Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying...

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Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics, University of New Mexico, Albuquerque, U.S.A. , Bruno Sans ´ o Department of Applied Mathematics and Statistics, University of California, Santa Cruz, U.S.A. , Summary. We propose a new approach for modeling extreme values that are measured in time and space. First we assume that the observations follow a Generalized Extreme Value (GEV) distribution for which the location, scale or shape parameters define the space-time structure. The temporal component is defined through a Dy- namic Linear Model (DLM) or state space representation that allows to estimate the trend or seasonality of the data in time. The spatial element is imposed through the evolution matrix of the DLM where we adopt a process convolution form. We show how to produce temporal and spatial estimates of our model via customized Markov Chain Monte Carlo (MCMC) simulation. We illustrate our methodology with extreme values of ozone levels pro- duced daily in the metropolitan area of Mexico City and with rainfall extremes measured at the Caribbean coast of Venezuela. Some key words: Spatio-temporal process, Extreme values, GEV distribution, Process convolutions, MCMC, ozone levels. 1. Introduction

Transcript of Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying...

Page 1: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

Time-Varying Models for Extreme Values

Gabriel Huerta

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, U.S.A.��������������� ���������� ��������� , ����� ��� ���������� ��������������������������Bruno Sanso

Department of Applied Mathematics and Statistics, University of California, Santa Cruz, U.S.A.� ����������� �!���#"���"!�$����� , ����� �%�� �!���#"���"!�$��������� � �������Summary. We propose a new approach for modeling extreme values that are measured in time and space. Firstwe assume that the observations follow a Generalized Extreme Value (GEV) distribution for which the location,scale or shape parameters define the space-time structure. The temporal component is defined through a Dy-namic Linear Model (DLM) or state space representation that allows to estimate the trend or seasonality of thedata in time. The spatial element is imposed through the evolution matrix of the DLM where we adopt a processconvolution form. We show how to produce temporal and spatial estimates of our model via customized MarkovChain Monte Carlo (MCMC) simulation. We illustrate our methodology with extreme values of ozone levels pro-duced daily in the metropolitan area of Mexico City and with rainfall extremes measured at the Caribbean coastof Venezuela.Some key words: Spatio-temporal process, Extreme values, GEV distribution, Process convolutions, MCMC,ozone levels.

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2 G. Huerta and B. Sanso

Time

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Fig. 1. Daily maxima of ozone concentration at station Merced in Mexico City from 1990 to 2002.

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Page 3: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

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Page 4: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

4 G. Huerta and B. Sanso

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Page 5: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

Time-Varying Models for Extreme Values 5

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Page 6: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

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Page 7: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

Time-Varying Models for Extreme Values 7

β

Den

sity

−4e−05 −2e−05 0e+00 2e−05

010

000

2000

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Fig. 5. Posterior distribution of�

for ozone data.

(a)

Time

1960 1970 1980 1990 2000

040

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0

rainfallµt

(b)

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NAO index

1960 1970 1980 1990 2000

−4

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02

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Fig. 6. (a) Maiquetıa monthly observations from 1961 and posterior mean of ��� . (b) Monthly values of NAO index from1961.

Page 8: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

8 G. Huerta and B. Sanso

time

β t

1960 1970 1980 1990

0.0

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Page 10: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

10 G. Huerta and B. Sanso

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Fig. 8. RAMA stations, kernel and interpolation grid positions. The stations are denoted with a three letters code. Thekernel centers are denoted with ��� . The grid points correspond to the locations used to obtain the values shown inFigure 11. The continuous line corresponds to the boundaries of Mexico City.

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Page 11: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

Time-Varying Models for Extreme Values 11

AZC

time

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time

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050.

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Fig. 9. Daily maxima for 1999 (dots) and posterior estimate of the �� � quantile of the GEV distribution at each timepoint. (continuous line). The quantile is based in a leave-one-out analysis. The chosen stations correspond to locationsin the NE, NW, SE and SW respectively, as shown in Figure 8.

AZC

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ple

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s

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Theoretical Quantiles

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ple

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Theoretical Quantiles

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Theoretical Quantiles

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ple

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Fig. 10.� � diagnostics for four of the stations using the posterior predictions based on leaving one station out at a time.

Page 12: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

12 G. Huerta and B. Sanso

−99.3 −99.2 −99.1 −99.0 −98.9

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day 63

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day 69

LAGTAC

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MER

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day 127

LAGTAC

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day 191

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day 263

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Fig. 11. Posterior estimate of the �� quantile of the space-time GEV distribution for a � � � resolution grid. Dotscorrespond to the locations of the stations. The boundary of Mexico City is shown as a reference. The contour linescorrespond to fixed ozone levels in ppm.

Page 13: Time-Varying Models for Extreme Valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · Time-Varying Models for Extreme Values Gabriel Huerta Department of Mathematics and Statistics,

Time-Varying Models for Extreme Values 13

u

χ(u)

TACXALTPNTAH

0.0 0.2 0.4 0.6 0.8 1.0

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