time-varying models for extreme valuesghuerta/papers/dyn_extrm.pdf · 2005-09-19 · time-varying...
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![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,](https://reader034.vdocument.in/reader034/viewer/2022050505/5f96a63ce0efad0e205b0ace/html5/thumbnails/1.jpg)
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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1990 1992 1994 1996 1998 2000 2002
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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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4 G. Huerta and B. Sanso
Time
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Time-Varying Models for Extreme Values 5
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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
8012
0
rainfallµt
(b)
Time
NAO index
1960 1970 1980 1990 2000
−4
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02
4
Fig. 6. (a) Maiquetıa monthly observations from 1961 and posterior mean of ��� . (b) Monthly values of NAO index from1961.
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8 G. Huerta and B. Sanso
time
β t
1960 1970 1980 1990
0.0
0.5
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Posterior median90% probability
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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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Time-Varying Models for Extreme Values 11
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12 G. Huerta and B. Sanso
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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.
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Time-Varying Models for Extreme Values 13
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