[PPT] - Space time analysis of extreme values a Gabriel Huerta Department PowerPoint Presentation

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Space time analysis of extreme values a

Gabriel Huerta

Department of Mathematics and Statistics University of New Mexico Albuquerque, NM, 87131, U.S.A. http://www.stat.unm.edu/∼ghuerta 4th EVA conference, Gothenburg, August 15-19 2005

aJoint paper with Bruno Sanso, University of California, Santa Cruz, U.S.A

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Summary Points

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Summary Points

Models for non-stationary extreme values.

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Summary Points

Models for non-stationary extreme values.
Space-time formulation for the GEV distribution.

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Summary Points

Models for non-stationary extreme values.
Space-time formulation for the GEV distribution.
Dynamic Linear Model (DLM) framework for temporal

components.

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Summary Points

Models for non-stationary extreme values.
Space-time formulation for the GEV distribution.
Dynamic Linear Model (DLM) framework for temporal

components.

Spatial elements through process convolutions.

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Summary Points

Models for non-stationary extreme values.
Space-time formulation for the GEV distribution.
Dynamic Linear Model (DLM) framework for temporal

components.

Spatial elements through process convolutions.
Model fitting via customized Markov chain Monte Carlo

(MCMC) methods.

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Summary Points

Models for non-stationary extreme values.
Space-time formulation for the GEV distribution.
Dynamic Linear Model (DLM) framework for temporal

components.

Spatial elements through process convolutions.
Model fitting via customized Markov chain Monte Carlo

(MCMC) methods.

Extreme values of ozone levels in Mexico City.

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Summary Points

Models for non-stationary extreme values.
Space-time formulation for the GEV distribution.
Dynamic Linear Model (DLM) framework for temporal

components.

Spatial elements through process convolutions.
Model fitting via customized Markov chain Monte Carlo

(MCMC) methods.

Extreme values of ozone levels in Mexico City.
Extreme values of rainfall in Venezuela.

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Figure 1: Daily maximum values of ozone levels.

Time Ozone 1990 1992 1994 1996 1998 2000 2002 0.0 0.1 0.2 0.3

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Extreme Value Modeling

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Extreme Value Modeling

The traditional approach is based on the Generalized Ex-

treme Value (GEV) distribution function: H(z) = exp

−
1 + ξ

z − µ σ −1/ξ

+

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Extreme Value Modeling

The traditional approach is based on the Generalized Ex-

treme Value (GEV) distribution function: H(z) = exp

−
1 + ξ

z − µ σ −1/ξ

+

– −∞ < µ < ∞; σ > 0; −∞ < ξ < ∞.

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Extreme Value Modeling

The traditional approach is based on the Generalized Ex-

treme Value (GEV) distribution function: H(z) = exp

−
1 + ξ

z − µ σ −1/ξ

+

– −∞ < µ < ∞; σ > 0; −∞ < ξ < ∞.

– + denotes the positive part of the argument.

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Extreme Value Modeling

The traditional approach is based on the Generalized Ex-

treme Value (GEV) distribution function: H(z) = exp

−
1 + ξ

z − µ σ −1/ξ

+

– −∞ < µ < ∞; σ > 0; −∞ < ξ < ∞.

– + denotes the positive part of the argument. – ξ > 0 Fr´ echet family; ξ < 0 the Weibull family; ξ → 0 Gumbel family.

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The book by Coles (2001) presents a very clear account
f statistical inference using the GEV.

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The book by Coles (2001) presents a very clear account
f statistical inference using the GEV.
A Bayesian analysis can be performed by imposing a prior
n (µ, σ, ξ) as in Coles and Tawn (1996).

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The book by Coles (2001) presents a very clear account
f statistical inference using the GEV.
A Bayesian analysis can be performed by imposing a prior
n (µ, σ, ξ) as in Coles and Tawn (1996).
Alternatively, models for exceedances over a high treshold

had been proposed.

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The book by Coles (2001) presents a very clear account
f statistical inference using the GEV.
A Bayesian analysis can be performed by imposing a prior
n (µ, σ, ξ) as in Coles and Tawn (1996).
Alternatively, models for exceedances over a high treshold

had been proposed.

This leads into the Generalized Pareto Distributions and

Point processes approaches. (Pickands 1971 and 1975).

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The book by Coles (2001) presents a very clear account
f statistical inference using the GEV.
A Bayesian analysis can be performed by imposing a prior
n (µ, σ, ξ) as in Coles and Tawn (1996).
Alternatively, models for exceedances over a high treshold

had been proposed.

This leads into the Generalized Pareto Distributions and

Point processes approaches. (Pickands 1971 and 1975).

These ideas had been developed into a Bayesian hierarchi-

cal modeling framework. Smith et al. (1997); Assuncao et al. (2004); Casson and Coles (1999); Gilleland et. al. (2004).

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Extremes for Non-Stationary Data

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Extremes for Non-Stationary Data

Coles (2001) mentions several possibilities.

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Extremes for Non-Stationary Data

Coles (2001) mentions several possibilities.
Approach: z1, z2, . . . , zm; zt ∼ GEV (µt, σ, ξ).

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Extremes for Non-Stationary Data

Coles (2001) mentions several possibilities.
Approach: z1, z2, . . . , zm; zt ∼ GEV (µt, σ, ξ).
Deterministic functions: µt = β0 + β1t; µt = β0 + β1 +

β2t + β3t2 or µt = β0 + β1Xt.

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Extremes for Non-Stationary Data

Coles (2001) mentions several possibilities.
Approach: z1, z2, . . . , zm; zt ∼ GEV (µt, σ, ξ).
Deterministic functions: µt = β0 + β1t; µt = β0 + β1 +

β2t + β3t2 or µt = β0 + β1Xt.

Non-stationarity can also be included for the shape

and/or scale parameters: σt = exp(β0 + β1t); ξt = β0 + β1t or ξt = β0 + β1t + β2t2.

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Extremes for Non-Stationary Data

Coles (2001) mentions several possibilities.
Approach: z1, z2, . . . , zm; zt ∼ GEV (µt, σ, ξ).
Deterministic functions: µt = β0 + β1t; µt = β0 + β1 +

β2t + β3t2 or µt = β0 + β1Xt.

Non-stationarity can also be included for the shape

and/or scale parameters: σt = exp(β0 + β1t); ξt = β0 + β1t or ξt = β0 + β1t + β2t2.

We propose the use of Dynamic Linear Models (DLM)

as in West and Harrison (1997) to model the parameter changes in time.

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GEV distribution with DLM’s

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GEV distribution with DLM’s

For z1, z2, . . . , zm, zt ∼ GEV (µt, σ, ξ)

Ht(zt) = exp

−[1 + ξ(zt − µt)/σ]−1/ξ

+

µt = θt + ǫt; ǫt ∼ N(0, V )

θt = θt−1 + ωt; ωt ∼ N(0, τV )

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GEV distribution with DLM’s

For z1, z2, . . . , zm, zt ∼ GEV (µt, σ, ξ)

Ht(zt) = exp

−[1 + ξ(zt − µt)/σ]−1/ξ

+

µt = θt + ǫt; ǫt ∼ N(0, V )

θt = θt−1 + ωt; ωt ∼ N(0, τV )

Parameters (t = 0) are assumed apriori independent.

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GEV distribution with DLM’s

For z1, z2, . . . , zm, zt ∼ GEV (µt, σ, ξ)

Ht(zt) = exp

−[1 + ξ(zt − µt)/σ]−1/ξ

+

µt = θt + ǫt; ǫt ∼ N(0, V )

θt = θt−1 + ωt; ωt ∼ N(0, τV )

Parameters (t = 0) are assumed apriori independent.
π(σ) ∼ LN(mσ, sσ) ; π(ξ) ∼ N(mξ, sξ).
θ0 ∼ N(m0, C0); V ∼ IG(αv, βv); τ ∼ IG(ατ, βτ)

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GEV distribution with DLM’s

For z1, z2, . . . , zm, zt ∼ GEV (µt, σ, ξ)

Ht(zt) = exp

−[1 + ξ(zt − µt)/σ]−1/ξ

+

µt = θt + ǫt; ǫt ∼ N(0, V )

θt = θt−1 + ωt; ωt ∼ N(0, τV )

Parameters (t = 0) are assumed apriori independent.
π(σ) ∼ LN(mσ, sσ) ; π(ξ) ∼ N(mξ, sξ).
θ0 ∼ N(m0, C0); V ∼ IG(αv, βv); τ ∼ IG(ατ, βτ) .
µt follows a first order polynomial DLM with state vector

θt.

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General DLM (Ft, V, Gt, W)

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General DLM (Ft, V, Gt, W)

µt = F

′

tθt + ǫt; ǫt ∼ N(0, V )

θt = Gtθt−1 + ωt; ωt ∼ N(0, W)

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General DLM (Ft, V, Gt, W)

µt = F

′

tθt + ǫt; ǫt ∼ N(0, V )

θt = Gtθt−1 + ωt; ωt ∼ N(0, W)

θt is a k × 1 state vector;
Ft is a k × 1 regressor vector;
Gt is a k × k evolution matrix;
V is an observational variance and
W is a k × k evolution covariance matrix.

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Posterior Inference for DLM-GEV models

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Posterior Inference for DLM-GEV models

Define Z = (z1, z2, . . . , zm); µ = (µ1, µ2, . . . , µm) and

θ = (θ1, θ2, . . . , θm).

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Posterior Inference for DLM-GEV models

Define Z = (z1, z2, . . . , zm); µ = (µ1, µ2, . . . , µm) and

θ = (θ1, θ2, . . . , θm).

p(µt|zt, σ, θt, V ); t

= 1, . . . , m is sampled with a Metropolis-Hastings step.

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Posterior Inference for DLM-GEV models

Define Z = (z1, z2, . . . , zm); µ = (µ1, µ2, . . . , µm) and

θ = (θ1, θ2, . . . , θm).

p(µt|zt, σ, θt, V ); t

= 1, . . . , m is sampled with a Metropolis-Hastings step.

p(σ|Z, µ, ξ) and p(ξ|Z, µ, σ) are also sampled via M-H.

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Posterior Inference for DLM-GEV models

Define Z = (z1, z2, . . . , zm); µ = (µ1, µ2, . . . , µm) and

θ = (θ1, θ2, . . . , θm).

p(µt|zt, σ, θt, V ); t

= 1, . . . , m is sampled with a Metropolis-Hastings step.

p(σ|Z, µ, ξ) and p(ξ|Z, µ, σ) are also sampled via M-H.
V and W are sampled from Inverse Gamma/Wishart dis-

tributions.

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Posterior Inference for DLM-GEV models

Define Z = (z1, z2, . . . , zm); µ = (µ1, µ2, . . . , µm) and

θ = (θ1, θ2, . . . , θm).

p(µt|zt, σ, θt, V ); t

= 1, . . . , m is sampled with a Metropolis-Hastings step.

p(σ|Z, µ, ξ) and p(ξ|Z, µ, σ) are also sampled via M-H.
V and W are sampled from Inverse Gamma/Wishart dis-

tributions.

For θt, we apply Forward Filtering Backward Simulation

(FFBS) as in Carter and Kohn or Fr¨ uhwirth-Schnatter (1994).

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– Forward in time, we obtain p(θt|Dt, V, W); t = 1, 2, . . . , m

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– Forward in time, we obtain p(θt|Dt, V, W); t = 1, 2, . . . , m – Backwards in time, we sample p(θm|Dm, V, W) and then, recursively we sample from p(θt|θt+1, Dm, V, W); t = m − 1, . . . , 1

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– Forward in time, we obtain p(θt|Dt, V, W); t = 1, 2, . . . , m – Backwards in time, we sample p(θm|Dm, V, W) and then, recursively we sample from p(θt|θt+1, Dm, V, W); t = m − 1, . . . , 1

A similar approach is discussed by Gaetan and Grigoletto

(2004) Extremes.

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– Forward in time, we obtain p(θt|Dt, V, W); t = 1, 2, . . . , m – Backwards in time, we sample p(θm|Dm, V, W) and then, recursively we sample from p(θt|θt+1, Dm, V, W); t = m − 1, . . . , 1

A similar approach is discussed by Gaetan and Grigoletto

(2004) Extremes. – Dynamics for scale/shape parameters. Sequential updating with Particle Filters.

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– Forward in time, we obtain p(θt|Dt, V, W); t = 1, 2, . . . , m – Backwards in time, we sample p(θm|Dm, V, W) and then, recursively we sample from p(θt|θt+1, Dm, V, W); t = m − 1, . . . , 1

A similar approach is discussed by Gaetan and Grigoletto

(2004) Extremes. – Dynamics for scale/shape parameters. Sequential updating with Particle Filters. – Observation variance equals zero.

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– Forward in time, we obtain p(θt|Dt, V, W); t = 1, 2, . . . , m – Backwards in time, we sample p(θm|Dm, V, W) and then, recursively we sample from p(θt|θt+1, Dm, V, W); t = m − 1, . . . , 1

A similar approach is discussed by Gaetan and Grigoletto

(2004) Extremes. – Dynamics for scale/shape parameters. Sequential updating with Particle Filters. – Observation variance equals zero. – No space or space-time structure.

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Figure 2: Posterior mean for µt, and 90% probability interval for θt; 1990-2002 data

Time

zone

1990 1992 1994 1996 1998 2000 2002 0.0 0.1 0.2 0.3 θt µt

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Detecting Trends

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Detecting Trends

Is there an overall decreasing trend in the maxima of the

previous figure?

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Detecting Trends

Is there an overall decreasing trend in the maxima of the

previous figure?

One possibility is to add an extra model parameter and

estimate incremental growth.

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Detecting Trends

Is there an overall decreasing trend in the maxima of the

previous figure?

One possibility is to add an extra model parameter and

estimate incremental growth.

Alternatively, we considered a regression

model on µt with time-varying intercept but a constant slope:

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Detecting Trends

Is there an overall decreasing trend in the maxima of the

previous figure?

One possibility is to add an extra model parameter and

estimate incremental growth.

Alternatively, we considered a regression

model on µt with time-varying intercept but a constant slope: zt ∼ GEV (µt, σ, ξ) µt = θt + β(t − ¯ t) + ǫt; ǫt ∼ N(0, V ) θt = θt−1 + ωt; ωt ∼ N(0, τV )

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where ¯ t = (1/T)(T

t=1 t) and β represents change of

level per unit of time.

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where ¯ t = (1/T)(T

t=1 t) and β represents change of

level per unit of time.

For model fitting, notice that:

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where ¯ t = (1/T)(T

t=1 t) and β represents change of

level per unit of time.

For model fitting, notice that:

– Conditional on θt, the difference µt − θt follows a regression model. – Conditional on β, µt − β(t − ¯ t) follows a first order polynomial DLM.

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where ¯ t = (1/T)(T

t=1 t) and β represents change of

level per unit of time.

For model fitting, notice that:

– Conditional on θt, the difference µt − θt follows a regression model. – Conditional on β, µt − β(t − ¯ t) follows a first order polynomial DLM. – This defines a Gibbs sampler scheme that produces posterior samples for β and θt.

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where ¯ t = (1/T)(T

t=1 t) and β represents change of

level per unit of time.

For model fitting, notice that:

– Conditional on θt, the difference µt − θt follows a regression model. – Conditional on β, µt − β(t − ¯ t) follows a first order polynomial DLM. – This defines a Gibbs sampler scheme that produces posterior samples for β and θt.

In fact, Pr(β < 0|Z) ≈ 0.79 indication of a decreasing

trend.

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Figure 3: Posterior distribution for β; ozone data 1990-2002.

β Density −4e−05 −2e−05 0e+00 2e−05 10000 20000 30000

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Maxima monthly rainfall values in Venezuela

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Maxima monthly rainfall values in Venezuela

Measurements were taken at the Maiquet´

ıa station lo- cated at the Simon Bolivar Airport near Caracas (Yt).

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Maxima monthly rainfall values in Venezuela

Measurements were taken at the Maiquet´

ıa station lo- cated at the Simon Bolivar Airport near Caracas (Yt).

The monthly North Atlantic Oscillation (NAO) index is

considered as a covariate (Xt).

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Maxima monthly rainfall values in Venezuela

Measurements were taken at the Maiquet´

ıa station lo- cated at the Simon Bolivar Airport near Caracas (Yt).

The monthly North Atlantic Oscillation (NAO) index is

considered as a covariate (Xt).

The model is:

Yt ∼ GEV (µt, σ, ξ) µt = θt + βtXt + ǫt θt = θt−1 + ω1t βt = βt−1 + ω2t

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Figure 4: Maxima monthly rainfall values in Venezuela and NAO index

(a)

Time 1960 1970 1980 1990 2000 40 80 120 rainfall µt

(b)

Time NAO index 1960 1970 1980 1990 2000 −4 −2 2 4

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Figure 5: Posterior median and 90% probability intervals for βt

time βt 1960 1970 1980 1990 0.0 0.5 1.0 Posterior median 90% probability

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Space-time model with Process Convolutions

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Space-time model with Process Convolutions

Consider data yt = (y1,t, . . . , ynt,t)′ which is recorded at

sites s1, . . . , snt.

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Space-time model with Process Convolutions

Consider data yt = (y1,t, . . . , ynt,t)′ which is recorded at

sites s1, . . . , snt.

A possible model (Higdon 2002) is

yt = Ktxt + ǫt xt = xt + νt

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Space-time model with Process Convolutions

Consider data yt = (y1,t, . . . , ynt,t)′ which is recorded at

sites s1, . . . , snt.

A possible model (Higdon 2002) is

yt = Ktxt + ǫt xt = xt + νt Kt is a nt × κ matrix given by Kt

ij = k(si − ωj), t = 1, . . . , m

ǫt ∼ N(0, σ2

ǫ), t = 1, . . . , m

νt ∼ N(0, σ2

ν), t = 1, . . . , m

x1 ∼ N(0, σ2

xIκ)

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k(· − ωj) defines a smoothing kernel.
ω1, . . . , ωκ are spatial sites where kernels are centered.
xt is interpreted as a latent process

y(s, t) =

j

k(s − ωj)xjt

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k(· − ωj) defines a smoothing kernel.
ω1, . . . , ωκ are spatial sites where kernels are centered.
xt is interpreted as a latent process

y(s, t) =

j

k(s − ωj)xjt

Possible kernels are:

– Gaussian: k(s) ∝ exp

−||s||2/2η
(η > 0).

– Exponential: k(s) ∝ exp {−||s||/η} (η > 0). – Spherical: k(s) ∝

1 − ||s||3

r3

3 I[s ≤ r].

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A Spatio-Temporal Model for the GEV dis- tribution

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A Spatio-Temporal Model for the GEV dis- tribution

Assume ys,t ∼ GEV (µs,t, σ, ξ); s = 1, . . . , S, t =

1, . . . , m

Hs,t(ys,t; µs,t, ξ, σ) = exp

−
1 + ξ

ys,t − µs,t σ −1/ξ

+

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A Spatio-Temporal Model for the GEV dis- tribution

Assume ys,t ∼ GEV (µs,t, σ, ξ); s = 1, . . . , S, t =

1, . . . , m

Hs,t(ys,t; µs,t, ξ, σ) = exp

−
1 + ξ

ys,t − µs,t σ −1/ξ

+

For each t, µt = (µ1,t, µ2,t, . . . , µS,t)′. (σ, ξ) constant in

time.

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A Spatio-Temporal Model for the GEV dis- tribution

Assume ys,t ∼ GEV (µs,t, σ, ξ); s = 1, . . . , S, t =

1, . . . , m

Hs,t(ys,t; µs,t, ξ, σ) = exp

−
1 + ξ

ys,t − µs,t σ −1/ξ

+

For each t, µt = (µ1,t, µ2,t, . . . , µS,t)′. (σ, ξ) constant in

time.

We define a DLM on µt:

µt = K′θt + ǫt; θt = θt−1 + νt

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θt

= (θt,1, . . . , θt,κ)′, ǫt = (ǫt,1, . . . , ǫt,κ)′, νt = (νt,1, . . . , νt,κ)′.

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θt

= (θt,1, . . . , θt,κ)′, ǫt = (ǫt,1, . . . , ǫt,κ)′, νt = (νt,1, . . . , νt,κ)′.

ǫt ∼ N(0, σ2

ǫIκ×κ); νt ∼ N(0, σ2 νIκ×κ)

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θt

= (θt,1, . . . , θt,κ)′, ǫt = (ǫt,1, . . . , ǫt,κ)′, νt = (νt,1, . . . , νt,κ)′.

ǫt ∼ N(0, σ2

ǫIκ×κ); νt ∼ N(0, σ2 νIκ×κ)

With a Gaussian kernel, K′ is an S × κ matrix with

entries: K′

ij = K(si − ωj);

K(si − ωj) ∝ exp(−d||si − ωj||2/2)

si is the position of station i.
ωj is the position of the kernel j = 1, . . . , κ.

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θt

= (θt,1, . . . , θt,κ)′, ǫt = (ǫt,1, . . . , ǫt,κ)′, νt = (νt,1, . . . , νt,κ)′.

ǫt ∼ N(0, σ2

ǫIκ×κ); νt ∼ N(0, σ2 νIκ×κ)

With a Gaussian kernel, K′ is an S × κ matrix with

entries: K′

ij = K(si − ωj);

K(si − ωj) ∝ exp(−d||si − ωj||2/2)

si is the position of station i.
ωj is the position of the kernel j = 1, . . . , κ.
d is a range parameter; d = cφ; 1/2 < c < 2; φ = knot

distance.

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1st stage priors:

π(σ) ∼ LN(µσ, sσ) and π(ξ) ∼ N(µξ, sξ) are th

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1st stage priors:

π(σ) ∼ LN(µσ, sσ) and π(ξ) ∼ N(µξ, sξ) are th

2nd stage priors:

θ0 ∼ N(0, σ2

θIκ×κ);

1/σ2

ǫ

∼ Gamma(αǫ, βǫ); 1/σ2

ν ∼ Gamma(αν, βν) and 1/σ2 θ ∼

Gamma(αθ, βθ).

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1st stage priors:

π(σ) ∼ LN(µσ, sσ) and π(ξ) ∼ N(µξ, sξ) are th

2nd stage priors:

θ0 ∼ N(0, σ2

θIκ×κ);

1/σ2

ǫ

∼ Gamma(αǫ, βǫ); 1/σ2

ν ∼ Gamma(αν, βν) and 1/σ2 θ ∼

Gamma(αθ, βθ).

The log-likelihood is equal to

l(θ) = −mS log σ −

m

t=1

S

s=1
1 + ξ

zs,t − µs,t σ −1/ξ

+

−

1 + 1

σ

m
t=1

S

s=1

log

1 + ξ

ys,t − µs,t σ

+

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Posterior Inference and Simulation for Space- Time Model

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Posterior Inference and Simulation for Space- Time Model

Follows similar lines as for the GEV time model.

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Posterior Inference and Simulation for Space- Time Model

Follows similar lines as for the GEV time model.
The full conditional for µt,s, σ and ξ are sampled through

a Metropolis-Hastings step.

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Posterior Inference and Simulation for Space- Time Model

Follows similar lines as for the GEV time model.
The full conditional for µt,s, σ and ξ are sampled through

a Metropolis-Hastings step.

θt is sampled with Forward Filtering Backward Simula-

tion.

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Posterior Inference and Simulation for Space- Time Model

Follows similar lines as for the GEV time model.
The full conditional for µt,s, σ and ξ are sampled through

a Metropolis-Hastings step.

θt is sampled with Forward Filtering Backward Simula-

tion.

The full conditionals of σ2

ǫ; σ2 ν and σ2 θ are sampled with

Inverse Gamma distributions.

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Posterior Inference and Simulation for Space- Time Model

Follows similar lines as for the GEV time model.
The full conditional for µt,s, σ and ξ are sampled through

a Metropolis-Hastings step.

θt is sampled with Forward Filtering Backward Simula-

tion.

The full conditionals of σ2

ǫ; σ2 ν and σ2 θ are sampled with

Inverse Gamma distributions.

The range parameter d is assumed fixed, d = cφ.

SLIDE 88

Figure 6: RAMA stations, kernel and interpolation grid posi- tions.

−99.3 −99.2 −99.1 −99.0 −98.9 19.25 19.30 19.35 19.40 19.45 19.50 19.55 longitude latitude ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9 ω10 ω11 ω12 ω13 ω14 ω15 ω16 LAG TAC EAC SAG AZC TLA XAL MER PED CES PLA HAN UIZ BJU TAH CUA TPN CHA TAX

SLIDE 89

Figure 7: Daily maxima for 1999 and posterior estimates of 0.5 quantile.

AZC

time

zone

50 100 150 200 250 300 0.05 0.10 0.15 0.20 0.25 Data Median

XAL

time

zone

50 100 150 200 250 300 0.05 0.10 0.15

TPN

time

zone

50 100 150 200 250 300 0.05 0.10 0.15 0.20

TAX

time

zone

50 100 150 200 250 300 0.05 0.10 0.15 0.20

SLIDE 90

Figure 8: Posterior estimate of the 0.5 quantile of the space- time GEV distribution for a 50 × 50 resolution grid.

−99.3 −99.2 −99.1 −99.0 −98.9 19.25 19.35 19.45 19.55

day 47