L'uso dei modelli geostatistici multivariati nelle ...€¦ · 1.Data 2.Model 3.Inference 4.Simulation 5.Appl 2.1.Model 2.2.Autocorrelation MultivariateModel-BasedGeostatistics Geostatistics

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L’uso dei modelli geostatistici multivariati

nelle applicazioni ambientali

Marco Minozzo

Dipartimento di Scienze EconomicheUniversita degli Studi di Verona

Giornata di StudioMetodi Statistici per l’Analisi di Dati Spaziali: Casi di Studio e Applicazioni

Dipartimento di Statistica – Universita degli Studi di Milano-BicoccaMilano, 13 gennaio 2011

Marco Minozzo Multivariate model-based geostatistics

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1 Geo-Referenced Multivariate Non-Gaussian DataPlankton in Lake TrasimenoRadionuclide Pollution at La Maddalena

2 Multivariate Model-Based GeostatisticsHierarchical Geostatistical Factor ModelSpatial Autocorrelation

3 InferenceEstimation With the Method of MomentsLikelihood Analysis

4 Simulation ExperimentsMonte Carlo Expectation Maximization

5 Applications to Real Data


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1.Data 2.Model 3.Inference 4.Simulation 5.Appl 1.1.Plankton 1.2.Uranium

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Plankton Data From Lake Trasimeno

Zooplankton species

Cyclops vicinus Daphnia galeata

Phytoplankton species

Peridinium cinctum Scenedesmus quadricauda



Plankton Data From Lake Trasimeno

5500

mm

ssiiddee

39 sampling sites: 3 sites (located at the vertices of an equilateraltriangle (50 m side)) for each of the 13 selected squares.

For each of the 39 sampling sites, counts are available of14 fitoplankton species and 64 zooplankton species, together withthe values of some physical parameters.



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Trans-Uranium Radionuclide Pollution at La Maddalena

The US Navy Submarine Base of Santo Stefano was located in thearchipelago of La Maddalena National Park.

In October 2003 a nuclear powered submarine ran aground ontogranitic shoals near the town of La Maddalena, three nautical milesfrom the submarine’s home base on the island of Santo Stefano.There followed a major explosion, damaging the submarine, whichled to fears of radioactive leakage from the submarine.




(Left) Typical alpha particle ‘hot spot’ from the surface of a darkbrown alga. The hot spot consists of roughly 100 tracks (whitespots) emitted from a single point which have pierced the 5 µthick red cellulose nitrate layer of the LR-115 film; some 150 tracks(black spots), inclined away from the center, have only partiallypenetrated the layer. Image size is approximately 0.2 mm across.(Right) Typical alpha particle nuclear tracks (white spots)randomly distributed on the layer.




Alpha track concentrations and occurrences of hot spots are higheralong the coastlines facing west/south west. The highestconcentrations are around the northern and eastern margins of theRada di S. Stefano, where all sites are looking into the wind andprevailing currents towards the nuclear submarine base on theeastern shore of the island of S. Stefano.


1.Data 2.Model 3.Inference 4.Simulation 5.Appl 2.1.Model 2.2.Autocorrelation

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Geo-Referenced Multivariate Data

For each of K (idealized) pointwise geographic locations xk ,consider the variables yi(xk)

Variable 1 . . . Variable i . . . Variable m

y1(xk) . . . yi (xk) . . . ym(xk)

2280000 2300000 2320000 2340000 2360000 2380000

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Traditional Statistical Approaches

Univariate geostatistical analyses: use of variograms(covariograms) and kriging.

Statistical methods for the analysis of ecological multivariatespatial data: Mantel nonparametric multivariate test; Mantelmultivariate correlogram; multivariate contiguity indices andmultivariate variograms based on dissimilarity metrics (Bourgaultet al., 1992); multivariate variograms (Mackas, 1984).

Principal component analysis: variogram analysis of the factorscores of the first few principal components; maps of predictedfactor scores of the first few principal components through kriging.

Proper multivariate geostatistical analyses: use of cross-variograms(cross-covariograms) and cokriging; linear coregionalization model(LCM) or intrinsic correlation model (Wackernagel, 1995); factorialkriging analysis (FKA) (Matheron, 1982).



Multivariate Model-Based Geostatistics

Geostatistics

variograms, covariograms,kriging(Krige 1951)

Model-BasedGeostatistics

(Diggle 1998)

Multivariate Geostatistics

cross-variograms, cokriging,linear coregionalization model (LCM),factorial kriging analysis (FKA)(Journel and Huijbregts 1978;Matheron 1982)

Multivariate

Model-Based

Geostatistics




Geostatistics



(Diggle 1998)



Multivariate

Model-Based

Geostatistics




Geostatistics



(Diggle 1998)



Multivariate

Model-Based

Geostatistics




Geostatistics



(Diggle 1998)



Multivariate

Model-Based

Geostatistics



Geostatistics

KrigingFor data y(xk), k = 1, . . . ,K , kriging provides linear unbiasedprediction at spatial location x0

Y (x0) =K∑

k=1

wk(x0) · y(xk),

where wk(x0) are weights which minimize the mean squareprediction error

E[(Y (x0)− Y (x0))

2].

For Gaussian data the linear prediction Y (x0) coincides with theconditional expectation

E[Y (x0)

∣∣∣y(xk) : k = 1, . . . ,K].



Model-Based Geostatistics

In all situations in which data cannot be assumed to be Gaussian,Diggle et al. (1998) proposed the following hierarchical(generalized linear mixed model) approach to geostatistics.They assumed for the observed spatial process

Y (x)|S(x) ∼ f (y |S(x)), x ∈ D ⊆ R2,

where S(x) is a stationary Gaussian process with zero mean andCov(S(x),S(x + h)) = σ2ρ(h).In particular they assumed that the conditional meanM(x) = E[Y (x)|S(x)] is dependent on S(x) through a link functionh(·), that is,

h(M(x)) = β + S(x),

where β is a parameter.




Proportional Covariance ModelThe m (intrinsically) stationary random functions Zi(x) satisfy themodel

Zi(x) =

P∑

p=1

aipFp(x),

where Fp(x) are P uncorrelated spatial common factors for which

E[Fp(x)] = 0 and Cov[Fp(x),Fp(x+ h)] = ζ2p · ρ(h).

If stationarity holds, the cross-covariance functions of the Zi(x) areall proportional to the same spatial autocorrelation function ρ(h),that is,

Cij(h) = σijρ(h),

where σij are covariances.




Linear Coregionalization Model (LCM)The m (intrinsically) stationary random functions Zi(x) satisfy

Zi(x) =

P∑

p=1

a0ipF0p (x) +

P∑

p=1

a1ipF1p (x) + · · ·+

P∑

p=1

aSipFSp (x),

where F up (x), u = 0, 1, . . . ,S , are uncorrelated spatial common

factors with E[F up (x)] = 0 and Cov[F u

p (x),Fup (x+ h)] = ζ2pu · ρu(h).

If stationarity holds, the cross-covariance functions are linearcombinations of the same set of spatial autocorrelation functionsρ0(h), ρ1(h), . . . , ρS(h), that is,

Cij(h) = b0ijρ0(h) + b1ijρ1(h) + · · ·+ bSij ρS(h),

where buij , u = 0, 1, . . . ,S , are covariance components.




CokrigingFor data yi (xk), i = 1, . . . ,m, k = 1, . . . ,K , cokriging provideslinear unbiased prediction at spatial location x0

Yi0(x0) =

m∑

i=1

K∑

k=1

w ik(x0) yi(xk),

where w ik(x0) are weights which minimize the mean square

prediction error

E[(Yi0(x0)− Yi0(x0))

2].




Factorial CokrigingFor data yi (xk), i = 1, . . . ,m, k = 1, . . . ,K , factorial cokrigingprovides linear unbiased prediction at spatial location x0

Fp(x0) =

m∑

i=1

K∑

k=1

w ik(x0) yi(xk),

where w ik(x0) are weights which minimize the mean square

prediction error

E[(Fp(x0)− Fp(x0))

2].

For Gaussian data, the linear predictor Fp(x0) coincides with theconditional expectation

E[Fp(x0)

∣∣∣yi(xk) : i = 1, . . . ,m; k = 1, . . . ,K],

and so it is ‘optimal’ even in the wider class of non-linearpredictors.



The Hierarchical Geostatistical Factor Model

We see these m regionalized variables yi (xk) as one (partial)realization of a set of m spatial random functions Yi(x),i = 1, . . . ,m, x ∈ D ⊆ R

2.Let Zi(x), i = 1, . . . ,m, be mean zero joint stationary Gaussianprocesses and let Yi(x) satisfy, for any x, for i 6= j ,

Yi(x)⊥⊥Yj(x)∣∣ Zi(x) and Yi (x)⊥⊥Zj(x)

∣∣ Zi(x),

and, for x′ 6= x′′, and j = 1, . . . ,m,

Yi(x′)⊥⊥Yj(x

′′)∣∣ Zi (x

′) and Yi (x′)⊥⊥Zj(x

′′)∣∣ Zi(x

′),

and have conditional distributions

Yi(x)∣∣ Zi(x) ∼ fi (y ;Mi(x)),

where Mi (x) = E[Yi(x)

∣∣Zi(x)], and

hi(Mi (x)) = βi + Zi (x).




Moreover, for Zi(x), i = 1, . . . ,m, assume the linear factor model

Z(x) = AF(x) + ξ(x),

that is, for i = 1, . . . ,m, and for P ≤ m,

Zi(x) =

P∑

p=1

aipFp(x) + ξi(x)

where common and unique factors Fp(x) and ξi(x) are stationaryGaussian processes with E

[Fp(x)

]= 0 and E

[ξi(x)

]= 0, and

Cov[Fp(x),Fp(x+ h)

]= ρ(h),

Cov[ξi(x), ξi (x+ h)

]= ψiρ(h),

where ρ(h) is a spatial correlation function with ρ(0) = 1 andρ(h) → 0, as |h| → ∞, and ψi ≥ 0 are parameters. Also, Fp(x)and ξi(x) have null cross-covariances.




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Directed acyclic graph representing, at any spatial location x, theconditional independence structure of the model. ‘Floors’ belowcommon and unique factors Fp(x) and ξi(x) show thecorresponding spatial autocovariance functions.



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Poisson Distribution: Spatial Autocorrelation

If fi(y ;Mi(x)) are Poisson distributions with means Mi(x) andhi (·) = ln(·), the variables Yi (x) are second order stationary:

meanµi = E

[Yi(x)

]= E

[E[Yi(x)

∣∣Zi (x)]]

= E[exp

{βi + Zi(x)

}]= τ∗i ,

varianceσ2i =Var

[Yi(x)

]=E

[Mi(x)

]+Var

[Mi(x)

]=τ∗i +(τ∗i )

2[exp

{ς2i}−1

],

covarianceCij(h) = Cov

[Yi(x),Yj (x+ h)

]= τ∗i τ

∗

j

[exp

{ςijρ(h)

}− 1

],

where τ∗i =exp{βi+ς

2i /2

}, ς2i =Var

[Zi(x)

], ςij =Cov

[Zi(x),Zj (x)

].

For the covariance function we have:Cij(h) = Cji (h);Cij(−h) = Cij(h), for ρ(h) = ρ(−h);Cij(∞) = 0;Cii (0) = σ2i 6= Cii(0

+).



Poisson Distribution: Variograms

For the variograms of the observable Yi(x) we have, for h 6= 0,

γii(h) =1

2E[(Yi (x+ h)− Yi(x))

2]

= τ∗i + (τ∗i )2[exp

{ς2i}− exp

{ς2i ρ(h)

}],

0Lag distance h

γii(h)

6

?τ∗

i

6

?

(τ ∗

i )2[

exp{

ς2i

}

− 1]

where Mi (x) = E[Yi(x)

∣∣Zi(x)], τ2i (x) = Var

[Yi(x)

∣∣Zi(x)],

τ∗i = exp{βi + ς2i /2

}, and ς2i = Var

[Zi(x)

].



Poisson Distribution: Cross-Variograms

For the cross-variograms of the Yi(x) we have, for i 6= j and h 6= 0,

γij(h) =1

2E[(Yi (x+ h)− Yi(x))(Yj (x+ h)− Yj(x))

]

=1

2Cov

[Mi(x),Mj (x)

]+

1

2Cov

[Mi(x+ h),Mj(x+ h)

]

−1

2Cov

[Mi(x),Mj (x+ h)

]−

1

2Cov

[Mi(x+ h),Mj (x)

]

= τ∗i τ∗

j

[exp

{ςij}− exp

{ςijρ(h)

}],

where ςij = Cov

[Zi(x),Zj (x)

].



Gamma Distribution: Variograms

Assume that, for a given i and for any given x, conditionally onZi(x), the random variables Yi(x) are Gamma distributed withconditional expectations

Mi(x) = E[Yi(x)

∣∣Zi (x)]= exp

{βi + Zi(x)

}= αb

(here hi(·) = ln(·)) and conditional variances given by

Var[Yi(x)

∣∣Zi(x)]= αb2 = α−1 exp

{2βi + 2Zi (x)

},

where α > 0 and b > 0 are parameters.Then the variogram of the observable Yi(x) is given by, for h 6= 0,

γii(h) =1

αe−2βi (τ∗i )

4 + (τ∗i )2[exp

{ς2i}− exp

{ς2i ρ(h)

}].

We see that the nugget is equal to α−1e−2βi (τ∗i )4 and the second

addendum is the same as in the Poisson case.



Gamma Distribution: Cross-Variograms

For the cross-variograms γij(h), when Yi and Yj are bothconditionally Gamma distributed, or when one is Gamma and theother is Poisson, we obtain the same form as in the case in whichthey are both Poisson.



Compound Gamma Distribution

We say that a random variable Y has a compound Gammadistribution if it has density

f (y) = P(N = 0)d0(y) +∞∑

j=1

P(N = j)y jα−1e−y/b

bjαΓ(jα),

for y ≥ 0, and f (y) = 0 otherwise, where d0(y) is the Dirac deltafunction at zero, N is Poisson distributed with mean λ > 0, andα > 0 and b > 0 are parameters.

This distribution can be seen as a mixtures of Gamma distributionsand has a probability mass at zero. It is easy to check that:

E(Y ) = αbλ

andVar(Y ) = α(1 + α)b2λ.



Compound Gamma Distribution: Variograms

For a given i and for any given x, assume that, conditionally onZi(x), the random variables Yi(x) are compound Gammadistributed with conditional expectations

Mi(x) = E[Yi(x)

∣∣Zi(x)]= exp

{βi + Zi(x)

}= αbλ

(here again hi(·) = ln(·)) and conditional variances

Var[Yi(x)

∣∣Zi(x)]= (1 + α)/(αλ) exp

{2βi + 2Zi(x)

}.

Then the variogram of the observable Yi(x) is given by, for h 6= 0,

γii(h) =1 + α

αλe−2βi (τ∗i )

4 + (τ∗i )2[exp

{ς2i}− exp

{ς2i ρ(h)

}].

As before, whereas the nugget is equal to (1 + α)/(αλ)e−2βi (τ∗i )4,

the second addendum is the same as in the Poisson case.



Compound Gamma Distribution: Cross-Variograms

For the cross-variograms, when Yi and Yj are both compoundGamma, or one is compound Gamma and the other is eitherPoisson or Gamma, we still obtain the same form as in the case inwhich they are both Poisson.


1.Data 2.Model 3.Inference 4.Simulation 5.Appl 3.1.Method of Moments 3.2.Likelihood Analysis

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Parameter Estimation: Method of Moments

• Sample means yi and sample variances s2i and covariances sij of

the variables Yi(xk) provide estimates of µi , Cii(0) and Cij(0).

• Estimates βi of βi and preliminary estimates ς2i and ςij , of ς2i

and ςij , can then be obtained.• The symmetric matrix ΣZ (with elements ς2i and ςij) is ingeneral not positive semidefinite.• Final estimates of ς2i and ςij are given by ΣZ = QΛ0QT ≥ 0,which minimizes

‖ ΣZ − ΣZ ‖=

√√√√m∑

i=1

m∑

j=1

(ςij − ςij)2 .

• Factor analysis of ΣZ ≥ 0 to find matrices A and Ψ (withvarimax rotation, etc.).• Estimation of empirical and theoretical variograms andcross-variograms γij(h) and choice of ρ(h).



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Parameter Estimation: Likelihood Analysis

The full likelihood is given by f (y1, . . . , ym, ξ1, . . . , ξm,F1, . . . ,FP)which (for Poisson data) can be written as

f (y1, . . . , ym|ξ1, . . . , ξm,F1, . . . ,FP) f (ξ1, . . . , ξm,F1, . . . ,FP)

=

[m∏

i=1

K∏

k=1

e−Mik (Mik)yik

yik !

]

︸︷︷︸βi aip

· f (ξ1) · · · f (ξm) · f (F1) · · · f (FP)︸︷︷︸K −multivariate normals

α δ ψ1 . . . ψm

,

where yi = (yi1, . . . , yiK ), ξi = (ξi1, . . . , ξiK ), Fp = (Fp1, . . . ,FpK ),for i = 1, . . . ,m and p = 1, . . . ,P , and

ln(Mik) = ln(E[Yi(xk)

∣∣Zi(xk)])

= βi +P∑

p=1

aipFpk + ξik .



Parameter Estimation: Monte Carlo EM Algorithm

Maximum likelihood estimates: the likelihood based on themarginal distribution of the observed Yi(xk) is not known.

EM algorithm: the analytical integration (averaging) of thelikelihood required for the E (expectation) step is difficult.

Stochastic versions (StEM, MCEM, etc.) of the EM algorithmreplace the integration requested at the E step by a numerical(Monte Carlo) approximation based on a random sample from theconditional distribution of the unobserved variablesξ1(xk), . . . , ξm(xk), F1(xk), . . . ,FP(xk), k = 1, . . . ,K , given theobservations y1(xk), . . . , ym(xk), k = 1, . . . ,K .




Being θr the current value of the parameter vector at the r th step,the MCEM algorithm supplies a new value θr+1 as follows:

S step – simulate (through MCMC) Sr samples(ξ1, . . . , ξm,F1, . . . ,FP)

(s), s = 1, . . . ,Sr from the conditionaldistribution f (ξ1, . . . , ξm,F1, . . . ,FP |y1, . . . , ym; θr );

E step – calculate

Qr+1(θ, θr ) =1

Sr

Sr∑

s=1

ln f (y1, . . . , ym, ξ1, . . . , ξm,F1, . . . ,FP ; θ);

M step – take as θr+1 the value of θ which maximizes Qr+1(θ, θr ).

Choosing Sr = 1 we obtain a stochastic EM (StEM) algorithm,whereas choosing Sr very big we approximate a (deterministic) EMalgorithm. Taking Sr → ∞ slowly, with r → ∞, we obtain a“simulated annealing” EM algorithm.




Maximization in the M step can be carried out numerically. For theStEM algorithm (Rs = 1), we have to maximize the fullloglikelihood

l(θ) =m∑

i=1

li (βi , ai ) +m∑

i=1

ln f (ξi ;ψi ,%) +P∑

p=1

ln f (Fp;%),

where, letting Pρ = ||ρ(xl − xk)||(k,l),

li(βi , ai ) =

K∑

k=1

[−Mik + yik ln(Mik)− ln(yik !)

],

ln f (ξi ;ψi ,%) = −K

2ln(2πψi )−

1

2ln |Pρ| −

1

2ψi

ξTi P−1ρ ξi ,

ln f (Fp;%) = −K

2ln(2π)−

1

2ln |Pρ| −

1

2FTp P

−1ρ Fp.




• Parameters βi and ai maximizing l(θ), for every i = 1, . . . ,m,are obtained from li (βi , ai ).

• Maximization with respect to ψi involves ln f (ξi ;ψi ,%),i=1,. . .,m, and leads to ψi = (1/K )ξTi P

−1ρ ξi , which depend on %.

• Substituting ψi into l(θ) leads to the profile likelihood

lρ(%) ∝ −K

2

m∑

i=1

ln( 1

KξTi P

−1ρ ξi

)−1

2

(m+P

)ln |Pρ|−

1

2

P∑

p=1

FTp P

−1ρ Fp.

(Usually we assume the parameter % known.)




In the case in which Rs > 1, the M step of the algorithm requiresthe maximization of the ‘average’ likelihood

Rs∑

r=1

l (r)(θ)=

m∑

i=1

Rs∑

r=1

l(r)i (βi , ai)+

m∑

i=1

Rs∑

r=1

ln f (ξ(r)i ;ψi ,%)+

P∑

p=1

Rs∑

r=1

ln f (F(r)p ;%),

and we can proceed similarly to the case of the StEM algorithm(Rs = 1).

For Rs small the algorithm does not converge pointwise andestimates of the parameters are given by summary statistics (e.g.arithmetic mean) of the values θs .For Rs large (or Rs increasing) the algorithm is guaranteed toconverge to a local maximum of the likelihood.Once estimates are obtained, different rotations of the matrix Amay be considered.



Prediction and Mapping

Assuming to know the full joint probability distribution of themodel (all parameters have been estimated), prediction of a givenfactor Fp(x) (or ξi (x)) at spatial location x0, x0 6∈ {x1, . . . , xK},can be performed by considering the conditional distribution

f(Fp(x0)

∣∣(yi (xk); i , k)).

Pointwise prediction is provided by

E[Fp(x0)

∣∣(yi (xk); i , k)]

and an evaluation of its accuracy by

Var[Fp(x0)

∣∣(yi (xk); i , k)].

Simulation of the whole conditional distribution can be performedby Markov chain Monte Carlo methods.


1.Data 2.Model 3.Inference 4.Simulation 5.Appl 4.1.MCEM

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MCEM: Simulation Experiments

4776000

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Grid of 8× 8 = 64 sites 1km apart. (The grid covers the area ofLake Trasimeno.) We assume Yi(x) are Poisson and β1 = −2.054,β2 = 0.002, β3 = 0.845, β4 = 1.531, ψ1 = 1.025, ψ2 = 0.237,ψ3 = 0.787, ψ4 = 0.027.




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Simulated values of Y1(x), Y2(x), Y3(x) and Y4(x) over the 8× 8grid. Values are: 0, 1, 2 (Y1(x)); 0, 1 to 11 (Y2(x));0, 1 to 40 (Y3(x)); 0, 1 to 20 (Y4(x)).




-2

-1,5

-1

-0,5

0

0,5

1

1,5

2

0 50 100 150 200 250 300

0

0,2

0,4

0,6

0,8

1

1,2

0 50 100 150 200 250 300

β1 = −1.788β2 = 0.142β3 = 0.992β4 = 1.497

ψ1 = 1.048ψ2 = 0.239ψ3 = 0.739ψ4 = 0.027

Run of 500 iterations of the MCEM algorithm (with 100 MCMCsamples at each S step) for the simulated data set. Estimates areobtained as the averages over the last 50 iterations.




-1,85

-1,83

-1,81

-1,79

-1,77

-1,75

-1,73

-1,710 50 100 150 200 250 300

β1 = −2

β1 = −2

β1 = 0

β1 = 1

β1 = 0.5

Convergence diagnostics for the estimation of β1 with the MCEMalgorithm using the simulated data set. Five runs of the MCEMalgorithm (500 iterations, each involving 100 MCMC samples) areshown, each starting from a different set of initial values.(True β1 = −2.054; estimated β1 = −1.788.)




βtrue1 = −2.054βtrue2 = 0.002βtrue3 = 0.845βtrue4 = 1.531

Initial valuesβblack =(−1.32, 0.57, 1.75, 1.67)βred = (0, 1,−2,−1)βgreen = (0, 0, 0, 0)

βblue = (−2,−2, 2, 2)

Convergence diagnostics for the estimation of β’s with the MCEMalgorithm using the simulated data set. Four runs of the MCEMalgorithm (500 iterations, each involving 100 MCMC samples) are shown,each starting from a different set of initial values.




0,9

0,95

1

1,05

1,1

0 50 100 150 200 250 300

ψ1 = 0.02

ψ1 = 0.8

ψ1 = 1

ψ1 = 0.5

ψ1 = 0.1

Convergence diagnostics for the estimation of ψ1 with the MCEMalgorithm using the simulated data set. Five runs of the MCEMalgorithm (500 iterations, each involving 100 MCMC samples) areshown, each starting from a different set of initial values.(True ψ1 = 1.025; estimated ψ1 = 1.048.)




ψtrue1 = 1.025ψtrue2 = 0.237ψtrue3 = 0.787ψtrue4 = 0.027

Initial valuesψblack = (0.5, 0.5, 0.5, 0.5)ψred = (0.5, 0.5, 0.5, 0.5)ψgreen = (1, 1, 1, 1)

ψblue = (1, 0.2, 5, 0.8)

Convergence diagnostics for the estimation of ψ’s with the MCEMalgorithm using the simulated data set. Four runs of the MCEMalgorithm (500 iterations, each involving 100 MCMC samples) are shown,each starting from a different set of initial values.




Distributions of the parameter estimates obtained from 50 differentsimulated data sets drawn fromYi (x) | Zi (x) ∼ Poisson(exp(βi + aiF + ξi (x)).

Red dashed lines are the means of distributions while the blue dotted

lines are the true parameters.




Maps of the underlying spatial factor.


1.Data 2.Model 3.Inference 4.Simulation 5.Appl

Outline








Zooplankton in Lake Trasimeno

Consider the four more common zooplankton species of theplankton community of Lake Trasimeno (Umbria):

• K = 39 sampling sites, pointwise geo-referenced by thegeographic coordinates xk , k = 1, . . . ,K ;

• zooplankton species counts yi(xk), i = 1, . . . ,m = 4;

• P = 1 common factor;

• autocorrelation ρ(h) = exp{−((αh)δ)}, α = 0.001, δ = 1.5.

i mean s2 sites βi ai ,1 li ,1 φiBosmina 1 4.3 244.2 16 0.16 −1.64 −0.99 1.706Cyclops 2 37.7 744.2 39 3.43 0.26 0.40 0.255Daphia 3 63.3 2559.9 36 3.91 0.70 0.84 0.0001Diaph. 4 1.9 3.3 29 0.51 0.64 0.72 0.246




Empirical and theoretical variograms for Bosmina longirostris,Cyclops sp. and Daphia galeata assuming ρ(h) = exp{−((αh)δ)},α = 0.001, δ = 1.5

Empirical and theoretical variograms for the factor scores of thefirst common factor extracted with standard PCA

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Lag Distance (m )

0

0 .2

0 .4

0 .6

0 .8

1

1 .2

1 .4

1 .6

1 .8

γ

39

45

45

117

72

72

82

93

23

5823

27

18

9

9




Maps of predictions of the first factor extracted (F1): standardPCA (kriging) and hierarchical model

260000 265000 270000

Easting (m)

4775

000

4780

000

4785

000

No

rth

ing

(m

)

-1.2-1-0.8-0.6-0.4-0.200.20.40.60.811.21.4

260000 265000 270000

Easting (m)

4775

000

4780

000

4785

000

No

rth

ing

(m

)

-2.5-2-1.5-1-0.500.511.522.533.54

Standard PCA: factor loadings −0.28, 0.68, 0.92, 0.35;eingenvalue 1.5; variance explained 0.38.Hierarchical model: factor loadings −0.990, 0.404, 0.843, 0.724;eingenvalue 2.44; variance explained 0.61.




0,00

20,00

40,00

60,00

80,00

100,00

120,00

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Lag distance h (meters)

Empirical (dashed line) and estimated theoretical (solid line)variograms for the number of tracks Y1(x). The theoreticalvariogram γ11(h) is obtained by assuming a conditional CompoundGamma distribution with logarithmic link function, andautocorrelation function ρ(h) = exp{−((αh)δ)}, with α = 0.0008,δ = 1.8.




532000 534000 536000 538000 540000

4558000

4560000

4562000

4564000

4566000

4568000

I. Maddalena

I. Caprera

I. S.Stefano

Palau

Map of predictions of common factor F1(x) representing theunderlying contamination level over the studies area, obtainedthrough MCMC simulation. Crosses (×) indicate the location ofthe sampling sites.


Conclusions Final Remarks Future Work Bibliography

Conclusions

Final Remarks

The multivariate geostatistical framework allows for commonfactors underlying more than one variable; we can modeljointly variables of different types.

Non identifiability: different rotations lead to different modelsand maps.

For multivariate geostatistical non-Gaussian models, MonteCarlo methods can be used for filtering (smoothing),prediction and inference.

With today computing power these Monte Carlo methods canbe used for reasonably large data sets.



Conclusions

Future Work

It would be desirable to have some more simulation results onthe sample properties of the estimation procedures.

It would be desirable to have some measures of overallgoodness of fit of the model and some more theoretical resultson the inferential procedures.

Other applications in geomarketing, retail industry, economics,insurance, epidemiology, etc.



Bibliography

J. P. Chiles and P. Delfiner (1999). Geostatistics: Modeling Spatial

Uncertainty. Wiley.

P. J. Diggle, R. A. Moyeed and J. A. Tawn (1998). Model-basedgeostatistics (with discussion). Applied Statistics, 47, 299–350.

P. J. Diggle and P. J. Ribeiro Jr. (2007). Model-Based Geostatistics.Springer.

O. F. Christensen (2004). Monte Carlo maximum likelihood inmodel-based geostatistics. Journal of Computational and Graphical

Statistics, 13, 702–718.

G. Fort and E. Moulines (2003). Convergence of the Monte Carloexpectation maximization for curved exponential families. The Annals of

Statistics, 31, 1220–1259.

M. Minozzo and D. Fruttini (2004). Loglinear spatial factor analysis: anapplication to diabetes mellitus complications. Environmetrics, 15,423–434.



Bibliography


Uncertainty. Wiley.






Statistics, 31, 1220–1259.




Bibliography


Uncertainty. Wiley.






Statistics, 31, 1220–1259.




Bibliography


Uncertainty. Wiley.






Statistics, 31, 1220–1259.




Bibliography


Uncertainty. Wiley.






Statistics, 31, 1220–1259.




Bibliography


Uncertainty. Wiley.






Statistics, 31, 1220–1259.



Documents

L'uso dei modelli geostatistici multivariati nelle ...€¦ · 1.Data 2.Model 3.Inference 4.Simulation 5.Appl 2.1.Model 2.2.Autocorrelation MultivariateModel-BasedGeostatistics Geostatistics