving the superscript In this application for fixed 119895 Δ 119894119895corresponds to 119884119894 in Section 2 119875119894 and 119860119867119894119895 correspond toscalar covariates1198831198942 and119879119894119895(119905) and119860119867119894119895 sdot119879119894119895(119905) correspond tofunctional covariates1198831198943(sdot) We first estimate the conditionaldistribution ofΔ 119894119895 given temperature119879119894119895(119905) average humidity119860119867119894119895 parity 119875119894 and interaction 119860119867119894119895 sdot 119879119894119895(119905) Specifically foreach of lactation days of interest (119895 = 3 10 and 17) we create aset of 100 equispaced grid of points between the fifth smallestand fifth largest values of Δ 119894119895rsquos and denote the grids withD = 119889ℓ ℓ = 1 100 Then we create artificial binaryresponses 1(Δ 119894119895 le 119889ℓ) ℓ = 1 100 and fit the followingmodel for 119865119894119895(119889ℓ) = 119864[1(Δ 119894119895 le 119889ℓ) | 119879119894119895(119905) 119860119867119894119895 119875119894]

119864 [1 (Δ 119894119895 le 119889ℓ) | 119879119894119895 (119905) 119860119867119894119895 119875119894] = 119892minus1 1205730 (119889ℓ)

+ 1205731 (119889ℓ) 119875119894 + 1205732 (119889ℓ) 119860119867119894119895 + int1205733 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

+ 119860119867119894119895 int1205734 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

(7)

where 1205730(sdot) is a smooth intercept 1205731(sdot) quantifies the smootheffect of young sows1205732(sdot) describes the effect of the humidityand 1205733(sdot 119905) and 1205734(sdot 119905) quantify the effect of the temperatureat time 119905 as well as the interaction between the temperature attime 119905 and average humidityWemodel1205730(sdot)using 20 univari-ate basis functions 1205731(sdot) and 1205732(sdot) using five univariate basisfunctions and 1205733(sdot sdot) and 1205734(sdot sdot) using tensor product of twounivariate bases functions (total of 25 functions)Throughoutthe analysis cubic B-spline bases are used and REML isused for estimating smoothing parameters The estimatedconditional distribution denoted by 119865119894119895(119889) is monotonizedby fitting isotonic regression to (119889ℓ 119865119894119895(119889ℓ)) ℓ = 10 90ten smallest and ten largest 119889ℓ and the corresponding valuesof 119865119895(119889ℓ) are removed to avoid boundary effects By abuse of

10 Journal of Probability and Statistics

2425

2627

2829

30

Temperature

6065

7075

80

Humidity

16H

20H

18H

22H

14H

12H

10H

14H8H4H0H 2H 6H

time of day

16H

20H

12H

22H

18H

10H

14H

14H 0H 6H2H 8H4H

time of day

Figure 1 Temperature (∘C) and humidity () observed profiles (dashed) for three randomly selected days and the corresponding smoothedones (solid) the x-axis begins at 14H (2PM)

2324

2526

2728

29

Lactation day 4

203040median607080

2324

2526

2728

29

Lactation day 11

203040median607080

2324

2526

2728

29

Lactation day 18

203040median607080

8H6H4H2H0H 12H

14H

10H

14H

22H

16H

20H

18H

time of day

8H4H2H0H 6H22H

12H

16H

20H

14H

14H

10H

18H

time of day

4H 6H0H 2H 8H20H

12H

14H

14H

22H

16H

18H

10H

time of day

Figure 2 Temperature curves with which prediction of quantiles is made Dashed black line is pointwise average of temperature curves andsolid lines are pointwise quartiles all curves are smoothed

notation 119865119894119895(119889) denotes the resulting monotonized estimateddistribution Finally we obtain estimated first quartiles iequantiles at 120591 = 025 level by inverting 119865119894119895(119889) namely 119876(120591 =025 | 119879119894119895(119905) 119860119867119894119895 119875119894) = inf119889 119865119894119895(119889) ge 025

To understand the relationship between the lactatingsows feed intake and the thermal condition of the farrowingroom we systematically compare and study the predictedquantiles of two responses at combinations of different valuesof temperature humidity and parity For each of threelactation days (119895 = 3 10 17) we consider three values ofaverage humidity (first quartile median and third quartile)and two levels of parity (0 for older sows and 1 for youngersows) Based on the experimentersrsquo interest for the functional

covariate119879119894119895(sdot)we consider seven smooth temperature curvesgiven in Figure 2 Each of these curves is obtained by firstcalculating pointwise quantiles of temperature at five-minuteintervals for a specific level and then smoothing it weconsidered quantiles levels 120578 = 02 03 and 08 In shortfor each of three lactation days we obtain the first quartileof two responses for 42 different combinations (3 humidityvalues times 2 parity levels times 7 temperature curves) using theproposed method To avoid extrapolation we ascertain that(i) there are reasonablymany observedmeasurements at eachof the combinations and (ii) bottom 25 of the responses arenot dominantly from one of the parity group see distributionof each response by the parity in Figure 3

Journal of Probability and Statistics 11

Response (2) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus10

00

10

20

Resp

onse

(2)

minus15

minus05

05

10

Resp

onse

(2)

minus2minus1

01

Resp

onse

(2)

Response (1) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus4minus2

02

46

Resp

onse

(1)

minus6minus4

minus20

24

6Re

spon

se (1

)

minus4minus2

02

46

Resp

onse

(1)

Figure 3 Top panels back-to-back boxplots of the absolute change in feed intake at a specific day by parity Bottom panels back-to-backboxplots of the relative change in feed intake at a specific day by parity

The resulting predicted quantiles are shown in Figure 4Here we focus our discussion on predicted quantile of Δ(2)

119894(119895+1)

at quantile level 120591 = 025 for lactation day 4 (119895 = 3)mdashthefirst plot of the second row in Figure 4 The results suggestthat the feed intake of older sows (parity 119875119894 = 0 greylines) is less affected by high temperatures than younger sows(black lines) this finding is in agreement with Bloemhof etal [42] We also observe that the effects of humidity andtemperature on feed intake change are strongly intertwinedFor illustration we focus on lactation day 4 (119895 = 3) again foryounger sows (black lines) For medium humidity (dashedlines) their feed intake stays pretty constant as temperatureincreases while for low and high humidity levels (solid anddotted lines respectively) it changes with an opposite direc-tion Specifically when temperature increases the predictedfirst quartile of Δ(2)

119894(119895+1)increases for low humidity (solid line)

whereas it decreases for high humidity (dotted line) Ourresults imply that high humidity (dotted line) is related toa negative impact of high temperature on feed intake whilelow humidity (solid line) alleviates it and this finding isconsistent with a previous study [45] The analysis resultsuggests to keep low humidity levels in order to maintainhealthy feed intake behavior when ambient temperature isabove 60th percentile high humidity levels are desirable forcooler ambient temperature

Interpretation of the other results is similar While theeffects of covariates on feed intake are less apparent toward

the end of lactation period we still observe similar patternacross all three lactation days For the 11th day (119895 = 10) the25th quantile of the feed intake is predicted to decrease whenthe temperature stays below the 40th percentile regardlessof humidity level and sows age However it starts increas-ing with low humidity while it continues decreasing withhigh humidity when the temperature rises above the 40thpercentile Similarly for the 18th day (119895 = 17) when thetemperature rises above the 60th percentile the predictedfirst quartile increases with low humidity while it decreaseswith high humidity The effect of temperature on feed intakeseems less obvious for lactation days 11 and 18 than for day 4while the effect may be due to lactation day it may also be aresult of other factors such asmore fluctuation and variabilityin temperature curves on day 4 than on other two days(see Figure 2) Overall we conclude that high humidity andtemperature affect the sows feed intake behavior negativelyand young sows (parity one) are more sensitive to heat stressthan older sows (higher parity) especially at the beginning oflactation period

6 Discussion

The proposed modeling framework opens up a couple offuture research directions A first research avenue is todevelop significance tests of null covariate effect Testingfor the null effect of a covariate on the conditional dis-tribution of the response is equivalent to testing that the

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

Journal of Probability and Statistics 3

Define 119885119894(119910) = 1(119884119894 le 119910) for 119910 isin R In practicefor each 119910 in a fine grid we view 119885119894(119910) as a binary-valued random functional variable It follows that model (1)can be written equivalently as a generalized function-on-function regression model through relating the ldquoartificialrdquobinary functional response 119885119894(119910) and the mixed covariatesX1198941 1198831198942 1198831198943(sdot)Thismodel can be fitted by using for examplethe ideas of Scheipl et al [22] which we briefly summarizenext

Model (1) can be represented as the following generalizedadditive model

119864 [119885119894 (119910) | X1198941 1198831198942 1198831198943 (sdot)] = 119892minus1 120578119894 (119910)

120578119894 (119910) =1205810

sum1198890=1

11986101198890 (119910) 12057901198890 + X11987911989411205731

+ 11988311989421205811

sum1198891=1

11986111198891 (119910) 12057911198891 +119898

sum119895=1

(119905119895+1 minus 119905119895)1198831198943 (119905119895)

sdot1205812119905

sum1198892=1

1205812119910

sum11988910158402=1

11986111990521198892 (119905119895) 119861119910

211988910158402(119910) 12057921198892 11988910158402

(2)

For convenience we use the notation 1198611198831198942 1198891(119910) =119883119894211986111198891(119910) 1198831198943(119905119895) = (119905119895+1 minus 119905119895)1198831198943(119905119895) We let B0(119910) =11986101(119910) 11986101205810(119910)119879 B1198941(119910) = 1198611198831198942 1(119910) 1198611198831198942 1205811(119910)119879B2119905(119905) = 11986111990521(119905) 11986111990521205812119905(119905)119879 B2119910(119910) = 11986111991021(119910) 11986111991021205812119910(119910)119879 1205790 = 12057901 12057901205810119879 1205791 = 12057911 12057911205811119879 andΘ2 = [12057921198892 11988910158402]1le1198892le1205812119905 1le11988910158402le1205812119910 is a coefficient matrix Then1205733(119905 119910) = B2119905(119905) otimesB2119910(119910)1198791205792 where 1205792 is the vectorizationof Θ2 We let X1198943(t) = 1198831198943(1199051) 1198831198943(119905119898)119879 isin R119898B2(t 119910) = B2119905(1199051) otimes B2119910(119910) B2119905(119905119898) otimes B2119910(119910)119879 isinR119898times12058121199051205812119910 and B1198942(119910) = B2(t 119910)119879X1198943(t) Now model (2) canbe written as

119864 [119885119894 (119910) | X1198941 1198831198942 1198831198943 (sdot)] = 119892minus1 X11987911989411205731 + B0 (119910)119879 1205790+ B1198941 (119910)119879 1205791 + B1198942 (119910)119879 1205792

(3)

The general idea is to set the bases dimensions 1205810 1205811 1205812119905and 1205812119910 to be sufficiently large to capture the complexity ofthe coefficient functions and control the smoothness of theestimator through some roughness penalties This approachof using roughness penalties has been widely used see forexample Eilers and Marx [23] Ruppert [24] Wood [25 26]among many others

It is important to emphasize that even in the case ofa single functional covariate our methodology differs from[21] in two directions (1) our proposed method is based onmodeling the unknown smooth coefficient functions usingprespecified basis function expansion and using penaltiesto control their roughness In contrast CM uses data-driven basis and chooses the number of basis functionsthrough the percentage of explained variance (PVE) of thefunctional predictors This key difference allows our methodto accommodate covariates of different types as well as

nonlinear effects (2) Our estimation approach is based on asingle step penalized function-on-function regression whileCM uses pointwise estimation based on functional principalcomponent bases and thus requires fitting multiple general-ized regressions This nice feature leads to an computationaladvantage

3 Estimation

31 Estimation via Penalized Log-Likelihood Let 119910ℓ ℓ =1 119871 be a set of equally spaced points in the range ofthe response variable 119884119894rsquos Conditioning on X1198941 1198831198942 1198831198943(sdot)we model 119885119894(119910ℓ) as independently distributed Bernoullivariables with mean 120583119894(119910ℓ) where 119892120583119894(119910ℓ) = 120578119894(119910ℓ) Thecoefficients 1205731 1205790 1205791 and 1205792 are estimated by minimizingthe penalized log-likelihood criterion

minus 2L (1205731 1205790 1205791 1205792 | 119885119894 (119910ℓ) forall119894 ℓ) + 12058201198750 (1205790)+ 12058211198751 (1205791) + 12058221199051198752119905 (1205792) + 12058221199101198752119910 (1205792)

(4)

where L is the log-likelihood function of data 119885119894(119910ℓ) ℓ =1 1198711le119894le119899 1205820 1205821 1205822119905 and 1205822119910 are smoothing parameterswhich control the balance between the model fit and itscomplexity and1198750(sdot)1198751(sdot)1198752119905(sdot) and1198752119910(sdot) are all penalties

There are several choices to define the penalty matrixin nonparametric regression see Eilers and Marx [23]and Wood [26] We use quadratic penalties which penal-ize the size of the curvature of the estimated coefficientfunctions Let 1198750(1205790) = int12059721205730(119910)12059711991022119889119910 = 1205791198790D01205790where D0 is of dimension 1205810 times 1205810 with its (119904 1199041015840) ele-ment equal to int12059721198610119904(119910)1205971199102120597211986101199041015840(119910)1205971199102119889119910 Similarly1198751(1205791) = int12059721205732(119910)12059711991022119889119910 = 1205791198791D11205791 where D1 isof dimension 1205811 times 1205811 with its (119904 1199041015840) element equal toint12059721198611119904(119910)1205971199102120597211986111199041015840(119910)1205971199102119889119910 As 1205733(sdot sdot) is a bivariatefunction the choice of penalty implies penalizing the sizeof curvature in each direction respectively 1198752119905(1205792) =int int12059721205733(119910 119905)12059711990522119889119910119889119905 = 1205791198792D21199051205792 where D2119905 = P2119905 otimesI1205812119910 is of dimension 12058121199101205812119905 times 12058121199101205812119905 with the (119904 1199041015840)element of P2119905 equal to int12059721198611199052119904(119905)1205971199052120597211986111990521199041015840(119905)1205971199052119889119905for some orthonormal spline bases Similarly 1198752119910(1205792) =int int12059721205733(119910 119905)12059711991022119889119910119889119905 = 1205791198792D21199101205792 where D2119910 = I1205812119905 otimesP2119910 is of dimension 12058121199101205812119905 times 12058121199101205812119905 with the (119904 1199041015840) elementof P2119910 equal to int12059721198611199102119904(119910)1205971199102120597211986111991021199041015840(119910)1205971199102119889119910

Criterion (4) can be viewed as a penalized quasilikelihood(PQL) of the corresponding generalized linear mixed model

119885119894 (119910ℓ) | 1205731 1205790 1205791 1205792 sim Bernoulli (120583119894 (119910ℓ)) ℓ = 1 119871

1205790 sim 119873(0 120582minus10 Dminus0 ) 1205791 sim 119873(0 120582minus11 Dminus1 ) 1205792 sim 119873 (0Dminus2 )

(5)

where Dminus0 is the generalized inverse matrix of D0 Dminus1 andDminus2 = (1205822119905D2119905 + 1205822119910D2119910)minus are defined similarly Wood [26]

4 Journal of Probability and Statistics

discusses an alternative way to deal with the rank-deficientmatrices in the context of restricted maximum likelihood(REML) estimation Here we do not account for the depen-dence over 119910 see Scheipl et al [22] for a general formulationSee also Ivanescu et al [27] who use the mixed modelrepresentation of a similar regression model to (5) but with aGaussian functional responseThe smoothing parameters areestimated using REML

32 Extension to Nonlinear Model One advantage of theproposed framework is that it can be easily extended toallow for more flexible effects ie extending the ideasto accommodate multiple covariates scalar or functionaland varied types of effects In particular the smooth effect11988311989421205732(119910) can be replaced by ℎ1(1198831198942 119910) and int1198831198943(119905)1205733(119905 119910)119889119905byint ℎ21198831198943(119905) 119905 119910119889119905 where ℎ1(sdot sdot) and ℎ2(sdot sdot sdot) are unknownbivariate and trivariate smooth functions respectively seeScheipl et al [22] and Kim et al [28] These changes requirelittle additional computational effort The modeling andestimation follow roughly similar ideas as Scheipl et al [22]We consider the nonlinear model in the simulation studyfor the case of having a scalar covariate only ie ℎ1(119883119894 119910)and the corresponding results are presented in Section S11of the Supplementary Materials The results show excellentprediction performance compared to the competitive nonlin-ear quantile regressionmethod namely constrained B-splinesmoothing (COBS) [29]

33 Extension to Sparse and Noisy Functional CovariatesIn practice the functional covariates are often observed atirregular times across the units and the measurements arepossibly corrupted by noises In such case one needs tofirst smooth and denoise the trajectories before fitting Whenthe sampling design of the functional covariate is dense thecommon approach is to smooth each trajectory using splinesor local polynomial smoothing as proposed in Ramsay andSilverman [3] and Zhang and Chen [30] When the designis sparse the smoothing can be done by pooling all thesubjects and following the PACE method proposed in Yaoet al [31] As recovering the trajectories has been extensivelydiscussed in the literatures we do not review the procedureshere Instead we discuss some available computing resourcesthat can be used to fit these methods In our numericalstudy we used fpcasc function in the refund R package[32] for recovering the latent trajectories irrespective of asampling design (dense or sparse) Alternatively one can usefpcaface [33] in refund for regular dense design andfacespare [34] in the R package face [35] for irregularsparse design Once the latent trajectories are estimated theycan be used in the fitting criterion (4)

34 Estimation of Conditional Quantile Let 1 0 1 and2 be parameter estimates in (5) It follows that the estimateddistribution function 119865119884119894|X1198941 11988311989421198831198943(sdot)(119910) can be obtained byplugging in the estimated coefficients The 120591th conditionalquantile is estimated by inverting the estimated distributionie 119876119884119894|X1198941 1198831198942 1198831198943(sdot)(120591) = inf119910 119865119884119894|X1198941 1198831198942 1198831198943(sdot)(119910) ge 120591 The

estimated distribution function is not a monotonic functionyet In practice we suggest to first apply a monotonizationmethod as described in Section 35 and then estimate theconditional quantiles by inverting the resulting estimateddistribution

35 Monotonization and Implementation While a condi-tional quantile function is nondecreasing the resulting esti-mated quantiles may not be Two approaches are widely usedone is to monotonize the estimated conditional distributionfunction and the other is to monotonize the estimatedconditional quantile function We choose the former as119865119884119894|X1198941 11988311989421198831198943(sdot)(119910) is readily available at dense grid points 119910ℓrsquosWeuse an isotonic regressionmodel [36] formonotonizationwhich imposes an order restriction this is done by using the Rfunction isoreg Other monotonization approaches includeChernozhukov et al [37] which was employed in Kato [16]

Our approach is implemented by first creating an artificialbinary response and then fitting a penalized function-on-function regression model and using the logit link functionFitting models in (4) can be done by extending the ideasof Ivanescu et al [27] for Gaussian functional responsethe extension of the model to the non-Gaussian functionalresponse has recently been studied and implemented byScheipl et al [22] as the pffr function in refund package[32]

4 Simulation Study

41 Simulation Setting In this section we evaluate theempirical performance of the proposed method We presentresults for the case when we have both functional and scalarcovariates additional results when there is only a singlescalar or a single functional covariate are discussed in theSupplementary Materials Section S1

Suppose the observed data for the 119894th subject are[119884119894 1198831119894 (1198821198941 1199051198941) (119882119894119898119894 119905119894119898119894)] 119905119894119895 isin [0 10] where

1198831119894 119894119894119889sim 119880119899119894119891(minus16 16)119882119894119895 = 1198832119894(119905119894119895) + 120598119894119895 for 1 le 119894 le 119899 1 le119895 le 119898119894 Let1198832119894(119905119894119895) = 120583(119905119894119895)+sum4119896=1 120585119894119896120601119896(119905119894119895)+120598119894119895 where 120583(119905) =119905 + sin(119905) 120601119896(119905) = cos(119896 + 1)12058711990510radic5 for odd values of 119896120601119896(119905) = sin11989612058711990510radic5 for even values of 119896 120585119894119896 119894119894119889sim 119873(0 120582119896)(1205821 1205822 1205823 1205824) = 16 9 756 506 and 120598119894119895 119894119894119889sim 119873(0 1205902120598 ) Weassume three cases for generating response 119884119894

(i) Gaussian119884119894 | 1198831119894 1198832119894(sdot) sim 119873(2 int1198832119894(119905)120573(119905)119889119905+2119883111989452) this corresponds to the quantile regressionmodel119876119884|1198831 1198832(sdot)(120591) = 2 int1198832119894(119905)120573(119905)119889119905 + 21198831119894 + 5Φminus1(120591)whereΦ(sdot) is the distribution function of the standardnormal

(ii) Mixture ofGaussians119884119894 | 11988311198941198832119894(sdot) sim 05119873(int1198832119894(119905)120573(119905)119889119905+119883111989412) + 05119873(3 int1198832119894(119905)120573(119905)119889119905+ 31198831119894 42) where thetrue quantiles can be approximated numerically byusing qnorMix function in the R package norMix

(iii) Gaussian with heterogeneous error 119884119894 | 11988311198941198832119894(sdot) sim 119873(2 int1198832119894(119905)120573(119905)119889119905 + 21198831119894 52 int1198832119894(119905)2119889119905

Journal of Probability and Statistics 5

sum4119896=1 120582119896) the true quantiles are given by119876119884|1198831 1198832(sdot)(120591) = 2 int1198832119894(119905)120573(119905)119889119905 + 21198831119894 +

5radicint1198832119894(119905)2119889119905sum4119896=1 120582119896Φminus1(120591)

Let 120573(119905) = sum4119896=1 120573119896120601119896(119905) where 120573119896 = 1 for 119896 = 1 4For each case we use different combinations of signal to

noise ratio (SNR) sample size and sampling designs to gen-erate 500 simulated datasets We define SNR asradicsum4119896=1 120582119896120590120598and we consider five levels of noise SNR = 150 10 5 2 1Two levels of sample size are 119899 = 100 and 119899 = 1000 Twosampling designs are considered (i) sparse design where 119905119894119895 119895 = 1 119898 are119898 = 15 randomly selected points from a setof 30 equispaced grids in [0 10] and (ii) dense design wherethe sampling points 119905119894119895 = 119905119895 119895 = 1 119898 are 119898 = 30equispaced time points in [0 10]

The performance is evaluated on a test set of 100 subjectsfor whichwe have 1198831119894lowast (119882119894lowast119895 119905119894lowast119895) 119895 = 1 119898 available interms of mean absolute error (MAE) for quantile levels 120591 =005 01 025 and 05

MAE (120591) = 1100100

sum119894lowast=1

10038161003816100381610038161003816119876119894lowast (120591) minus 119876119894lowast (120591)10038161003816100381610038161003816 (6)

42 CompetingMethods Wedenote the proposedmethod byJoint QR to emphasize the single step estimation approachWe compare our method with two alternative approaches(1) a variant of our proposed approach using pointwiseestimation denoted by Pointwise QRThis approach consistsof fittingmultiple regressionmodels with binomial link func-tion as implemented by the penalized functional regressionpfr developed byGoldsmith et al [7] of therefundpackagefor generalized scalar responses (2) a modified version ofthe CM method denoted by Mod CM that we developedto account for additional scalar covariates and which fitsmultiple generalized linear models with scalar covariates andfPC scores as predictors (3) a linear quantile regressionapproach using the quantile loss function and the partialquantile regression bases for functional covariates proposedby Yu et al [19] and denoted by PQRNotice that although theformulation of the first twomethods implicitly accounts for avarying effect of the covariates on the response distributionthey do not ensure that this effect is smooth The thirdapproach can only estimate a specific quantile rather than theentire conditional distribution Note that all the competingmethods are monotonized for a fair comparison

The R function pfr can incorporate both scalarvectorand functional predictors by adopting a mixed effects modelframework The functional covariates are presmoothed byfPC analysis [31] Throughout the simulation study we fixPVE as 095 for fPC analysis to determine the number ofprincipal components and use REML to select the smoothingparameters for our proposed methods Other basis settingsare set to their default values We use 100 equally distancedpoints between the minimum andmaximum of the observed119884119894rsquos to set the grid 119910ℓ ℓ = 1 119871 for the conditionaldistribution function

43 Simulation Results Tables 1 and 2 show the accuracyof the quantile prediction for the two cases (normal andmixture) when the functional covariate is observed sparselyand the sample size is 119899 = 100 (Table 1) and 119899 = 1000(Table 2) Table 3 presents the prediction accuracy for thecase of heteroskedasticity with sparsely observed functionalcovariates The results based on dense sampling design showsimilar patterns and thus are relegated to the supplement seeSection S13 The comparison of running times is presentedin Table 4

For the case when the response is Gaussian Tables 1and 2 suggest that the Joint QR typically outperforms itscompetitors especially for lower quantile levels (120591 = 005and 120591 = 01) For very small noise level (SNR = 150) PQRperforms the best followed closely by the proposed Joint QRThe variant Pointwise QR which has a poorer performanceis generally better than the modified CM approach Asexpected as the sample size increases (119899 = 1000) all theaccuracy results improve the proposed Joint QR continuesto yield most accurate quantiles for the low quantile levelsFor mixture of Gaussians the results are somewhat similarThe accuracy of the quantile estimators with the PointwiseQR improves greatly in fact the Joint QR and PointwiseQR outperform the other approaches for quantile levels 120591 =005 01 025 irrespective of the SNR Finally Table 3 showsthat the results for Gaussian with heterogeneous error areclose to those for the case of Gaussian Again the proposedmethod has competitive performance in terms of predictionaccuracy

Table 4 compares the three methods that involve esti-mating the conditional distribution in terms of the runningtime required for fitting The times are reported based ona computer with a 23 GHz CPU and 8GB of RAM Notsurprisingly by fitting the model a single time Joint QR is thefastest in some cases being order ofmagnitude faster than therest Pointwise QR can be up to twice as fast as Mod CM

For completeness we also compare our proposedmethodto the appropriate competitivemethods for the cases (1) whenthere is a single scalar covariate and (2) when there is a singlefunctional covariate In the SupplementaryMaterials SectionS11 discusses the former case and compares Joint QR andPointwise QR with the linear quantile regression (LQR) [12]implemented by rq function in the R package quantreg andthe constrained B-splines nonparametric regression quantiles(COBS) implemented by the cobs function in the R pack-age COBS [29] in an extensive simulation experiment thatinvolves both linear quantile settings and nonlinear quantilesettings Overall the results show that the proposed methodshave similar behavior as LQR see Table S1 Furthermore weconsider the proposed method and its variant with nonlinearmodeling of the conditional distribution as discussed inSection 32 which we denote with Joint QR (NL) for jointfitting andPointwiseQR (NL) for pointwise fittingNonlinearversions of the proposed methods have an excellent MAEperformance which is comparable to or better than that ofthe COBS method

Finally Section S12 in the Supplementary Materialsdiscusses the simulation study for the case of having a singlefunctional covariate and compares the proposed methods

6 Journal of Probability and Statistics

Table 1 Average MAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 100Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 367 (003) 353 (003) 330 (002) 317 (002)Pointwise QR 496 (003) 461 (003) 422 (002) 418 (002)Mod CM 604 (003) 581 (003) 555 (003) 514 (003)PQR 317 (004) 271 (003) 231 (002) 216 (002)

Normal 10

Joint QR 632 (003) 600 (003) 576 (002) 573 (002)Pointwise QR 744 (004) 685 (003) 639 (003) 628 (003)Mod CM 820 (004) 810 (004) 804 (004) 801 (004)PQR 682 (005) 611 (004) 534 (003) 509 (002)

Normal 5

Joint QR 784 (004) 734 (003) 693 (003) 684 (003)Pointwise QR 891 (004) 812 (004) 745 (003) 726 (003)Mod CM 934 (004) 923 (004) 914 (005) 906 (005)PQR 868 (006) 781 (005) 674 (003) 634 (002)

Normal 2

Joint QR 1005 (005) 922 (004) 847 (003) 828 (003)Pointwise QR 1091 (006) 986 (004) 887 (004) 854 (003)Mod CM 1085 (005) 1055 (005) 1034 (006) 1021 (006)PQR 1121 (008) 1003 (006) 856 (004) 796 (003)

Normal 1

Joint QR 1150 (006) 1041 (005) 940 (004) 911 (003)Pointwise QR 1212 (006) 1088 (005) 970 (004) 930 (003)Mod CM 1195 (006) 1146 (006) 1107 (006) 1105 (007 )PQR 1282 (008) 1138 (006) 960 (004) 886 (003)

Mixture 150

Joint QR 692 (006) 623 (006) 616 (006) 481 (006)Pointwise QR 810 (008) 680 (006) 666 (006) 525 (006)Mod CM 918 (007) 899 (007) 893 (007) 790 (007)PQR 843 (006) 718 (004) 622 (004) 548 (014)

Mixture 10

Joint QR 902 (006) 795 (005) 785 (005) 619 (006)Pointwise QR 1011 (007) 852 (005) 795 (005) 642 (006)Mod CM 1133 (007) 1095 (007) 1079 (007) 980 (008)PQR 1072 (008) 899 (005) 763 (004) 533 (009)

Mixture 5

Joint QR 1018 (006) 891 (005) 853 (005) 682 (005)Pointwise QR 1118 (007) 938 (005) 858 (005) 698 (005)Mod CM 1219 (007) 1175 (007) 1152 (007) 1047 (008)PQR 1212 (009) 1012 (006) 840 (004) 573 (007)

Mixture 2

Joint QR 1193 (007) 1026 (005) 946 (005) 761 (005)Pointwise QR 1268 (008) 1063 (006) 951 (005) 770 (005)Mod CM 1333 (008) 1260 (008) 1227 (008) 1108 (010)PQR 1416 (010) 1172 (006) 955 (004) 641 (005)

Mixture 1

Joint QR 1317 (008) 1119 (005) 1006 (005) 804 (005)Pointwise QR 1371 (009) 1144 (006) 1010 (005) 813 (005)Mod CM 1416 (008) 1322 (008) 1271 (009) 1144 (011)PQR 1544 (011) 1284 (007) 1023 (004) 689 (004)

with CM in terms of MAE as well as computation timesee results displayed in Tables S2 and S3 The results showthat the proposed Joint QR is comparable to CM in termsof the prediction accuracy and has less computation timeIn our simulation study we also consider the joint fitting ofthe model by treating the binary response as normal and usepffr [27] with Gaussian link denoted by Joint QR (G)

5 Sow Data Application

Our motivating application is an experimental study carriedout at a commercial farm in Oklahoma from July 21 2013 toAugust 19 2013 [38] The study comprises 480 lactating sowsof different parities (ie number of previous pregnancieswhich serves as a surrogate for age and body weight) that

were observed during their first 21 lactation days theirfeed intake was recorded daily as the difference betweenthe feed offer and the feed refusal In addition the studycontains information on the temperature and humidity of thefarrowing rooms each recorded at five minute intervals Thefinal dataset we used for the analysis consists of 475 sows afterfive sows with unreliable measurements were removed by theexperimenters

The experiment was conducted to gain better insightsinto the way that the ambient temperature and humidity ofthe farrowing room affect the feed intake of lactating sowsPrevious studies seem to suggest a reduction in the sowrsquosfeed intake due to heat stress above 29∘C sows decrease feedintake by 05 kg per additional degree in temperature [9]Studying the effect of heat stress on lactating sows is a very

Journal of Probability and Statistics 7

Table 2 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 1000Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 168 (001) 165 (001) 162 (001) 160 (001)Pointwise QR 194 (001) 192 (001) 188 (001) 181 (001)Mod CM 193 (001) 188 (001) 187 (001) 187 (001)PQR 172 (001) 161 (001) 151 (001) 148 (001)

Normal 10

Joint QR 545 (002) 497 (002) 466 (002) 465 (002)Pointwise QR 564 (002) 513 (002) 478 (002) 475 (002)Mod CM 569 (002) 521 (002) 487 (002) 485 (002)PQR 585 (002) 537 (002) 481 (002) 460 (002)

Normal 5

Joint QR 734 (003) 654 (002) 594 (002) 585 (002)Pointwise QR 753 (003) 669 (002) 604 (002) 594 (002)Mod CM 753 (003) 677 (002) 618 (002) 606 (002)PQR 784 (003) 705 (002) 616 (002) 581 (002)

Normal 2

Joint QR 997 (003) 870 (003) 762 (002) 738 (002)Pointwise QR 1015 (003) 885 (003) 771 (002) 745 (002)Mod CM 1012 (003) 893 (003) 787 (003) 760 (003)PQR 1055 (004) 934 (003) 791 (003) 734 (002)

Normal 1

Joint QR 1169 (004) 1010 (003) 868 (003) 832 (003)Pointwise QR 1188 (004) 1025 (004) 877 (003) 839 (003)Mod CM 1185 (004) 1037 (004) 896 (003) 857 (003)PQR 1229 (004) 1077 (004) 902 (003) 829 (003)

Mixture 150

Joint QR 456 (002) 444 (002) 433 (003) 368 (003)Pointwise QR 434 (003) 424 (002) 420 (003) 366 (004)Mod CM 468 (003) 459 (002) 438 (003) 401 (003)PQR 784 (003) 645 (002) 529 (002) 319 (001)

Mixture 10

Joint QR 756 (003) 679 (002) 614 (003) 502 (003)Pointwise QR 749 (003) 661 (002) 588 (003) 502 (003)Mod CM 767 (004) 697 (003) 617 (003) 552 (004)PQR 1011 (004) 829 (003) 671 (002) 362 (002)

Mixture 5

Joint QR 920 (004) 713 (003) 694 (003) 56 (003)Pointwise QR 917 (004) 701 (003) 671 (003) 558 (003)Mod CM 932 (004) 737 (003) 708 (003) 618 (004)PQR 1146 (004) 936 (003) 749 (003) 432 (002)

Mixture 2

Joint QR 1157 (005) 902 (003) 805 (003) 635 (003)Pointwise QR 1158 (005) 898 (004) 790 (003) 631 (003)Mod CM 1169 (005) 938 (004) 838 (004) 701 (004)PQR 1349 (005) 1095 (004) 861 (003) 532 (002)

Mixture 1

Joint QR 1318 (005) 1031 (004) 879 (004) 679 (003)Pointwise QR 1321 (005) 1028 (004) 869 (004) 673 (003)Mod CM 1326 (005) 1067 (004) 921 (004) 747 (004)PQR 1493 (006) 1205 (004) 935 (004) 596 (002)

important scientific question because of a couple of reasonsFirst the reduction of feed intake of the lactating sows isassociated with a decrease in both their bodyweight (BW)and milk production as well as the weight gain of their litter[10 39 40] Sows with poor feed intake during lactationcontinue the subsequent reproductive period with negativeenergy balance [41] which leads to preventing the onset of anew reproductive cycle Second heat stress reduces farrowingrate (number of sows that deliver a new litter) and number

of piglets born [42] the reduction in reproduction due toseasonality is estimated to cost 300 million dollars per yearfor the swine industry [11] Economic losses are estimated toincrease [43] because high temperatures are likely to occurmore frequently due to global warming [44]

Our primary goal is to understand the thermal needsof the lactating sows for proper feeding behavior duringthe lactation time We are interested in how the interplaybetween the temperature and humidity of the farrowing room

8 Journal of Probability and Statistics

Table 3 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of heteroskedasticitywith a scalar covariateand a sparsely observed functional covariate

Sample size SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

100 150

Joint QR 441 (003) 399 (002) 348 (002) 325 (002)Pointwise QR 540 (003) 484 (003) 426 (002) 421 (003)Mod CM 600 (003) 571 (003) 542 (003) 539 (003)PQR 438 (005) 343 (003) 252 (002) 214 (002)

100 10

Joint QR 693 (003) 640 (003) 589 (002) 576 (002)Pointwise QR 794 (004) 720 (003) 648 (003) 630 (003)Mod CM 841 (004) 822 (004) 813 (004) 805 (004)PQR 776 (006) 657 (004) 547 (003) 508 (002)

100 5

Joint QR 844 (004) 772 (003) 705 (003) 686 (003)Pointwise QR 944 (005) 850 (004) 757 (003) 726 (003)Mod CM 965 (005) 940 (005) 936 (005) 926 (005)PQR 951 (007) 825 (005) 687 (003) 633 (002)

100 2

Joint QR 1068 (005) 963 (004) 860 (003) 830 (003)Pointwise QR 1155 (006) 1029 (005) 900 (004) 854 (003)Mod CM 1127 (005) 1080 (005) 1043 (005) 1040 (006)PQR 1207 (008) 1053 (006) 866 (004) 792 (003)

100 1

Joint QR 1223 (006) 1091 (005) 954 (004) 911 (003)Pointwise QR 1287 (007) 1139 (005) 985 (004) 929 (003)Mod CM 1243 (006) 1180 (006) 1119 (006) 1109 (007)PQR 1370 (009) 1191 (006) 970 (004) 884 (003)

1000 150

Joint QR 287 (001) 242 (001) 189 (001) 165 (001)Pointwise QR 306 (001) 260 (001) 207 (001) 186 (001)Mod CM 313 (001) 265 (001) 210 (001) 191 (001)PQR 310 (001) 247 (001) 178 (001) 148 (001)

1000 10

Joint QR 621 (002) 545 (002) 479 (002) 466 (002)Pointwise QR 638 (002) 559 (002) 490 (002) 477 (002)Mod CM 646 (002) 570 (002) 501 (002) 486 (002)PQR 668 (003) 582 (002) 492 (002) 460 (002)

1000 5

Joint QR 808 (003) 701 (002) 607 (002) 585 (002)Pointwise QR 827 (003) 715 (002) 616 (002) 594 (002)Mod CM 830 (003) 726 (002) 631 (002) 608 (002)PQR 864 (003) 750 (003) 627 (002) 581 (002)

1000 2

Joint QR 1076 (003) 921 (003) 776 (002) 738 (002)Pointwise QR 1095 (004) 934 (003) 785 (003) 745 (002)Mod CM 1092 (003) 945 (003) 801 (003) 760 (003)PQR 1138 (004) 979 (003) 803 (003) 734 (002)

1000 1

Joint QR 1254 (004) 1065 (003) 883 (003) 832 (003)Pointwise QR 1274 (004) 1081 (004) 892 (003) 839 (003)Mod CM 1270 (004) 1091 (004) 910 (003) 856 (003)PQR 1316 (005) 1126 (004) 913 (003) 829 (003)

Table 4 Average computing time (in seconds) of the three approaches that involve estimating the conditional distribution for the case ofhaving a scalar covariate and a densely observed functional covariate

Distribution Method 119899 = 100 119899 = 1000

NormalJoint QR 12 133

Pointwise QR 148 271Mod CM 278 511

MixtureJoint QR 13 154

Pointwise QR 151 296Mod CM 327 532

Journal of Probability and Statistics 9

affects the feed intake demeanor of lactating sows of differentparities We focus on three specific time points during thelactation periodmdashbeginning (lactation day 4) middle (day11) and end (day 18)mdashand the analyses are done separatelyfor each time point We consider two types of responses thatare meant to assess the feed intake behavior using the currentand the previous lactation days The first one quantifies theabsolute change in the feed intake over two consecutive daysand the second one quantifies the relative change and takesinto account the usual sowrsquos feed intake We define themformally after introducing some notation

Let 119865119868119894119895 be the 119895th measurement of the feed intakeobserved for the 119894th sow and denote by the lactation day 119871119863119894119895when 119865119868119894119895 is measured here 119895 = 1 119899119894 Most sows areobserved every day within the first 21 lactation days and thushave 119899119894 = 21 First define the absolute change in the feedintake between two consecutive days asΔ(1)

119894(119895+1)= 119865119868119894(119895+1)minus119865119868119894119895

for 119895 that satisfies 119871119863119894(119895+1)minus119871119863119894119895 = 1 For instanceΔ(1)119894(119895+1) = 0means there was no change in feed intake of sow 119894 betweenthe current day and the previous day while Δ(1)

119894(119895+1)lt 0means

that the feed intake consumed by the 119894th sow in the currentday is smaller than the feed intake consumed in the previousday However the same amount of change in the feed intakemay reflect some stress level for a sow who typically eats a lotand a more serious stress level for a sow that usually has alower appetite For this we define the relative change in thefeed intake by Δ(2)

119894(119895+1)= (119865119868119894(119895+1) minus 119865119868119894119895)(119871119863119894(119895+1) minus 119871119863119894119895) sdot

119879119860 119894 where 119879119860 119894 is the trimmed average of feed intake of 119894thsow calculated as the average feed intake after removing thelowest 20 and highest 20 of the feed intake measurements1198651198681198941 119865119868119894119899119894 taken for the corresponding sow Here 119879119860 119894 issurrogate for the usual amount of feed intake of the 119894th sowTrimmed average is used instead of the common average toremove outliers of very low feed intakes in first few lactatingdays For example consider the situation of two sows sow119894 that typically consumes 10lb food per day and sow 1198941015840 thatconsumes 5lb food per day A reduction of 5lb in the feedintake over two consecutive days corresponds to Δ(2)

119894(119895+1)=

minus50 for the 119894th sow and Δ(2)1198941015840(119895+1)

= minus100 for the 1198941015840th sowClearly both sows are stressed (negative value) but the secondsow is stressed more as its absolute relative change is largerin view of this we refer to the second response as the stressindex Due to the construction of the two types of responsesthe data size varies for lactation days 4 (119895 = 3) 11 (119895 = 10)and 18 (119895 = 17) for the first response Δ(1)

119894(119895+1) we have sample

sizes of 233 350 and 278 whereas for Δ(2)119894(119895+1)

the sample sizesare 362 373 and 336 for the respective lactation days

In this analysis we center the attention on the effect ofthe ambient temperature and humidity on the 1st quartile ofthe proxy stress measures and gain more understanding ofthe food consumption of sows that are most susceptible toheat stress While the association between the feed intake oflactating sows and the ambient conditions of the farrowingroom has been an active research area for some timeaccounting for the temperature daily profile has not beenconsidered yet hitherto Figure 1 displays the temperature

and humidity daily profiles recorded at a frequency of 5-minute window intervals for three different days Preliminaryinvestigation reveals that temperature is negatively correlatedwith humidity at each time this phenomenon is causedbecause the farm uses cool cell panels and fans to controlthe ambient temperature Furthermore it appears that thereis a strong pointwise correlation between temperature andhumidity In view of these observations in our analysis weconsider the daily average of humidity Exploratory analysisof the feed intake behavior of the sows suggests similaritiesfor the sows with parity greater than older sows (ones whoare at their third pregnancy or higher) thus we use a parityindicator instead of the actual parity of the sow The parityindicator 119875119894 is defined as one if the 119894th sow has parity oneand zero otherwise

For the analysis we smooth daily temperature measure-ments of each sow using univariate smoother with 15 cubicregression bases and quadratic penalty REML is used toestimate smoothing parameter The smoothed temperaturecurve for sow 119894rsquos 119895th repeatedmeasure is denoted by119879119894119895(119905) 119905 isin[0 24) and the corresponding daily average humidity isdenoted by119860119867119894119895 Both temperature and average humidity arecentered before being used in the analysis

For convenience we denote the response with Δ 119894119895 byremoving the superscript In this application for fixed 119895 Δ 119894119895corresponds to 119884119894 in Section 2 119875119894 and 119860119867119894119895 correspond toscalar covariates1198831198942 and119879119894119895(119905) and119860119867119894119895 sdot119879119894119895(119905) correspond tofunctional covariates1198831198943(sdot) We first estimate the conditionaldistribution ofΔ 119894119895 given temperature119879119894119895(119905) average humidity119860119867119894119895 parity 119875119894 and interaction 119860119867119894119895 sdot 119879119894119895(119905) Specifically foreach of lactation days of interest (119895 = 3 10 and 17) we create aset of 100 equispaced grid of points between the fifth smallestand fifth largest values of Δ 119894119895rsquos and denote the grids withD = 119889ℓ ℓ = 1 100 Then we create artificial binaryresponses 1(Δ 119894119895 le 119889ℓ) ℓ = 1 100 and fit the followingmodel for 119865119894119895(119889ℓ) = 119864[1(Δ 119894119895 le 119889ℓ) | 119879119894119895(119905) 119860119867119894119895 119875119894]

119864 [1 (Δ 119894119895 le 119889ℓ) | 119879119894119895 (119905) 119860119867119894119895 119875119894] = 119892minus1 1205730 (119889ℓ)

+ 1205731 (119889ℓ) 119875119894 + 1205732 (119889ℓ) 119860119867119894119895 + int1205733 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

+ 119860119867119894119895 int1205734 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

(7)

where 1205730(sdot) is a smooth intercept 1205731(sdot) quantifies the smootheffect of young sows1205732(sdot) describes the effect of the humidityand 1205733(sdot 119905) and 1205734(sdot 119905) quantify the effect of the temperatureat time 119905 as well as the interaction between the temperature attime 119905 and average humidityWemodel1205730(sdot)using 20 univari-ate basis functions 1205731(sdot) and 1205732(sdot) using five univariate basisfunctions and 1205733(sdot sdot) and 1205734(sdot sdot) using tensor product of twounivariate bases functions (total of 25 functions)Throughoutthe analysis cubic B-spline bases are used and REML isused for estimating smoothing parameters The estimatedconditional distribution denoted by 119865119894119895(119889) is monotonizedby fitting isotonic regression to (119889ℓ 119865119894119895(119889ℓ)) ℓ = 10 90ten smallest and ten largest 119889ℓ and the corresponding valuesof 119865119895(119889ℓ) are removed to avoid boundary effects By abuse of

10 Journal of Probability and Statistics

2425

2627

2829

30

Temperature

6065

7075

80

Humidity

16H

20H

18H

22H

14H

12H

10H

14H8H4H0H 2H 6H

time of day

16H

20H

12H

22H

18H

10H

14H

14H 0H 6H2H 8H4H

time of day

Figure 1 Temperature (∘C) and humidity () observed profiles (dashed) for three randomly selected days and the corresponding smoothedones (solid) the x-axis begins at 14H (2PM)

2324

2526

2728

29

Lactation day 4

203040median607080

2324

2526

2728

29

Lactation day 11

203040median607080

2324

2526

2728

29

Lactation day 18

203040median607080

8H6H4H2H0H 12H

14H

10H

14H

22H

16H

20H

18H

time of day

8H4H2H0H 6H22H

12H

16H

20H

14H

14H

10H

18H

time of day

4H 6H0H 2H 8H20H

12H

14H

14H

22H

16H

18H

10H

time of day

Figure 2 Temperature curves with which prediction of quantiles is made Dashed black line is pointwise average of temperature curves andsolid lines are pointwise quartiles all curves are smoothed

notation 119865119894119895(119889) denotes the resulting monotonized estimateddistribution Finally we obtain estimated first quartiles iequantiles at 120591 = 025 level by inverting 119865119894119895(119889) namely 119876(120591 =025 | 119879119894119895(119905) 119860119867119894119895 119875119894) = inf119889 119865119894119895(119889) ge 025

To understand the relationship between the lactatingsows feed intake and the thermal condition of the farrowingroom we systematically compare and study the predictedquantiles of two responses at combinations of different valuesof temperature humidity and parity For each of threelactation days (119895 = 3 10 17) we consider three values ofaverage humidity (first quartile median and third quartile)and two levels of parity (0 for older sows and 1 for youngersows) Based on the experimentersrsquo interest for the functional

covariate119879119894119895(sdot)we consider seven smooth temperature curvesgiven in Figure 2 Each of these curves is obtained by firstcalculating pointwise quantiles of temperature at five-minuteintervals for a specific level and then smoothing it weconsidered quantiles levels 120578 = 02 03 and 08 In shortfor each of three lactation days we obtain the first quartileof two responses for 42 different combinations (3 humidityvalues times 2 parity levels times 7 temperature curves) using theproposed method To avoid extrapolation we ascertain that(i) there are reasonablymany observedmeasurements at eachof the combinations and (ii) bottom 25 of the responses arenot dominantly from one of the parity group see distributionof each response by the parity in Figure 3

Journal of Probability and Statistics 11

Response (2) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus10

00

10

20

Resp

onse

(2)

minus15

minus05

05

10

Resp

onse

(2)

minus2minus1

01

Resp

onse

(2)

Response (1) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus4minus2

02

46

Resp

onse

(1)

minus6minus4

minus20

24

6Re

spon

se (1

)

minus4minus2

02

46

Resp

onse

(1)

Figure 3 Top panels back-to-back boxplots of the absolute change in feed intake at a specific day by parity Bottom panels back-to-backboxplots of the relative change in feed intake at a specific day by parity

The resulting predicted quantiles are shown in Figure 4Here we focus our discussion on predicted quantile of Δ(2)

119894(119895+1)

at quantile level 120591 = 025 for lactation day 4 (119895 = 3)mdashthefirst plot of the second row in Figure 4 The results suggestthat the feed intake of older sows (parity 119875119894 = 0 greylines) is less affected by high temperatures than younger sows(black lines) this finding is in agreement with Bloemhof etal [42] We also observe that the effects of humidity andtemperature on feed intake change are strongly intertwinedFor illustration we focus on lactation day 4 (119895 = 3) again foryounger sows (black lines) For medium humidity (dashedlines) their feed intake stays pretty constant as temperatureincreases while for low and high humidity levels (solid anddotted lines respectively) it changes with an opposite direc-tion Specifically when temperature increases the predictedfirst quartile of Δ(2)

119894(119895+1)increases for low humidity (solid line)

whereas it decreases for high humidity (dotted line) Ourresults imply that high humidity (dotted line) is related toa negative impact of high temperature on feed intake whilelow humidity (solid line) alleviates it and this finding isconsistent with a previous study [45] The analysis resultsuggests to keep low humidity levels in order to maintainhealthy feed intake behavior when ambient temperature isabove 60th percentile high humidity levels are desirable forcooler ambient temperature

Interpretation of the other results is similar While theeffects of covariates on feed intake are less apparent toward

the end of lactation period we still observe similar patternacross all three lactation days For the 11th day (119895 = 10) the25th quantile of the feed intake is predicted to decrease whenthe temperature stays below the 40th percentile regardlessof humidity level and sows age However it starts increas-ing with low humidity while it continues decreasing withhigh humidity when the temperature rises above the 40thpercentile Similarly for the 18th day (119895 = 17) when thetemperature rises above the 60th percentile the predictedfirst quartile increases with low humidity while it decreaseswith high humidity The effect of temperature on feed intakeseems less obvious for lactation days 11 and 18 than for day 4while the effect may be due to lactation day it may also be aresult of other factors such asmore fluctuation and variabilityin temperature curves on day 4 than on other two days(see Figure 2) Overall we conclude that high humidity andtemperature affect the sows feed intake behavior negativelyand young sows (parity one) are more sensitive to heat stressthan older sows (higher parity) especially at the beginning oflactation period

6 Discussion

The proposed modeling framework opens up a couple offuture research directions A first research avenue is todevelop significance tests of null covariate effect Testingfor the null effect of a covariate on the conditional dis-tribution of the response is equivalent to testing that the

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

4 Journal of Probability and Statistics

discusses an alternative way to deal with the rank-deficientmatrices in the context of restricted maximum likelihood(REML) estimation Here we do not account for the depen-dence over 119910 see Scheipl et al [22] for a general formulationSee also Ivanescu et al [27] who use the mixed modelrepresentation of a similar regression model to (5) but with aGaussian functional responseThe smoothing parameters areestimated using REML

32 Extension to Nonlinear Model One advantage of theproposed framework is that it can be easily extended toallow for more flexible effects ie extending the ideasto accommodate multiple covariates scalar or functionaland varied types of effects In particular the smooth effect11988311989421205732(119910) can be replaced by ℎ1(1198831198942 119910) and int1198831198943(119905)1205733(119905 119910)119889119905byint ℎ21198831198943(119905) 119905 119910119889119905 where ℎ1(sdot sdot) and ℎ2(sdot sdot sdot) are unknownbivariate and trivariate smooth functions respectively seeScheipl et al [22] and Kim et al [28] These changes requirelittle additional computational effort The modeling andestimation follow roughly similar ideas as Scheipl et al [22]We consider the nonlinear model in the simulation studyfor the case of having a scalar covariate only ie ℎ1(119883119894 119910)and the corresponding results are presented in Section S11of the Supplementary Materials The results show excellentprediction performance compared to the competitive nonlin-ear quantile regressionmethod namely constrained B-splinesmoothing (COBS) [29]

33 Extension to Sparse and Noisy Functional CovariatesIn practice the functional covariates are often observed atirregular times across the units and the measurements arepossibly corrupted by noises In such case one needs tofirst smooth and denoise the trajectories before fitting Whenthe sampling design of the functional covariate is dense thecommon approach is to smooth each trajectory using splinesor local polynomial smoothing as proposed in Ramsay andSilverman [3] and Zhang and Chen [30] When the designis sparse the smoothing can be done by pooling all thesubjects and following the PACE method proposed in Yaoet al [31] As recovering the trajectories has been extensivelydiscussed in the literatures we do not review the procedureshere Instead we discuss some available computing resourcesthat can be used to fit these methods In our numericalstudy we used fpcasc function in the refund R package[32] for recovering the latent trajectories irrespective of asampling design (dense or sparse) Alternatively one can usefpcaface [33] in refund for regular dense design andfacespare [34] in the R package face [35] for irregularsparse design Once the latent trajectories are estimated theycan be used in the fitting criterion (4)

34 Estimation of Conditional Quantile Let 1 0 1 and2 be parameter estimates in (5) It follows that the estimateddistribution function 119865119884119894|X1198941 11988311989421198831198943(sdot)(119910) can be obtained byplugging in the estimated coefficients The 120591th conditionalquantile is estimated by inverting the estimated distributionie 119876119884119894|X1198941 1198831198942 1198831198943(sdot)(120591) = inf119910 119865119884119894|X1198941 1198831198942 1198831198943(sdot)(119910) ge 120591 The

estimated distribution function is not a monotonic functionyet In practice we suggest to first apply a monotonizationmethod as described in Section 35 and then estimate theconditional quantiles by inverting the resulting estimateddistribution

35 Monotonization and Implementation While a condi-tional quantile function is nondecreasing the resulting esti-mated quantiles may not be Two approaches are widely usedone is to monotonize the estimated conditional distributionfunction and the other is to monotonize the estimatedconditional quantile function We choose the former as119865119884119894|X1198941 11988311989421198831198943(sdot)(119910) is readily available at dense grid points 119910ℓrsquosWeuse an isotonic regressionmodel [36] formonotonizationwhich imposes an order restriction this is done by using the Rfunction isoreg Other monotonization approaches includeChernozhukov et al [37] which was employed in Kato [16]

Our approach is implemented by first creating an artificialbinary response and then fitting a penalized function-on-function regression model and using the logit link functionFitting models in (4) can be done by extending the ideasof Ivanescu et al [27] for Gaussian functional responsethe extension of the model to the non-Gaussian functionalresponse has recently been studied and implemented byScheipl et al [22] as the pffr function in refund package[32]

4 Simulation Study

41 Simulation Setting In this section we evaluate theempirical performance of the proposed method We presentresults for the case when we have both functional and scalarcovariates additional results when there is only a singlescalar or a single functional covariate are discussed in theSupplementary Materials Section S1

Suppose the observed data for the 119894th subject are[119884119894 1198831119894 (1198821198941 1199051198941) (119882119894119898119894 119905119894119898119894)] 119905119894119895 isin [0 10] where

1198831119894 119894119894119889sim 119880119899119894119891(minus16 16)119882119894119895 = 1198832119894(119905119894119895) + 120598119894119895 for 1 le 119894 le 119899 1 le119895 le 119898119894 Let1198832119894(119905119894119895) = 120583(119905119894119895)+sum4119896=1 120585119894119896120601119896(119905119894119895)+120598119894119895 where 120583(119905) =119905 + sin(119905) 120601119896(119905) = cos(119896 + 1)12058711990510radic5 for odd values of 119896120601119896(119905) = sin11989612058711990510radic5 for even values of 119896 120585119894119896 119894119894119889sim 119873(0 120582119896)(1205821 1205822 1205823 1205824) = 16 9 756 506 and 120598119894119895 119894119894119889sim 119873(0 1205902120598 ) Weassume three cases for generating response 119884119894

(i) Gaussian119884119894 | 1198831119894 1198832119894(sdot) sim 119873(2 int1198832119894(119905)120573(119905)119889119905+2119883111989452) this corresponds to the quantile regressionmodel119876119884|1198831 1198832(sdot)(120591) = 2 int1198832119894(119905)120573(119905)119889119905 + 21198831119894 + 5Φminus1(120591)whereΦ(sdot) is the distribution function of the standardnormal

(ii) Mixture ofGaussians119884119894 | 11988311198941198832119894(sdot) sim 05119873(int1198832119894(119905)120573(119905)119889119905+119883111989412) + 05119873(3 int1198832119894(119905)120573(119905)119889119905+ 31198831119894 42) where thetrue quantiles can be approximated numerically byusing qnorMix function in the R package norMix

(iii) Gaussian with heterogeneous error 119884119894 | 11988311198941198832119894(sdot) sim 119873(2 int1198832119894(119905)120573(119905)119889119905 + 21198831119894 52 int1198832119894(119905)2119889119905

Journal of Probability and Statistics 5

sum4119896=1 120582119896) the true quantiles are given by119876119884|1198831 1198832(sdot)(120591) = 2 int1198832119894(119905)120573(119905)119889119905 + 21198831119894 +

5radicint1198832119894(119905)2119889119905sum4119896=1 120582119896Φminus1(120591)

Let 120573(119905) = sum4119896=1 120573119896120601119896(119905) where 120573119896 = 1 for 119896 = 1 4For each case we use different combinations of signal to

noise ratio (SNR) sample size and sampling designs to gen-erate 500 simulated datasets We define SNR asradicsum4119896=1 120582119896120590120598and we consider five levels of noise SNR = 150 10 5 2 1Two levels of sample size are 119899 = 100 and 119899 = 1000 Twosampling designs are considered (i) sparse design where 119905119894119895 119895 = 1 119898 are119898 = 15 randomly selected points from a setof 30 equispaced grids in [0 10] and (ii) dense design wherethe sampling points 119905119894119895 = 119905119895 119895 = 1 119898 are 119898 = 30equispaced time points in [0 10]

The performance is evaluated on a test set of 100 subjectsfor whichwe have 1198831119894lowast (119882119894lowast119895 119905119894lowast119895) 119895 = 1 119898 available interms of mean absolute error (MAE) for quantile levels 120591 =005 01 025 and 05

MAE (120591) = 1100100

sum119894lowast=1

10038161003816100381610038161003816119876119894lowast (120591) minus 119876119894lowast (120591)10038161003816100381610038161003816 (6)

42 CompetingMethods Wedenote the proposedmethod byJoint QR to emphasize the single step estimation approachWe compare our method with two alternative approaches(1) a variant of our proposed approach using pointwiseestimation denoted by Pointwise QRThis approach consistsof fittingmultiple regressionmodels with binomial link func-tion as implemented by the penalized functional regressionpfr developed byGoldsmith et al [7] of therefundpackagefor generalized scalar responses (2) a modified version ofthe CM method denoted by Mod CM that we developedto account for additional scalar covariates and which fitsmultiple generalized linear models with scalar covariates andfPC scores as predictors (3) a linear quantile regressionapproach using the quantile loss function and the partialquantile regression bases for functional covariates proposedby Yu et al [19] and denoted by PQRNotice that although theformulation of the first twomethods implicitly accounts for avarying effect of the covariates on the response distributionthey do not ensure that this effect is smooth The thirdapproach can only estimate a specific quantile rather than theentire conditional distribution Note that all the competingmethods are monotonized for a fair comparison

The R function pfr can incorporate both scalarvectorand functional predictors by adopting a mixed effects modelframework The functional covariates are presmoothed byfPC analysis [31] Throughout the simulation study we fixPVE as 095 for fPC analysis to determine the number ofprincipal components and use REML to select the smoothingparameters for our proposed methods Other basis settingsare set to their default values We use 100 equally distancedpoints between the minimum andmaximum of the observed119884119894rsquos to set the grid 119910ℓ ℓ = 1 119871 for the conditionaldistribution function

43 Simulation Results Tables 1 and 2 show the accuracyof the quantile prediction for the two cases (normal andmixture) when the functional covariate is observed sparselyand the sample size is 119899 = 100 (Table 1) and 119899 = 1000(Table 2) Table 3 presents the prediction accuracy for thecase of heteroskedasticity with sparsely observed functionalcovariates The results based on dense sampling design showsimilar patterns and thus are relegated to the supplement seeSection S13 The comparison of running times is presentedin Table 4

For the case when the response is Gaussian Tables 1and 2 suggest that the Joint QR typically outperforms itscompetitors especially for lower quantile levels (120591 = 005and 120591 = 01) For very small noise level (SNR = 150) PQRperforms the best followed closely by the proposed Joint QRThe variant Pointwise QR which has a poorer performanceis generally better than the modified CM approach Asexpected as the sample size increases (119899 = 1000) all theaccuracy results improve the proposed Joint QR continuesto yield most accurate quantiles for the low quantile levelsFor mixture of Gaussians the results are somewhat similarThe accuracy of the quantile estimators with the PointwiseQR improves greatly in fact the Joint QR and PointwiseQR outperform the other approaches for quantile levels 120591 =005 01 025 irrespective of the SNR Finally Table 3 showsthat the results for Gaussian with heterogeneous error areclose to those for the case of Gaussian Again the proposedmethod has competitive performance in terms of predictionaccuracy

Table 4 compares the three methods that involve esti-mating the conditional distribution in terms of the runningtime required for fitting The times are reported based ona computer with a 23 GHz CPU and 8GB of RAM Notsurprisingly by fitting the model a single time Joint QR is thefastest in some cases being order ofmagnitude faster than therest Pointwise QR can be up to twice as fast as Mod CM

For completeness we also compare our proposedmethodto the appropriate competitivemethods for the cases (1) whenthere is a single scalar covariate and (2) when there is a singlefunctional covariate In the SupplementaryMaterials SectionS11 discusses the former case and compares Joint QR andPointwise QR with the linear quantile regression (LQR) [12]implemented by rq function in the R package quantreg andthe constrained B-splines nonparametric regression quantiles(COBS) implemented by the cobs function in the R pack-age COBS [29] in an extensive simulation experiment thatinvolves both linear quantile settings and nonlinear quantilesettings Overall the results show that the proposed methodshave similar behavior as LQR see Table S1 Furthermore weconsider the proposed method and its variant with nonlinearmodeling of the conditional distribution as discussed inSection 32 which we denote with Joint QR (NL) for jointfitting andPointwiseQR (NL) for pointwise fittingNonlinearversions of the proposed methods have an excellent MAEperformance which is comparable to or better than that ofthe COBS method

Finally Section S12 in the Supplementary Materialsdiscusses the simulation study for the case of having a singlefunctional covariate and compares the proposed methods

6 Journal of Probability and Statistics

Table 1 Average MAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 100Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 367 (003) 353 (003) 330 (002) 317 (002)Pointwise QR 496 (003) 461 (003) 422 (002) 418 (002)Mod CM 604 (003) 581 (003) 555 (003) 514 (003)PQR 317 (004) 271 (003) 231 (002) 216 (002)

Normal 10

Joint QR 632 (003) 600 (003) 576 (002) 573 (002)Pointwise QR 744 (004) 685 (003) 639 (003) 628 (003)Mod CM 820 (004) 810 (004) 804 (004) 801 (004)PQR 682 (005) 611 (004) 534 (003) 509 (002)

Normal 5

Joint QR 784 (004) 734 (003) 693 (003) 684 (003)Pointwise QR 891 (004) 812 (004) 745 (003) 726 (003)Mod CM 934 (004) 923 (004) 914 (005) 906 (005)PQR 868 (006) 781 (005) 674 (003) 634 (002)

Normal 2

Joint QR 1005 (005) 922 (004) 847 (003) 828 (003)Pointwise QR 1091 (006) 986 (004) 887 (004) 854 (003)Mod CM 1085 (005) 1055 (005) 1034 (006) 1021 (006)PQR 1121 (008) 1003 (006) 856 (004) 796 (003)

Normal 1

Joint QR 1150 (006) 1041 (005) 940 (004) 911 (003)Pointwise QR 1212 (006) 1088 (005) 970 (004) 930 (003)Mod CM 1195 (006) 1146 (006) 1107 (006) 1105 (007 )PQR 1282 (008) 1138 (006) 960 (004) 886 (003)

Mixture 150

Joint QR 692 (006) 623 (006) 616 (006) 481 (006)Pointwise QR 810 (008) 680 (006) 666 (006) 525 (006)Mod CM 918 (007) 899 (007) 893 (007) 790 (007)PQR 843 (006) 718 (004) 622 (004) 548 (014)

Mixture 10

Joint QR 902 (006) 795 (005) 785 (005) 619 (006)Pointwise QR 1011 (007) 852 (005) 795 (005) 642 (006)Mod CM 1133 (007) 1095 (007) 1079 (007) 980 (008)PQR 1072 (008) 899 (005) 763 (004) 533 (009)

Mixture 5

Joint QR 1018 (006) 891 (005) 853 (005) 682 (005)Pointwise QR 1118 (007) 938 (005) 858 (005) 698 (005)Mod CM 1219 (007) 1175 (007) 1152 (007) 1047 (008)PQR 1212 (009) 1012 (006) 840 (004) 573 (007)

Mixture 2

Joint QR 1193 (007) 1026 (005) 946 (005) 761 (005)Pointwise QR 1268 (008) 1063 (006) 951 (005) 770 (005)Mod CM 1333 (008) 1260 (008) 1227 (008) 1108 (010)PQR 1416 (010) 1172 (006) 955 (004) 641 (005)

Mixture 1

Joint QR 1317 (008) 1119 (005) 1006 (005) 804 (005)Pointwise QR 1371 (009) 1144 (006) 1010 (005) 813 (005)Mod CM 1416 (008) 1322 (008) 1271 (009) 1144 (011)PQR 1544 (011) 1284 (007) 1023 (004) 689 (004)

with CM in terms of MAE as well as computation timesee results displayed in Tables S2 and S3 The results showthat the proposed Joint QR is comparable to CM in termsof the prediction accuracy and has less computation timeIn our simulation study we also consider the joint fitting ofthe model by treating the binary response as normal and usepffr [27] with Gaussian link denoted by Joint QR (G)

5 Sow Data Application

Our motivating application is an experimental study carriedout at a commercial farm in Oklahoma from July 21 2013 toAugust 19 2013 [38] The study comprises 480 lactating sowsof different parities (ie number of previous pregnancieswhich serves as a surrogate for age and body weight) that

were observed during their first 21 lactation days theirfeed intake was recorded daily as the difference betweenthe feed offer and the feed refusal In addition the studycontains information on the temperature and humidity of thefarrowing rooms each recorded at five minute intervals Thefinal dataset we used for the analysis consists of 475 sows afterfive sows with unreliable measurements were removed by theexperimenters

The experiment was conducted to gain better insightsinto the way that the ambient temperature and humidity ofthe farrowing room affect the feed intake of lactating sowsPrevious studies seem to suggest a reduction in the sowrsquosfeed intake due to heat stress above 29∘C sows decrease feedintake by 05 kg per additional degree in temperature [9]Studying the effect of heat stress on lactating sows is a very

Journal of Probability and Statistics 7

Table 2 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 1000Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 168 (001) 165 (001) 162 (001) 160 (001)Pointwise QR 194 (001) 192 (001) 188 (001) 181 (001)Mod CM 193 (001) 188 (001) 187 (001) 187 (001)PQR 172 (001) 161 (001) 151 (001) 148 (001)

Normal 10

Joint QR 545 (002) 497 (002) 466 (002) 465 (002)Pointwise QR 564 (002) 513 (002) 478 (002) 475 (002)Mod CM 569 (002) 521 (002) 487 (002) 485 (002)PQR 585 (002) 537 (002) 481 (002) 460 (002)

Normal 5

Joint QR 734 (003) 654 (002) 594 (002) 585 (002)Pointwise QR 753 (003) 669 (002) 604 (002) 594 (002)Mod CM 753 (003) 677 (002) 618 (002) 606 (002)PQR 784 (003) 705 (002) 616 (002) 581 (002)

Normal 2

Joint QR 997 (003) 870 (003) 762 (002) 738 (002)Pointwise QR 1015 (003) 885 (003) 771 (002) 745 (002)Mod CM 1012 (003) 893 (003) 787 (003) 760 (003)PQR 1055 (004) 934 (003) 791 (003) 734 (002)

Normal 1

Joint QR 1169 (004) 1010 (003) 868 (003) 832 (003)Pointwise QR 1188 (004) 1025 (004) 877 (003) 839 (003)Mod CM 1185 (004) 1037 (004) 896 (003) 857 (003)PQR 1229 (004) 1077 (004) 902 (003) 829 (003)

Mixture 150

Joint QR 456 (002) 444 (002) 433 (003) 368 (003)Pointwise QR 434 (003) 424 (002) 420 (003) 366 (004)Mod CM 468 (003) 459 (002) 438 (003) 401 (003)PQR 784 (003) 645 (002) 529 (002) 319 (001)

Mixture 10

Joint QR 756 (003) 679 (002) 614 (003) 502 (003)Pointwise QR 749 (003) 661 (002) 588 (003) 502 (003)Mod CM 767 (004) 697 (003) 617 (003) 552 (004)PQR 1011 (004) 829 (003) 671 (002) 362 (002)

Mixture 5

Joint QR 920 (004) 713 (003) 694 (003) 56 (003)Pointwise QR 917 (004) 701 (003) 671 (003) 558 (003)Mod CM 932 (004) 737 (003) 708 (003) 618 (004)PQR 1146 (004) 936 (003) 749 (003) 432 (002)

Mixture 2

Joint QR 1157 (005) 902 (003) 805 (003) 635 (003)Pointwise QR 1158 (005) 898 (004) 790 (003) 631 (003)Mod CM 1169 (005) 938 (004) 838 (004) 701 (004)PQR 1349 (005) 1095 (004) 861 (003) 532 (002)

Mixture 1

Joint QR 1318 (005) 1031 (004) 879 (004) 679 (003)Pointwise QR 1321 (005) 1028 (004) 869 (004) 673 (003)Mod CM 1326 (005) 1067 (004) 921 (004) 747 (004)PQR 1493 (006) 1205 (004) 935 (004) 596 (002)

important scientific question because of a couple of reasonsFirst the reduction of feed intake of the lactating sows isassociated with a decrease in both their bodyweight (BW)and milk production as well as the weight gain of their litter[10 39 40] Sows with poor feed intake during lactationcontinue the subsequent reproductive period with negativeenergy balance [41] which leads to preventing the onset of anew reproductive cycle Second heat stress reduces farrowingrate (number of sows that deliver a new litter) and number

of piglets born [42] the reduction in reproduction due toseasonality is estimated to cost 300 million dollars per yearfor the swine industry [11] Economic losses are estimated toincrease [43] because high temperatures are likely to occurmore frequently due to global warming [44]

Our primary goal is to understand the thermal needsof the lactating sows for proper feeding behavior duringthe lactation time We are interested in how the interplaybetween the temperature and humidity of the farrowing room

8 Journal of Probability and Statistics

Table 3 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of heteroskedasticitywith a scalar covariateand a sparsely observed functional covariate

Sample size SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

100 150

Joint QR 441 (003) 399 (002) 348 (002) 325 (002)Pointwise QR 540 (003) 484 (003) 426 (002) 421 (003)Mod CM 600 (003) 571 (003) 542 (003) 539 (003)PQR 438 (005) 343 (003) 252 (002) 214 (002)

100 10

Joint QR 693 (003) 640 (003) 589 (002) 576 (002)Pointwise QR 794 (004) 720 (003) 648 (003) 630 (003)Mod CM 841 (004) 822 (004) 813 (004) 805 (004)PQR 776 (006) 657 (004) 547 (003) 508 (002)

100 5

Joint QR 844 (004) 772 (003) 705 (003) 686 (003)Pointwise QR 944 (005) 850 (004) 757 (003) 726 (003)Mod CM 965 (005) 940 (005) 936 (005) 926 (005)PQR 951 (007) 825 (005) 687 (003) 633 (002)

100 2

Joint QR 1068 (005) 963 (004) 860 (003) 830 (003)Pointwise QR 1155 (006) 1029 (005) 900 (004) 854 (003)Mod CM 1127 (005) 1080 (005) 1043 (005) 1040 (006)PQR 1207 (008) 1053 (006) 866 (004) 792 (003)

100 1

Joint QR 1223 (006) 1091 (005) 954 (004) 911 (003)Pointwise QR 1287 (007) 1139 (005) 985 (004) 929 (003)Mod CM 1243 (006) 1180 (006) 1119 (006) 1109 (007)PQR 1370 (009) 1191 (006) 970 (004) 884 (003)

1000 150

Joint QR 287 (001) 242 (001) 189 (001) 165 (001)Pointwise QR 306 (001) 260 (001) 207 (001) 186 (001)Mod CM 313 (001) 265 (001) 210 (001) 191 (001)PQR 310 (001) 247 (001) 178 (001) 148 (001)

1000 10

Joint QR 621 (002) 545 (002) 479 (002) 466 (002)Pointwise QR 638 (002) 559 (002) 490 (002) 477 (002)Mod CM 646 (002) 570 (002) 501 (002) 486 (002)PQR 668 (003) 582 (002) 492 (002) 460 (002)

1000 5

Joint QR 808 (003) 701 (002) 607 (002) 585 (002)Pointwise QR 827 (003) 715 (002) 616 (002) 594 (002)Mod CM 830 (003) 726 (002) 631 (002) 608 (002)PQR 864 (003) 750 (003) 627 (002) 581 (002)

1000 2

Joint QR 1076 (003) 921 (003) 776 (002) 738 (002)Pointwise QR 1095 (004) 934 (003) 785 (003) 745 (002)Mod CM 1092 (003) 945 (003) 801 (003) 760 (003)PQR 1138 (004) 979 (003) 803 (003) 734 (002)

1000 1

Joint QR 1254 (004) 1065 (003) 883 (003) 832 (003)Pointwise QR 1274 (004) 1081 (004) 892 (003) 839 (003)Mod CM 1270 (004) 1091 (004) 910 (003) 856 (003)PQR 1316 (005) 1126 (004) 913 (003) 829 (003)

Table 4 Average computing time (in seconds) of the three approaches that involve estimating the conditional distribution for the case ofhaving a scalar covariate and a densely observed functional covariate

Distribution Method 119899 = 100 119899 = 1000

NormalJoint QR 12 133

Pointwise QR 148 271Mod CM 278 511

MixtureJoint QR 13 154

Pointwise QR 151 296Mod CM 327 532

Journal of Probability and Statistics 9

affects the feed intake demeanor of lactating sows of differentparities We focus on three specific time points during thelactation periodmdashbeginning (lactation day 4) middle (day11) and end (day 18)mdashand the analyses are done separatelyfor each time point We consider two types of responses thatare meant to assess the feed intake behavior using the currentand the previous lactation days The first one quantifies theabsolute change in the feed intake over two consecutive daysand the second one quantifies the relative change and takesinto account the usual sowrsquos feed intake We define themformally after introducing some notation

Let 119865119868119894119895 be the 119895th measurement of the feed intakeobserved for the 119894th sow and denote by the lactation day 119871119863119894119895when 119865119868119894119895 is measured here 119895 = 1 119899119894 Most sows areobserved every day within the first 21 lactation days and thushave 119899119894 = 21 First define the absolute change in the feedintake between two consecutive days asΔ(1)

119894(119895+1)= 119865119868119894(119895+1)minus119865119868119894119895

for 119895 that satisfies 119871119863119894(119895+1)minus119871119863119894119895 = 1 For instanceΔ(1)119894(119895+1) = 0means there was no change in feed intake of sow 119894 betweenthe current day and the previous day while Δ(1)

119894(119895+1)lt 0means

that the feed intake consumed by the 119894th sow in the currentday is smaller than the feed intake consumed in the previousday However the same amount of change in the feed intakemay reflect some stress level for a sow who typically eats a lotand a more serious stress level for a sow that usually has alower appetite For this we define the relative change in thefeed intake by Δ(2)

119894(119895+1)= (119865119868119894(119895+1) minus 119865119868119894119895)(119871119863119894(119895+1) minus 119871119863119894119895) sdot

119879119860 119894 where 119879119860 119894 is the trimmed average of feed intake of 119894thsow calculated as the average feed intake after removing thelowest 20 and highest 20 of the feed intake measurements1198651198681198941 119865119868119894119899119894 taken for the corresponding sow Here 119879119860 119894 issurrogate for the usual amount of feed intake of the 119894th sowTrimmed average is used instead of the common average toremove outliers of very low feed intakes in first few lactatingdays For example consider the situation of two sows sow119894 that typically consumes 10lb food per day and sow 1198941015840 thatconsumes 5lb food per day A reduction of 5lb in the feedintake over two consecutive days corresponds to Δ(2)

119894(119895+1)=

minus50 for the 119894th sow and Δ(2)1198941015840(119895+1)

= minus100 for the 1198941015840th sowClearly both sows are stressed (negative value) but the secondsow is stressed more as its absolute relative change is largerin view of this we refer to the second response as the stressindex Due to the construction of the two types of responsesthe data size varies for lactation days 4 (119895 = 3) 11 (119895 = 10)and 18 (119895 = 17) for the first response Δ(1)

119894(119895+1) we have sample

sizes of 233 350 and 278 whereas for Δ(2)119894(119895+1)

the sample sizesare 362 373 and 336 for the respective lactation days

In this analysis we center the attention on the effect ofthe ambient temperature and humidity on the 1st quartile ofthe proxy stress measures and gain more understanding ofthe food consumption of sows that are most susceptible toheat stress While the association between the feed intake oflactating sows and the ambient conditions of the farrowingroom has been an active research area for some timeaccounting for the temperature daily profile has not beenconsidered yet hitherto Figure 1 displays the temperature

and humidity daily profiles recorded at a frequency of 5-minute window intervals for three different days Preliminaryinvestigation reveals that temperature is negatively correlatedwith humidity at each time this phenomenon is causedbecause the farm uses cool cell panels and fans to controlthe ambient temperature Furthermore it appears that thereis a strong pointwise correlation between temperature andhumidity In view of these observations in our analysis weconsider the daily average of humidity Exploratory analysisof the feed intake behavior of the sows suggests similaritiesfor the sows with parity greater than older sows (ones whoare at their third pregnancy or higher) thus we use a parityindicator instead of the actual parity of the sow The parityindicator 119875119894 is defined as one if the 119894th sow has parity oneand zero otherwise

For the analysis we smooth daily temperature measure-ments of each sow using univariate smoother with 15 cubicregression bases and quadratic penalty REML is used toestimate smoothing parameter The smoothed temperaturecurve for sow 119894rsquos 119895th repeatedmeasure is denoted by119879119894119895(119905) 119905 isin[0 24) and the corresponding daily average humidity isdenoted by119860119867119894119895 Both temperature and average humidity arecentered before being used in the analysis

For convenience we denote the response with Δ 119894119895 byremoving the superscript In this application for fixed 119895 Δ 119894119895corresponds to 119884119894 in Section 2 119875119894 and 119860119867119894119895 correspond toscalar covariates1198831198942 and119879119894119895(119905) and119860119867119894119895 sdot119879119894119895(119905) correspond tofunctional covariates1198831198943(sdot) We first estimate the conditionaldistribution ofΔ 119894119895 given temperature119879119894119895(119905) average humidity119860119867119894119895 parity 119875119894 and interaction 119860119867119894119895 sdot 119879119894119895(119905) Specifically foreach of lactation days of interest (119895 = 3 10 and 17) we create aset of 100 equispaced grid of points between the fifth smallestand fifth largest values of Δ 119894119895rsquos and denote the grids withD = 119889ℓ ℓ = 1 100 Then we create artificial binaryresponses 1(Δ 119894119895 le 119889ℓ) ℓ = 1 100 and fit the followingmodel for 119865119894119895(119889ℓ) = 119864[1(Δ 119894119895 le 119889ℓ) | 119879119894119895(119905) 119860119867119894119895 119875119894]

119864 [1 (Δ 119894119895 le 119889ℓ) | 119879119894119895 (119905) 119860119867119894119895 119875119894] = 119892minus1 1205730 (119889ℓ)

+ 1205731 (119889ℓ) 119875119894 + 1205732 (119889ℓ) 119860119867119894119895 + int1205733 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

+ 119860119867119894119895 int1205734 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

(7)

where 1205730(sdot) is a smooth intercept 1205731(sdot) quantifies the smootheffect of young sows1205732(sdot) describes the effect of the humidityand 1205733(sdot 119905) and 1205734(sdot 119905) quantify the effect of the temperatureat time 119905 as well as the interaction between the temperature attime 119905 and average humidityWemodel1205730(sdot)using 20 univari-ate basis functions 1205731(sdot) and 1205732(sdot) using five univariate basisfunctions and 1205733(sdot sdot) and 1205734(sdot sdot) using tensor product of twounivariate bases functions (total of 25 functions)Throughoutthe analysis cubic B-spline bases are used and REML isused for estimating smoothing parameters The estimatedconditional distribution denoted by 119865119894119895(119889) is monotonizedby fitting isotonic regression to (119889ℓ 119865119894119895(119889ℓ)) ℓ = 10 90ten smallest and ten largest 119889ℓ and the corresponding valuesof 119865119895(119889ℓ) are removed to avoid boundary effects By abuse of

10 Journal of Probability and Statistics

2425

2627

2829

30

Temperature

6065

7075

80

Humidity

16H

20H

18H

22H

14H

12H

10H

14H8H4H0H 2H 6H

time of day

16H

20H

12H

22H

18H

10H

14H

14H 0H 6H2H 8H4H

time of day

Figure 1 Temperature (∘C) and humidity () observed profiles (dashed) for three randomly selected days and the corresponding smoothedones (solid) the x-axis begins at 14H (2PM)

2324

2526

2728

29

Lactation day 4

203040median607080

2324

2526

2728

29

Lactation day 11

203040median607080

2324

2526

2728

29

Lactation day 18

203040median607080

8H6H4H2H0H 12H

14H

10H

14H

22H

16H

20H

18H

time of day

8H4H2H0H 6H22H

12H

16H

20H

14H

14H

10H

18H

time of day

4H 6H0H 2H 8H20H

12H

14H

14H

22H

16H

18H

10H

time of day

Figure 2 Temperature curves with which prediction of quantiles is made Dashed black line is pointwise average of temperature curves andsolid lines are pointwise quartiles all curves are smoothed

notation 119865119894119895(119889) denotes the resulting monotonized estimateddistribution Finally we obtain estimated first quartiles iequantiles at 120591 = 025 level by inverting 119865119894119895(119889) namely 119876(120591 =025 | 119879119894119895(119905) 119860119867119894119895 119875119894) = inf119889 119865119894119895(119889) ge 025

To understand the relationship between the lactatingsows feed intake and the thermal condition of the farrowingroom we systematically compare and study the predictedquantiles of two responses at combinations of different valuesof temperature humidity and parity For each of threelactation days (119895 = 3 10 17) we consider three values ofaverage humidity (first quartile median and third quartile)and two levels of parity (0 for older sows and 1 for youngersows) Based on the experimentersrsquo interest for the functional

covariate119879119894119895(sdot)we consider seven smooth temperature curvesgiven in Figure 2 Each of these curves is obtained by firstcalculating pointwise quantiles of temperature at five-minuteintervals for a specific level and then smoothing it weconsidered quantiles levels 120578 = 02 03 and 08 In shortfor each of three lactation days we obtain the first quartileof two responses for 42 different combinations (3 humidityvalues times 2 parity levels times 7 temperature curves) using theproposed method To avoid extrapolation we ascertain that(i) there are reasonablymany observedmeasurements at eachof the combinations and (ii) bottom 25 of the responses arenot dominantly from one of the parity group see distributionof each response by the parity in Figure 3

Journal of Probability and Statistics 11

Response (2) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus10

00

10

20

Resp

onse

(2)

minus15

minus05

05

10

Resp

onse

(2)

minus2minus1

01

Resp

onse

(2)

Response (1) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus4minus2

02

46

Resp

onse

(1)

minus6minus4

minus20

24

6Re

spon

se (1

)

minus4minus2

02

46

Resp

onse

(1)

Figure 3 Top panels back-to-back boxplots of the absolute change in feed intake at a specific day by parity Bottom panels back-to-backboxplots of the relative change in feed intake at a specific day by parity

The resulting predicted quantiles are shown in Figure 4Here we focus our discussion on predicted quantile of Δ(2)

119894(119895+1)

at quantile level 120591 = 025 for lactation day 4 (119895 = 3)mdashthefirst plot of the second row in Figure 4 The results suggestthat the feed intake of older sows (parity 119875119894 = 0 greylines) is less affected by high temperatures than younger sows(black lines) this finding is in agreement with Bloemhof etal [42] We also observe that the effects of humidity andtemperature on feed intake change are strongly intertwinedFor illustration we focus on lactation day 4 (119895 = 3) again foryounger sows (black lines) For medium humidity (dashedlines) their feed intake stays pretty constant as temperatureincreases while for low and high humidity levels (solid anddotted lines respectively) it changes with an opposite direc-tion Specifically when temperature increases the predictedfirst quartile of Δ(2)

119894(119895+1)increases for low humidity (solid line)

whereas it decreases for high humidity (dotted line) Ourresults imply that high humidity (dotted line) is related toa negative impact of high temperature on feed intake whilelow humidity (solid line) alleviates it and this finding isconsistent with a previous study [45] The analysis resultsuggests to keep low humidity levels in order to maintainhealthy feed intake behavior when ambient temperature isabove 60th percentile high humidity levels are desirable forcooler ambient temperature

Interpretation of the other results is similar While theeffects of covariates on feed intake are less apparent toward

the end of lactation period we still observe similar patternacross all three lactation days For the 11th day (119895 = 10) the25th quantile of the feed intake is predicted to decrease whenthe temperature stays below the 40th percentile regardlessof humidity level and sows age However it starts increas-ing with low humidity while it continues decreasing withhigh humidity when the temperature rises above the 40thpercentile Similarly for the 18th day (119895 = 17) when thetemperature rises above the 60th percentile the predictedfirst quartile increases with low humidity while it decreaseswith high humidity The effect of temperature on feed intakeseems less obvious for lactation days 11 and 18 than for day 4while the effect may be due to lactation day it may also be aresult of other factors such asmore fluctuation and variabilityin temperature curves on day 4 than on other two days(see Figure 2) Overall we conclude that high humidity andtemperature affect the sows feed intake behavior negativelyand young sows (parity one) are more sensitive to heat stressthan older sows (higher parity) especially at the beginning oflactation period

6 Discussion

The proposed modeling framework opens up a couple offuture research directions A first research avenue is todevelop significance tests of null covariate effect Testingfor the null effect of a covariate on the conditional dis-tribution of the response is equivalent to testing that the

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

Journal of Probability and Statistics 5

sum4119896=1 120582119896) the true quantiles are given by119876119884|1198831 1198832(sdot)(120591) = 2 int1198832119894(119905)120573(119905)119889119905 + 21198831119894 +

5radicint1198832119894(119905)2119889119905sum4119896=1 120582119896Φminus1(120591)

Let 120573(119905) = sum4119896=1 120573119896120601119896(119905) where 120573119896 = 1 for 119896 = 1 4For each case we use different combinations of signal to

noise ratio (SNR) sample size and sampling designs to gen-erate 500 simulated datasets We define SNR asradicsum4119896=1 120582119896120590120598and we consider five levels of noise SNR = 150 10 5 2 1Two levels of sample size are 119899 = 100 and 119899 = 1000 Twosampling designs are considered (i) sparse design where 119905119894119895 119895 = 1 119898 are119898 = 15 randomly selected points from a setof 30 equispaced grids in [0 10] and (ii) dense design wherethe sampling points 119905119894119895 = 119905119895 119895 = 1 119898 are 119898 = 30equispaced time points in [0 10]

The performance is evaluated on a test set of 100 subjectsfor whichwe have 1198831119894lowast (119882119894lowast119895 119905119894lowast119895) 119895 = 1 119898 available interms of mean absolute error (MAE) for quantile levels 120591 =005 01 025 and 05

MAE (120591) = 1100100

sum119894lowast=1

10038161003816100381610038161003816119876119894lowast (120591) minus 119876119894lowast (120591)10038161003816100381610038161003816 (6)

42 CompetingMethods Wedenote the proposedmethod byJoint QR to emphasize the single step estimation approachWe compare our method with two alternative approaches(1) a variant of our proposed approach using pointwiseestimation denoted by Pointwise QRThis approach consistsof fittingmultiple regressionmodels with binomial link func-tion as implemented by the penalized functional regressionpfr developed byGoldsmith et al [7] of therefundpackagefor generalized scalar responses (2) a modified version ofthe CM method denoted by Mod CM that we developedto account for additional scalar covariates and which fitsmultiple generalized linear models with scalar covariates andfPC scores as predictors (3) a linear quantile regressionapproach using the quantile loss function and the partialquantile regression bases for functional covariates proposedby Yu et al [19] and denoted by PQRNotice that although theformulation of the first twomethods implicitly accounts for avarying effect of the covariates on the response distributionthey do not ensure that this effect is smooth The thirdapproach can only estimate a specific quantile rather than theentire conditional distribution Note that all the competingmethods are monotonized for a fair comparison

The R function pfr can incorporate both scalarvectorand functional predictors by adopting a mixed effects modelframework The functional covariates are presmoothed byfPC analysis [31] Throughout the simulation study we fixPVE as 095 for fPC analysis to determine the number ofprincipal components and use REML to select the smoothingparameters for our proposed methods Other basis settingsare set to their default values We use 100 equally distancedpoints between the minimum andmaximum of the observed119884119894rsquos to set the grid 119910ℓ ℓ = 1 119871 for the conditionaldistribution function

43 Simulation Results Tables 1 and 2 show the accuracyof the quantile prediction for the two cases (normal andmixture) when the functional covariate is observed sparselyand the sample size is 119899 = 100 (Table 1) and 119899 = 1000(Table 2) Table 3 presents the prediction accuracy for thecase of heteroskedasticity with sparsely observed functionalcovariates The results based on dense sampling design showsimilar patterns and thus are relegated to the supplement seeSection S13 The comparison of running times is presentedin Table 4

For the case when the response is Gaussian Tables 1and 2 suggest that the Joint QR typically outperforms itscompetitors especially for lower quantile levels (120591 = 005and 120591 = 01) For very small noise level (SNR = 150) PQRperforms the best followed closely by the proposed Joint QRThe variant Pointwise QR which has a poorer performanceis generally better than the modified CM approach Asexpected as the sample size increases (119899 = 1000) all theaccuracy results improve the proposed Joint QR continuesto yield most accurate quantiles for the low quantile levelsFor mixture of Gaussians the results are somewhat similarThe accuracy of the quantile estimators with the PointwiseQR improves greatly in fact the Joint QR and PointwiseQR outperform the other approaches for quantile levels 120591 =005 01 025 irrespective of the SNR Finally Table 3 showsthat the results for Gaussian with heterogeneous error areclose to those for the case of Gaussian Again the proposedmethod has competitive performance in terms of predictionaccuracy

Table 4 compares the three methods that involve esti-mating the conditional distribution in terms of the runningtime required for fitting The times are reported based ona computer with a 23 GHz CPU and 8GB of RAM Notsurprisingly by fitting the model a single time Joint QR is thefastest in some cases being order ofmagnitude faster than therest Pointwise QR can be up to twice as fast as Mod CM

For completeness we also compare our proposedmethodto the appropriate competitivemethods for the cases (1) whenthere is a single scalar covariate and (2) when there is a singlefunctional covariate In the SupplementaryMaterials SectionS11 discusses the former case and compares Joint QR andPointwise QR with the linear quantile regression (LQR) [12]implemented by rq function in the R package quantreg andthe constrained B-splines nonparametric regression quantiles(COBS) implemented by the cobs function in the R pack-age COBS [29] in an extensive simulation experiment thatinvolves both linear quantile settings and nonlinear quantilesettings Overall the results show that the proposed methodshave similar behavior as LQR see Table S1 Furthermore weconsider the proposed method and its variant with nonlinearmodeling of the conditional distribution as discussed inSection 32 which we denote with Joint QR (NL) for jointfitting andPointwiseQR (NL) for pointwise fittingNonlinearversions of the proposed methods have an excellent MAEperformance which is comparable to or better than that ofthe COBS method

Finally Section S12 in the Supplementary Materialsdiscusses the simulation study for the case of having a singlefunctional covariate and compares the proposed methods

6 Journal of Probability and Statistics

Table 1 Average MAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 100Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 367 (003) 353 (003) 330 (002) 317 (002)Pointwise QR 496 (003) 461 (003) 422 (002) 418 (002)Mod CM 604 (003) 581 (003) 555 (003) 514 (003)PQR 317 (004) 271 (003) 231 (002) 216 (002)

Normal 10

Joint QR 632 (003) 600 (003) 576 (002) 573 (002)Pointwise QR 744 (004) 685 (003) 639 (003) 628 (003)Mod CM 820 (004) 810 (004) 804 (004) 801 (004)PQR 682 (005) 611 (004) 534 (003) 509 (002)

Normal 5

Joint QR 784 (004) 734 (003) 693 (003) 684 (003)Pointwise QR 891 (004) 812 (004) 745 (003) 726 (003)Mod CM 934 (004) 923 (004) 914 (005) 906 (005)PQR 868 (006) 781 (005) 674 (003) 634 (002)

Normal 2

Joint QR 1005 (005) 922 (004) 847 (003) 828 (003)Pointwise QR 1091 (006) 986 (004) 887 (004) 854 (003)Mod CM 1085 (005) 1055 (005) 1034 (006) 1021 (006)PQR 1121 (008) 1003 (006) 856 (004) 796 (003)

Normal 1

Joint QR 1150 (006) 1041 (005) 940 (004) 911 (003)Pointwise QR 1212 (006) 1088 (005) 970 (004) 930 (003)Mod CM 1195 (006) 1146 (006) 1107 (006) 1105 (007 )PQR 1282 (008) 1138 (006) 960 (004) 886 (003)

Mixture 150

Joint QR 692 (006) 623 (006) 616 (006) 481 (006)Pointwise QR 810 (008) 680 (006) 666 (006) 525 (006)Mod CM 918 (007) 899 (007) 893 (007) 790 (007)PQR 843 (006) 718 (004) 622 (004) 548 (014)

Mixture 10

Joint QR 902 (006) 795 (005) 785 (005) 619 (006)Pointwise QR 1011 (007) 852 (005) 795 (005) 642 (006)Mod CM 1133 (007) 1095 (007) 1079 (007) 980 (008)PQR 1072 (008) 899 (005) 763 (004) 533 (009)

Mixture 5

Joint QR 1018 (006) 891 (005) 853 (005) 682 (005)Pointwise QR 1118 (007) 938 (005) 858 (005) 698 (005)Mod CM 1219 (007) 1175 (007) 1152 (007) 1047 (008)PQR 1212 (009) 1012 (006) 840 (004) 573 (007)

Mixture 2

Joint QR 1193 (007) 1026 (005) 946 (005) 761 (005)Pointwise QR 1268 (008) 1063 (006) 951 (005) 770 (005)Mod CM 1333 (008) 1260 (008) 1227 (008) 1108 (010)PQR 1416 (010) 1172 (006) 955 (004) 641 (005)

Mixture 1

Joint QR 1317 (008) 1119 (005) 1006 (005) 804 (005)Pointwise QR 1371 (009) 1144 (006) 1010 (005) 813 (005)Mod CM 1416 (008) 1322 (008) 1271 (009) 1144 (011)PQR 1544 (011) 1284 (007) 1023 (004) 689 (004)

with CM in terms of MAE as well as computation timesee results displayed in Tables S2 and S3 The results showthat the proposed Joint QR is comparable to CM in termsof the prediction accuracy and has less computation timeIn our simulation study we also consider the joint fitting ofthe model by treating the binary response as normal and usepffr [27] with Gaussian link denoted by Joint QR (G)

5 Sow Data Application

Our motivating application is an experimental study carriedout at a commercial farm in Oklahoma from July 21 2013 toAugust 19 2013 [38] The study comprises 480 lactating sowsof different parities (ie number of previous pregnancieswhich serves as a surrogate for age and body weight) that

were observed during their first 21 lactation days theirfeed intake was recorded daily as the difference betweenthe feed offer and the feed refusal In addition the studycontains information on the temperature and humidity of thefarrowing rooms each recorded at five minute intervals Thefinal dataset we used for the analysis consists of 475 sows afterfive sows with unreliable measurements were removed by theexperimenters

The experiment was conducted to gain better insightsinto the way that the ambient temperature and humidity ofthe farrowing room affect the feed intake of lactating sowsPrevious studies seem to suggest a reduction in the sowrsquosfeed intake due to heat stress above 29∘C sows decrease feedintake by 05 kg per additional degree in temperature [9]Studying the effect of heat stress on lactating sows is a very

Journal of Probability and Statistics 7

Table 2 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 1000Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 168 (001) 165 (001) 162 (001) 160 (001)Pointwise QR 194 (001) 192 (001) 188 (001) 181 (001)Mod CM 193 (001) 188 (001) 187 (001) 187 (001)PQR 172 (001) 161 (001) 151 (001) 148 (001)

Normal 10

Joint QR 545 (002) 497 (002) 466 (002) 465 (002)Pointwise QR 564 (002) 513 (002) 478 (002) 475 (002)Mod CM 569 (002) 521 (002) 487 (002) 485 (002)PQR 585 (002) 537 (002) 481 (002) 460 (002)

Normal 5

Joint QR 734 (003) 654 (002) 594 (002) 585 (002)Pointwise QR 753 (003) 669 (002) 604 (002) 594 (002)Mod CM 753 (003) 677 (002) 618 (002) 606 (002)PQR 784 (003) 705 (002) 616 (002) 581 (002)

Normal 2

Joint QR 997 (003) 870 (003) 762 (002) 738 (002)Pointwise QR 1015 (003) 885 (003) 771 (002) 745 (002)Mod CM 1012 (003) 893 (003) 787 (003) 760 (003)PQR 1055 (004) 934 (003) 791 (003) 734 (002)

Normal 1

Joint QR 1169 (004) 1010 (003) 868 (003) 832 (003)Pointwise QR 1188 (004) 1025 (004) 877 (003) 839 (003)Mod CM 1185 (004) 1037 (004) 896 (003) 857 (003)PQR 1229 (004) 1077 (004) 902 (003) 829 (003)

Mixture 150

Joint QR 456 (002) 444 (002) 433 (003) 368 (003)Pointwise QR 434 (003) 424 (002) 420 (003) 366 (004)Mod CM 468 (003) 459 (002) 438 (003) 401 (003)PQR 784 (003) 645 (002) 529 (002) 319 (001)

Mixture 10

Joint QR 756 (003) 679 (002) 614 (003) 502 (003)Pointwise QR 749 (003) 661 (002) 588 (003) 502 (003)Mod CM 767 (004) 697 (003) 617 (003) 552 (004)PQR 1011 (004) 829 (003) 671 (002) 362 (002)

Mixture 5

Joint QR 920 (004) 713 (003) 694 (003) 56 (003)Pointwise QR 917 (004) 701 (003) 671 (003) 558 (003)Mod CM 932 (004) 737 (003) 708 (003) 618 (004)PQR 1146 (004) 936 (003) 749 (003) 432 (002)

Mixture 2

Joint QR 1157 (005) 902 (003) 805 (003) 635 (003)Pointwise QR 1158 (005) 898 (004) 790 (003) 631 (003)Mod CM 1169 (005) 938 (004) 838 (004) 701 (004)PQR 1349 (005) 1095 (004) 861 (003) 532 (002)

Mixture 1

Joint QR 1318 (005) 1031 (004) 879 (004) 679 (003)Pointwise QR 1321 (005) 1028 (004) 869 (004) 673 (003)Mod CM 1326 (005) 1067 (004) 921 (004) 747 (004)PQR 1493 (006) 1205 (004) 935 (004) 596 (002)

important scientific question because of a couple of reasonsFirst the reduction of feed intake of the lactating sows isassociated with a decrease in both their bodyweight (BW)and milk production as well as the weight gain of their litter[10 39 40] Sows with poor feed intake during lactationcontinue the subsequent reproductive period with negativeenergy balance [41] which leads to preventing the onset of anew reproductive cycle Second heat stress reduces farrowingrate (number of sows that deliver a new litter) and number

of piglets born [42] the reduction in reproduction due toseasonality is estimated to cost 300 million dollars per yearfor the swine industry [11] Economic losses are estimated toincrease [43] because high temperatures are likely to occurmore frequently due to global warming [44]

Our primary goal is to understand the thermal needsof the lactating sows for proper feeding behavior duringthe lactation time We are interested in how the interplaybetween the temperature and humidity of the farrowing room

8 Journal of Probability and Statistics

Table 3 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of heteroskedasticitywith a scalar covariateand a sparsely observed functional covariate

Sample size SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

100 150

Joint QR 441 (003) 399 (002) 348 (002) 325 (002)Pointwise QR 540 (003) 484 (003) 426 (002) 421 (003)Mod CM 600 (003) 571 (003) 542 (003) 539 (003)PQR 438 (005) 343 (003) 252 (002) 214 (002)

100 10

Joint QR 693 (003) 640 (003) 589 (002) 576 (002)Pointwise QR 794 (004) 720 (003) 648 (003) 630 (003)Mod CM 841 (004) 822 (004) 813 (004) 805 (004)PQR 776 (006) 657 (004) 547 (003) 508 (002)

100 5

Joint QR 844 (004) 772 (003) 705 (003) 686 (003)Pointwise QR 944 (005) 850 (004) 757 (003) 726 (003)Mod CM 965 (005) 940 (005) 936 (005) 926 (005)PQR 951 (007) 825 (005) 687 (003) 633 (002)

100 2

Joint QR 1068 (005) 963 (004) 860 (003) 830 (003)Pointwise QR 1155 (006) 1029 (005) 900 (004) 854 (003)Mod CM 1127 (005) 1080 (005) 1043 (005) 1040 (006)PQR 1207 (008) 1053 (006) 866 (004) 792 (003)

100 1

Joint QR 1223 (006) 1091 (005) 954 (004) 911 (003)Pointwise QR 1287 (007) 1139 (005) 985 (004) 929 (003)Mod CM 1243 (006) 1180 (006) 1119 (006) 1109 (007)PQR 1370 (009) 1191 (006) 970 (004) 884 (003)

1000 150

Joint QR 287 (001) 242 (001) 189 (001) 165 (001)Pointwise QR 306 (001) 260 (001) 207 (001) 186 (001)Mod CM 313 (001) 265 (001) 210 (001) 191 (001)PQR 310 (001) 247 (001) 178 (001) 148 (001)

1000 10

Joint QR 621 (002) 545 (002) 479 (002) 466 (002)Pointwise QR 638 (002) 559 (002) 490 (002) 477 (002)Mod CM 646 (002) 570 (002) 501 (002) 486 (002)PQR 668 (003) 582 (002) 492 (002) 460 (002)

1000 5

Joint QR 808 (003) 701 (002) 607 (002) 585 (002)Pointwise QR 827 (003) 715 (002) 616 (002) 594 (002)Mod CM 830 (003) 726 (002) 631 (002) 608 (002)PQR 864 (003) 750 (003) 627 (002) 581 (002)

1000 2

Joint QR 1076 (003) 921 (003) 776 (002) 738 (002)Pointwise QR 1095 (004) 934 (003) 785 (003) 745 (002)Mod CM 1092 (003) 945 (003) 801 (003) 760 (003)PQR 1138 (004) 979 (003) 803 (003) 734 (002)

1000 1

Joint QR 1254 (004) 1065 (003) 883 (003) 832 (003)Pointwise QR 1274 (004) 1081 (004) 892 (003) 839 (003)Mod CM 1270 (004) 1091 (004) 910 (003) 856 (003)PQR 1316 (005) 1126 (004) 913 (003) 829 (003)

Table 4 Average computing time (in seconds) of the three approaches that involve estimating the conditional distribution for the case ofhaving a scalar covariate and a densely observed functional covariate

Distribution Method 119899 = 100 119899 = 1000

NormalJoint QR 12 133

Pointwise QR 148 271Mod CM 278 511

MixtureJoint QR 13 154

Pointwise QR 151 296Mod CM 327 532

Journal of Probability and Statistics 9

affects the feed intake demeanor of lactating sows of differentparities We focus on three specific time points during thelactation periodmdashbeginning (lactation day 4) middle (day11) and end (day 18)mdashand the analyses are done separatelyfor each time point We consider two types of responses thatare meant to assess the feed intake behavior using the currentand the previous lactation days The first one quantifies theabsolute change in the feed intake over two consecutive daysand the second one quantifies the relative change and takesinto account the usual sowrsquos feed intake We define themformally after introducing some notation

Let 119865119868119894119895 be the 119895th measurement of the feed intakeobserved for the 119894th sow and denote by the lactation day 119871119863119894119895when 119865119868119894119895 is measured here 119895 = 1 119899119894 Most sows areobserved every day within the first 21 lactation days and thushave 119899119894 = 21 First define the absolute change in the feedintake between two consecutive days asΔ(1)

119894(119895+1)= 119865119868119894(119895+1)minus119865119868119894119895

for 119895 that satisfies 119871119863119894(119895+1)minus119871119863119894119895 = 1 For instanceΔ(1)119894(119895+1) = 0means there was no change in feed intake of sow 119894 betweenthe current day and the previous day while Δ(1)

119894(119895+1)lt 0means

that the feed intake consumed by the 119894th sow in the currentday is smaller than the feed intake consumed in the previousday However the same amount of change in the feed intakemay reflect some stress level for a sow who typically eats a lotand a more serious stress level for a sow that usually has alower appetite For this we define the relative change in thefeed intake by Δ(2)

119894(119895+1)= (119865119868119894(119895+1) minus 119865119868119894119895)(119871119863119894(119895+1) minus 119871119863119894119895) sdot

119879119860 119894 where 119879119860 119894 is the trimmed average of feed intake of 119894thsow calculated as the average feed intake after removing thelowest 20 and highest 20 of the feed intake measurements1198651198681198941 119865119868119894119899119894 taken for the corresponding sow Here 119879119860 119894 issurrogate for the usual amount of feed intake of the 119894th sowTrimmed average is used instead of the common average toremove outliers of very low feed intakes in first few lactatingdays For example consider the situation of two sows sow119894 that typically consumes 10lb food per day and sow 1198941015840 thatconsumes 5lb food per day A reduction of 5lb in the feedintake over two consecutive days corresponds to Δ(2)

119894(119895+1)=

minus50 for the 119894th sow and Δ(2)1198941015840(119895+1)

= minus100 for the 1198941015840th sowClearly both sows are stressed (negative value) but the secondsow is stressed more as its absolute relative change is largerin view of this we refer to the second response as the stressindex Due to the construction of the two types of responsesthe data size varies for lactation days 4 (119895 = 3) 11 (119895 = 10)and 18 (119895 = 17) for the first response Δ(1)

119894(119895+1) we have sample

sizes of 233 350 and 278 whereas for Δ(2)119894(119895+1)

the sample sizesare 362 373 and 336 for the respective lactation days

In this analysis we center the attention on the effect ofthe ambient temperature and humidity on the 1st quartile ofthe proxy stress measures and gain more understanding ofthe food consumption of sows that are most susceptible toheat stress While the association between the feed intake oflactating sows and the ambient conditions of the farrowingroom has been an active research area for some timeaccounting for the temperature daily profile has not beenconsidered yet hitherto Figure 1 displays the temperature

and humidity daily profiles recorded at a frequency of 5-minute window intervals for three different days Preliminaryinvestigation reveals that temperature is negatively correlatedwith humidity at each time this phenomenon is causedbecause the farm uses cool cell panels and fans to controlthe ambient temperature Furthermore it appears that thereis a strong pointwise correlation between temperature andhumidity In view of these observations in our analysis weconsider the daily average of humidity Exploratory analysisof the feed intake behavior of the sows suggests similaritiesfor the sows with parity greater than older sows (ones whoare at their third pregnancy or higher) thus we use a parityindicator instead of the actual parity of the sow The parityindicator 119875119894 is defined as one if the 119894th sow has parity oneand zero otherwise

For the analysis we smooth daily temperature measure-ments of each sow using univariate smoother with 15 cubicregression bases and quadratic penalty REML is used toestimate smoothing parameter The smoothed temperaturecurve for sow 119894rsquos 119895th repeatedmeasure is denoted by119879119894119895(119905) 119905 isin[0 24) and the corresponding daily average humidity isdenoted by119860119867119894119895 Both temperature and average humidity arecentered before being used in the analysis

For convenience we denote the response with Δ 119894119895 byremoving the superscript In this application for fixed 119895 Δ 119894119895corresponds to 119884119894 in Section 2 119875119894 and 119860119867119894119895 correspond toscalar covariates1198831198942 and119879119894119895(119905) and119860119867119894119895 sdot119879119894119895(119905) correspond tofunctional covariates1198831198943(sdot) We first estimate the conditionaldistribution ofΔ 119894119895 given temperature119879119894119895(119905) average humidity119860119867119894119895 parity 119875119894 and interaction 119860119867119894119895 sdot 119879119894119895(119905) Specifically foreach of lactation days of interest (119895 = 3 10 and 17) we create aset of 100 equispaced grid of points between the fifth smallestand fifth largest values of Δ 119894119895rsquos and denote the grids withD = 119889ℓ ℓ = 1 100 Then we create artificial binaryresponses 1(Δ 119894119895 le 119889ℓ) ℓ = 1 100 and fit the followingmodel for 119865119894119895(119889ℓ) = 119864[1(Δ 119894119895 le 119889ℓ) | 119879119894119895(119905) 119860119867119894119895 119875119894]

119864 [1 (Δ 119894119895 le 119889ℓ) | 119879119894119895 (119905) 119860119867119894119895 119875119894] = 119892minus1 1205730 (119889ℓ)

+ 1205731 (119889ℓ) 119875119894 + 1205732 (119889ℓ) 119860119867119894119895 + int1205733 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

+ 119860119867119894119895 int1205734 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

(7)

where 1205730(sdot) is a smooth intercept 1205731(sdot) quantifies the smootheffect of young sows1205732(sdot) describes the effect of the humidityand 1205733(sdot 119905) and 1205734(sdot 119905) quantify the effect of the temperatureat time 119905 as well as the interaction between the temperature attime 119905 and average humidityWemodel1205730(sdot)using 20 univari-ate basis functions 1205731(sdot) and 1205732(sdot) using five univariate basisfunctions and 1205733(sdot sdot) and 1205734(sdot sdot) using tensor product of twounivariate bases functions (total of 25 functions)Throughoutthe analysis cubic B-spline bases are used and REML isused for estimating smoothing parameters The estimatedconditional distribution denoted by 119865119894119895(119889) is monotonizedby fitting isotonic regression to (119889ℓ 119865119894119895(119889ℓ)) ℓ = 10 90ten smallest and ten largest 119889ℓ and the corresponding valuesof 119865119895(119889ℓ) are removed to avoid boundary effects By abuse of

10 Journal of Probability and Statistics

2425

2627

2829

30

Temperature

6065

7075

80

Humidity

16H

20H

18H

22H

14H

12H

10H

14H8H4H0H 2H 6H

time of day

16H

20H

12H

22H

18H

10H

14H

14H 0H 6H2H 8H4H

time of day

Figure 1 Temperature (∘C) and humidity () observed profiles (dashed) for three randomly selected days and the corresponding smoothedones (solid) the x-axis begins at 14H (2PM)

2324

2526

2728

29

Lactation day 4

203040median607080

2324

2526

2728

29

Lactation day 11

203040median607080

2324

2526

2728

29

Lactation day 18

203040median607080

8H6H4H2H0H 12H

14H

10H

14H

22H

16H

20H

18H

time of day

8H4H2H0H 6H22H

12H

16H

20H

14H

14H

10H

18H

time of day

4H 6H0H 2H 8H20H

12H

14H

14H

22H

16H

18H

10H

time of day

Figure 2 Temperature curves with which prediction of quantiles is made Dashed black line is pointwise average of temperature curves andsolid lines are pointwise quartiles all curves are smoothed

notation 119865119894119895(119889) denotes the resulting monotonized estimateddistribution Finally we obtain estimated first quartiles iequantiles at 120591 = 025 level by inverting 119865119894119895(119889) namely 119876(120591 =025 | 119879119894119895(119905) 119860119867119894119895 119875119894) = inf119889 119865119894119895(119889) ge 025

To understand the relationship between the lactatingsows feed intake and the thermal condition of the farrowingroom we systematically compare and study the predictedquantiles of two responses at combinations of different valuesof temperature humidity and parity For each of threelactation days (119895 = 3 10 17) we consider three values ofaverage humidity (first quartile median and third quartile)and two levels of parity (0 for older sows and 1 for youngersows) Based on the experimentersrsquo interest for the functional

covariate119879119894119895(sdot)we consider seven smooth temperature curvesgiven in Figure 2 Each of these curves is obtained by firstcalculating pointwise quantiles of temperature at five-minuteintervals for a specific level and then smoothing it weconsidered quantiles levels 120578 = 02 03 and 08 In shortfor each of three lactation days we obtain the first quartileof two responses for 42 different combinations (3 humidityvalues times 2 parity levels times 7 temperature curves) using theproposed method To avoid extrapolation we ascertain that(i) there are reasonablymany observedmeasurements at eachof the combinations and (ii) bottom 25 of the responses arenot dominantly from one of the parity group see distributionof each response by the parity in Figure 3

Journal of Probability and Statistics 11

Response (2) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus10

00

10

20

Resp

onse

(2)

minus15

minus05

05

10

Resp

onse

(2)

minus2minus1

01

Resp

onse

(2)

Response (1) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus4minus2

02

46

Resp

onse

(1)

minus6minus4

minus20

24

6Re

spon

se (1

)

minus4minus2

02

46

Resp

onse

(1)

Figure 3 Top panels back-to-back boxplots of the absolute change in feed intake at a specific day by parity Bottom panels back-to-backboxplots of the relative change in feed intake at a specific day by parity

The resulting predicted quantiles are shown in Figure 4Here we focus our discussion on predicted quantile of Δ(2)

119894(119895+1)

at quantile level 120591 = 025 for lactation day 4 (119895 = 3)mdashthefirst plot of the second row in Figure 4 The results suggestthat the feed intake of older sows (parity 119875119894 = 0 greylines) is less affected by high temperatures than younger sows(black lines) this finding is in agreement with Bloemhof etal [42] We also observe that the effects of humidity andtemperature on feed intake change are strongly intertwinedFor illustration we focus on lactation day 4 (119895 = 3) again foryounger sows (black lines) For medium humidity (dashedlines) their feed intake stays pretty constant as temperatureincreases while for low and high humidity levels (solid anddotted lines respectively) it changes with an opposite direc-tion Specifically when temperature increases the predictedfirst quartile of Δ(2)

119894(119895+1)increases for low humidity (solid line)

whereas it decreases for high humidity (dotted line) Ourresults imply that high humidity (dotted line) is related toa negative impact of high temperature on feed intake whilelow humidity (solid line) alleviates it and this finding isconsistent with a previous study [45] The analysis resultsuggests to keep low humidity levels in order to maintainhealthy feed intake behavior when ambient temperature isabove 60th percentile high humidity levels are desirable forcooler ambient temperature

Interpretation of the other results is similar While theeffects of covariates on feed intake are less apparent toward

the end of lactation period we still observe similar patternacross all three lactation days For the 11th day (119895 = 10) the25th quantile of the feed intake is predicted to decrease whenthe temperature stays below the 40th percentile regardlessof humidity level and sows age However it starts increas-ing with low humidity while it continues decreasing withhigh humidity when the temperature rises above the 40thpercentile Similarly for the 18th day (119895 = 17) when thetemperature rises above the 60th percentile the predictedfirst quartile increases with low humidity while it decreaseswith high humidity The effect of temperature on feed intakeseems less obvious for lactation days 11 and 18 than for day 4while the effect may be due to lactation day it may also be aresult of other factors such asmore fluctuation and variabilityin temperature curves on day 4 than on other two days(see Figure 2) Overall we conclude that high humidity andtemperature affect the sows feed intake behavior negativelyand young sows (parity one) are more sensitive to heat stressthan older sows (higher parity) especially at the beginning oflactation period

6 Discussion

The proposed modeling framework opens up a couple offuture research directions A first research avenue is todevelop significance tests of null covariate effect Testingfor the null effect of a covariate on the conditional dis-tribution of the response is equivalent to testing that the

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

6 Journal of Probability and Statistics

Table 1 Average MAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 100Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 367 (003) 353 (003) 330 (002) 317 (002)Pointwise QR 496 (003) 461 (003) 422 (002) 418 (002)Mod CM 604 (003) 581 (003) 555 (003) 514 (003)PQR 317 (004) 271 (003) 231 (002) 216 (002)

Normal 10

Joint QR 632 (003) 600 (003) 576 (002) 573 (002)Pointwise QR 744 (004) 685 (003) 639 (003) 628 (003)Mod CM 820 (004) 810 (004) 804 (004) 801 (004)PQR 682 (005) 611 (004) 534 (003) 509 (002)

Normal 5

Joint QR 784 (004) 734 (003) 693 (003) 684 (003)Pointwise QR 891 (004) 812 (004) 745 (003) 726 (003)Mod CM 934 (004) 923 (004) 914 (005) 906 (005)PQR 868 (006) 781 (005) 674 (003) 634 (002)

Normal 2

Joint QR 1005 (005) 922 (004) 847 (003) 828 (003)Pointwise QR 1091 (006) 986 (004) 887 (004) 854 (003)Mod CM 1085 (005) 1055 (005) 1034 (006) 1021 (006)PQR 1121 (008) 1003 (006) 856 (004) 796 (003)

Normal 1

Joint QR 1150 (006) 1041 (005) 940 (004) 911 (003)Pointwise QR 1212 (006) 1088 (005) 970 (004) 930 (003)Mod CM 1195 (006) 1146 (006) 1107 (006) 1105 (007 )PQR 1282 (008) 1138 (006) 960 (004) 886 (003)

Mixture 150

Joint QR 692 (006) 623 (006) 616 (006) 481 (006)Pointwise QR 810 (008) 680 (006) 666 (006) 525 (006)Mod CM 918 (007) 899 (007) 893 (007) 790 (007)PQR 843 (006) 718 (004) 622 (004) 548 (014)

Mixture 10

Joint QR 902 (006) 795 (005) 785 (005) 619 (006)Pointwise QR 1011 (007) 852 (005) 795 (005) 642 (006)Mod CM 1133 (007) 1095 (007) 1079 (007) 980 (008)PQR 1072 (008) 899 (005) 763 (004) 533 (009)

Mixture 5

Joint QR 1018 (006) 891 (005) 853 (005) 682 (005)Pointwise QR 1118 (007) 938 (005) 858 (005) 698 (005)Mod CM 1219 (007) 1175 (007) 1152 (007) 1047 (008)PQR 1212 (009) 1012 (006) 840 (004) 573 (007)

Mixture 2

Joint QR 1193 (007) 1026 (005) 946 (005) 761 (005)Pointwise QR 1268 (008) 1063 (006) 951 (005) 770 (005)Mod CM 1333 (008) 1260 (008) 1227 (008) 1108 (010)PQR 1416 (010) 1172 (006) 955 (004) 641 (005)

Mixture 1

Joint QR 1317 (008) 1119 (005) 1006 (005) 804 (005)Pointwise QR 1371 (009) 1144 (006) 1010 (005) 813 (005)Mod CM 1416 (008) 1322 (008) 1271 (009) 1144 (011)PQR 1544 (011) 1284 (007) 1023 (004) 689 (004)

with CM in terms of MAE as well as computation timesee results displayed in Tables S2 and S3 The results showthat the proposed Joint QR is comparable to CM in termsof the prediction accuracy and has less computation timeIn our simulation study we also consider the joint fitting ofthe model by treating the binary response as normal and usepffr [27] with Gaussian link denoted by Joint QR (G)

5 Sow Data Application

Our motivating application is an experimental study carriedout at a commercial farm in Oklahoma from July 21 2013 toAugust 19 2013 [38] The study comprises 480 lactating sowsof different parities (ie number of previous pregnancieswhich serves as a surrogate for age and body weight) that

were observed during their first 21 lactation days theirfeed intake was recorded daily as the difference betweenthe feed offer and the feed refusal In addition the studycontains information on the temperature and humidity of thefarrowing rooms each recorded at five minute intervals Thefinal dataset we used for the analysis consists of 475 sows afterfive sows with unreliable measurements were removed by theexperimenters

The experiment was conducted to gain better insightsinto the way that the ambient temperature and humidity ofthe farrowing room affect the feed intake of lactating sowsPrevious studies seem to suggest a reduction in the sowrsquosfeed intake due to heat stress above 29∘C sows decrease feedintake by 05 kg per additional degree in temperature [9]Studying the effect of heat stress on lactating sows is a very

Journal of Probability and Statistics 7

Table 2 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 1000Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 168 (001) 165 (001) 162 (001) 160 (001)Pointwise QR 194 (001) 192 (001) 188 (001) 181 (001)Mod CM 193 (001) 188 (001) 187 (001) 187 (001)PQR 172 (001) 161 (001) 151 (001) 148 (001)

Normal 10

Joint QR 545 (002) 497 (002) 466 (002) 465 (002)Pointwise QR 564 (002) 513 (002) 478 (002) 475 (002)Mod CM 569 (002) 521 (002) 487 (002) 485 (002)PQR 585 (002) 537 (002) 481 (002) 460 (002)

Normal 5

Joint QR 734 (003) 654 (002) 594 (002) 585 (002)Pointwise QR 753 (003) 669 (002) 604 (002) 594 (002)Mod CM 753 (003) 677 (002) 618 (002) 606 (002)PQR 784 (003) 705 (002) 616 (002) 581 (002)

Normal 2

Joint QR 997 (003) 870 (003) 762 (002) 738 (002)Pointwise QR 1015 (003) 885 (003) 771 (002) 745 (002)Mod CM 1012 (003) 893 (003) 787 (003) 760 (003)PQR 1055 (004) 934 (003) 791 (003) 734 (002)

Normal 1

Joint QR 1169 (004) 1010 (003) 868 (003) 832 (003)Pointwise QR 1188 (004) 1025 (004) 877 (003) 839 (003)Mod CM 1185 (004) 1037 (004) 896 (003) 857 (003)PQR 1229 (004) 1077 (004) 902 (003) 829 (003)

Mixture 150

Joint QR 456 (002) 444 (002) 433 (003) 368 (003)Pointwise QR 434 (003) 424 (002) 420 (003) 366 (004)Mod CM 468 (003) 459 (002) 438 (003) 401 (003)PQR 784 (003) 645 (002) 529 (002) 319 (001)

Mixture 10

Joint QR 756 (003) 679 (002) 614 (003) 502 (003)Pointwise QR 749 (003) 661 (002) 588 (003) 502 (003)Mod CM 767 (004) 697 (003) 617 (003) 552 (004)PQR 1011 (004) 829 (003) 671 (002) 362 (002)

Mixture 5

Joint QR 920 (004) 713 (003) 694 (003) 56 (003)Pointwise QR 917 (004) 701 (003) 671 (003) 558 (003)Mod CM 932 (004) 737 (003) 708 (003) 618 (004)PQR 1146 (004) 936 (003) 749 (003) 432 (002)

Mixture 2

Joint QR 1157 (005) 902 (003) 805 (003) 635 (003)Pointwise QR 1158 (005) 898 (004) 790 (003) 631 (003)Mod CM 1169 (005) 938 (004) 838 (004) 701 (004)PQR 1349 (005) 1095 (004) 861 (003) 532 (002)

Mixture 1

Joint QR 1318 (005) 1031 (004) 879 (004) 679 (003)Pointwise QR 1321 (005) 1028 (004) 869 (004) 673 (003)Mod CM 1326 (005) 1067 (004) 921 (004) 747 (004)PQR 1493 (006) 1205 (004) 935 (004) 596 (002)

important scientific question because of a couple of reasonsFirst the reduction of feed intake of the lactating sows isassociated with a decrease in both their bodyweight (BW)and milk production as well as the weight gain of their litter[10 39 40] Sows with poor feed intake during lactationcontinue the subsequent reproductive period with negativeenergy balance [41] which leads to preventing the onset of anew reproductive cycle Second heat stress reduces farrowingrate (number of sows that deliver a new litter) and number

of piglets born [42] the reduction in reproduction due toseasonality is estimated to cost 300 million dollars per yearfor the swine industry [11] Economic losses are estimated toincrease [43] because high temperatures are likely to occurmore frequently due to global warming [44]

Our primary goal is to understand the thermal needsof the lactating sows for proper feeding behavior duringthe lactation time We are interested in how the interplaybetween the temperature and humidity of the farrowing room

8 Journal of Probability and Statistics

Table 3 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of heteroskedasticitywith a scalar covariateand a sparsely observed functional covariate

Sample size SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

100 150

Joint QR 441 (003) 399 (002) 348 (002) 325 (002)Pointwise QR 540 (003) 484 (003) 426 (002) 421 (003)Mod CM 600 (003) 571 (003) 542 (003) 539 (003)PQR 438 (005) 343 (003) 252 (002) 214 (002)

100 10

Joint QR 693 (003) 640 (003) 589 (002) 576 (002)Pointwise QR 794 (004) 720 (003) 648 (003) 630 (003)Mod CM 841 (004) 822 (004) 813 (004) 805 (004)PQR 776 (006) 657 (004) 547 (003) 508 (002)

100 5

Joint QR 844 (004) 772 (003) 705 (003) 686 (003)Pointwise QR 944 (005) 850 (004) 757 (003) 726 (003)Mod CM 965 (005) 940 (005) 936 (005) 926 (005)PQR 951 (007) 825 (005) 687 (003) 633 (002)

100 2

Joint QR 1068 (005) 963 (004) 860 (003) 830 (003)Pointwise QR 1155 (006) 1029 (005) 900 (004) 854 (003)Mod CM 1127 (005) 1080 (005) 1043 (005) 1040 (006)PQR 1207 (008) 1053 (006) 866 (004) 792 (003)

100 1

Joint QR 1223 (006) 1091 (005) 954 (004) 911 (003)Pointwise QR 1287 (007) 1139 (005) 985 (004) 929 (003)Mod CM 1243 (006) 1180 (006) 1119 (006) 1109 (007)PQR 1370 (009) 1191 (006) 970 (004) 884 (003)

1000 150

Joint QR 287 (001) 242 (001) 189 (001) 165 (001)Pointwise QR 306 (001) 260 (001) 207 (001) 186 (001)Mod CM 313 (001) 265 (001) 210 (001) 191 (001)PQR 310 (001) 247 (001) 178 (001) 148 (001)

1000 10

Joint QR 621 (002) 545 (002) 479 (002) 466 (002)Pointwise QR 638 (002) 559 (002) 490 (002) 477 (002)Mod CM 646 (002) 570 (002) 501 (002) 486 (002)PQR 668 (003) 582 (002) 492 (002) 460 (002)

1000 5

Joint QR 808 (003) 701 (002) 607 (002) 585 (002)Pointwise QR 827 (003) 715 (002) 616 (002) 594 (002)Mod CM 830 (003) 726 (002) 631 (002) 608 (002)PQR 864 (003) 750 (003) 627 (002) 581 (002)

1000 2

Joint QR 1076 (003) 921 (003) 776 (002) 738 (002)Pointwise QR 1095 (004) 934 (003) 785 (003) 745 (002)Mod CM 1092 (003) 945 (003) 801 (003) 760 (003)PQR 1138 (004) 979 (003) 803 (003) 734 (002)

1000 1

Joint QR 1254 (004) 1065 (003) 883 (003) 832 (003)Pointwise QR 1274 (004) 1081 (004) 892 (003) 839 (003)Mod CM 1270 (004) 1091 (004) 910 (003) 856 (003)PQR 1316 (005) 1126 (004) 913 (003) 829 (003)

Table 4 Average computing time (in seconds) of the three approaches that involve estimating the conditional distribution for the case ofhaving a scalar covariate and a densely observed functional covariate

Distribution Method 119899 = 100 119899 = 1000

NormalJoint QR 12 133

Pointwise QR 148 271Mod CM 278 511

MixtureJoint QR 13 154

Pointwise QR 151 296Mod CM 327 532

Journal of Probability and Statistics 9

affects the feed intake demeanor of lactating sows of differentparities We focus on three specific time points during thelactation periodmdashbeginning (lactation day 4) middle (day11) and end (day 18)mdashand the analyses are done separatelyfor each time point We consider two types of responses thatare meant to assess the feed intake behavior using the currentand the previous lactation days The first one quantifies theabsolute change in the feed intake over two consecutive daysand the second one quantifies the relative change and takesinto account the usual sowrsquos feed intake We define themformally after introducing some notation

Let 119865119868119894119895 be the 119895th measurement of the feed intakeobserved for the 119894th sow and denote by the lactation day 119871119863119894119895when 119865119868119894119895 is measured here 119895 = 1 119899119894 Most sows areobserved every day within the first 21 lactation days and thushave 119899119894 = 21 First define the absolute change in the feedintake between two consecutive days asΔ(1)

119894(119895+1)= 119865119868119894(119895+1)minus119865119868119894119895

for 119895 that satisfies 119871119863119894(119895+1)minus119871119863119894119895 = 1 For instanceΔ(1)119894(119895+1) = 0means there was no change in feed intake of sow 119894 betweenthe current day and the previous day while Δ(1)

119894(119895+1)lt 0means

that the feed intake consumed by the 119894th sow in the currentday is smaller than the feed intake consumed in the previousday However the same amount of change in the feed intakemay reflect some stress level for a sow who typically eats a lotand a more serious stress level for a sow that usually has alower appetite For this we define the relative change in thefeed intake by Δ(2)

119894(119895+1)= (119865119868119894(119895+1) minus 119865119868119894119895)(119871119863119894(119895+1) minus 119871119863119894119895) sdot

119879119860 119894 where 119879119860 119894 is the trimmed average of feed intake of 119894thsow calculated as the average feed intake after removing thelowest 20 and highest 20 of the feed intake measurements1198651198681198941 119865119868119894119899119894 taken for the corresponding sow Here 119879119860 119894 issurrogate for the usual amount of feed intake of the 119894th sowTrimmed average is used instead of the common average toremove outliers of very low feed intakes in first few lactatingdays For example consider the situation of two sows sow119894 that typically consumes 10lb food per day and sow 1198941015840 thatconsumes 5lb food per day A reduction of 5lb in the feedintake over two consecutive days corresponds to Δ(2)

119894(119895+1)=

minus50 for the 119894th sow and Δ(2)1198941015840(119895+1)

= minus100 for the 1198941015840th sowClearly both sows are stressed (negative value) but the secondsow is stressed more as its absolute relative change is largerin view of this we refer to the second response as the stressindex Due to the construction of the two types of responsesthe data size varies for lactation days 4 (119895 = 3) 11 (119895 = 10)and 18 (119895 = 17) for the first response Δ(1)

119894(119895+1) we have sample

sizes of 233 350 and 278 whereas for Δ(2)119894(119895+1)

the sample sizesare 362 373 and 336 for the respective lactation days

In this analysis we center the attention on the effect ofthe ambient temperature and humidity on the 1st quartile ofthe proxy stress measures and gain more understanding ofthe food consumption of sows that are most susceptible toheat stress While the association between the feed intake oflactating sows and the ambient conditions of the farrowingroom has been an active research area for some timeaccounting for the temperature daily profile has not beenconsidered yet hitherto Figure 1 displays the temperature

and humidity daily profiles recorded at a frequency of 5-minute window intervals for three different days Preliminaryinvestigation reveals that temperature is negatively correlatedwith humidity at each time this phenomenon is causedbecause the farm uses cool cell panels and fans to controlthe ambient temperature Furthermore it appears that thereis a strong pointwise correlation between temperature andhumidity In view of these observations in our analysis weconsider the daily average of humidity Exploratory analysisof the feed intake behavior of the sows suggests similaritiesfor the sows with parity greater than older sows (ones whoare at their third pregnancy or higher) thus we use a parityindicator instead of the actual parity of the sow The parityindicator 119875119894 is defined as one if the 119894th sow has parity oneand zero otherwise

For the analysis we smooth daily temperature measure-ments of each sow using univariate smoother with 15 cubicregression bases and quadratic penalty REML is used toestimate smoothing parameter The smoothed temperaturecurve for sow 119894rsquos 119895th repeatedmeasure is denoted by119879119894119895(119905) 119905 isin[0 24) and the corresponding daily average humidity isdenoted by119860119867119894119895 Both temperature and average humidity arecentered before being used in the analysis

For convenience we denote the response with Δ 119894119895 byremoving the superscript In this application for fixed 119895 Δ 119894119895corresponds to 119884119894 in Section 2 119875119894 and 119860119867119894119895 correspond toscalar covariates1198831198942 and119879119894119895(119905) and119860119867119894119895 sdot119879119894119895(119905) correspond tofunctional covariates1198831198943(sdot) We first estimate the conditionaldistribution ofΔ 119894119895 given temperature119879119894119895(119905) average humidity119860119867119894119895 parity 119875119894 and interaction 119860119867119894119895 sdot 119879119894119895(119905) Specifically foreach of lactation days of interest (119895 = 3 10 and 17) we create aset of 100 equispaced grid of points between the fifth smallestand fifth largest values of Δ 119894119895rsquos and denote the grids withD = 119889ℓ ℓ = 1 100 Then we create artificial binaryresponses 1(Δ 119894119895 le 119889ℓ) ℓ = 1 100 and fit the followingmodel for 119865119894119895(119889ℓ) = 119864[1(Δ 119894119895 le 119889ℓ) | 119879119894119895(119905) 119860119867119894119895 119875119894]

119864 [1 (Δ 119894119895 le 119889ℓ) | 119879119894119895 (119905) 119860119867119894119895 119875119894] = 119892minus1 1205730 (119889ℓ)

+ 1205731 (119889ℓ) 119875119894 + 1205732 (119889ℓ) 119860119867119894119895 + int1205733 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

+ 119860119867119894119895 int1205734 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

(7)

where 1205730(sdot) is a smooth intercept 1205731(sdot) quantifies the smootheffect of young sows1205732(sdot) describes the effect of the humidityand 1205733(sdot 119905) and 1205734(sdot 119905) quantify the effect of the temperatureat time 119905 as well as the interaction between the temperature attime 119905 and average humidityWemodel1205730(sdot)using 20 univari-ate basis functions 1205731(sdot) and 1205732(sdot) using five univariate basisfunctions and 1205733(sdot sdot) and 1205734(sdot sdot) using tensor product of twounivariate bases functions (total of 25 functions)Throughoutthe analysis cubic B-spline bases are used and REML isused for estimating smoothing parameters The estimatedconditional distribution denoted by 119865119894119895(119889) is monotonizedby fitting isotonic regression to (119889ℓ 119865119894119895(119889ℓ)) ℓ = 10 90ten smallest and ten largest 119889ℓ and the corresponding valuesof 119865119895(119889ℓ) are removed to avoid boundary effects By abuse of

10 Journal of Probability and Statistics

2425

2627

2829

30

Temperature

6065

7075

80

Humidity

16H

20H

18H

22H

14H

12H

10H

14H8H4H0H 2H 6H

time of day

16H

20H

12H

22H

18H

10H

14H

14H 0H 6H2H 8H4H

time of day

Figure 1 Temperature (∘C) and humidity () observed profiles (dashed) for three randomly selected days and the corresponding smoothedones (solid) the x-axis begins at 14H (2PM)

2324

2526

2728

29

Lactation day 4

203040median607080

2324

2526

2728

29

Lactation day 11

203040median607080

2324

2526

2728

29

Lactation day 18

203040median607080

8H6H4H2H0H 12H

14H

10H

14H

22H

16H

20H

18H

time of day

8H4H2H0H 6H22H

12H

16H

20H

14H

14H

10H

18H

time of day

4H 6H0H 2H 8H20H

12H

14H

14H

22H

16H

18H

10H

time of day

Figure 2 Temperature curves with which prediction of quantiles is made Dashed black line is pointwise average of temperature curves andsolid lines are pointwise quartiles all curves are smoothed

notation 119865119894119895(119889) denotes the resulting monotonized estimateddistribution Finally we obtain estimated first quartiles iequantiles at 120591 = 025 level by inverting 119865119894119895(119889) namely 119876(120591 =025 | 119879119894119895(119905) 119860119867119894119895 119875119894) = inf119889 119865119894119895(119889) ge 025

To understand the relationship between the lactatingsows feed intake and the thermal condition of the farrowingroom we systematically compare and study the predictedquantiles of two responses at combinations of different valuesof temperature humidity and parity For each of threelactation days (119895 = 3 10 17) we consider three values ofaverage humidity (first quartile median and third quartile)and two levels of parity (0 for older sows and 1 for youngersows) Based on the experimentersrsquo interest for the functional

covariate119879119894119895(sdot)we consider seven smooth temperature curvesgiven in Figure 2 Each of these curves is obtained by firstcalculating pointwise quantiles of temperature at five-minuteintervals for a specific level and then smoothing it weconsidered quantiles levels 120578 = 02 03 and 08 In shortfor each of three lactation days we obtain the first quartileof two responses for 42 different combinations (3 humidityvalues times 2 parity levels times 7 temperature curves) using theproposed method To avoid extrapolation we ascertain that(i) there are reasonablymany observedmeasurements at eachof the combinations and (ii) bottom 25 of the responses arenot dominantly from one of the parity group see distributionof each response by the parity in Figure 3

Journal of Probability and Statistics 11

Response (2) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus10

00

10

20

Resp

onse

(2)

minus15

minus05

05

10

Resp

onse

(2)

minus2minus1

01

Resp

onse

(2)

Response (1) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus4minus2

02

46

Resp

onse

(1)

minus6minus4

minus20

24

6Re

spon

se (1

)

minus4minus2

02

46

Resp

onse

(1)

Figure 3 Top panels back-to-back boxplots of the absolute change in feed intake at a specific day by parity Bottom panels back-to-backboxplots of the relative change in feed intake at a specific day by parity

The resulting predicted quantiles are shown in Figure 4Here we focus our discussion on predicted quantile of Δ(2)

119894(119895+1)

at quantile level 120591 = 025 for lactation day 4 (119895 = 3)mdashthefirst plot of the second row in Figure 4 The results suggestthat the feed intake of older sows (parity 119875119894 = 0 greylines) is less affected by high temperatures than younger sows(black lines) this finding is in agreement with Bloemhof etal [42] We also observe that the effects of humidity andtemperature on feed intake change are strongly intertwinedFor illustration we focus on lactation day 4 (119895 = 3) again foryounger sows (black lines) For medium humidity (dashedlines) their feed intake stays pretty constant as temperatureincreases while for low and high humidity levels (solid anddotted lines respectively) it changes with an opposite direc-tion Specifically when temperature increases the predictedfirst quartile of Δ(2)

119894(119895+1)increases for low humidity (solid line)

whereas it decreases for high humidity (dotted line) Ourresults imply that high humidity (dotted line) is related toa negative impact of high temperature on feed intake whilelow humidity (solid line) alleviates it and this finding isconsistent with a previous study [45] The analysis resultsuggests to keep low humidity levels in order to maintainhealthy feed intake behavior when ambient temperature isabove 60th percentile high humidity levels are desirable forcooler ambient temperature

Interpretation of the other results is similar While theeffects of covariates on feed intake are less apparent toward

the end of lactation period we still observe similar patternacross all three lactation days For the 11th day (119895 = 10) the25th quantile of the feed intake is predicted to decrease whenthe temperature stays below the 40th percentile regardlessof humidity level and sows age However it starts increas-ing with low humidity while it continues decreasing withhigh humidity when the temperature rises above the 40thpercentile Similarly for the 18th day (119895 = 17) when thetemperature rises above the 60th percentile the predictedfirst quartile increases with low humidity while it decreaseswith high humidity The effect of temperature on feed intakeseems less obvious for lactation days 11 and 18 than for day 4while the effect may be due to lactation day it may also be aresult of other factors such asmore fluctuation and variabilityin temperature curves on day 4 than on other two days(see Figure 2) Overall we conclude that high humidity andtemperature affect the sows feed intake behavior negativelyand young sows (parity one) are more sensitive to heat stressthan older sows (higher parity) especially at the beginning oflactation period

6 Discussion

The proposed modeling framework opens up a couple offuture research directions A first research avenue is todevelop significance tests of null covariate effect Testingfor the null effect of a covariate on the conditional dis-tribution of the response is equivalent to testing that the

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

Journal of Probability and Statistics 7

Table 2 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of having a scalar covariate and a sparselyobserved functional covariate Sample size 119899 = 1000Distribution SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

Normal 150

Joint QR 168 (001) 165 (001) 162 (001) 160 (001)Pointwise QR 194 (001) 192 (001) 188 (001) 181 (001)Mod CM 193 (001) 188 (001) 187 (001) 187 (001)PQR 172 (001) 161 (001) 151 (001) 148 (001)

Normal 10

Joint QR 545 (002) 497 (002) 466 (002) 465 (002)Pointwise QR 564 (002) 513 (002) 478 (002) 475 (002)Mod CM 569 (002) 521 (002) 487 (002) 485 (002)PQR 585 (002) 537 (002) 481 (002) 460 (002)

Normal 5

Joint QR 734 (003) 654 (002) 594 (002) 585 (002)Pointwise QR 753 (003) 669 (002) 604 (002) 594 (002)Mod CM 753 (003) 677 (002) 618 (002) 606 (002)PQR 784 (003) 705 (002) 616 (002) 581 (002)

Normal 2

Joint QR 997 (003) 870 (003) 762 (002) 738 (002)Pointwise QR 1015 (003) 885 (003) 771 (002) 745 (002)Mod CM 1012 (003) 893 (003) 787 (003) 760 (003)PQR 1055 (004) 934 (003) 791 (003) 734 (002)

Normal 1

Joint QR 1169 (004) 1010 (003) 868 (003) 832 (003)Pointwise QR 1188 (004) 1025 (004) 877 (003) 839 (003)Mod CM 1185 (004) 1037 (004) 896 (003) 857 (003)PQR 1229 (004) 1077 (004) 902 (003) 829 (003)

Mixture 150

Joint QR 456 (002) 444 (002) 433 (003) 368 (003)Pointwise QR 434 (003) 424 (002) 420 (003) 366 (004)Mod CM 468 (003) 459 (002) 438 (003) 401 (003)PQR 784 (003) 645 (002) 529 (002) 319 (001)

Mixture 10

Joint QR 756 (003) 679 (002) 614 (003) 502 (003)Pointwise QR 749 (003) 661 (002) 588 (003) 502 (003)Mod CM 767 (004) 697 (003) 617 (003) 552 (004)PQR 1011 (004) 829 (003) 671 (002) 362 (002)

Mixture 5

Joint QR 920 (004) 713 (003) 694 (003) 56 (003)Pointwise QR 917 (004) 701 (003) 671 (003) 558 (003)Mod CM 932 (004) 737 (003) 708 (003) 618 (004)PQR 1146 (004) 936 (003) 749 (003) 432 (002)

Mixture 2

Joint QR 1157 (005) 902 (003) 805 (003) 635 (003)Pointwise QR 1158 (005) 898 (004) 790 (003) 631 (003)Mod CM 1169 (005) 938 (004) 838 (004) 701 (004)PQR 1349 (005) 1095 (004) 861 (003) 532 (002)

Mixture 1

Joint QR 1318 (005) 1031 (004) 879 (004) 679 (003)Pointwise QR 1321 (005) 1028 (004) 869 (004) 673 (003)Mod CM 1326 (005) 1067 (004) 921 (004) 747 (004)PQR 1493 (006) 1205 (004) 935 (004) 596 (002)

important scientific question because of a couple of reasonsFirst the reduction of feed intake of the lactating sows isassociated with a decrease in both their bodyweight (BW)and milk production as well as the weight gain of their litter[10 39 40] Sows with poor feed intake during lactationcontinue the subsequent reproductive period with negativeenergy balance [41] which leads to preventing the onset of anew reproductive cycle Second heat stress reduces farrowingrate (number of sows that deliver a new litter) and number

of piglets born [42] the reduction in reproduction due toseasonality is estimated to cost 300 million dollars per yearfor the swine industry [11] Economic losses are estimated toincrease [43] because high temperatures are likely to occurmore frequently due to global warming [44]

Our primary goal is to understand the thermal needsof the lactating sows for proper feeding behavior duringthe lactation time We are interested in how the interplaybetween the temperature and humidity of the farrowing room

8 Journal of Probability and Statistics

Table 3 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of heteroskedasticitywith a scalar covariateand a sparsely observed functional covariate

Sample size SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

100 150

Joint QR 441 (003) 399 (002) 348 (002) 325 (002)Pointwise QR 540 (003) 484 (003) 426 (002) 421 (003)Mod CM 600 (003) 571 (003) 542 (003) 539 (003)PQR 438 (005) 343 (003) 252 (002) 214 (002)

100 10

Joint QR 693 (003) 640 (003) 589 (002) 576 (002)Pointwise QR 794 (004) 720 (003) 648 (003) 630 (003)Mod CM 841 (004) 822 (004) 813 (004) 805 (004)PQR 776 (006) 657 (004) 547 (003) 508 (002)

100 5

Joint QR 844 (004) 772 (003) 705 (003) 686 (003)Pointwise QR 944 (005) 850 (004) 757 (003) 726 (003)Mod CM 965 (005) 940 (005) 936 (005) 926 (005)PQR 951 (007) 825 (005) 687 (003) 633 (002)

100 2

Joint QR 1068 (005) 963 (004) 860 (003) 830 (003)Pointwise QR 1155 (006) 1029 (005) 900 (004) 854 (003)Mod CM 1127 (005) 1080 (005) 1043 (005) 1040 (006)PQR 1207 (008) 1053 (006) 866 (004) 792 (003)

100 1

Joint QR 1223 (006) 1091 (005) 954 (004) 911 (003)Pointwise QR 1287 (007) 1139 (005) 985 (004) 929 (003)Mod CM 1243 (006) 1180 (006) 1119 (006) 1109 (007)PQR 1370 (009) 1191 (006) 970 (004) 884 (003)

1000 150

Joint QR 287 (001) 242 (001) 189 (001) 165 (001)Pointwise QR 306 (001) 260 (001) 207 (001) 186 (001)Mod CM 313 (001) 265 (001) 210 (001) 191 (001)PQR 310 (001) 247 (001) 178 (001) 148 (001)

1000 10

Joint QR 621 (002) 545 (002) 479 (002) 466 (002)Pointwise QR 638 (002) 559 (002) 490 (002) 477 (002)Mod CM 646 (002) 570 (002) 501 (002) 486 (002)PQR 668 (003) 582 (002) 492 (002) 460 (002)

1000 5

Joint QR 808 (003) 701 (002) 607 (002) 585 (002)Pointwise QR 827 (003) 715 (002) 616 (002) 594 (002)Mod CM 830 (003) 726 (002) 631 (002) 608 (002)PQR 864 (003) 750 (003) 627 (002) 581 (002)

1000 2

Joint QR 1076 (003) 921 (003) 776 (002) 738 (002)Pointwise QR 1095 (004) 934 (003) 785 (003) 745 (002)Mod CM 1092 (003) 945 (003) 801 (003) 760 (003)PQR 1138 (004) 979 (003) 803 (003) 734 (002)

1000 1

Joint QR 1254 (004) 1065 (003) 883 (003) 832 (003)Pointwise QR 1274 (004) 1081 (004) 892 (003) 839 (003)Mod CM 1270 (004) 1091 (004) 910 (003) 856 (003)PQR 1316 (005) 1126 (004) 913 (003) 829 (003)

Table 4 Average computing time (in seconds) of the three approaches that involve estimating the conditional distribution for the case ofhaving a scalar covariate and a densely observed functional covariate

Distribution Method 119899 = 100 119899 = 1000

NormalJoint QR 12 133

Pointwise QR 148 271Mod CM 278 511

MixtureJoint QR 13 154

Pointwise QR 151 296Mod CM 327 532

Journal of Probability and Statistics 9

affects the feed intake demeanor of lactating sows of differentparities We focus on three specific time points during thelactation periodmdashbeginning (lactation day 4) middle (day11) and end (day 18)mdashand the analyses are done separatelyfor each time point We consider two types of responses thatare meant to assess the feed intake behavior using the currentand the previous lactation days The first one quantifies theabsolute change in the feed intake over two consecutive daysand the second one quantifies the relative change and takesinto account the usual sowrsquos feed intake We define themformally after introducing some notation

Let 119865119868119894119895 be the 119895th measurement of the feed intakeobserved for the 119894th sow and denote by the lactation day 119871119863119894119895when 119865119868119894119895 is measured here 119895 = 1 119899119894 Most sows areobserved every day within the first 21 lactation days and thushave 119899119894 = 21 First define the absolute change in the feedintake between two consecutive days asΔ(1)

119894(119895+1)= 119865119868119894(119895+1)minus119865119868119894119895

for 119895 that satisfies 119871119863119894(119895+1)minus119871119863119894119895 = 1 For instanceΔ(1)119894(119895+1) = 0means there was no change in feed intake of sow 119894 betweenthe current day and the previous day while Δ(1)

119894(119895+1)lt 0means

that the feed intake consumed by the 119894th sow in the currentday is smaller than the feed intake consumed in the previousday However the same amount of change in the feed intakemay reflect some stress level for a sow who typically eats a lotand a more serious stress level for a sow that usually has alower appetite For this we define the relative change in thefeed intake by Δ(2)

119894(119895+1)= (119865119868119894(119895+1) minus 119865119868119894119895)(119871119863119894(119895+1) minus 119871119863119894119895) sdot

119879119860 119894 where 119879119860 119894 is the trimmed average of feed intake of 119894thsow calculated as the average feed intake after removing thelowest 20 and highest 20 of the feed intake measurements1198651198681198941 119865119868119894119899119894 taken for the corresponding sow Here 119879119860 119894 issurrogate for the usual amount of feed intake of the 119894th sowTrimmed average is used instead of the common average toremove outliers of very low feed intakes in first few lactatingdays For example consider the situation of two sows sow119894 that typically consumes 10lb food per day and sow 1198941015840 thatconsumes 5lb food per day A reduction of 5lb in the feedintake over two consecutive days corresponds to Δ(2)

119894(119895+1)=

minus50 for the 119894th sow and Δ(2)1198941015840(119895+1)

= minus100 for the 1198941015840th sowClearly both sows are stressed (negative value) but the secondsow is stressed more as its absolute relative change is largerin view of this we refer to the second response as the stressindex Due to the construction of the two types of responsesthe data size varies for lactation days 4 (119895 = 3) 11 (119895 = 10)and 18 (119895 = 17) for the first response Δ(1)

119894(119895+1) we have sample

sizes of 233 350 and 278 whereas for Δ(2)119894(119895+1)

the sample sizesare 362 373 and 336 for the respective lactation days

In this analysis we center the attention on the effect ofthe ambient temperature and humidity on the 1st quartile ofthe proxy stress measures and gain more understanding ofthe food consumption of sows that are most susceptible toheat stress While the association between the feed intake oflactating sows and the ambient conditions of the farrowingroom has been an active research area for some timeaccounting for the temperature daily profile has not beenconsidered yet hitherto Figure 1 displays the temperature

and humidity daily profiles recorded at a frequency of 5-minute window intervals for three different days Preliminaryinvestigation reveals that temperature is negatively correlatedwith humidity at each time this phenomenon is causedbecause the farm uses cool cell panels and fans to controlthe ambient temperature Furthermore it appears that thereis a strong pointwise correlation between temperature andhumidity In view of these observations in our analysis weconsider the daily average of humidity Exploratory analysisof the feed intake behavior of the sows suggests similaritiesfor the sows with parity greater than older sows (ones whoare at their third pregnancy or higher) thus we use a parityindicator instead of the actual parity of the sow The parityindicator 119875119894 is defined as one if the 119894th sow has parity oneand zero otherwise

For the analysis we smooth daily temperature measure-ments of each sow using univariate smoother with 15 cubicregression bases and quadratic penalty REML is used toestimate smoothing parameter The smoothed temperaturecurve for sow 119894rsquos 119895th repeatedmeasure is denoted by119879119894119895(119905) 119905 isin[0 24) and the corresponding daily average humidity isdenoted by119860119867119894119895 Both temperature and average humidity arecentered before being used in the analysis

For convenience we denote the response with Δ 119894119895 byremoving the superscript In this application for fixed 119895 Δ 119894119895corresponds to 119884119894 in Section 2 119875119894 and 119860119867119894119895 correspond toscalar covariates1198831198942 and119879119894119895(119905) and119860119867119894119895 sdot119879119894119895(119905) correspond tofunctional covariates1198831198943(sdot) We first estimate the conditionaldistribution ofΔ 119894119895 given temperature119879119894119895(119905) average humidity119860119867119894119895 parity 119875119894 and interaction 119860119867119894119895 sdot 119879119894119895(119905) Specifically foreach of lactation days of interest (119895 = 3 10 and 17) we create aset of 100 equispaced grid of points between the fifth smallestand fifth largest values of Δ 119894119895rsquos and denote the grids withD = 119889ℓ ℓ = 1 100 Then we create artificial binaryresponses 1(Δ 119894119895 le 119889ℓ) ℓ = 1 100 and fit the followingmodel for 119865119894119895(119889ℓ) = 119864[1(Δ 119894119895 le 119889ℓ) | 119879119894119895(119905) 119860119867119894119895 119875119894]

119864 [1 (Δ 119894119895 le 119889ℓ) | 119879119894119895 (119905) 119860119867119894119895 119875119894] = 119892minus1 1205730 (119889ℓ)

+ 1205731 (119889ℓ) 119875119894 + 1205732 (119889ℓ) 119860119867119894119895 + int1205733 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

+ 119860119867119894119895 int1205734 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

(7)

where 1205730(sdot) is a smooth intercept 1205731(sdot) quantifies the smootheffect of young sows1205732(sdot) describes the effect of the humidityand 1205733(sdot 119905) and 1205734(sdot 119905) quantify the effect of the temperatureat time 119905 as well as the interaction between the temperature attime 119905 and average humidityWemodel1205730(sdot)using 20 univari-ate basis functions 1205731(sdot) and 1205732(sdot) using five univariate basisfunctions and 1205733(sdot sdot) and 1205734(sdot sdot) using tensor product of twounivariate bases functions (total of 25 functions)Throughoutthe analysis cubic B-spline bases are used and REML isused for estimating smoothing parameters The estimatedconditional distribution denoted by 119865119894119895(119889) is monotonizedby fitting isotonic regression to (119889ℓ 119865119894119895(119889ℓ)) ℓ = 10 90ten smallest and ten largest 119889ℓ and the corresponding valuesof 119865119895(119889ℓ) are removed to avoid boundary effects By abuse of

10 Journal of Probability and Statistics

2425

2627

2829

30

Temperature

6065

7075

80

Humidity

16H

20H

18H

22H

14H

12H

10H

14H8H4H0H 2H 6H

time of day

16H

20H

12H

22H

18H

10H

14H

14H 0H 6H2H 8H4H

time of day

Figure 1 Temperature (∘C) and humidity () observed profiles (dashed) for three randomly selected days and the corresponding smoothedones (solid) the x-axis begins at 14H (2PM)

2324

2526

2728

29

Lactation day 4

203040median607080

2324

2526

2728

29

Lactation day 11

203040median607080

2324

2526

2728

29

Lactation day 18

203040median607080

8H6H4H2H0H 12H

14H

10H

14H

22H

16H

20H

18H

time of day

8H4H2H0H 6H22H

12H

16H

20H

14H

14H

10H

18H

time of day

4H 6H0H 2H 8H20H

12H

14H

14H

22H

16H

18H

10H

time of day

Figure 2 Temperature curves with which prediction of quantiles is made Dashed black line is pointwise average of temperature curves andsolid lines are pointwise quartiles all curves are smoothed

notation 119865119894119895(119889) denotes the resulting monotonized estimateddistribution Finally we obtain estimated first quartiles iequantiles at 120591 = 025 level by inverting 119865119894119895(119889) namely 119876(120591 =025 | 119879119894119895(119905) 119860119867119894119895 119875119894) = inf119889 119865119894119895(119889) ge 025

To understand the relationship between the lactatingsows feed intake and the thermal condition of the farrowingroom we systematically compare and study the predictedquantiles of two responses at combinations of different valuesof temperature humidity and parity For each of threelactation days (119895 = 3 10 17) we consider three values ofaverage humidity (first quartile median and third quartile)and two levels of parity (0 for older sows and 1 for youngersows) Based on the experimentersrsquo interest for the functional

covariate119879119894119895(sdot)we consider seven smooth temperature curvesgiven in Figure 2 Each of these curves is obtained by firstcalculating pointwise quantiles of temperature at five-minuteintervals for a specific level and then smoothing it weconsidered quantiles levels 120578 = 02 03 and 08 In shortfor each of three lactation days we obtain the first quartileof two responses for 42 different combinations (3 humidityvalues times 2 parity levels times 7 temperature curves) using theproposed method To avoid extrapolation we ascertain that(i) there are reasonablymany observedmeasurements at eachof the combinations and (ii) bottom 25 of the responses arenot dominantly from one of the parity group see distributionof each response by the parity in Figure 3

Journal of Probability and Statistics 11

Response (2) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus10

00

10

20

Resp

onse

(2)

minus15

minus05

05

10

Resp

onse

(2)

minus2minus1

01

Resp

onse

(2)

Response (1) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus4minus2

02

46

Resp

onse

(1)

minus6minus4

minus20

24

6Re

spon

se (1

)

minus4minus2

02

46

Resp

onse

(1)

Figure 3 Top panels back-to-back boxplots of the absolute change in feed intake at a specific day by parity Bottom panels back-to-backboxplots of the relative change in feed intake at a specific day by parity

The resulting predicted quantiles are shown in Figure 4Here we focus our discussion on predicted quantile of Δ(2)

119894(119895+1)

at quantile level 120591 = 025 for lactation day 4 (119895 = 3)mdashthefirst plot of the second row in Figure 4 The results suggestthat the feed intake of older sows (parity 119875119894 = 0 greylines) is less affected by high temperatures than younger sows(black lines) this finding is in agreement with Bloemhof etal [42] We also observe that the effects of humidity andtemperature on feed intake change are strongly intertwinedFor illustration we focus on lactation day 4 (119895 = 3) again foryounger sows (black lines) For medium humidity (dashedlines) their feed intake stays pretty constant as temperatureincreases while for low and high humidity levels (solid anddotted lines respectively) it changes with an opposite direc-tion Specifically when temperature increases the predictedfirst quartile of Δ(2)

119894(119895+1)increases for low humidity (solid line)

whereas it decreases for high humidity (dotted line) Ourresults imply that high humidity (dotted line) is related toa negative impact of high temperature on feed intake whilelow humidity (solid line) alleviates it and this finding isconsistent with a previous study [45] The analysis resultsuggests to keep low humidity levels in order to maintainhealthy feed intake behavior when ambient temperature isabove 60th percentile high humidity levels are desirable forcooler ambient temperature

Interpretation of the other results is similar While theeffects of covariates on feed intake are less apparent toward

the end of lactation period we still observe similar patternacross all three lactation days For the 11th day (119895 = 10) the25th quantile of the feed intake is predicted to decrease whenthe temperature stays below the 40th percentile regardlessof humidity level and sows age However it starts increas-ing with low humidity while it continues decreasing withhigh humidity when the temperature rises above the 40thpercentile Similarly for the 18th day (119895 = 17) when thetemperature rises above the 60th percentile the predictedfirst quartile increases with low humidity while it decreaseswith high humidity The effect of temperature on feed intakeseems less obvious for lactation days 11 and 18 than for day 4while the effect may be due to lactation day it may also be aresult of other factors such asmore fluctuation and variabilityin temperature curves on day 4 than on other two days(see Figure 2) Overall we conclude that high humidity andtemperature affect the sows feed intake behavior negativelyand young sows (parity one) are more sensitive to heat stressthan older sows (higher parity) especially at the beginning oflactation period

6 Discussion

The proposed modeling framework opens up a couple offuture research directions A first research avenue is todevelop significance tests of null covariate effect Testingfor the null effect of a covariate on the conditional dis-tribution of the response is equivalent to testing that the

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

8 Journal of Probability and Statistics

Table 3 AverageMAE (standard error in parentheses) of the predicted 120591-level quantile for the case of heteroskedasticitywith a scalar covariateand a sparsely observed functional covariate

Sample size SNR Method 120591 = 005 120591 = 01 120591 = 025 120591 = 05

100 150

Joint QR 441 (003) 399 (002) 348 (002) 325 (002)Pointwise QR 540 (003) 484 (003) 426 (002) 421 (003)Mod CM 600 (003) 571 (003) 542 (003) 539 (003)PQR 438 (005) 343 (003) 252 (002) 214 (002)

100 10

Joint QR 693 (003) 640 (003) 589 (002) 576 (002)Pointwise QR 794 (004) 720 (003) 648 (003) 630 (003)Mod CM 841 (004) 822 (004) 813 (004) 805 (004)PQR 776 (006) 657 (004) 547 (003) 508 (002)

100 5

Joint QR 844 (004) 772 (003) 705 (003) 686 (003)Pointwise QR 944 (005) 850 (004) 757 (003) 726 (003)Mod CM 965 (005) 940 (005) 936 (005) 926 (005)PQR 951 (007) 825 (005) 687 (003) 633 (002)

100 2

Joint QR 1068 (005) 963 (004) 860 (003) 830 (003)Pointwise QR 1155 (006) 1029 (005) 900 (004) 854 (003)Mod CM 1127 (005) 1080 (005) 1043 (005) 1040 (006)PQR 1207 (008) 1053 (006) 866 (004) 792 (003)

100 1

Joint QR 1223 (006) 1091 (005) 954 (004) 911 (003)Pointwise QR 1287 (007) 1139 (005) 985 (004) 929 (003)Mod CM 1243 (006) 1180 (006) 1119 (006) 1109 (007)PQR 1370 (009) 1191 (006) 970 (004) 884 (003)

1000 150

Joint QR 287 (001) 242 (001) 189 (001) 165 (001)Pointwise QR 306 (001) 260 (001) 207 (001) 186 (001)Mod CM 313 (001) 265 (001) 210 (001) 191 (001)PQR 310 (001) 247 (001) 178 (001) 148 (001)

1000 10

Joint QR 621 (002) 545 (002) 479 (002) 466 (002)Pointwise QR 638 (002) 559 (002) 490 (002) 477 (002)Mod CM 646 (002) 570 (002) 501 (002) 486 (002)PQR 668 (003) 582 (002) 492 (002) 460 (002)

1000 5

Joint QR 808 (003) 701 (002) 607 (002) 585 (002)Pointwise QR 827 (003) 715 (002) 616 (002) 594 (002)Mod CM 830 (003) 726 (002) 631 (002) 608 (002)PQR 864 (003) 750 (003) 627 (002) 581 (002)

1000 2

Joint QR 1076 (003) 921 (003) 776 (002) 738 (002)Pointwise QR 1095 (004) 934 (003) 785 (003) 745 (002)Mod CM 1092 (003) 945 (003) 801 (003) 760 (003)PQR 1138 (004) 979 (003) 803 (003) 734 (002)

1000 1

Joint QR 1254 (004) 1065 (003) 883 (003) 832 (003)Pointwise QR 1274 (004) 1081 (004) 892 (003) 839 (003)Mod CM 1270 (004) 1091 (004) 910 (003) 856 (003)PQR 1316 (005) 1126 (004) 913 (003) 829 (003)

Table 4 Average computing time (in seconds) of the three approaches that involve estimating the conditional distribution for the case ofhaving a scalar covariate and a densely observed functional covariate

Distribution Method 119899 = 100 119899 = 1000

NormalJoint QR 12 133

Pointwise QR 148 271Mod CM 278 511

MixtureJoint QR 13 154

Pointwise QR 151 296Mod CM 327 532

Journal of Probability and Statistics 9

affects the feed intake demeanor of lactating sows of differentparities We focus on three specific time points during thelactation periodmdashbeginning (lactation day 4) middle (day11) and end (day 18)mdashand the analyses are done separatelyfor each time point We consider two types of responses thatare meant to assess the feed intake behavior using the currentand the previous lactation days The first one quantifies theabsolute change in the feed intake over two consecutive daysand the second one quantifies the relative change and takesinto account the usual sowrsquos feed intake We define themformally after introducing some notation

Let 119865119868119894119895 be the 119895th measurement of the feed intakeobserved for the 119894th sow and denote by the lactation day 119871119863119894119895when 119865119868119894119895 is measured here 119895 = 1 119899119894 Most sows areobserved every day within the first 21 lactation days and thushave 119899119894 = 21 First define the absolute change in the feedintake between two consecutive days asΔ(1)

119894(119895+1)= 119865119868119894(119895+1)minus119865119868119894119895

for 119895 that satisfies 119871119863119894(119895+1)minus119871119863119894119895 = 1 For instanceΔ(1)119894(119895+1) = 0means there was no change in feed intake of sow 119894 betweenthe current day and the previous day while Δ(1)

119894(119895+1)lt 0means

that the feed intake consumed by the 119894th sow in the currentday is smaller than the feed intake consumed in the previousday However the same amount of change in the feed intakemay reflect some stress level for a sow who typically eats a lotand a more serious stress level for a sow that usually has alower appetite For this we define the relative change in thefeed intake by Δ(2)

119894(119895+1)= (119865119868119894(119895+1) minus 119865119868119894119895)(119871119863119894(119895+1) minus 119871119863119894119895) sdot

119879119860 119894 where 119879119860 119894 is the trimmed average of feed intake of 119894thsow calculated as the average feed intake after removing thelowest 20 and highest 20 of the feed intake measurements1198651198681198941 119865119868119894119899119894 taken for the corresponding sow Here 119879119860 119894 issurrogate for the usual amount of feed intake of the 119894th sowTrimmed average is used instead of the common average toremove outliers of very low feed intakes in first few lactatingdays For example consider the situation of two sows sow119894 that typically consumes 10lb food per day and sow 1198941015840 thatconsumes 5lb food per day A reduction of 5lb in the feedintake over two consecutive days corresponds to Δ(2)

119894(119895+1)=

minus50 for the 119894th sow and Δ(2)1198941015840(119895+1)

= minus100 for the 1198941015840th sowClearly both sows are stressed (negative value) but the secondsow is stressed more as its absolute relative change is largerin view of this we refer to the second response as the stressindex Due to the construction of the two types of responsesthe data size varies for lactation days 4 (119895 = 3) 11 (119895 = 10)and 18 (119895 = 17) for the first response Δ(1)

119894(119895+1) we have sample

sizes of 233 350 and 278 whereas for Δ(2)119894(119895+1)

the sample sizesare 362 373 and 336 for the respective lactation days

In this analysis we center the attention on the effect ofthe ambient temperature and humidity on the 1st quartile ofthe proxy stress measures and gain more understanding ofthe food consumption of sows that are most susceptible toheat stress While the association between the feed intake oflactating sows and the ambient conditions of the farrowingroom has been an active research area for some timeaccounting for the temperature daily profile has not beenconsidered yet hitherto Figure 1 displays the temperature

and humidity daily profiles recorded at a frequency of 5-minute window intervals for three different days Preliminaryinvestigation reveals that temperature is negatively correlatedwith humidity at each time this phenomenon is causedbecause the farm uses cool cell panels and fans to controlthe ambient temperature Furthermore it appears that thereis a strong pointwise correlation between temperature andhumidity In view of these observations in our analysis weconsider the daily average of humidity Exploratory analysisof the feed intake behavior of the sows suggests similaritiesfor the sows with parity greater than older sows (ones whoare at their third pregnancy or higher) thus we use a parityindicator instead of the actual parity of the sow The parityindicator 119875119894 is defined as one if the 119894th sow has parity oneand zero otherwise

For the analysis we smooth daily temperature measure-ments of each sow using univariate smoother with 15 cubicregression bases and quadratic penalty REML is used toestimate smoothing parameter The smoothed temperaturecurve for sow 119894rsquos 119895th repeatedmeasure is denoted by119879119894119895(119905) 119905 isin[0 24) and the corresponding daily average humidity isdenoted by119860119867119894119895 Both temperature and average humidity arecentered before being used in the analysis

For convenience we denote the response with Δ 119894119895 byremoving the superscript In this application for fixed 119895 Δ 119894119895corresponds to 119884119894 in Section 2 119875119894 and 119860119867119894119895 correspond toscalar covariates1198831198942 and119879119894119895(119905) and119860119867119894119895 sdot119879119894119895(119905) correspond tofunctional covariates1198831198943(sdot) We first estimate the conditionaldistribution ofΔ 119894119895 given temperature119879119894119895(119905) average humidity119860119867119894119895 parity 119875119894 and interaction 119860119867119894119895 sdot 119879119894119895(119905) Specifically foreach of lactation days of interest (119895 = 3 10 and 17) we create aset of 100 equispaced grid of points between the fifth smallestand fifth largest values of Δ 119894119895rsquos and denote the grids withD = 119889ℓ ℓ = 1 100 Then we create artificial binaryresponses 1(Δ 119894119895 le 119889ℓ) ℓ = 1 100 and fit the followingmodel for 119865119894119895(119889ℓ) = 119864[1(Δ 119894119895 le 119889ℓ) | 119879119894119895(119905) 119860119867119894119895 119875119894]

119864 [1 (Δ 119894119895 le 119889ℓ) | 119879119894119895 (119905) 119860119867119894119895 119875119894] = 119892minus1 1205730 (119889ℓ)

+ 1205731 (119889ℓ) 119875119894 + 1205732 (119889ℓ) 119860119867119894119895 + int1205733 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

+ 119860119867119894119895 int1205734 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

(7)

where 1205730(sdot) is a smooth intercept 1205731(sdot) quantifies the smootheffect of young sows1205732(sdot) describes the effect of the humidityand 1205733(sdot 119905) and 1205734(sdot 119905) quantify the effect of the temperatureat time 119905 as well as the interaction between the temperature attime 119905 and average humidityWemodel1205730(sdot)using 20 univari-ate basis functions 1205731(sdot) and 1205732(sdot) using five univariate basisfunctions and 1205733(sdot sdot) and 1205734(sdot sdot) using tensor product of twounivariate bases functions (total of 25 functions)Throughoutthe analysis cubic B-spline bases are used and REML isused for estimating smoothing parameters The estimatedconditional distribution denoted by 119865119894119895(119889) is monotonizedby fitting isotonic regression to (119889ℓ 119865119894119895(119889ℓ)) ℓ = 10 90ten smallest and ten largest 119889ℓ and the corresponding valuesof 119865119895(119889ℓ) are removed to avoid boundary effects By abuse of

10 Journal of Probability and Statistics

2425

2627

2829

30

Temperature

6065

7075

80

Humidity

16H

20H

18H

22H

14H

12H

10H

14H8H4H0H 2H 6H

time of day

16H

20H

12H

22H

18H

10H

14H

14H 0H 6H2H 8H4H

time of day

Figure 1 Temperature (∘C) and humidity () observed profiles (dashed) for three randomly selected days and the corresponding smoothedones (solid) the x-axis begins at 14H (2PM)

2324

2526

2728

29

Lactation day 4

203040median607080

2324

2526

2728

29

Lactation day 11

203040median607080

2324

2526

2728

29

Lactation day 18

203040median607080

8H6H4H2H0H 12H

14H

10H

14H

22H

16H

20H

18H

time of day

8H4H2H0H 6H22H

12H

16H

20H

14H

14H

10H

18H

time of day

4H 6H0H 2H 8H20H

12H

14H

14H

22H

16H

18H

10H

time of day

Figure 2 Temperature curves with which prediction of quantiles is made Dashed black line is pointwise average of temperature curves andsolid lines are pointwise quartiles all curves are smoothed

notation 119865119894119895(119889) denotes the resulting monotonized estimateddistribution Finally we obtain estimated first quartiles iequantiles at 120591 = 025 level by inverting 119865119894119895(119889) namely 119876(120591 =025 | 119879119894119895(119905) 119860119867119894119895 119875119894) = inf119889 119865119894119895(119889) ge 025

To understand the relationship between the lactatingsows feed intake and the thermal condition of the farrowingroom we systematically compare and study the predictedquantiles of two responses at combinations of different valuesof temperature humidity and parity For each of threelactation days (119895 = 3 10 17) we consider three values ofaverage humidity (first quartile median and third quartile)and two levels of parity (0 for older sows and 1 for youngersows) Based on the experimentersrsquo interest for the functional

covariate119879119894119895(sdot)we consider seven smooth temperature curvesgiven in Figure 2 Each of these curves is obtained by firstcalculating pointwise quantiles of temperature at five-minuteintervals for a specific level and then smoothing it weconsidered quantiles levels 120578 = 02 03 and 08 In shortfor each of three lactation days we obtain the first quartileof two responses for 42 different combinations (3 humidityvalues times 2 parity levels times 7 temperature curves) using theproposed method To avoid extrapolation we ascertain that(i) there are reasonablymany observedmeasurements at eachof the combinations and (ii) bottom 25 of the responses arenot dominantly from one of the parity group see distributionof each response by the parity in Figure 3

Journal of Probability and Statistics 11

Response (2) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus10

00

10

20

Resp

onse

(2)

minus15

minus05

05

10

Resp

onse

(2)

minus2minus1

01

Resp

onse

(2)

Response (1) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus4minus2

02

46

Resp

onse

(1)

minus6minus4

minus20

24

6Re

spon

se (1

)

minus4minus2

02

46

Resp

onse

(1)

Figure 3 Top panels back-to-back boxplots of the absolute change in feed intake at a specific day by parity Bottom panels back-to-backboxplots of the relative change in feed intake at a specific day by parity

The resulting predicted quantiles are shown in Figure 4Here we focus our discussion on predicted quantile of Δ(2)

119894(119895+1)

at quantile level 120591 = 025 for lactation day 4 (119895 = 3)mdashthefirst plot of the second row in Figure 4 The results suggestthat the feed intake of older sows (parity 119875119894 = 0 greylines) is less affected by high temperatures than younger sows(black lines) this finding is in agreement with Bloemhof etal [42] We also observe that the effects of humidity andtemperature on feed intake change are strongly intertwinedFor illustration we focus on lactation day 4 (119895 = 3) again foryounger sows (black lines) For medium humidity (dashedlines) their feed intake stays pretty constant as temperatureincreases while for low and high humidity levels (solid anddotted lines respectively) it changes with an opposite direc-tion Specifically when temperature increases the predictedfirst quartile of Δ(2)

119894(119895+1)increases for low humidity (solid line)

whereas it decreases for high humidity (dotted line) Ourresults imply that high humidity (dotted line) is related toa negative impact of high temperature on feed intake whilelow humidity (solid line) alleviates it and this finding isconsistent with a previous study [45] The analysis resultsuggests to keep low humidity levels in order to maintainhealthy feed intake behavior when ambient temperature isabove 60th percentile high humidity levels are desirable forcooler ambient temperature

Interpretation of the other results is similar While theeffects of covariates on feed intake are less apparent toward

the end of lactation period we still observe similar patternacross all three lactation days For the 11th day (119895 = 10) the25th quantile of the feed intake is predicted to decrease whenthe temperature stays below the 40th percentile regardlessof humidity level and sows age However it starts increas-ing with low humidity while it continues decreasing withhigh humidity when the temperature rises above the 40thpercentile Similarly for the 18th day (119895 = 17) when thetemperature rises above the 60th percentile the predictedfirst quartile increases with low humidity while it decreaseswith high humidity The effect of temperature on feed intakeseems less obvious for lactation days 11 and 18 than for day 4while the effect may be due to lactation day it may also be aresult of other factors such asmore fluctuation and variabilityin temperature curves on day 4 than on other two days(see Figure 2) Overall we conclude that high humidity andtemperature affect the sows feed intake behavior negativelyand young sows (parity one) are more sensitive to heat stressthan older sows (higher parity) especially at the beginning oflactation period

6 Discussion

The proposed modeling framework opens up a couple offuture research directions A first research avenue is todevelop significance tests of null covariate effect Testingfor the null effect of a covariate on the conditional dis-tribution of the response is equivalent to testing that the

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

Journal of Probability and Statistics 9

affects the feed intake demeanor of lactating sows of differentparities We focus on three specific time points during thelactation periodmdashbeginning (lactation day 4) middle (day11) and end (day 18)mdashand the analyses are done separatelyfor each time point We consider two types of responses thatare meant to assess the feed intake behavior using the currentand the previous lactation days The first one quantifies theabsolute change in the feed intake over two consecutive daysand the second one quantifies the relative change and takesinto account the usual sowrsquos feed intake We define themformally after introducing some notation

Let 119865119868119894119895 be the 119895th measurement of the feed intakeobserved for the 119894th sow and denote by the lactation day 119871119863119894119895when 119865119868119894119895 is measured here 119895 = 1 119899119894 Most sows areobserved every day within the first 21 lactation days and thushave 119899119894 = 21 First define the absolute change in the feedintake between two consecutive days asΔ(1)

119894(119895+1)= 119865119868119894(119895+1)minus119865119868119894119895

for 119895 that satisfies 119871119863119894(119895+1)minus119871119863119894119895 = 1 For instanceΔ(1)119894(119895+1) = 0means there was no change in feed intake of sow 119894 betweenthe current day and the previous day while Δ(1)

119894(119895+1)lt 0means

that the feed intake consumed by the 119894th sow in the currentday is smaller than the feed intake consumed in the previousday However the same amount of change in the feed intakemay reflect some stress level for a sow who typically eats a lotand a more serious stress level for a sow that usually has alower appetite For this we define the relative change in thefeed intake by Δ(2)

119894(119895+1)= (119865119868119894(119895+1) minus 119865119868119894119895)(119871119863119894(119895+1) minus 119871119863119894119895) sdot

119879119860 119894 where 119879119860 119894 is the trimmed average of feed intake of 119894thsow calculated as the average feed intake after removing thelowest 20 and highest 20 of the feed intake measurements1198651198681198941 119865119868119894119899119894 taken for the corresponding sow Here 119879119860 119894 issurrogate for the usual amount of feed intake of the 119894th sowTrimmed average is used instead of the common average toremove outliers of very low feed intakes in first few lactatingdays For example consider the situation of two sows sow119894 that typically consumes 10lb food per day and sow 1198941015840 thatconsumes 5lb food per day A reduction of 5lb in the feedintake over two consecutive days corresponds to Δ(2)

119894(119895+1)=

minus50 for the 119894th sow and Δ(2)1198941015840(119895+1)

= minus100 for the 1198941015840th sowClearly both sows are stressed (negative value) but the secondsow is stressed more as its absolute relative change is largerin view of this we refer to the second response as the stressindex Due to the construction of the two types of responsesthe data size varies for lactation days 4 (119895 = 3) 11 (119895 = 10)and 18 (119895 = 17) for the first response Δ(1)

119894(119895+1) we have sample

sizes of 233 350 and 278 whereas for Δ(2)119894(119895+1)

the sample sizesare 362 373 and 336 for the respective lactation days

In this analysis we center the attention on the effect ofthe ambient temperature and humidity on the 1st quartile ofthe proxy stress measures and gain more understanding ofthe food consumption of sows that are most susceptible toheat stress While the association between the feed intake oflactating sows and the ambient conditions of the farrowingroom has been an active research area for some timeaccounting for the temperature daily profile has not beenconsidered yet hitherto Figure 1 displays the temperature

and humidity daily profiles recorded at a frequency of 5-minute window intervals for three different days Preliminaryinvestigation reveals that temperature is negatively correlatedwith humidity at each time this phenomenon is causedbecause the farm uses cool cell panels and fans to controlthe ambient temperature Furthermore it appears that thereis a strong pointwise correlation between temperature andhumidity In view of these observations in our analysis weconsider the daily average of humidity Exploratory analysisof the feed intake behavior of the sows suggests similaritiesfor the sows with parity greater than older sows (ones whoare at their third pregnancy or higher) thus we use a parityindicator instead of the actual parity of the sow The parityindicator 119875119894 is defined as one if the 119894th sow has parity oneand zero otherwise

For the analysis we smooth daily temperature measure-ments of each sow using univariate smoother with 15 cubicregression bases and quadratic penalty REML is used toestimate smoothing parameter The smoothed temperaturecurve for sow 119894rsquos 119895th repeatedmeasure is denoted by119879119894119895(119905) 119905 isin[0 24) and the corresponding daily average humidity isdenoted by119860119867119894119895 Both temperature and average humidity arecentered before being used in the analysis

For convenience we denote the response with Δ 119894119895 byremoving the superscript In this application for fixed 119895 Δ 119894119895corresponds to 119884119894 in Section 2 119875119894 and 119860119867119894119895 correspond toscalar covariates1198831198942 and119879119894119895(119905) and119860119867119894119895 sdot119879119894119895(119905) correspond tofunctional covariates1198831198943(sdot) We first estimate the conditionaldistribution ofΔ 119894119895 given temperature119879119894119895(119905) average humidity119860119867119894119895 parity 119875119894 and interaction 119860119867119894119895 sdot 119879119894119895(119905) Specifically foreach of lactation days of interest (119895 = 3 10 and 17) we create aset of 100 equispaced grid of points between the fifth smallestand fifth largest values of Δ 119894119895rsquos and denote the grids withD = 119889ℓ ℓ = 1 100 Then we create artificial binaryresponses 1(Δ 119894119895 le 119889ℓ) ℓ = 1 100 and fit the followingmodel for 119865119894119895(119889ℓ) = 119864[1(Δ 119894119895 le 119889ℓ) | 119879119894119895(119905) 119860119867119894119895 119875119894]

119864 [1 (Δ 119894119895 le 119889ℓ) | 119879119894119895 (119905) 119860119867119894119895 119875119894] = 119892minus1 1205730 (119889ℓ)

+ 1205731 (119889ℓ) 119875119894 + 1205732 (119889ℓ) 119860119867119894119895 + int1205733 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

+ 119860119867119894119895 int1205734 (119889ℓ 119905) 119879119894119895 (119905) 119889119905

(7)

where 1205730(sdot) is a smooth intercept 1205731(sdot) quantifies the smootheffect of young sows1205732(sdot) describes the effect of the humidityand 1205733(sdot 119905) and 1205734(sdot 119905) quantify the effect of the temperatureat time 119905 as well as the interaction between the temperature attime 119905 and average humidityWemodel1205730(sdot)using 20 univari-ate basis functions 1205731(sdot) and 1205732(sdot) using five univariate basisfunctions and 1205733(sdot sdot) and 1205734(sdot sdot) using tensor product of twounivariate bases functions (total of 25 functions)Throughoutthe analysis cubic B-spline bases are used and REML isused for estimating smoothing parameters The estimatedconditional distribution denoted by 119865119894119895(119889) is monotonizedby fitting isotonic regression to (119889ℓ 119865119894119895(119889ℓ)) ℓ = 10 90ten smallest and ten largest 119889ℓ and the corresponding valuesof 119865119895(119889ℓ) are removed to avoid boundary effects By abuse of

10 Journal of Probability and Statistics

2425

2627

2829

30

Temperature

6065

7075

80

Humidity

16H

20H

18H

22H

14H

12H

10H

14H8H4H0H 2H 6H

time of day

16H

20H

12H

22H

18H

10H

14H

14H 0H 6H2H 8H4H

time of day

Figure 1 Temperature (∘C) and humidity () observed profiles (dashed) for three randomly selected days and the corresponding smoothedones (solid) the x-axis begins at 14H (2PM)

2324

2526

2728

29

Lactation day 4

203040median607080

2324

2526

2728

29

Lactation day 11

203040median607080

2324

2526

2728

29

Lactation day 18

203040median607080

8H6H4H2H0H 12H

14H

10H

14H

22H

16H

20H

18H

time of day

8H4H2H0H 6H22H

12H

16H

20H

14H

14H

10H

18H

time of day

4H 6H0H 2H 8H20H

12H

14H

14H

22H

16H

18H

10H

time of day

Figure 2 Temperature curves with which prediction of quantiles is made Dashed black line is pointwise average of temperature curves andsolid lines are pointwise quartiles all curves are smoothed

notation 119865119894119895(119889) denotes the resulting monotonized estimateddistribution Finally we obtain estimated first quartiles iequantiles at 120591 = 025 level by inverting 119865119894119895(119889) namely 119876(120591 =025 | 119879119894119895(119905) 119860119867119894119895 119875119894) = inf119889 119865119894119895(119889) ge 025

To understand the relationship between the lactatingsows feed intake and the thermal condition of the farrowingroom we systematically compare and study the predictedquantiles of two responses at combinations of different valuesof temperature humidity and parity For each of threelactation days (119895 = 3 10 17) we consider three values ofaverage humidity (first quartile median and third quartile)and two levels of parity (0 for older sows and 1 for youngersows) Based on the experimentersrsquo interest for the functional

covariate119879119894119895(sdot)we consider seven smooth temperature curvesgiven in Figure 2 Each of these curves is obtained by firstcalculating pointwise quantiles of temperature at five-minuteintervals for a specific level and then smoothing it weconsidered quantiles levels 120578 = 02 03 and 08 In shortfor each of three lactation days we obtain the first quartileof two responses for 42 different combinations (3 humidityvalues times 2 parity levels times 7 temperature curves) using theproposed method To avoid extrapolation we ascertain that(i) there are reasonablymany observedmeasurements at eachof the combinations and (ii) bottom 25 of the responses arenot dominantly from one of the parity group see distributionof each response by the parity in Figure 3

Journal of Probability and Statistics 11

Response (2) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus10

00

10

20

Resp

onse

(2)

minus15

minus05

05

10

Resp

onse

(2)

minus2minus1

01

Resp

onse

(2)

Response (1) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus4minus2

02

46

Resp

onse

(1)

minus6minus4

minus20

24

6Re

spon

se (1

)

minus4minus2

02

46

Resp

onse

(1)

Figure 3 Top panels back-to-back boxplots of the absolute change in feed intake at a specific day by parity Bottom panels back-to-backboxplots of the relative change in feed intake at a specific day by parity

The resulting predicted quantiles are shown in Figure 4Here we focus our discussion on predicted quantile of Δ(2)

119894(119895+1)

at quantile level 120591 = 025 for lactation day 4 (119895 = 3)mdashthefirst plot of the second row in Figure 4 The results suggestthat the feed intake of older sows (parity 119875119894 = 0 greylines) is less affected by high temperatures than younger sows(black lines) this finding is in agreement with Bloemhof etal [42] We also observe that the effects of humidity andtemperature on feed intake change are strongly intertwinedFor illustration we focus on lactation day 4 (119895 = 3) again foryounger sows (black lines) For medium humidity (dashedlines) their feed intake stays pretty constant as temperatureincreases while for low and high humidity levels (solid anddotted lines respectively) it changes with an opposite direc-tion Specifically when temperature increases the predictedfirst quartile of Δ(2)

119894(119895+1)increases for low humidity (solid line)

whereas it decreases for high humidity (dotted line) Ourresults imply that high humidity (dotted line) is related toa negative impact of high temperature on feed intake whilelow humidity (solid line) alleviates it and this finding isconsistent with a previous study [45] The analysis resultsuggests to keep low humidity levels in order to maintainhealthy feed intake behavior when ambient temperature isabove 60th percentile high humidity levels are desirable forcooler ambient temperature

Interpretation of the other results is similar While theeffects of covariates on feed intake are less apparent toward

the end of lactation period we still observe similar patternacross all three lactation days For the 11th day (119895 = 10) the25th quantile of the feed intake is predicted to decrease whenthe temperature stays below the 40th percentile regardlessof humidity level and sows age However it starts increas-ing with low humidity while it continues decreasing withhigh humidity when the temperature rises above the 40thpercentile Similarly for the 18th day (119895 = 17) when thetemperature rises above the 60th percentile the predictedfirst quartile increases with low humidity while it decreaseswith high humidity The effect of temperature on feed intakeseems less obvious for lactation days 11 and 18 than for day 4while the effect may be due to lactation day it may also be aresult of other factors such asmore fluctuation and variabilityin temperature curves on day 4 than on other two days(see Figure 2) Overall we conclude that high humidity andtemperature affect the sows feed intake behavior negativelyand young sows (parity one) are more sensitive to heat stressthan older sows (higher parity) especially at the beginning oflactation period

6 Discussion

The proposed modeling framework opens up a couple offuture research directions A first research avenue is todevelop significance tests of null covariate effect Testingfor the null effect of a covariate on the conditional dis-tribution of the response is equivalent to testing that the

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

10 Journal of Probability and Statistics

2425

2627

2829

30

Temperature

6065

7075

80

Humidity

16H

20H

18H

22H

14H

12H

10H

14H8H4H0H 2H 6H

time of day

16H

20H

12H

22H

18H

10H

14H

14H 0H 6H2H 8H4H

time of day

Figure 1 Temperature (∘C) and humidity () observed profiles (dashed) for three randomly selected days and the corresponding smoothedones (solid) the x-axis begins at 14H (2PM)

2324

2526

2728

29

Lactation day 4

203040median607080

2324

2526

2728

29

Lactation day 11

203040median607080

2324

2526

2728

29

Lactation day 18

203040median607080

8H6H4H2H0H 12H

14H

10H

14H

22H

16H

20H

18H

time of day

8H4H2H0H 6H22H

12H

16H

20H

14H

14H

10H

18H

time of day

4H 6H0H 2H 8H20H

12H

14H

14H

22H

16H

18H

10H

time of day

Figure 2 Temperature curves with which prediction of quantiles is made Dashed black line is pointwise average of temperature curves andsolid lines are pointwise quartiles all curves are smoothed

notation 119865119894119895(119889) denotes the resulting monotonized estimateddistribution Finally we obtain estimated first quartiles iequantiles at 120591 = 025 level by inverting 119865119894119895(119889) namely 119876(120591 =025 | 119879119894119895(119905) 119860119867119894119895 119875119894) = inf119889 119865119894119895(119889) ge 025

To understand the relationship between the lactatingsows feed intake and the thermal condition of the farrowingroom we systematically compare and study the predictedquantiles of two responses at combinations of different valuesof temperature humidity and parity For each of threelactation days (119895 = 3 10 17) we consider three values ofaverage humidity (first quartile median and third quartile)and two levels of parity (0 for older sows and 1 for youngersows) Based on the experimentersrsquo interest for the functional

covariate119879119894119895(sdot)we consider seven smooth temperature curvesgiven in Figure 2 Each of these curves is obtained by firstcalculating pointwise quantiles of temperature at five-minuteintervals for a specific level and then smoothing it weconsidered quantiles levels 120578 = 02 03 and 08 In shortfor each of three lactation days we obtain the first quartileof two responses for 42 different combinations (3 humidityvalues times 2 parity levels times 7 temperature curves) using theproposed method To avoid extrapolation we ascertain that(i) there are reasonablymany observedmeasurements at eachof the combinations and (ii) bottom 25 of the responses arenot dominantly from one of the parity group see distributionof each response by the parity in Figure 3

Journal of Probability and Statistics 11

Response (2) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus10

00

10

20

Resp

onse

(2)

minus15

minus05

05

10

Resp

onse

(2)

minus2minus1

01

Resp

onse

(2)

Response (1) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus4minus2

02

46

Resp

onse

(1)

minus6minus4

minus20

24

6Re

spon

se (1

)

minus4minus2

02

46

Resp

onse

(1)

Figure 3 Top panels back-to-back boxplots of the absolute change in feed intake at a specific day by parity Bottom panels back-to-backboxplots of the relative change in feed intake at a specific day by parity

The resulting predicted quantiles are shown in Figure 4Here we focus our discussion on predicted quantile of Δ(2)

119894(119895+1)

at quantile level 120591 = 025 for lactation day 4 (119895 = 3)mdashthefirst plot of the second row in Figure 4 The results suggestthat the feed intake of older sows (parity 119875119894 = 0 greylines) is less affected by high temperatures than younger sows(black lines) this finding is in agreement with Bloemhof etal [42] We also observe that the effects of humidity andtemperature on feed intake change are strongly intertwinedFor illustration we focus on lactation day 4 (119895 = 3) again foryounger sows (black lines) For medium humidity (dashedlines) their feed intake stays pretty constant as temperatureincreases while for low and high humidity levels (solid anddotted lines respectively) it changes with an opposite direc-tion Specifically when temperature increases the predictedfirst quartile of Δ(2)

119894(119895+1)increases for low humidity (solid line)

whereas it decreases for high humidity (dotted line) Ourresults imply that high humidity (dotted line) is related toa negative impact of high temperature on feed intake whilelow humidity (solid line) alleviates it and this finding isconsistent with a previous study [45] The analysis resultsuggests to keep low humidity levels in order to maintainhealthy feed intake behavior when ambient temperature isabove 60th percentile high humidity levels are desirable forcooler ambient temperature

Interpretation of the other results is similar While theeffects of covariates on feed intake are less apparent toward

the end of lactation period we still observe similar patternacross all three lactation days For the 11th day (119895 = 10) the25th quantile of the feed intake is predicted to decrease whenthe temperature stays below the 40th percentile regardlessof humidity level and sows age However it starts increas-ing with low humidity while it continues decreasing withhigh humidity when the temperature rises above the 40thpercentile Similarly for the 18th day (119895 = 17) when thetemperature rises above the 60th percentile the predictedfirst quartile increases with low humidity while it decreaseswith high humidity The effect of temperature on feed intakeseems less obvious for lactation days 11 and 18 than for day 4while the effect may be due to lactation day it may also be aresult of other factors such asmore fluctuation and variabilityin temperature curves on day 4 than on other two days(see Figure 2) Overall we conclude that high humidity andtemperature affect the sows feed intake behavior negativelyand young sows (parity one) are more sensitive to heat stressthan older sows (higher parity) especially at the beginning oflactation period

6 Discussion

The proposed modeling framework opens up a couple offuture research directions A first research avenue is todevelop significance tests of null covariate effect Testingfor the null effect of a covariate on the conditional dis-tribution of the response is equivalent to testing that the

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

Journal of Probability and Statistics 11

Response (2) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus10

00

10

20

Resp

onse

(2)

minus15

minus05

05

10

Resp

onse

(2)

minus2minus1

01

Resp

onse

(2)

Response (1) by ParityLactation day 4

ParityParity gt 2 Parity = 1

Lactation day 11

ParityParity gt 2 Parity = 1

Lactation day 18

ParityParity gt 2 Parity = 1

minus4minus2

02

46

Resp

onse

(1)

minus6minus4

minus20

24

6Re

spon

se (1

)

minus4minus2

02

46

Resp

onse

(1)

Figure 3 Top panels back-to-back boxplots of the absolute change in feed intake at a specific day by parity Bottom panels back-to-backboxplots of the relative change in feed intake at a specific day by parity

The resulting predicted quantiles are shown in Figure 4Here we focus our discussion on predicted quantile of Δ(2)

119894(119895+1)

at quantile level 120591 = 025 for lactation day 4 (119895 = 3)mdashthefirst plot of the second row in Figure 4 The results suggestthat the feed intake of older sows (parity 119875119894 = 0 greylines) is less affected by high temperatures than younger sows(black lines) this finding is in agreement with Bloemhof etal [42] We also observe that the effects of humidity andtemperature on feed intake change are strongly intertwinedFor illustration we focus on lactation day 4 (119895 = 3) again foryounger sows (black lines) For medium humidity (dashedlines) their feed intake stays pretty constant as temperatureincreases while for low and high humidity levels (solid anddotted lines respectively) it changes with an opposite direc-tion Specifically when temperature increases the predictedfirst quartile of Δ(2)

119894(119895+1)increases for low humidity (solid line)

whereas it decreases for high humidity (dotted line) Ourresults imply that high humidity (dotted line) is related toa negative impact of high temperature on feed intake whilelow humidity (solid line) alleviates it and this finding isconsistent with a previous study [45] The analysis resultsuggests to keep low humidity levels in order to maintainhealthy feed intake behavior when ambient temperature isabove 60th percentile high humidity levels are desirable forcooler ambient temperature

Interpretation of the other results is similar While theeffects of covariates on feed intake are less apparent toward

the end of lactation period we still observe similar patternacross all three lactation days For the 11th day (119895 = 10) the25th quantile of the feed intake is predicted to decrease whenthe temperature stays below the 40th percentile regardlessof humidity level and sows age However it starts increas-ing with low humidity while it continues decreasing withhigh humidity when the temperature rises above the 40thpercentile Similarly for the 18th day (119895 = 17) when thetemperature rises above the 60th percentile the predictedfirst quartile increases with low humidity while it decreaseswith high humidity The effect of temperature on feed intakeseems less obvious for lactation days 11 and 18 than for day 4while the effect may be due to lactation day it may also be aresult of other factors such asmore fluctuation and variabilityin temperature curves on day 4 than on other two days(see Figure 2) Overall we conclude that high humidity andtemperature affect the sows feed intake behavior negativelyand young sows (parity one) are more sensitive to heat stressthan older sows (higher parity) especially at the beginning oflactation period

6 Discussion

The proposed modeling framework opens up a couple offuture research directions A first research avenue is todevelop significance tests of null covariate effect Testingfor the null effect of a covariate on the conditional dis-tribution of the response is equivalent to testing that the

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

12 Journal of Probability and Statistics

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (1)minus2

5minus2

0minus1

5minus1

0minus0

50

0

Lactation day 4 Lactation day 11 Lactation day 18

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

minus25

minus20

minus15

minus10

minus05

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Parity gt 2 Low HumidityParity gt 2 Medium HumidityParity gt 2 High Humidity

Parity = 1 Low HumidityParity = 1 Medium HumidityParity = 1 High Humidity

Predicted 1st Quartile of Response (2)

minus05

minus04

minus03

minus02

minus01

00

Lactation day 4

minus05

minus04

minus03

minus02

minus01

00

Lactation day 11 Lactation day 18

minus05

minus04

minus03

minus02

minus01

00

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

30th 40th 50th 60th 70th 80th20th

pointwise percentiles of temperature curves

Figure 4 Displayed are the predicted quantiles of Δ(1)119894(119895+1)

and Δ(2)119894(119895+1)

for different parities average humidity and temperature levels In eachof all six panels black thick lines correspond to the young sows (119875119894 = 1) and grey thin lines correspond to the old sows (119875119894 = 0) Line typesindicate different average humidity levels solid dashed and dotted correspond to low medium and high average humidity levels (given bythe first quartile median and the third quartiles of119860119867119894119895) respectivelyThe seven grids in 119909-axis of each panel correspond to the 7 temperaturecurves given in the respective panel of Figure 2

corresponding regression coefficient function is equal tozero in the associated function-on-function mean regressionmodel Such significance tests have been studied when thefunctional response is continuous [30 46] however theirstudy for binary-valued functional responses is an openproblem in functional data literature and only recently hasbeen considered in Chen et al [47] Another research avenueis to do variable selection in the setting where there are manyscalar covariates and functional covariates Many currentapplications collect data with increasing number of mixedcovariates and selecting the ones that have an effect on the

conditional distribution of the response is very importantThis problem is an active research area in functional meanregression where the response is normal [48 49] Theproposed modeling framework has the potential to facilitatestudying such problem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

Journal of Probability and Statistics 13

Conflicts of Interest

The authors declare that they have no conflicts of interests

Acknowledgments

The data used originated from work supported in part by theNorth Carolina Agricultural Foundation Raleigh NC Theauthors acknowledge Zhen Han for preparing simulationsStaicursquos research was supported by National Science Founda-tion DMS 0454942 and DMS 1454942 and National Institutesof Health grants R01 NS085211 R01 MH086633 and 5P01CA142538-09

Supplementary Materials

Section S1 provides additional simulation settings and resultsfor the cases of having either a single scalar covariate or asingle functional covariate Additional results for the case ofhaving a scalar covariate and a densely observed functionalcovariate are also included Section S2 presents additionaldata analysis using the proposed method on the bike sharingdataset [50 51] (Supplementary Materials)

References

[1] H Cardot F Ferraty and P Sarda ldquoFunctional linear modelrdquoStatistics and Probability Letters vol 45 no 1 pp 11ndash22 1999

[2] G M James ldquoGeneralized linear models with functionalpredictorsrdquo Journal of the Royal Statistical Society Series B(Statistical Methodology) vol 64 no 3 pp 411ndash432 2002

[3] J Ramsay and B Silverman Functional Data Analysis SpringerNew York NY USA 2005

[4] J Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[5] F Ferraty and P VieuNonparametric Functional Data AnalysisSpringer New York NY USA 2006

[6] F Ferraty and P Vieu ldquoAdditive prediction and boosting forfunctional datardquo Computational Statistics amp Data Analysis vol53 no 4 pp 1400ndash1413 2009

[7] J Goldsmith J Bobb CM Crainiceanu B Caffo andD ReichldquoPenalized functional regressionrdquo Journal of Computational andGraphical Statistics vol 20 no 4 pp 830ndash851 2011

[8] M W McLean G Hooker A-M Staicu F Scheipl and DRuppert ldquoFunctional generalized additive modelsrdquo Journal ofComputational and Graphical Statistics vol 23 no 1 pp 249ndash269 2014

[9] N Quiniou and J Noblet ldquoInfluence of high ambient tempera-tures on performance of multiparous lactating sowsrdquo Journal ofAnimal Science vol 77 no 8 pp 2124ndash2134 1999

[10] D Renaudeau and J Noblet ldquoEffects of exposure to highambient temperature and dietary protein level on sow milkproduction and performance of pigletsrdquo Journal of AnimalScience vol 79 no 6 pp 1540ndash1548 2001

[11] N St-Pierre B Cobanov and G Schnitkey ldquoEconomic lossesfrom heat stress by US livestock industriesrdquo Journal of DairyScience vol 86 pp E52ndashE77 2003

[12] R Koenker andG Bassett Jr ldquoRegression quantilesrdquo Economet-rica vol 46 no 1 pp 33ndash50 1978

[13] R Koenker Quantile Regression vol 38 Cambridge UniversityPress 2015

[14] H D Bondell B J Reich and H Wang ldquoNoncrossing quantileregression curve estimationrdquo Biometrika vol 97 no 4 pp 825ndash838 2010

[15] H Cardot C Crambes and P Sarda ldquoQuantile regressionwhenthe covariates are functionsrdquo Journal ofNonparametric Statisticsvol 17 no 7 pp 841ndash856 2005

[16] K Kato ldquoEstimation in functional linear quantile regressionrdquo13e Annals of Statistics vol 40 no 6 pp 3108ndash3136 2012

[17] Q Tang and L Cheng ldquoPartial functional linear quantileregressionrdquo Science ChinaMathematics vol 57 no 12 pp 2589ndash2608 2014

[18] Y Lu J Du and Z Sun ldquoFunctional partially linear quantileregression modelrdquoMetrika vol 77 no 2 pp 317ndash332 2014

[19] D Yu L Kong and IMizera ldquoPartial functional linear quantileregression for neuroimaging data analysisrdquo Neurocomputingvol 195 pp 74ndash87 2016

[20] F Ferraty A Rabhi and P Vieu ldquoConditional quantiles fordependent functional data with application to the climatic rsquoelninorsquo phenomenonrdquo Sankhya 13e Indian Journal of Statisticsvol 67 no 2 pp 378ndash398 2005

[21] K Chen and H-G Muller ldquoConditional quantile analysiswhen covariates are functions with application to growth datardquoJournal of the Royal Statistical Society Series B (StatisticalMethodology) vol 74 no 1 pp 67ndash89 2012

[22] F Scheipl J Gertheiss and S Greven ldquoGeneralized functionaladditive mixed modelsrdquo Electronic Journal of Statistics vol 10no 1 pp 1455ndash1492 2016

[23] P H Eilers and B D Marx ldquoFlexible smoothing with b-splinesand penaltiesrdquo Statistical Science pp 89ndash102 1996

[24] D Ruppert ldquoSelecting the number of knots for penalizedsplinesrdquo Journal of Computational and Graphical Statistics vol11 no 4 pp 735ndash757 2002

[25] S N Wood ldquoThin plate regression splinesrdquo Journal of the RoyalStatistical Society B Statistical Methodology vol 65 no 1 pp95ndash114 2003

[26] S Wood Generalized Additive Models An Introduction with RChapman and HallCRC Boca Raton FL USA 2006

[27] A E Ivanescu A-M Staicu F Scheipl and S Greven ldquoPenal-ized function-on-function regressionrdquoComputational Statisticsvol 30 no 2 pp 539ndash568 2015

[28] J S Kim A-M Staicu A Maity R J Carroll and D RuppertldquoAdditive function-on-function regressionrdquo Journal of Compu-tational and Graphical Statistics vol 27 no 1 pp 234ndash244 2018

[29] P Ng and M Maechler ldquoA fast and efficient implementation ofqualitatively constrained quantile smoothing splinesrdquo StatisticalModelling An International Journal vol 7 no 4 pp 315ndash3282007

[30] J T Zhang and J Chen ldquoStatistical inferences for functionaldatardquo13e Annals of Statistics vol 35 no 3 pp 1052ndash1079 2007

[31] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[32] L Huang F Scheipl J Goldsmith et al ldquoR package refundmethodology for regression with functional data (version01-13)rdquo 2015 httpscranr-projectorgwebpackagesrefundindexhtml

[33] L Xiao V Zipunnikov D Ruppert and C Crainiceanu ldquoFastcovariance estimation for high-dimensional functional datardquoStatistics and Computing vol 26 no 1-2 pp 409ndash421 2016

14 Journal of Probability and Statistics

[34] L Xiao C LiWCheckley andCCrainiceanu ldquoFast covarianceestimation for sparse functional datardquo Statistics and Computingvol 28 no 3 pp 511ndash522 2018

[35] L Xiao C LiWCheckley andC Crainiceanu ldquoR package facefast covariance estimation for sparse functional data (version01-4)rdquo 2018 httpcranr-projectorgwebpackagesfaceindexhtml

[36] R E Barlow D J Bartholomew J M Bremner and H DBrunk Statistical inference under order restrictions 13e theoryand application of isotonic regression (No 04 QA278 7 B3)New York Wiley 1972

[37] V Chernozhukov I Fernandez-Val and A Galichon ldquoImprov-ing point and interval estimators of monotone functions byrearrangementrdquo Biometrika vol 96 no 3 pp 559ndash575 2009

[38] D S Rosero R D Boyd M McCulley J Odle and E vanHeugten ldquoEssential fatty acid supplementation during lactationis required to maximize the subsequent reproductive perfor-mance of the modern sowrdquo Animal Reproduction Science vol168 pp 151ndash163 2016

[39] L J Johnston M Ellis G W Libal et al ldquoEffect of room tem-perature and dietary amino acid concentration on performanceof lactating sows ncr-89 committee on swine managementrdquoJournal of Animal Science vol 77 no 7 pp 1638ndash1644 1999

[40] D Renaudeau N Quiniou and J Noblet ldquoEffects of exposureto high ambient temperature and dietary protein level onperformance of multiparous lactating sowsrdquo Journal of AnimalScience vol 79 no 5 pp 1240ndash1249 2001

[41] J L Black B PMullanM L Lorschy andL RGiles ldquoLactationin the sow during heat stressrdquo Livestock Production Science vol35 no 1-2 pp 153ndash170 1993

[42] S Bloemhof P K Mathur E F Knol and E H van der WaaijldquoEffect of daily environmental temperature on farrowing rateand total born in dam line sowsrdquo Journal of Animal Science vol91 no 6 pp 2667ndash2679 2013

[43] G C Nelson M W Rosegrant J Koo et al ldquoClimate changeimpact on agriculture and costs of adaptationrdquo InternationalFood Policy Research Institute vol 21 2009

[44] J M Melillo Climate Change Impacts in the United States Usnational climate assessment Highlights 2014

[45] R Bergsma and S Hermesch ldquoExploring breeding opportuni-ties for reduced thermal sensitivity of feed intake in the lactatingsowrdquo Journal of Animal Science vol 90 no 1 pp 85ndash98 2012

[46] Q Shen and J Faraway ldquoAn f test for linear models withfunctional responsesrdquo Statistica Sinica pp 1239ndash1257 2004

[47] S Chen L Xiao and A-M Staicu ldquoTesting for general-ized scalar-on-function linear modelsrdquo httpsarxivorgabs190604889 2018

[48] J Gertheiss A Maity and A-M Staicu ldquoVariable selection ingeneralized functional linear modelsrdquo Stat vol 2 no 1 pp 86ndash101 2013

[49] Y Chen J Goldsmith and R T Ogden ldquoVariable selection infunction-on-scalar regressionrdquo Stat vol 5 no 1 pp 88ndash1012016

[50] H Fanaee-T and J Gama ldquoEvent labeling combining ensembledetectors and background knowledgerdquo Progress in ArtificialIntelligence vol 2 no 2-3 pp 113ndash127 2014

[51] M Lichman ldquoUCI machine learning repositoryrdquo 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom