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T HE STATISTICAL ANALYSIS OF WET AND DRY SPELLS BY BINARY DARMA (1,1) MODEL IN S PLIT, C ROATIA

T HE STATISTICAL ANALYSIS OF WET AND DRY SPELLS BY BINARY DARMA (1,1) MODEL IN S PLIT, C ROATIA. Ksenija Cindrić Meteorological and Hydrological Service of Croatia BALWOIS 2006 Conference, Ohrid. 1. Introduction 2. The binary DARMA(1,1) model

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T HE STATISTICAL ANALYSIS OF WET AND DRY SPELLS BY BINARY DARMA (1,1) MODEL IN S PLIT, C ROATIA

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  1. THE STATISTICAL ANALYSIS OF WET AND DRY SPELLS BYBINARY DARMA (1,1) MODEL IN SPLIT,CROATIA Ksenija Cindrić Meteorological and Hydrological Service of Croatia BALWOIS 2006 Conference, Ohrid

  2. 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

  3. 1. Introduction • extreme precipitation events - floods and droughtsare of the vital interest • great spatial and temporal variability of the precipitation amount - precipitationanalysiscomplicated(large number of statistical values) • daily precipitation amount datavs. 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

  4.  = 43°31‘ • = 16°26' h = 122 m

  5. 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

  6. Yn, w.p. b An-1, w.p. 1- b • 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.

  7. Estimation of parametres • empirical distribution of run lengths of dry and wet spells=> mean run lengths m0 and m1 p(0)=1-p1

  8. The distribution of run lengths • probability distribution of the run lengths of zeros (T0): P(T0 = n) = P{Xk = 0 for all k  [1,n] and Xn+1 = 1  X0 = 1, X1 = 0}, n = 1,2,… = {P( X0 = 1, X1 = 0,…, Xn = 0) – P(X0 = 1, X1 = 0,…, Xn = 0, Xn+1 = 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)

  9. 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 <1 mm May June

  10. Table 1.Mean run lengths of wet and dry spells and estimated parameters for the binary DARMA(1,1) model. Split – Marjan, 1948-2000

  11. Figure1. Annual course of the first-order autocorrelation coefficient (c) for Observatory Split-Marjan, 1948-2000

  12. Figure 2. Empirical and theoreticalcumulative frequencies of dry spells in for I, III, VII and XII. Observatory Split – Marjan, 1948-2000

  13. 4. Conclusions Split-Marjan (1948-2000) • acc one of the most important parameter of precipitation regime (measure of persistence) cold periodwarm period future work analyse spatial distribution of acc to the whole Croatian or even Balcanic territory

  14. 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.

  15. The End Thank you for your attention

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