# State-space models for maxima precipitation

Journal de la société française de statistique (2007)

- Volume: 148, Issue: 1, page 107-120
- ISSN: 1962-5197

## Access Full Article

top## Abstract

top## How to cite

topNaveau, Philippe, and Poncet, Paul. "State-space models for maxima precipitation." Journal de la société française de statistique 148.1 (2007): 107-120. <http://eudml.org/doc/93452>.

@article{Naveau2007,

abstract = {A very active research field in atmospheric sciences is centered around the modeling of weather extremes. This is mainly due to the large economic and human impacts of such extreme events. In this paper, we focus on the statistical temporal modeling of precipitation maxima because daily and monthly maxima have been recorded for many decades and at various sites.
Our goal is to propose two new state-space models whose distributional foundations lie in Extreme Value Theory (EVT). Our first model takes advantage of max-stable processes, previously studied by Davis and Resnick (1989), among others. It can be viewed as a “translation“ of the gaussian linear Kalman filter into a Fréchet-type world in which the classical addition $a+b$ has been replaced by the max operator $a \vee b = \max (a,b)$ and the noise component is from a heavy-tailed distribution instead of being gaussian. Our second state-space model is built from the mixture extremes framework proposed by Fougères et al., (2006). Its strong points are its flexibility and richness with respect to applications. In addition to addressing the theoretical questions brought by our models, the main benefit of introducing them is to propose simple and powerful connections between EVT and data assimilation communities. The latter term “data assimilation” regroups statistical/dynamical techniques extensively used in climate studies. These procedures involve the combination of observational data with the underlying dynamical principles governing the physical system under observation. Hence, improving our knowledge about the representation of extremes in a state-space model framework is of strong interest from a data assimilation point of view.},

author = {Naveau, Philippe, Poncet, Paul},

journal = {Journal de la société française de statistique},

keywords = {data assimilation; Kalman filter; extreme value theory; generalized extreme value distribution; max-stable state-space model; GEV state-space model},

language = {eng},

number = {1},

pages = {107-120},

publisher = {Société française de statistique},

title = {State-space models for maxima precipitation},

url = {http://eudml.org/doc/93452},

volume = {148},

year = {2007},

}

TY - JOUR

AU - Naveau, Philippe

AU - Poncet, Paul

TI - State-space models for maxima precipitation

JO - Journal de la société française de statistique

PY - 2007

PB - Société française de statistique

VL - 148

IS - 1

SP - 107

EP - 120

AB - A very active research field in atmospheric sciences is centered around the modeling of weather extremes. This is mainly due to the large economic and human impacts of such extreme events. In this paper, we focus on the statistical temporal modeling of precipitation maxima because daily and monthly maxima have been recorded for many decades and at various sites.
Our goal is to propose two new state-space models whose distributional foundations lie in Extreme Value Theory (EVT). Our first model takes advantage of max-stable processes, previously studied by Davis and Resnick (1989), among others. It can be viewed as a “translation“ of the gaussian linear Kalman filter into a Fréchet-type world in which the classical addition $a+b$ has been replaced by the max operator $a \vee b = \max (a,b)$ and the noise component is from a heavy-tailed distribution instead of being gaussian. Our second state-space model is built from the mixture extremes framework proposed by Fougères et al., (2006). Its strong points are its flexibility and richness with respect to applications. In addition to addressing the theoretical questions brought by our models, the main benefit of introducing them is to propose simple and powerful connections between EVT and data assimilation communities. The latter term “data assimilation” regroups statistical/dynamical techniques extensively used in climate studies. These procedures involve the combination of observational data with the underlying dynamical principles governing the physical system under observation. Hence, improving our knowledge about the representation of extremes in a state-space model framework is of strong interest from a data assimilation point of view.

LA - eng

KW - data assimilation; Kalman filter; extreme value theory; generalized extreme value distribution; max-stable state-space model; GEV state-space model

UR - http://eudml.org/doc/93452

ER -

## References

top- [1] Buishand T.A. (1991) Extreme rainfall estimation by combining data from several sites. Hydrolog. Sci. J. 36(4):345-365.
- [2] Chevallier F., Lopez P., Tompkins A.M., Janiskov M., and Moreau E. (2004) The capability of 4D-Var systems to assimilate cloud-affected satellite infrared radiances. Quart. J. Roy. Meteor. Soc. 130:917-932.
- [3] Coles S.G. (2001) An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics. Springer-Verlag London Ltd., London. Zbl0980.62043MR1932132
- [4] Cooley D., Naveau P., Jomelli V., Rabatel A., and Grancher D. (2005) A bayesian hierarchical extreme value model for lichenometry. Environmetrics 16:1-20.
- [5] Davis R.A. and Resnick S.I. (1989) Basic Properties and Prediction of Max-Arma Processes. Adv. Appl. Probab. 21:781-803. Zbl0716.62098MR1039628
- [6] Dharssi I., Lorenc A.C., and Ingleby N.B. (1992) Treatment of gross errors using maximum probability theory. Quart. J. Roy. Meteor. Soc. 118(507), Part B:1017-1036(20).
- [7] Embrechts P., Klüppelberg C., and Mikosch T. (1997) Modelling Extremal Events for Insurance and Finance, Volume 33 of Applications of Mathematics. Springer-Verlag, Berlin. Zbl0873.62116MR1458613
- [8] Evensen G. (1994) Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 5:10143-10162.
- [9] Fisher R.A. and Tippett L.H.C. (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Cambridge. Philos. Soc. 24:180-190. Zbl54.0560.05JFM54.0560.05
- [10] Fougères A.L., Nolan J.P., and Rootzén H. (2006) Mixture Models for Extremes. Submitted..
- [11] Guo W., Wang Y., and Brown M. (1999) A signal extraction approach to modeling hormone time series with pulses and a changing baseline. J. Amer. Stat. Assoc. 94:746-756.
- [12] Helland I.S. and Nilsen T.S. (1976) On a General Random Exchange Model. J. Appl. Probab. 13(4):781-790. Zbl0349.60066MR431437
- [13] Hosking J.R.M., Wallis J.R., and Wood E.F. (1985) Estimation of the Generalized Extreme-Value Distribution by the Method of Probability-Weighted Moments. Technometrics, 27(3):251-261. MR797563
- [14] Katz R.W., Parlange M.B., and Naveau P. (2002) Statistics of extremes in hydrology. Adv. Water Resour. 25:1287-1304.
- [15] Kharin V. and Zwiers F. (2000) Changes in the extremes in an ensemble of transient climate simulations with a coupled atmosphere-ocean gcm. J. Climate 13:3760-3788.
- [16] Koutsoyiannis D. and Baloutsos G. (2000) Analysis of a Long Record of Annual Maximum Rainfall in Athens, Greece, and Design Rainfall Inferences. Natural Hazards 22(1):31-51.
- [17] Naveau P., Genton M.G., and Shen X. (2005) A skewed Kalman filter. J. Multiv. Anal. 94(2):382-400. Zbl1066.62091MR2167921
- [18] Rodgers C.D. (2000) Inverse methods for atmospheric sounding. Theory and practice. Series on Atmospheric, Oceanic and Planetery Physics, Volume 2. World Scientific. Singapore-New Jersey-London-Hong-Kong. Zbl0962.86002MR1788828
- [19] Shephard N. (1994) Partially Non-Gaussian State-space Models. Biometrika 81:115-131. Zbl0796.62079MR1279661
- [20] Stuck B.W. (1977) Minimum Error Dispersion Linear Filtering of Scalar Symmetric Stable Processes. IEEE Trans. Automat. Contr. AC-23(3):507-509. Zbl0377.93051
- [21] Tawn J. (1990) Modelling multivariate extreme value distributions. Biometrika 77:245-253. Zbl0716.62051
- [22] West M. and Harrison J. (1997) Bayesian forecasting and dynamic models. Springer, New York. Zbl0871.62026MR1482232
- [23] Wilson P.S. and Toumi R. (2005) A fundamental probability distribution for heavy rainfall. Geophys. Res. Lett. 32(14).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.