Bayesian State Space Modeling for Spatio-Temporal Rainfall Disaggregation · 2014. 9. 23. · many parameters in the temporal disaggregation model (Keller, 2007). Further, Makhnin

Bayesian State Space Modeling for Spatio-Temporal Rainfall Disaggregation

S. Astutik1, N. Iriawan2, G. Nair3, Suhartono2 and Sutikno2

1Statistics Department, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia;

& Brawijaya University, Malang, Indonesia Email: [email protected]

2Statistics Department, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Email: [email protected]; [email protected]; [email protected]

3School of Mathematics and Statistics, The University of Western Australia, Perth, Australia Email: [email protected]

ABSTRACT

With the aim of generating a finer time scale data from a coarser time scale observations, the paper develops a rainfall disaggregation method as a combination of Bayesian state-space modeling and the adjusting procedure. The method uses spatio-tempral model incorporating the cross-covariance structure between spatial observation sites. The paper develops algorithms for estimating the parameters in terms of Bayesian method, and for generating finer time scale data with and without adjusting procedure. The model and the computation methods are applied to spatio-temporal rainfall observation data from two rain gages in Sampean Watershed, Bondowoso, Indonesia. Simulation study demonstrates that the Bayesian state space model with adjusting procedure performs better than the model without adjusting procedure, in terms of preserving some of the important data characteristics.

Keywords: Bayesian state space model; adjusting procedure; spatio-temporal model;

disaggregation algorithm; rainfall data.

Mathematics Subject Classification: 62F15

1. INTRODUCTION

Rainfall disaggregation, which enables to generate a finer resolution data from a given coarser

resolution time/spatial observations, addresses some of the common problems encountered in

hydrological studies. One of such problems is the limited availability of rainfall data at suitable finer

resolution of time and/or space. Also, in hydrological simulation studies, often one requires data at

finer scale, than the observation scale or the scale at which synthetic data is generated, as input

variable in subsequent computations. Rainfall disaggregation method, due its ability to provide

synthetic data at higher resolution in such a way that hydrological structures are preserved at all

levels, can be used to determine storm duration, distribution of rainfall intensity, evaporation rate,

flood prediction, change of soil moisturize, etc.

International Journal of Applied Mathematics and Statistics,Int. J. Appl. Math. Stat.; Vol. 37; Issue No. 7; Year 2013, ISSN 0973-1377 (Print), ISSN 0973-7545 (Online) Copyright © 2013 by CESER Publications

www.ceser.in/ijamas.html www.ceserp.com/cp-jour www.ceserpublications.com

Spatio-temporal disaggregation method is a multivariate generalization of temporal disaggregation at

single site to multiple sites. This method incorporates spatio-temporal correlation in the model, in

addition to spatial and temporal correlations. Koutsoyiannis, et al. (2003) have developed a spatio-

temporal rainfall disaggregation method by using Multivariate Autoregressive, MAR(1), model for

rainfall in multiple locations. This method has a spatial correlation relation among the sites built in as a

part of the model, and the paper combines the Bartlett Lewis rainfall model methods and adjusting

procedure (coupling transformation) in a software package called MuDRain. This disaggregation

method has been applied, by Fytilas, et al., 2003, to rainfall data from Brue catchment area located in

South-Western England and Tyber river Basin, Italy.

Cowpertwait, et al., (2006) presented a Poisson model approach to disaggregating spatio-temporal

rainfall data. They used seven years hourly rainfall data collected across Auckland city to infer that

their method of disaggregation method works well, and can be used with confidence for many similar

applications. Segond, et al. (2006) combined generalized linear models (GLMs) with a single-site

disaggregation model based on Poisson cluster processes, and validated their disaggregation method

by applying to observed rainfall data. The above mentioned studies show that the disaggregation

methods, except for extreme situations, preserve many statistical properties of rainfall such as

marginal moments, temporal and spatial correlations, along with proportions and lengths of dry

intervals. Hence the methods give an effective way of generating finer scale data, which are extremely

useful in many application areas. These finding are further advanced by the use Bayesian techniques

in disaggregation methods.

Bayesian approach had tremendous influence in many statistical applications in many disciplines. The

main advantage of this method is its ability to deliver prediction accurately, especially for sparse data

(Withers, 2002). It can also overcome the difficulties of missing values and complex hierarchical

structures. Its ability to incorporate prior knowledge without hindering the assumptions of classical

approaches has made Bayesian inference as a powerful prediction tool for many fields (Withers,

2002). In rainfall studies, Bayesian approach is used to predict frequency of rainfall event, duration,

and rainfall intensity (Chandler, et al., 2000, Chandler, et al., 2007) and tackle difficulties of estimating

many parameters in the temporal disaggregation model (Keller, 2007).

Further, Makhnin and Allister (2009) used Gibbs sampler in developing a Bayesian approach in state

space (dynamic) linear modeling for jointly generating daily occurrence and intensity of rainfall. Sigrist,

et al. (2011) considered Bayesian approach in their spatio-temporal model for precipitation, and

applied the method to model three-hourly rainfall data in Switzerland. These approaches can

overcome issues such as negative values (missing values or records correspond to long period of no

rainfall) and under estimation of autocorrelation and cross correlation lag 1 of the model, due to zero

rain cases. These methods involve only fewer parameters. However, the methods have difficulty in

creating synthetic rainfall data in dense grids of locations (regionalization), and were designed to

generate synthetic data at the observational scale rather than to generate data at a finer time scale

than the observation time scale.

International Journal of Applied Mathematics and Statistics

27

In this paper we develop a Bayesian spatio-temporal disaggregation method for multi-site by

combining the stochastic spatial-temporal disaggregation model (Sigrist et al., 2011, Shaddick and

Wakefield, 2001) and the adjusting procedure (Koutsoyiannis, et al., 1996, Koutsoyiannis, et al., 2002,

Koutsoyiannis, et al., 2003). Benefit of combining Bayesian state space model with coupling

transformation (adjusting procedure) is demonstrated by applying the methodology to a rainfall data at

Sampean Watershed, Bondowoso, Indonesia. WinBUGS is used to run the algorithm to generate

synthetic (generation) lower scale spatio-temporal data that preserve the statistical properties of the

observed higher scale spatio-temporal data. Section 2 details the spatio-tempoal model along with the

required Bayesian elements and estimation procedures; Section 3 presents the results of the methods

applied to rainfall data, and comarsion of the Bayesian state space model with adjusting procedure;

Section 4 summaries the findings and benefits of the method.

2. MATERIALS AND METHODS In this section we detail the spatio-temporal model, its Bayesian formulation, adjusting procedure,

estimation and computational issues. The posterrior distribution for the model is analytically

interactible and hence MCMC produres are given along with a numerical example to explain the main

principles.

2.1. Bayesian State Space Model for Spatio-Temporal Data Here we follow the hierarchical dynamic (state space) linear model, given in Shaddick and Wakefield,

(2002), which is represented in three stages.

First Stage: Observed Data Model

Let �� denote the spatio-temporal rainfall data at time � � �� and locations �� .

Suppose �� ,where ��, t = 1, ..., T; s = 1, …, S are the measurement errors, assumed to be independent with

distribution �� , �� is the spatial effect at site s, and �� is the error term for temporal

dependency which is detailed in second stage (b).

Second Stage (a): Spatial Model:

The random effect, � � �� is assumed to have the multivariate normal distribution

�� where � is an � vector of zeros, �� is the between-site variance, �! is the correlation

matrix, in which element �� "� represents the correlation between sites � and �#and is assumed to be

isotropic, i.e.

$%&�� '( � )*+ ,-%'&��( .�


28

where &��/ is the distance between sites � and �"�� " � �� , and ' 0 � represents the

correlation strength between sites.

When the site-specific levels are assumed to be (conditionally) independent, the model for the random

effect simplifies to �1�� 2Second Stage (b): Temporal Model

The temporal error model, which induces dependency between amounts of rainfalls at different times,

is assumed to be a limiting form of the first-order autoregressive model. This is represented as �� 3� � 4��where 4��%�� 5�(, for t=1,2, …, T, and are independent.

Third Stage: Hyperpriors Determination as a part of Bayesian approach

Based on Shaddick and Wakefield (2002), priors for the precision parameters, ��3� and �53� , are

specified as a Gamma distribution. Specifically, ��3��67�7�� 8�� and �53��67�75� 85�. Prior for ' is

taken as uniformly distributed based on the relation between correlation and distance. For example,

when '�&�� - 9:;%<��( (isotropic case), where <�� is correlation between sites � and �#, the prior for

' is set to be '�=��>&��.With the above model and prior specifications, the distribution of ? � �� @� at Stage 2 can be

represented as

where A� and �B� are the sum and the mean, respectively, of the neighbouring values of ��. The prior

distribution for �� can be represented as

C%��D�3�� 5�(�EFGFH �%��I�� 5�(� �� JKLMNIKLON� � PQR� S � �� T� � - ��%��3�� 5�(� � � �

where �3� denotes the vector of �� without ��.The joint posterior distribution of ?, ��, and �5� can be written as

C%�� 5�UV( � C�V�3� WXC�V�U�� @�Y�

XC��U��3�� 5��@�Y�

Z C��C��C��5��In this paper estimation of the parameters of the above analytically intractable distribution is achieved

by generating samples using Markov Chain Monte Carlo (MCMC) algorithm (Smith and Roberts,

1993).

� � � �

� �

� �

2 21

21

2

2

2

2

2

2

| | ,

1exp2

1exp2

T

t t

T

t

t

t

t t

T

t t t t

p p

n

� �

�

�

� � � � �

� ��

� � ��

�

�

�

�

�

��

�

�

�


29

2.2. Bayesian Spatio-Temporal Disaggregation Algorithm

The disaggregation algorithm is developed in two stages. In the first stage data is generated using

Bayesian state space model given in the above section. The MCMC method is used to estimate the

Bayesian state space model parameters (see Shaddick and Wakefield (2002) for details), with hourly

time scale and by using the observed hourly rainfall. Then lower scale synthetic data (hourly), [\]^, is

generated using the estimated model. We refer this synthetic data as M1 data or as data from M1

In the second stage the well-known proportional adjusting procedure proposed by Koutsoyianis et al.

(1996) is applied to the data from M1 in order to preserve consistency between the generated lower

scale data and the observed higher scale data. To be precise, let _] denote the higher scale rainfall

value at time h, and []^� ` � �� a, denote the lower scale rainfalls corresponding to the time h, for

1,...,h H� . Here each higher scale (day for our data) period is divided to a lower scale (hour)

periods. The additive (consistency) relation between _] and []^ can be written as

��b[c]^d^Y�

� _]��where [c]^ is the � of estimated vector of rainfalls for the lower scale period ` at the higher scale

period e2 In the proportional adjusting procedure the disaggregation value [c]^ is computed as

�f]^� � g]�gh]� �h]^�� e � �� i� j �` � �� aj � � �� where �h]^� is the generated lower scale rainfall at site � for the lower scale period ` and the higher

scale period e� and gh]� � k �h]^�d̂Y� . Note that the above adjusting procedure guarantees that the

additive (consistency) relation between the data at the two time scales is preserved. The synthetic

data obtained after the adjusting procedure will be referred as M2 data or as data from M2

2.3. Numerical Example In this section we apply the model and the desegregation algorithm to the spatio-temporal rainfall data

at Sampean Watershed. The measurements are from Tlogo and Sentral rain gages at Bondowoso,

Indonesia (see Figure 1). Hidayah, et al. (2010) considered the data, in their analysis they used auto-

regressive model to generate disaggregated data at single site. We apply the multi-site Bayesian

state model to generate disaggregated data for the two sites, and validate the method.

The data is the hourly spatio-temporal rainfall that recorded during January 2006 and January 2007.

The number of hourly observations is 1488 from each rain gage. This gives 62 observations of daily

rainfall at each location, and this is used in the adjusting procedure. Bayesian state space model is

applied to the observed hourly rainfall data, and then synthetic hourly data is generated form the

model, giving M1 data. Then the proportional adjusting procedure is applied to the M1 data, using the

observed daily data, to give M2 data. In order to determine which method of disaggregation is better,

properties of the generated hourly M1 and M2 data sets are compared with that of the observed

(historical) hourly data.


30

Figure 1. Rain gage location at Sampean Watershed, Bondowoso

The method can be summarized in the following steps:

1. Explore the data. This step is to fit an appropriate distribution and describe the data

characteristics (Astutik, et al., 2011a, Astutik, et al., 2011b).

2. Specify the state space model, as explained in Section 2.

3. Estimate and examine the model parameters by implementing MCMC using WinBUGS. The

parameter estimation is done by applying Gibbs sampling coupled with MCMC method, and

using full conditional posterior of each parameter (Congdon, 2006, Ntzoufras, 2009, Iriawan,

et al., 2010).

4. Generate data using Bayesian state space model procedure (as a part of the spatio-temporal

rainfall disaggregation algorithm) to form M1 and M2 synthetic data sets, and graphically

characterize the statistics properties of these data sets.

5. Evaluate model (Model validation). The two models are compared by looking at the

differences between the synthetic data sets and the observed data (both hourly and daily) at

the two rain gages. Mean Absolute Error (MAE) is used as the error measure for the

validation of the models and it is given by

lmn � �ia bbbD�]^� - �f]^�D� � � �� j ` � �� 2 2 � aj e � �� 2 2 � i��Y�

d^Y�

o]Y�

where �]^� and �f]^� are respectively the observed and the synthetic lower scale (hours in our case)

data at the period e (day), the subperiod �` (hour) and location �2 The smaller the MAE values

generated, the better the model. In step 1, Minitab 1.4 and Excel are used to explore the data, and

WinBUGS 1.4 was used to implement the modeling steps 3 and 4.

3. RESULTS Estimation of Bayesian state space model for the hourly rainfall observations from the two rain gages

at Sampean Watershed, Bondowoso, Indonesia is done by WinBUGS 1.4 MCMC algorithm with its

Gibbs sampler, which is used to sample the parameters of model from the posterior distributions. The


31

simulated posterior observations are used to estimate the model parameters. Each parameter is

estimated by averaging the sample values from the corresponding posterior distribution.

3.1 Influence of adjusting procedure on Bayesian state space model Here we explore the influence of adjusting procedure on the Bayesian state space model

observations. A comparison of two models, namely, the Bayesian state space model without adjusting

procedure (M1) and the Bayesian state space model with adjusting procedure (M2) are detailed below.

For our case the Bayesian state space model without adjusting procedure (Shaddick and Wakefield,

2001), given in Section 2.1, takes the form �]^� � �]^ � �� ]^�� e � �� pTj ` � ��Tqj � � �� T�with state equation �]^ � �]^3� � 4]^�where ��1�� ]^�� 4e`��%�� 4T(2� The prior distributions are specified as

��67��2�� and �5��67��2��. In the estimation phase of the computation convergence is

checked using history plot of MCMC series in WinBUGS, which is run 10000 iterations (Figure 2),

excluding the first 500 iterations are discarded as burn in. Figures 2 shows that, during the estimation

process, series of the Markov chain estimation process are mixing very well in both cases, and hence

the estimation algorithms converge.

Figure 2. History plot of the MCMC Algorithm: left for Phi1 and right for Tau.m.

The results of Bayesian parameter estimation are given in Table1. Here Phi1 is the estimate of the correlation between the two sites, and Tau.m is the estimate of �>��Table 1. Result of parameter estimation of Bayesian state space model

Node mean sd MC error 2.5% median 97.5% start sample

Phi1 0.05853 0.032 0.0008205 0.007339 0.05794 0.112 501 9500

Tau.m 131.4 107.5 3.774 10.58 103.2 417.8 501 9500

Once the Bayesian state space model parameters are estimated, synthetic hourly data is generated

from the model for the two sites and for the observation time period (62 days), producing M1 data.

Then the adjusting procedure, given in Section 2.2, is applied to this data to obtain M2 data. We use

the observed daily spatio-temporal rainfall data in the adjusting procedure. As explained before, the


32

adjusting procedure is applied to make sure that the synthetic hourly data preserves some of the

statistical properties of the daily data. In order to compare the performance of M1 and M2, MAE (given

in Section 2.3) is calculated. The results of the comparation between M1 and M2 are given in Table

2. The Table gives summay statistics associated with MAE(M1) and MAE(M2), the MAE between

observed hourly data and, repectively, M1 (without adjusting procedure) and M2 (with adjusting

procedure) data sets. The summary is based on 9500 iterations.The estimation and synthetic data

generation is run using computer with processor: Intel(R) Core (TM) 2 Duo CPU T5250 1.50 GHz,

RAM 2.50 GB, System type : 32-bit Operating System. The entire computation up to calculation of

MAE (M1) took 507 seconds of CPU time, and the adjusting procedure and then computing MAE (M2)

took additional 264 seconds.

Table 2. The comparation between M1 and M2

Node mean sd MC error 2.5% median 97.5% start sample

MAE(M1) 0.01921 0.009073 0.0009083 0.00907 0.01671 0.04771 501 9500

MAE(M2) 0.01767 0.007559 0.0007.553 0.008396 0.01561 0.04072 501 9500

3.2. Model Performance Goodness of fit of the two models can be evaluated through a comparison of the statistical properties

such as rainfall depth, standard deviation, and coefficient of skewness of the generated (synthetic)

data and the observed (historical) data from the two stations. The comparisons observed for the three

data series are similar for the two rain gages. Hence here mainly give details of the comparisons for

the data at the Tlogo rain gage.

Figure 3 gives histograms of the hourly rainfall depth corresponding to the observed, M1 and M2 data

sets at Tlogo rain gage. It shows that in all three cases, the number of observations either zero or very

close to zero, are similar. However, it should be noted that all hourly M1 data points are positive,

irrespective of whether observed hourly values or zero or not.

(a) (b)


33

60544842363024181261

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Time (Daily)

Rai

nfal

l Dep

th D

iffer

ence

(m

m)

TlogoSentral

(c)

Figure 3. Comparison of hourly spatio-temporal rainfall data at Tlogo Rain Gage: (a) Historical, (b) Bayesian state space model without adjusting procedure (M1), and (c) Bayesian state space model with adjusting procedure (M2).

Figure 4 gives plots of the differences (simulated – observed) between the observed series and a

simulated M1 data. Figure on the left is the plot of difference of the hourly series for the Tlogo rain

station. It is clear from the plot that the in most of the cases simulated hourly rainfalls are

overestimates of the observed rainfalls. Similar characteristic is observed for the hourly series for the

Sentral rain gage.

Figure 4. Plot of the series “simulated M1data –observed data”. Left: hourly difference series for Tlogo rain gage; Right: daily difference series for both and Tlogo and Sentral rain gages.

The right side plot in Figure 3 gives the difference (simulated –observed) of the daily observed series

and the simulated daily M1 data for the two rain station gages. Daily M1 series is obtained by adding

the corresponding simulated hourly data. The plot indicates that the simulated daily rainfalls are

overestimates of the corresponding observed daily rainfalls. Since, in our case, the daily M1 data

overestimate corresponding daily observed rainfall, M2 hourly data can be thought of as a scaling of

M1 data with each day scaling factor less than 1. Note that if the observed total rainfall on particular

1341119210438947455964472981491

0.20

0.15

0.10

0.05

0.00

-0.05

-0.10

Time (hourly)

Rai

nfal

l Dep

th D

iffer

ence

(m

m)


34

day is zero, then the all the hourly rainfall values for M2 are zero for the day, irrespective of the

corresponding values for M1. Hence the size of the overestimation is reduced in the M2 hourly data.

By the adjusting procedure, it also follows that M2 daily data is exactly the same as the observed daily

values.

Figure 5 (left) shows daily series of the standard deviation for the hourly M1 data and observed rainfall

at the Tlogo rain gage. The standard deviations for the M1 daily data are similar to that of the

observed daily data, except on days when the observed daily rainfalls are zero. The simulated M1

daily rainfall, corresponding to an observed zero daily rainfall, is always positive, and gives positive

skewness for the day. By the adjusting procedure, thinking it as scaling with scaling factor less than

1, the daily standard deviations of the M2 series can be smaller than that of the M1 series.

Figure 5. Comparison of standard deviation (left) and coefficient of skewness (right) of historical and generated hourly rainfall data at Tlogo rain gage.

Figure 5 (right) gives that daily skewness for the three series. We noted that the skewness values of

M1 series are higher than that of the observed series. Again, one of the reasons for this could the

positive values of M1 series corresponding to the zero values of the observed series. By the

adjusting procedure, it follows that skewness values for M1 and M2 are the same for days when the

observed daily total take positive values. Other days daily skewness values M2 and observed data

are taken as zero.

4. DISCUSSIONS AND CONCLUSION

The Bayesian estimation results in Table 1 show that the estimated parameters for the spatio-

temporal model are good. In addition, From Figure 2 it is evident that the Markov chain series in the

MCMC estimation process are mixing very well, ensuring convergence of parameter estimation

algorithm. In regard to comparting the two methods of disaggregation, the MAE value of M2 is

relatively smaller than that of M1, and the extra computation time for M2 is not much. Hence overall,

as far as the model diagnostics are concerned, it can be inferred that M2 is better than M1.

Figure 3 shows that both models are almost perfectly imitating the pattern of observed hourly rainfall

data. To see how, for each day, M1 & M2 hourly rainfall series preserves some of the characteristics

of hourly observed rainfalls, we calculated the total, standard deviation and skewness from the hourly

data for each of the 62 days and for reach of the three data sets. From Figures 4 (right) it can be seen

that series of the daily total always overestimates the observed daily rainfall. The adjusting procedure

60544842363024181261

5

4

3

2

1

0

Time (Daily)

Skew

ness

Coe

ffic

ient

of

Rai

nfal

l Dep

th (

mm

)

HistoricalModel 1


35

forces the daily total for the M2 series to be the same as the observed daily total. The daily standard

deviations of M2 data are smaller than that of the M1 data. The daily skewness values for M1 and M2

are the same for the days when the observed daily rainfalls are positive, whereas the skewness

values for M1 data are positive on other days. Hence this analysis shows that M2 is better than M1 in

preserving some of the each day hourly rainfall properties, and it preserves the daily total rainfall

values. Aim is, as a part of future research in this area, to apply the multi-site methodology discussed

in this paper to rainfall data at other rain gages at Sampean Watershed, Bondowoso, Indonesia, and

to compare the performance of the two models.

5. ACKNOWLEDGMNET

This research was funded by the Directorate General of Higher Education. The Ministry of Education

and Culture of the Republic of Indonesia funded the Sandwich Program with the University of Western

Australia. The authors would like to thank to Dr. Entin Hidayah for providing the data and her help in

analyzing the data.

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Astutik, S., Iriawan, N., Suhartono, Sutikno, 2011b, Spatio-Temporal Rainfall Disaggregation using MUDRAIN on DAS Sampean Baru, Proceeding of The International Conference on Mathematics and Sciences (ICOMSc). ITS, Surabaya.

Bojilova, E. K. , 2004, Disaggregation Modelling of Spring Discharges. International Journal of Speleol3 (1/4), 65-72.

Chandler, R. E., Mackay, N. G., Wheater, H. S., Onof, C. J., 2000, Bayesian Image Analysis and Disaggregation of Rainfall. Journal of Atmospheric and Oceanic Technology 17, 641-650.

Chandler, R. E., Isham, V., Wheather, H. S., Onof, C. J., Leith, N., Frost, A., 2007, Spatial Temporal Rainfall Modelling with Climate Change Scenarios. R&D Technical Report FD2113/TR.

Cowpertwait, P., Lockie, T., Davis, M., 2006, A Stochastic Spatial-temporal Disaggregation Model for Rainfall. Journal of Hydrology(NZ) 45(1), 1-12.

Congdon, P., 2006, Bayesian Statistical Modelling (2 ed.). USA: John Wiley & Sons.

Fytilas, P., Koutsoyiannis, D., Napolitano, F., 2002, Multivariate Rainfall Disaggregation at A Fine Tme Scale, Diploma Thesis, University of Rome "La Sapienza".

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Bayesian State Space Modeling for Spatio-Temporal Rainfall Disaggregation · 2014. 9. 23. · many parameters in the temporal disaggregation model (Keller, 2007). Further, Makhnin