Dec,2008 ThaisCOdaFonsecaJointworkwithProfMarkFJSteel Non-gaussianspatiotemporalmodeling

Non-gaussian spatiotemporal modeling

Thais C O da Fonseca Joint work with Prof Mark F J Steel

Department of Statistics University of Warwick

Dec, 2008

1 Introduction Motivation

2 Spatiotemporal modeling Nonseparable models Heavy tailed processes

3 Simulation Results Data 1

Data 2 Data 3

4 Temperature data

Non-gaussian spatiotemporal modeling Results

Spatiotemporal data

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

Maximum temperature data - Spanish Basque Country (67 stations)

Spatiotemporal data

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

Maximum temperature data - Spanish Basque Country (67 stations)

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Example

31 time points (july 2006)

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maximum temperature

Typical problem

Given: observationsZ(si,tj)at a finite number locationssi, i=1, . . . ,I and time pointstj, j=1, . . . ,J.

Desired: predictive distribution for the unknown valueZ(s0,t₀)at the space-time coordinate(s₀,t₀).

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

Z(s,t), (s,t)∈D×T, whereD⊆ <^d, T⊆ <

Typical problem

Given: observationsZ(si,tj)at a finite number locationssi, i=1, . . . ,I and time pointstj, j=1, . . . ,J.

Desired: predictive distribution for the unknown valueZ(s0,t₀)at the space-time coordinate(s₀,t₀).

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

Z(s,t), (s,t)∈D×T, whereD⊆ <^d, T⊆ <

Typical problem

Given: observationsZ(si,tj)at a finite number locationssi, i=1, . . . ,I and time pointstj, j=1, . . . ,J.

Desired: predictive distribution for the unknown valueZ(s0,t₀)at the space-time coordinate(s₀,t₀).

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

Z(s,t), (s,t)∈D×T, whereD⊆ <^d, T⊆ <

General modeling formulation

The uncertainty of the unobserved parts of the process can be expressed probabilistically by arandom functionin space and time:

{Z(s,t); (s,t)∈D×T}.

We need to specify avalidcovariance structure for the process.

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

General modeling formulation

The uncertainty of the unobserved parts of the process can be expressed probabilistically by arandom functionin space and time:

{Z(s,t); (s,t)∈D×T}.

We need to specify avalidcovariance structure for the process.

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

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)C_t(t₁,t2)

I Gaussianity: The process has finite dimensional Gaussian distribution.

Models based on Gaussianity will not perform well (poor predictions) if

I the data are contaminated by outliers;

I there are regions with larger observational variance;

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)C_t(t₁,t2)

I Gaussianity: The process has finite dimensional Gaussian distribution.

Models based on Gaussianity will not perform well (poor predictions) if

I the data are contaminated by outliers;

I there are regions with larger observational variance;

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)C_t(t₁,t2)

I Gaussianity: The process has finite dimensional Gaussian distribution.

Models based on Gaussianity will not perform well (poor predictions) if

I the data are contaminated by outliers;

I there are regions with larger observational variance;

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)C_t(t₁,t2)

I Gaussianity: The process has finite dimensional Gaussian distribution.

Models based on Gaussianity will not perform well (poor predictions) if

I the data are contaminated by outliers;

I there are regions with larger observational variance;

Example

Maximum temperature data - Spanish Basque Country

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empirical variance

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

Example

Maximum temperature data - Spanish Basque Country

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empirical variance

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empirical variance

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

We will consider processes that are stationary

isotropic nonseparable non-Gaussian

Continuous mixture

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

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

Nonseparable model

Z(s,t) =Z₁(s;U)Z₂(t;V) (1)

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

Unconditional covariance C(s,t) =

Z

C₁(s;u)C₂(t;v)dF(u,v) (2) C(s,t)is valid and nonseparable.

Continuous mixture

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

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

Nonseparable model

Z(s,t) =Z₁(s;U)Z₂(t;V) (1) (U,V)is a bivariate random vector with correlationc.

Unconditional covariance C(s,t) =

Z

C₁(s;u)C₂(t;v)dF(u,v) (2) C(s,t)is valid and nonseparable.

Continuous mixture

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

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

Nonseparable model

Z(s,t) =Z₁(s;U)Z₂(t;V) (1) (U,V)is a bivariate random vector with correlationc.

Unconditional covariance C(s,t) =

Z

C₁(s;u)C₂(t;v)dF(u,v) (2) C(s,t)is valid and nonseparable.

In particular, ifC₁(s;u) =σ1exp{−γ₁(s)u}andC₂(t;v) =σ2exp{−γ₂(t)v}

andU=X₀+X₁andV=X₀+X₂, whereX_i has finite moment generating functionMi, then

Proposition

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

whereγ₁(s)andγ₂(t)are spatial and temporal variograms.

For instance,γ₁(s) =||s/a||^αandγ₂(t) =|t/b|^β.

Notice thatc=corr(U,V)measures separability andc∈[0,1].

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

In particular, ifC₁(s;u) =σ1exp{−γ₁(s)u}andC₂(t;v) =σ2exp{−γ₂(t)v}

andU=X₀+X₁andV=X₀+X₂, whereX_i has finite moment generating functionMi, then

Proposition

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

whereγ₁(s)andγ₂(t)are spatial and temporal variograms.

For instance,γ₁(s) =||s/a||^αandγ₂(t) =|t/b|^β.

Notice thatc=corr(U,V)measures separability andc∈[0,1].

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

In particular, ifC₁(s;u) =σ1exp{−γ₁(s)u}andC₂(t;v) =σ2exp{−γ₂(t)v}

andU=X₀+X₁andV=X₀+X₂, whereX_i has finite moment generating functionMi, then

Proposition

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

whereγ₁(s)andγ₂(t)are spatial and temporal variograms.

For instance,γ₁(s) =||s/a||^αandγ₂(t) =|t/b|^β.

Notice thatc=corr(U,V)measures separability andc∈[0,1].

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

In particular, ifC₁(s;u) =σ1exp{−γ₁(s)u}andC₂(t;v) =σ2exp{−γ₂(t)v}

andU=X₀+X₁andV=X₀+X₂, whereX_i has finite moment generating functionMi, then

Proposition

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

whereγ₁(s)andγ₂(t)are spatial and temporal variograms.

For instance,γ₁(s) =||s/a||^αandγ₂(t) =|t/b|^β.

Notice thatc=corr(U,V)measures separability andc∈[0,1].

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

Mixing in space and time

We consider the process

Z(s,˜ t) = ˜Z₁(s;U)˜Z₂(t;V), (4)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s)

ph(s) (5) Mixing in time

Z˜₂(t;V) = Z₂(t;V)

pλ₂(t) (6)

Mixing in space and time

We consider the process

Z(s,˜ t) = ˜Z₁(s;U)˜Z₂(t;V), (4)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s)

ph(s) (5) Mixing in time

Z˜₂(t;V) = Z₂(t;V)

pλ₂(t) (6)

Mixing in space and time

We consider the process

Z(s,˜ t) = ˜Z₁(s;U)˜Z₂(t;V), (4)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s)

ph(s) (5) Mixing in time

Z˜₂(t;V) = Z₂(t;V)

pλ₂(t) (6)

Process λ

(s)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s) ph(s)

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

λ₁(s)needs to be correlated to induce m.s. continuity ofZ˜₁(s;U), this is equivalent toE[λ^−1/2₁ (si)λ^−1/2₁ (si⁰)]→E[λ⁻¹₁ (si)]assi→si⁰.

This is satisfied byλ₁(s) =λ,∀s⇒student-t process. But is does not account for regions with larger variance.

This is also satisfied by the glg process where{ln(λ₁(s));s∈D}is a gaussian process with mean−^ν₂ and covariance structureνC₁(.).

[Palacios and Steel, 2006]

Process λ

(s)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s) ph(s)

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

λ₁(s)needs to be correlated to induce m.s. continuity ofZ˜₁(s;U), this is equivalent toE[λ^−1/2₁ (si)λ^−1/2₁ (si⁰)]→E[λ⁻¹₁ (si)]assi→si⁰.

This is satisfied byλ₁(s) =λ,∀s⇒student-t process. But is does not account for regions with larger variance.

This is also satisfied by the glg process where{ln(λ₁(s));s∈D}is a gaussian process with mean−^ν₂ and covariance structureνC₁(.).

[Palacios and Steel, 2006]

Process λ

(s)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s) ph(s)

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

λ₁(s)needs to be correlated to induce m.s. continuity ofZ˜₁(s;U), this is equivalent toE[λ^−1/2₁ (si)λ^−1/2₁ (si⁰)]→E[λ⁻¹₁ (si)]assi→si⁰.

This is satisfied byλ₁(s) =λ,∀s⇒student-t process. But is does not account for regions with larger variance.

This is also satisfied by the glg process where{ln(λ₁(s));s∈D}is a gaussian process with mean−^ν₂ and covariance structureνC₁(.).

[Palacios and Steel, 2006]

Process λ

(s)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s) ph(s)

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

λ₁(s)needs to be correlated to induce m.s. continuity ofZ˜₁(s;U), this is equivalent toE[λ^−1/2₁ (si)λ^−1/2₁ (si⁰)]→E[λ⁻¹₁ (si)]assi→si⁰.

This is satisfied byλ₁(s) =λ,∀s⇒student-t process. But is does not account for regions with larger variance.

This is also satisfied by the glg process where{ln(λ₁(s));s∈D}is a gaussian process with mean−^ν₂ and covariance structureνC₁(.).

[Palacios and Steel, 2006]

Process λ

(s)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s) ph(s)

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

λ₁(s)needs to be correlated to induce m.s. continuity ofZ˜₁(s;U), this is equivalent toE[λ^−1/2₁ (si)λ^−1/2₁ (si⁰)]→E[λ⁻¹₁ (si)]assi→si⁰.

This is satisfied byλ₁(s) =λ,∀s⇒student-t process. But is does not account for regions with larger variance.

This is also satisfied by the glg process where{ln(λ₁(s));s∈D}is a gaussian process with mean−^ν₂ and covariance structureνC₁(.).

[Palacios and Steel, 2006]

Process h(s)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s) ph(s)

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

We consider the detection of outliers jointly in the estimation procedure and the variablehi =h(si),i=1, . . . ,Iare considered latent variables Their posterior distribution indicate outlying observations (hiclose to 0).

We consider

I log(hi)∼N(−νh/2, νh).

I hi∼Ga(1/νh,1/νh).

Process h(s)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s) ph(s)

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

We consider the detection of outliers jointly in the estimation procedure and the variablehi =h(si),i=1, . . . ,Iare considered latent variables Their posterior distribution indicate outlying observations (hiclose to 0).

We consider

I log(hi)∼N(−νh/2, νh).

I hi∼Ga(1/νh,1/νh).

Process h(s)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s) ph(s)

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

We consider the detection of outliers jointly in the estimation procedure and the variablehi =h(si),i=1, . . . ,Iare considered latent variables Their posterior distribution indicate outlying observations (hiclose to 0).

We consider

I log(hi)∼N(−νh/2, νh).

I hi∼Ga(1/νh,1/νh).

Process h(s)

Mixing in space

Z˜₁(s;U) =p

1−τ²Z₁(s;U)

pλ₁(s) +τ (s) ph(s)

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

We consider the detection of outliers jointly in the estimation procedure and the variablehi =h(si),i=1, . . . ,Iare considered latent variables Their posterior distribution indicate outlying observations (hiclose to 0).

We consider

I log(hi)∼N(−νh/2, νh).

I hi∼Ga(1/νh,1/νh).

Process λ

(t)

Mixing in time

Z˜₂(t;V) = Z₂(t;V) pλ₂(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 a gaussian process with mean−^ν₂² and covariance structureν₂C₂(.).

Process λ

(t)

Mixing in time

Z˜₂(t;V) = Z₂(t;V) pλ₂(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 a gaussian process with mean−^ν₂² and covariance structureν₂C₂(.).

Process λ

(t)

Mixing in time

Z˜₂(t;V) = Z₂(t;V) pλ₂(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 a gaussian process with mean−^ν₂² and covariance structureν₂C₂(.).

Predictions

(λ_1i,h_i, λ_2j)are considered latent variables and sampled in our MCMC sampler.

Given(λ1i,hi, λ2j)the process is gaussian and we can predict at unobserved 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) +²_ξ(x−q2)I(x>q2). We useξ=0.05 resulting in a 95%credible interval.

Predictions

(λ_1i,h_i, λ_2j)are considered latent variables and sampled in our MCMC sampler.

Given(λ1i,hi, λ2j)the process is gaussian and we can predict at unobserved 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) +²_ξ(x−q2)I(x>q2). We useξ=0.05 resulting in a 95%credible interval.

Predictions

(λ_1i,h_i, λ_2j)are considered latent variables and sampled in our MCMC sampler.

Given(λ1i,hi, λ2j)the process is gaussian and we can predict at unobserved 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) +²_ξ(x−q2)I(x>q2). We useξ=0.05 resulting in a 95%credible interval.

Predictions

(λ_1i,h_i, λ_2j)are considered latent variables and sampled in our MCMC sampler.

Given(λ1i,hi, λ2j)the process is gaussian and we can predict at unobserved 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) +²_ξ(x−q2)I(x>q2). We useξ=0.05 resulting in a 95%credible interval.

Predictions

(λ_1i,h_i, λ_2j)are considered latent variables and sampled in our MCMC sampler.

Given(λ1i,hi, λ2j)the process is gaussian and we can predict at unobserved 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) +²_ξ(x−q2)I(x>q2). We useξ=0.05 resulting in a 95%credible interval.

Data

This data set hasI=30 locations andJ =30 time points generated from a Gaussian model with no nugget effect (τ²=0).

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

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

Data

This data set hasI=30 locations andJ =30 time points generated from a Gaussian model with no nugget effect (τ²=0).

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

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

Data

This data set hasI=30 locations andJ =30 time points generated from a Gaussian model with no nugget effect (τ²=0).

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

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

Spatial domain

0.0 0.1 0.2 0.3 0.4 0.5

0.00.10.20.30.40.5

s1

s2

1

2 3

4 5

6

7

8 9

10 11

12 13

14 15 16

17

18

19 20

21

22 23 24

25 26 27

28

29

30

The proposal forλ1i,hi, i=1, . . . ,Iin the MCMC sampler is

constructed by dividing the observations in blocks defined by position in the spatial domain.

Spatial domain

0.0 0.1 0.2 0.3 0.4 0.5

0.00.10.20.30.40.5

s1

s2

1

2 3

4 5

6

7

8 9

10 11

12 13

14 15 16

17

18

19 20

21

22 23 24

25 26 27

28

29

30

The proposal forλ1i,hi, i=1, . . . ,Iin the MCMC sampler is

constructed by dividing the observations in blocks defined by position in the spatial domain.

Description and BF

One location was selected at random (location 7) and a random

increment from Unif(1.0,1.5)times the standard deviation was added to each 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

Description and BF

One location was selected at random (location 7) and a random

increment from Unif(1.0,1.5)times the standard deviation was added to each 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

Estimated correlation function - t

= 1

0.0 0.1 0.2 0.3 0.4 0.5

0.00.10.20.30.40.50.6

Distance in Space

Correlation

0.0 0.1 0.2 0.3 0.4 0.5

0.00.10.20.30.40.50.6

Distance in Space

Correlation

(a) Gaussian (b) Nongaussian withλ1

0.0 0.1 0.2 0.3 0.4 0.5

0.00.10.20.30.40.50.6

Distance in Space

Correlation

0.0 0.1 0.2 0.3 0.4 0.5

0.00.10.20.30.40.50.6

Distance in Space

Correlation

(c) Nongaussian withhandλ₁ (d) Gaussian (Uncontaminated data)

Nongaussian model with λ

1 4 7 10 14 18 22 26 30

0.00.51.01.52.0

Observation indexes

Variance

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0.0 0.1 0.2 0.3 0.4

0.40.60.81.01.2

Distance from observation 07

Median

(a) Variance for each location. (b) Median ofσ_i²vs. distance from obs. 7.

Nongaussian model with h (lognormal)

1 4 7 10 14 18 22 26 30

0.00.20.40.60.81.0

Observation indexes

Variance

1 4 7 10 14 18 22 26 30

0.00.20.40.60.81.0

Observation indexes ττ2hi

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

Nongaussian model with λ

and h

1 4 7 10 14 18 22 26 30

0.00.20.40.60.81.0

Observation indexes

Variance

1 4 7 10 14 18 22 26 30

0.00.20.40.60.81.0

Observation indexes ττ2hi

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

1 4 7 10 14 18 22 26 30

0.81.01.21.41.61.8

Observation indexes λλi

1 4 7 10 14 18 22 26 30

01234567

Observation indexes hi

(c)λ_1i,i=1, . . . ,30. (d)hi,i=1, . . . ,30.

Description and BF

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

0.0 0.1 0.2 0.3 0.4 0.5

0.00.10.20.30.40.5

s1

s2

1

2 3

4 5

6

7

8 9

10 11

12 13

14 15

16

17

18

19 20

21

22 23 24

25 26 27

28

29

30 164

21 27

The logarithm of the BF using Shifted-Gamma (λ=0.98) estimators:

nug. h(lognormal) h(gamma) λ₁ λ₁&h(lognormal)

Gaussian 44 70 72 75 110

Description and BF

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

0.0 0.1 0.2 0.3 0.4 0.5

0.00.10.20.30.40.5

s1

s2

1

2 3

4 5

6

7

8 9

10 11

12 13

14 15

16

17

18

19 20

21

22 23 24

25 26 27

28

29

30 164

21 27

The logarithm of the BF using Shifted-Gamma (λ=0.98) estimators:

nug. h(lognormal) h(gamma) λ₁ λ₁&h(lognormal)

Gaussian 44 70 72 75 110

Nongaussian model with λ

and h

1 4 7 10 14 18 22 26 30

0.00.20.40.60.81.01.2

Observation indexes

Variance

1 4 7 10 14 18 22 26 30

0.000.020.040.060.080.10

Observation indexes ττ2hi

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

1 4 7 10 14 18 22 26 30

0.51.01.52.0

Observation indexes λλi

1 4 7 10 14 18 22 26 30

0.01.02.03.0

Observation indexes hi

(c)λ_1i,i=1, . . . ,30. (d)hi,i=1, . . . ,30.

Data 2

- Description and BF

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

The logarithm of the BF using Shifted-Gamma (λ=0.98) estimators:

nug. h(lognormal) h(gamma) λ₁ λ₁&h(lognormal)

Gaussian -2 -4 -4 24 20

Data 2

- Description and BF

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

The logarithm of the BF using Shifted-Gamma (λ=0.98) estimators:

nug. h(lognormal) h(gamma) λ₁ λ₁&h(lognormal)

Gaussian -2 -4 -4 24 20

Description and BF

The observations at time points 11 to 15 were contaminated by adding a random increment from Unif(0.5,1.5)times the standard deviation to each 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

Description and BF

The observations at time points 11 to 15 were contaminated by adding a random increment from Unif(0.5,1.5)times the standard deviation to each 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

Nongaussian models

1 4 7 10 14 18 22 26 30

0.00.20.40.60.81.0

Observation indexes

Variance

1 4 7 10 14 18 22 26 30

0.00.20.40.60.81.0

Observation indexes

Variance

(a) Model with lognormalh(s). (b) Model with lognormalh(s)andλ₁(s).

Nongaussian model with λ

and λ

1 4 7 10 14 18 22 26 30

01234

location 1

Time indexes

Variance

1 4 7 10 14 18 22 26 30

01234

location 29

Time indexes

Variance

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

1 4 7 10 14 18 22 26 30

0.60.81.01.21.41.6

Location indexes λλ1

1 4 7 10 14 18 22 26 30

0.01.02.03.0

Time indexes λλ2

(c)λ_1i,i=1, . . . ,I. (d)λ_2j,j=1, . . . ,J.