Department of Statistics University of Warwickleg.est.ufpr.br/lib/exe/fetch.php/seminarios:apbeamer.pdf · Outline Introduction Spatiotemporal modeling Simulation Results Temperature

Outline Introduction Spatiotemporal modeling Simulation Results Temperature data

Non-gaussian spatiotemporal modeling

Thais C O da FonsecaJoint work with Prof Mark F J Steel

Department of StatisticsUniversity of Warwick

Dec, 2008

Dec, 2008 1/ 37


1 IntroductionMotivation

2 Spatiotemporal modelingNonseparable modelsHeavy tailed processes

3 Simulation ResultsData 1Data 2Data 3

4 Temperature dataNon-gaussian spatiotemporal modelingResults

Dec, 2008 2/ 37


Motivation

Spatiotemporal data

Due to the proliferation of data sets that are indexed in both space andtime, spatiotemporal models have received an increased attention in theliterature.

Maximum temperature data - Spanish Basque Country (67 stations)

Dec, 2008 3/ 37


Motivation

Spatiotemporal data

Due to the proliferation of data sets that are indexed in both space andtime, spatiotemporal models have received an increased attention in theliterature.Maximum temperature data - Spanish Basque Country (67 stations)

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Dec, 2008 3/ 37


Motivation

Example

31 time points (july 2006)

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Dec, 2008 4/ 37


Motivation

Typical problem

Given: observations Z(si, tj) at a finite number locations si, i = 1, . . . , Iand time points tj, j = 1, . . . , J.

Desired: predictive distribution for the unknown value Z(s0, t0) at thespace-time coordinate (s0, t0).

Focus: continuous space and continuous time which allow for predictionand interpolation at any location and any time.

Z(s, t), (s, t) ∈ D× T, where D ⊆ <d, T ⊆ <

Dec, 2008 5/ 37


Motivation

Typical problem





Dec, 2008 5/ 37


Motivation

Typical problem





Dec, 2008 5/ 37


Motivation

General modeling formulation

The uncertainty of the unobserved parts of the process can be expressedprobabilistically by a random function in space and time:

{Z(s, t); (s, t) ∈ D× T}.

We need to specify a valid covariance structure for the process.

C(s1, s2; t1, t2) = Cov(Z(s1, t1),Z(s2, t2))

Dec, 2008 6/ 37


Motivation

General modeling formulation

The uncertainty of the unobserved parts of the process can be expressedprobabilistically by a random function in space and time:

{Z(s, t); (s, t) ∈ D× T}.

We need to specify a valid covariance structure for the process.

C(s1, s2; t1, t2) = Cov(Z(s1, t1),Z(s2, t2))

Dec, 2008 6/ 37


Motivation

Non-gaussian spatiotemporal models

But building adequate models for these processes is not an easy task.

One observation of the process⇒ simplifying assumptions:I Stationarity: Cov(Z(s1, t1),Z(s2, t2)) = C(s1 − s2, t1 − t2)I Isotropy: Cov(Z(s1, t1),Z(s2, t2)) = C(||s1 − s2||, |t1 − t2|)I Separability: Cov(Z(s1, t1),Z(s2, t2)) = Cs(s1, s2)Ct(t1, t2)I Gaussianity: The process has finite dimensional Gaussian distribution.

Models based on Gaussianity will not perform well (poor predictions) ifI the data are contaminated by outliers;I there are regions with larger observational variance;

Dec, 2008 7/ 37


Motivation





Dec, 2008 7/ 37


Motivation





Dec, 2008 7/ 37


Motivation





Dec, 2008 7/ 37


Motivation

Example

Maximum temperature data - Spanish Basque Country

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For this reason, we consider spatiotemporal processes with heavy tailedfinite dimensional distributions.

Dec, 2008 8/ 37


Motivation

Example

Maximum temperature data - Spanish Basque Country

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For this reason, we consider spatiotemporal processes with heavy tailedfinite dimensional distributions.

Dec, 2008 8/ 37


We will consider processes that are

stationary

isotropic

nonseparable

non-Gaussian

Dec, 2008 9/ 37


Nonseparable models

Continuous mixture

Idea: Continuous mixture of separable covariance functions [Ma, 2002].

It takes advantage of the well known theory developed for purely spatialand purely temporal processes.

Nonseparable model

Z(s, t) = Z1(s; U)Z2(t; V) (1)

(U,V) is a bivariate random vector with correlation c.

Unconditional covariance

C(s, t) =∫

C1(s; u)C2(t; v)dF(u, v) (2)

C(s, t) is valid and nonseparable.

Dec, 2008 10/ 37


Nonseparable models

Continuous mixture



Nonseparable model

Z(s, t) = Z1(s; U)Z2(t; V) (1)



C(s, t) =∫

C1(s; u)C2(t; v)dF(u, v) (2)


Dec, 2008 10/ 37


Nonseparable models

Continuous mixture



Nonseparable model

Z(s, t) = Z1(s; U)Z2(t; V) (1)



C(s, t) =∫

C1(s; u)C2(t; v)dF(u, v) (2)


Dec, 2008 10/ 37


Nonseparable models

In particular, if C1(s; u) = σ1 exp{−γ1(s)u} and C2(t; v) = σ2 exp{−γ2(t)v}and U = X0 + X1 and V = X0 + X2, where Xi has finite moment generatingfunction Mi, then

Proposition

C(s, t) = σ2M0(−(γ1(s) + γ2(t))) M1(−γ1(s)) M2(−γ2(t)), (s, t) ∈ D× T, (3)

where γ1(s) and γ2(t) are spatial and temporal variograms.

For instance, γ1(s) = ||s/a||α and γ2(t) = |t/b|β .

Notice that c = corr(U,V) measures separability and c ∈ [0, 1].

See [Fonseca and Steel, 2008] for more details.

Dec, 2008 11/ 37


Nonseparable models


Proposition






Dec, 2008 11/ 37


Nonseparable models


Proposition






Dec, 2008 11/ 37


Nonseparable models


Proposition






Dec, 2008 11/ 37


Heavy tailed processes

Mixing in space and time

We consider the process

Z(s, t) = Z1(s; U)Z2(t; V), (4)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)

(5)

Mixing in time

Z2(t; V) =Z2(t; V)√λ2(t)

(6)

Dec, 2008 12/ 37





Z(s, t) = Z1(s; U)Z2(t; V), (4)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)

(5)

Mixing in time

Z2(t; V) =Z2(t; V)√λ2(t)

(6)

Dec, 2008 12/ 37





Z(s, t) = Z1(s; U)Z2(t; V), (4)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)

(5)

Mixing in time

Z2(t; V) =Z2(t; V)√λ2(t)

(6)

Dec, 2008 12/ 37



Process λ1(s)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)

λ1(s) accounts for regions in space with larger observational variance.

λ1(s) needs to be correlated to induce m.s. continuity of Z1(s; U), this isequivalent to E[λ−1/2

1 (si)λ−1/21 (si′)]→ E[λ−1

1 (si)] as si → si′ .

This is satisfied by λ1(s) = λ, ∀s⇒ student-t process. But is does notaccount for regions with larger variance.

This is also satisfied by the glg process where {ln(λ1(s)); s ∈ D} is agaussian process with mean −ν

2 and covariance structure νC1(.).[Palacios and Steel, 2006]

Dec, 2008 13/ 37



Process λ1(s)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)



1 (si)λ−1/21 (si′)]→ E[λ−1

1 (si)] as si → si′ .




Dec, 2008 13/ 37



Process λ1(s)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)



1 (si)λ−1/21 (si′)]→ E[λ−1

1 (si)] as si → si′ .




Dec, 2008 13/ 37



Process λ1(s)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)



1 (si)λ−1/21 (si′)]→ E[λ−1

1 (si)] as si → si′ .




Dec, 2008 13/ 37



Process λ1(s)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)



1 (si)λ−1/21 (si′)]→ E[λ−1

1 (si)] as si → si′ .




Dec, 2008 13/ 37



Process h(s)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)

h(s) accounts for traditional outliers (different nugget effects).

We consider the detection of outliers jointly in the estimation procedureand the variable hi = h(si), i = 1, . . . , I are considered latent variables

Their posterior distribution indicate outlying observations (hi close to 0).

We considerI log(hi) ∼ N(−νh/2, νh).I hi ∼ Ga(1/νh, 1/νh).

Dec, 2008 14/ 37



Process h(s)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)





Dec, 2008 14/ 37



Process h(s)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)





Dec, 2008 14/ 37



Process h(s)

Mixing in space

Z1(s; U) =√

1− τ 2 Z1(s; U)√λ1(s)

+ τε(s)√h(s)





Dec, 2008 14/ 37



Process λ2(t)

Mixing in time

Z2(t; V) =Z2(t; V)√λ2(t)

λ2(t) accounts for sections in time with larger observational variance.

This can be seen as a way to adress the issue of volatility clustering,which is common in finantial time series data.

We consider the log gaussian process where {ln(λ2(t)); t ∈ T} is agaussian process with mean −ν2

2 and covariance structure ν2C2(.).

Dec, 2008 15/ 37



Process λ2(t)

Mixing in time

Z2(t; V) =Z2(t; V)√λ2(t)





Dec, 2008 15/ 37



Process λ2(t)

Mixing in time

Z2(t; V) =Z2(t; V)√λ2(t)





Dec, 2008 15/ 37



Predictions

(λ1i, hi, λ2j) are considered latent variables and sampled in our MCMCsampler.

Given (λ1i, hi, λ2j) the process is gaussian and we can predict atunobserved locations and time points.

We compare the predictive performance using proper scoring rules[Gneiting and Raftery, 2008]:

I LPS(p, x) = −log(p(x))I IS(q1, q2; x) = (q2 − q1) + 2

ξ (q1 − x)I(x < q1) + 2ξ (x− q2)I(x > q2). We

use ξ = 0.05 resulting in a 95% credible interval.

Dec, 2008 16/ 37



Predictions





ξ (q1 − x)I(x < q1) + 2ξ (x− q2)I(x > q2). We


Dec, 2008 16/ 37



Predictions





ξ (q1 − x)I(x < q1) + 2ξ (x− q2)I(x > q2). We


Dec, 2008 16/ 37



Predictions





ξ (q1 − x)I(x < q1) + 2ξ (x− q2)I(x > q2). We


Dec, 2008 16/ 37



Predictions





ξ (q1 − x)I(x < q1) + 2ξ (x− q2)I(x > q2). We


Dec, 2008 16/ 37


Data

This data set has I = 30 locations and J = 30 time points generated froma Gaussian model with no nugget effect (τ 2 = 0).

The covariance model is nonseparable Cauchy (Xi ∼ Ga(λi, 1),i = 0, 1, 2) in space and time with c = 0.5.

We contaminated this data set with different kinds of ”outliers” in orderto see the performance of the proposed models in each situation.

Dec, 2008 17/ 37


Data




Dec, 2008 17/ 37


Data




Dec, 2008 17/ 37


Spatial domain

0.0 0.1 0.2 0.3 0.4 0.5

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The proposal for λ1i, hi, i = 1, . . . , I in the MCMC sampler isconstructed by dividing the observations in blocks defined by position inthe spatial domain.

Dec, 2008 18/ 37


Spatial domain

0.0 0.1 0.2 0.3 0.4 0.5

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s2

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The proposal for λ1i, hi, i = 1, . . . , I in the MCMC sampler isconstructed by dividing the observations in blocks defined by position inthe spatial domain.

Dec, 2008 18/ 37


Data 1

Description and BF

One location was selected at random (location 7) and a randomincrement from Unif(1.0, 1.5) times the standard deviation was added toeach observation for this location for the first 20 time points.

The logarithm of the BF using Shifted-Gamma (λ = 0.98) estimators:nug. h (lognormal) h (gamma) λ1 λ1 & h (lognormal)

Gaussian -1 101 98 78 109

Dec, 2008 19/ 37


Data 1

Description and BF

One location was selected at random (location 7) and a randomincrement from Unif(1.0, 1.5) times the standard deviation was added toeach observation for this location for the first 20 time points.


Gaussian -1 101 98 78 109

Dec, 2008 19/ 37


Data 1

Estimated correlation function - t0 = 1

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Distance in Space

Cor

rela

tion

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Distance in Space

Cor

rela

tion

(a) Gaussian (b) Nongaussian with λ1

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Distance in Space

Cor

rela

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0.0 0.1 0.2 0.3 0.4 0.5

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0.2

0.3

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0.6

Distance in Space

Cor

rela

tion

(c) Nongaussian with h and λ1 (d) Gaussian (Uncontaminated data)Dec, 2008 20/ 37


Data 1

Nongaussian model with λ1

1 4 7 10 14 18 22 26 30

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Observation indexes

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0.0 0.1 0.2 0.3 0.4

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0.8

1.0

1.2

Distance from observation 07M

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(a) Variance for each location. (b) Median of σ2i vs. distance from obs. 7.

Dec, 2008 21/ 37


Data 1

Nongaussian model with h (lognormal)

1 4 7 10 14 18 22 26 30

0.0

0.2

0.4

0.6

0.8

1.0

Observation indexes

Var

ianc

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1 4 7 10 14 18 22 26 30

0.0

0.2

0.4

0.6

0.8

1.0

Observation indexes

ττ2h i

(a) Variance for each location. (b) Nugget for each location.

Dec, 2008 22/ 37


Data 1

Nongaussian model with λ1 and h

1 4 7 10 14 18 22 26 30

0.0

0.2

0.4

0.6

0.8

1.0

Observation indexes

Var

ianc

e

1 4 7 10 14 18 22 26 30

0.0

0.2

0.4

0.6

0.8

1.0

Observation indexes

ττ2h i


1 4 7 10 14 18 22 26 30

0.8

1.0

1.2

1.4

1.6

1.8

Observation indexes

λλ i

1 4 7 10 14 18 22 26 30

01

23

45

67

Observation indexes

h i

(c) λ1i, i = 1, . . . , 30. (d) hi, i = 1, . . . , 30.Dec, 2008 23/ 37


Data 2

Description and BF

A region was selected and an increment from Unif(0.5, 1.5) times thestandard deviation was added to each observation for the first 10 timepoints.

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

s1

s2

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41621

27


Gaussian 44 70 72 75 110

Dec, 2008 24/ 37


Data 2

Description and BF

A region was selected and an increment from Unif(0.5, 1.5) times thestandard deviation was added to each observation for the first 10 timepoints.

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

s1

s2

1

2

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41621

27


Gaussian 44 70 72 75 110

Dec, 2008 24/ 37


Data 2

Nongaussian model with λ1 and h

1 4 7 10 14 18 22 26 30

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Observation indexes

Var

ianc

e

1 4 7 10 14 18 22 26 30

0.00

0.02

0.04

0.06

0.08

0.10

Observation indexes

ττ2h i


1 4 7 10 14 18 22 26 30

0.5

1.0

1.5

2.0

Observation indexes

λλ i

1 4 7 10 14 18 22 26 30

0.0

1.0

2.0

3.0

Observation indexes

h i

(c) λ1i, i = 1, . . . , 30. (d) hi, i = 1, . . . , 30.Dec, 2008 25/ 37


Data 2

Data 2∗ - Description and BF

A region of the spatial domain was selected (locations 4, 16, 21 and 27)and the same random increment from Unif(0.5, 1.5) times the standarddeviation was added to each observation for the first 10 time points.


Gaussian -2 -4 -4 24 20

Dec, 2008 26/ 37


Data 2

Data 2∗ - Description and BF

A region of the spatial domain was selected (locations 4, 16, 21 and 27)and the same random increment from Unif(0.5, 1.5) times the standarddeviation was added to each observation for the first 10 time points.


Gaussian -2 -4 -4 24 20

Dec, 2008 26/ 37


Data 3

Description and BF

The observations at time points 11 to 15 were contaminated by adding arandom increment from Unif(0.5, 1.5) times the standard deviation toeach observation for all spatial locations.

The logarithm of the BF using Shifted-Gamma (λ = 0.98) estimators:nug. h (lognormal) λ1 λ2 λ1&λ2 λ1&λ2&h

Gaussian 18 44 28 76 112 111

Dec, 2008 27/ 37


Data 3

Description and BF

The observations at time points 11 to 15 were contaminated by adding arandom increment from Unif(0.5, 1.5) times the standard deviation toeach observation for all spatial locations.

The logarithm of the BF using Shifted-Gamma (λ = 0.98) estimators:nug. h (lognormal) λ1 λ2 λ1&λ2 λ1&λ2&h

Gaussian 18 44 28 76 112 111

Dec, 2008 27/ 37


Data 3

Nongaussian models

1 4 7 10 14 18 22 26 30

0.0

0.2

0.4

0.6

0.8

1.0

Observation indexes

Var

ianc

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1 4 7 10 14 18 22 26 30

0.0

0.2

0.4

0.6

0.8

1.0

Observation indexesV

aria

nce

(a) Model with lognormal h(s). (b) Model with lognormal h(s) and λ1(s).

Dec, 2008 28/ 37


Data 3

Nongaussian model with λ1 and λ2

1 4 7 10 14 18 22 26 30

01

23

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location 1

Time indexes

Var

ianc

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1 4 7 10 14 18 22 26 30

01

23

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location 29

Time indexes

Var

ianc

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(a) Variance for each time. (b) Variance for each time.

1 4 7 10 14 18 22 26 30

0.6

0.8

1.0

1.2

1.4

1.6

Location indexes

λλ 1

1 4 7 10 14 18 22 26 30

0.0

1.0

2.0

3.0

Time indexes

λλ 2

(c) λ1i, i = 1, . . . , I. (d) λ2j, j = 1, . . . , J.Dec, 2008 29/ 37



Data

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● ●●●●●●●●●●●

●

●

4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.047

.247

.447

.647

.848

.0

s2

s 1

1

2

34

5

67

8

9

10

11

12

13

14

15

16

17 18

19

20

21

22

23

24

25

26

27

28

29 30

31

32

33

34

35

36

37

38

3940 4142

43 44

45

4647

4849

50

51

5253

5455

56

57

58

596061

62

63

64 65

66

67

12

3

**

*

(a) Spain and France Map. (b) Basque Country (Zoom).

Dec, 2008 30/ 37



Model

Mean function:

µ(s, t) = δ0 + δ1s1 + δ2s2 + δ3h + δ4t + δ5t2

Cauchy covariance function: Xi ∼ Ga(λi, 1)

C(s, t) =(

11 + ||s/a||α

)λ1(

11 + |t/b|β

)λ2(

11 + ||s/a||α + |t/b|β

)λ0

λ1 = λ2 = 1 and c = λ0/(1 + λ0).

Dec, 2008 31/ 37



Model

Mean function:

µ(s, t) = δ0 + δ1s1 + δ2s2 + δ3h + δ4t + δ5t2

Cauchy covariance function: Xi ∼ Ga(λi, 1)

C(s, t) =(

11 + ||s/a||α

)λ1(

11 + |t/b|β

)λ2(

11 + ||s/a||α + |t/b|β

)λ0

λ1 = λ2 = 1 and c = λ0/(1 + λ0).

Dec, 2008 31/ 37



Likelihood

In order to calculate the likelihood function we need to invert a matrixwith dimension 2077× 2077.

We approximate the likelihood by using conditional distributions.

We consider a partition of Z into subvectors Z1, ...,Z31 whereZj = (Z(s1, tj), . . . ,Z(s67, tj))′ and we define Z(j) = (Zj−L+1, ...,Zj).Then

p(z|φ) ≈ p(z1|φ)31∏

j=2

p(zj|z(j−1), φ). (7)

This means the distribution of Zj will only depend on the observations inspace for the previous L time points.

In this application we used L = 5 to make the MCMC feasible.

Dec, 2008 32/ 37


Results

Bayes Factor

h λ1 λ1 & h λ2 λ2 & h λ1 & λ2 λ1, h & λ2

Shifted gamma 172 148 345 138 279 417 547

Table: The natural logarithm of the Bayes factor in favor of the model in the columnversus Gaussian model using Shifted-Gamma (λ = 0.98) estimator for the predictivedensity of z.

Dec, 2008 33/ 37


Results

Model with h and λ2

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

010

2030

4050

σσ2 ττ2h

1 4 7 11 16 21 26 31 36 41 46 51 56 61 66

01

23

45

6

σσ2 ττ2h

(a) σ2(1− τ 2)/λ2. (b) σ2τ 2/h.

Dec, 2008 34/ 37


Results

Predicted temperature at the out-of-sample stations

*

*

***

* **

*

**

* *

*

*

*

*

*

* *

**

**

*

*

***

*

*

18 20 22 24 26 28 30 32

2025

3035

Station 1*

observed

pred

icte

d

*

*

*

**

*** ***

**

*

*

**

*

***

*

****

**

*

*

*

16 18 20 22 24 26 28

1015

2025

Station 2*

observed

pred

icte

d

*

*

*

* *

***

*

**

**

*

*

*

*

*

**

**

** *

*

** *

*

*

22 24 26 28 30 32 34

2025

3035

Station 3*

observed

pred

icte

d

(a) Gaussian Model. (b) Gaussian Model. (c) Gaussian Model.

*

*

*

**

* **

*

**

* **

*

*

*

*

* *

**

**

*

*

***

*

*

18 20 22 24 26 28 30 32

2025

3035

Station 1*

observed

pred

icte

d

*

*

*

***** ***

***

*

**

*

***

*

****

**

*

*

*

16 18 20 22 24 26 28

1015

2025

30

Station 2*

observed

pred

icte

d

*

*

** *

***

*

**

***

*

*

*

*

**

**

** **

** *

*

*

22 24 26 28 30 32 34

2025

3035

Station 3*

observed

pred

icte

d

(d) Model with λ2 & h. (e) Model with λ2 & h. (f) Model with λ2 & h.

Dec, 2008 35/ 37


Results

Model comparison

model Average width IS LPSGaussian 3.78 4.35 103.81

h 3.83 4.34 102.04λ1 3.74 4.36 105.09

λ1 & h 3.75 4.48 103.79λ2 3.73 3.94 87.33

λ2 & h 3.73 3.87 86.57λ1 & λ2 4.51 4.65 85.89λ1, h & λ2 3.84 4.02 83.78

Dec, 2008 36/ 37


Results

References

T Fonseca and M F J SteelA New Class of Nonseparable Spatiotemporal ModelsCRiSM Working Paper 08-13. 2008.

T Gneiting and A E RafteryStrictly proper scoring rules, prediction and estimationJASA. (102) 360–378, 2007.

C MaSpatio-temporal covariance functions generated by mixturesMathematical geology. (34) 965–975, 2002.

M B Palacios and M F J SteelNon-Gaussian Bayesian Geostatistical ModelingJASA. (101) 604–618, 2006.

Dec, 2008 37/ 37

Documents

Department of Statistics University of Warwickleg.est.ufpr.br/lib/exe/fetch.php/seminarios:apbeamer.pdf · Outline Introduction Spatiotemporal modeling Simulation Results Temperature