THE STATISTICAL ANALYSIS OF WET AND DRY SPELLS BY BINARY DARMA 1,1 MODEL IN SPLIT,CROATIA

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THE STATISTICAL ANALYSIS OF WET AND DRY SPELLS
BYBINARY DARMA (1,1) MODEL IN SPLIT,CROATIA

• Ksenija Cindric
• Meteorological and Hydrological Service of
  Croatia
• BALWOIS 2006 Conference, Ohrid

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• 1. Introduction
• 2. The binary DARMA(1,1) model
• Some properties of the DAR(1) and DARMA(1,1)
  models
• Estimation of parametres
• The distribution of run lengths
• 3. Wet and dry sequences in Split
• 4. Conclusions

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1. Introduction

• extreme precipitation events - floods and
  droughts are of the vital interest
• great spatial and temporal variability of the
  precipitation amount - precipitation analysis
  complicated (large number of statistical values)
• daily precipitation amount data vs. wet and dry
  spells in estimating dry(wet)ness of particular
  month
• DAR(1) (First-order Markov chain) and DARMA(1,1)
  models are used and compared
• autocorrelation coefficient (acc) - annual
  course, Split Marjan,1948-2000 period

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• ? 4331
• 1626'
• h 122 m

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2. The binary DARMA(1,1) modelSome properties
of the DAR(1) and DARMA(1,1)

• special cases of the more general class of
  DARMA(p,q)
• DARMA sequence Xn is formed by a probabilistic
  linear combination of sequence Yn of i.i.d. rv
  in such a way that its marginal distribution is
  given by
• p(k) P(Yn k), k 0,1,
• DAR(1) r.v. is defined
• Xn
• rk r1k , k 1

An-1, w.p. r Yn, w.p. 1- r
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• DARMA(1,1) r.v. is defined
• Xn
• rk crk-1 , k 1
• - first-order acc c (1-b) (r b -2 r b)
• acf of the daily precipitation sequence - measure
  of persistence - one of the most important
  statistical property when considering dry and wet
  spells length
• c can be taken as a model parameter rather
  than b.

Yn, w.p. b An-1, w.p. 1- b
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Estimation of parametres

• empirical distribution of run lengths of dry and
  wet spells gt mean run lengths m0 and m1
• 
  

p(0)1-p1
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The distribution of run lengths

• probability distribution of the run lengths of
  zeros (T0)
• P(T0 n) PXk 0 for all k ? 1,n and Xn1
  1 ? X0 1, X1 0, n 1,2,
• P( X0 1, X1 0,, Xn 0)
  
• P(X0 1, X1 0,, Xn 0, Xn1 0) / P X0
  1, X1 0
• P(T1 n) analogous
• - transition probabilities are 2x2 matrices and
  can be obtained by models parameters, Chang et
  al.(1984)

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3. Wet and dry sequences in Split

• wet dry day
• Xn
• Example ...0 1 1 1 1 1 1 1 0...

1, if Rn 1 mm 0, if Rn lt 1 mm
May
June
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Table 1. Mean run lengths of wet and dry spells
and estimated parameters for the binary
DARMA(1,1) model. Split Marjan, 1948-2000
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Figure 1. Annual course of the first-order
autocorrelation coefficient (c) for Observatory
Split-Marjan, 1948-2000
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Figure 2. Empirical and theoretical cumulative
frequencies of dry spells in for I, III, VII and
XII. Observatory Split Marjan, 1948-2000
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4. Conclusions
Split-Marjan (1948-2000)

• acc one of the most important parameter of
  precipitation regime (measure of persistence)
• cold period warm period
• future work analyse spatial distribution of
  acc to the whole Croatian or even Balcanic
  territory

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Acknowledgements - wish to thank to Dr. J. Juras
for many valuable suggestions

• References
• Buishand, R.A., 1978 The binary DARMA(1,1)
  process as a model for wet and dry sequence.
  Dept. of Math., Agricultural Univrsity,
  Wageningen, 49 pp.
• Chang, T. J., M. L. Kavvas and J. W. Delleur,
  1984a Daily precipitation modelling by discrete
  autoregressive moving average processes. Water
  Resour. Res., 20, 565 580.
• Chang, T. J., M. L. Kavvas and J. W. Delleur,
  1984b Modeling of sequences of wet and dry days
  by binary discrete autoregressive moving average
  processes. J. Clim. Appl. Meteor., 23, 1367
  1378.
• Gabriel, K. R., and J. Neumann, 1962 A Markov
  chain model for daily rainfall occurrence at Tel
  Aviv. Quart. J. Roy. Meteor. Soc., 88, 90 95.
• Juras, J. 1989 On modelling binary
  meteorological sequences with special emphasis on
  frequencies of warm and cold spells. Rasprave,
  24, 29 37.
• Juras, J. 1995 Methods for estimating the
  temporal variability of rainfall. Ph.D. Thesis,
  University of Zagreb, 160 pp.

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• The End
• Thank you for
• your attention