L ászló Márkus and Péter Elek Dept. Probability Theory and Statistics Eötvös Loránd University

1. Uni- and multivariate modelling of runoffs and their effect on the extremes László Márkus and Péter Elek Dept. Probability Theory and Statistics Eötvös Loránd University Budapest, Hungary

2. River Tisza and its aquifer

3. Water discharge at Vásárosnamény(We have 5 more monitoring sites)from1901-2000

4. Empirical and smoothed seasonal components

5. Autocorrelation function is slowly decaying

6. Indicators of long memory • Nonparametric statistics • Rescaled adjusted range or R/S • Classical • Lo’s (test) • Taqqu’s graphical (robust) • Variance plot • Log-periodogram (Geweke-Porter Hudak)

9. Linear long-memory model : fractional ARIMA-process(Montanari et al., Lago Maggiore, 1997) • Fractional ARIMA-model: • Fitting is done by Whittle-estimator: • based on the empirical and theoretical periodogram • quite robust: consistent and asymptotically normal for linear processes driven by innovatons with finite forth moments (Giraitis and Surgailis, 1990)

10. Results of fractional ARIMAfit • H=0.846 (standard error: 0.014) • p-value: 0.558 (indicates goodness of fit) • Innovations can be reconstructed using a linear filter (the inverse of the filter above)

11. Reconstruct the innovation from the fitted model

12. Reconstructed innovations are uncorrelated... But not independent

13. Simulations using i.i.d. innovations • If we assume that innovations are i.i.d, we can generate synthetic series: • Use resampling to generate synthetic innovations • Apply then the linear filter • Add the sesonal components to get a synthetic streamflow series • But: these series do not approximate well the high quantiles of the original series

14. But: they fail to catch the densities and underestimate the high quantiles of the original series

15. Innovations can be regarded as shocks to the linear system • Few properties: • Squared and absolute values are autocorrelated • Skewed and peaked marginal distribution • There are periods of high and low variance • All these point to a GARCH-type model • The classical GARCH is far too heavy tailed to our purposes

16. Simulations: Generate i.i.d. series from the estimated GARCH-residuals Then simulate the GARCH(1,1) process using these residuals Apply the linear filter and the seasonalities The simulated series are much heavier-tailed than the original series Simulation from the GARCH-process

17. A smooth transition GARCH-model

18. ACF of GARCH-residuals

19. Results of simulationsat Vásárosnamény

20. Back to the original GARCH philosophy • The above described GARCH model is somewhat artificial, and hard to find heuristic explanations for it: • why does the conditional variance depend on the innovations of the linear filter? • in the original GARCH-context the variance is dependent on the lagged values of the process itself. • Possible solution: condition the variance on the lagged discharge process instead ! • Theoretical problems (e.g. on stationarity) arise but heuristically clear explanation canbe given more easily

21. Estimated variance of innovations plotted against the lagged discharge • Spectacularly linear relationship • Distorted at sites with damming (lower row) • This motivates the next modelling attempt

22. The variance is not conditional on the lagged innovation but it is conditional on the lagged water discharge. Estimation is carried out by normal-based maximum likelihood. (This is not uncommon in the GARCH-context, even if the residuals are non-Gaussian. See McNeil and Frey, 2000)

23. How to simulate the residuals of the new GARCH-typemodel • Residuals are highly skewed and peaked. • Simulation: • Use resampling to simulate from the central quantiles of the distribution • Use Generalized Pareto distribution to simulate from upper and lower quantiles • Use periodic monthly densities

24. The simulation process resampling and GPD Zt GARCH-type model t FARIMA filter Xt Seasonal filter

25. Evaluating the model fit • Independence of residual series ACF, extremal clustering • Fit of probability density and high quantiles • Variance – lagged discharge relationship • Extremal index • Consistence of parameter estimates

26. ACF of original and squared innovation series – residual series

27. Results of new simulationsat Vásárosnamény

28. Densities and quantiles at all 6 locations

29. The seasonal appearance of the highest values (upper 1%) of the simulated processes follows closely the same for the observed one. Seasonalities of extremes

30. Estimated extremal indices displayed

31. Multivariate modelling • Final aim: to model the runoff processes simultaneously • Nonlinear interdependence and non-Gaussianity should be addressed here, too • First, the joint behaviour of the discharges inflowing into Hungary should be modelled • Differential equation-oriented models of conventional hydrology may be used to describedownstream evolution of runoffs • Now we concentrate on joint modelling of two rivers: Tisza (at Tivadar) and Szamos (at Csenger)

32. Issues of joint modelling • Measures of linear interdependences (the cross-correlations) are likely to be insufficient. • High runoffs appear to be more synchronized on the two rivers than small ones • The reason may be the common generating weather patterns for high flows • This requires a non-conventional analysis of the dependence structure of the observed series

33. Basic statistics of Tivadar (Tisza) and Csenger (Szamos) • The model described previously was applied to both rivers • Correlations between the series of raw values, innovations and residuals are highest when either series at Tivadar are lagged by one day • Correlations: • Raw discharges: 0.79 • Deseasonalized data: 0.77 • Innovations: 0.40 • Residuals: 0.48 • Conditional variances: 0.84

34. Displaying the nature of interdependence • The joint plot may not be informative because of the highly non-Gaussian distributions • Transform the marginals into uniform distributions (produce the so-called copula), • then the scatterplot is more informative on the joint behaviour • The strange behaviour of the copula of the innovations is characterized by the concentration of points • 1. at the main diagonal, especially at the upper right corner (tail dependence) • 2. at the upper left (and the lower right) corner(s) • Taking into account these properties is crucial during joint simulation • The GARCH-residuals lack the second type of irregularity 2 1

35. A possible explanation of this type of interdependence • The cond. variance process is essentially common for the two rivers (correlation = 0.84) • This gives a hint to explain the interdependence of the innovations: • Generate two interdependent residual series (correlation=0.48) • Multiply by a common standard deviation process(distributed as Gamma) • The obtained copula is very similar to the observed copula of the innovations • This justifies the hypothesis that the common variance causes the interdependence of the given type

36. Thank you for your attention!