Ismaila Ba MSc Student, Department of Mathematics and Statistics Université de Moncton

Transcription

1 Discrimination between statistical distributions for hydrometeorological frequency modeling Ismaila Ba MSc Student, Department of Mathematics and Statistics Université de Moncton

2 INTRODUCTION The identification of a statistical distribution to model the frequency of occurrence of extreme hydro-meteorological events is important. Two-parameter distributions such as the Generalized Pareto, lognormal, gamma or Weibull are useful in fitting datasets in areas such as POT extreme value modeling. Three-parameter distributions are also very important, such as for fitting annual maximum flood or precipitation series. 2

3 INTRODUCTION We recommend some methods of discrimination between distributions. The discriminations considered are between: Generalized Pareto (GP) and Kappa, Gumbel and some alternative frequency models, and model pairs belonging to the group {generalized extreme value (GEV), Pearson type (P), generalized logistic (GLO)}. Four discrimination methods are compared by Monte Carlo Simulation in terms of their discrimination power and discriminaton bias.

4 DISCRIMINATION STATISTICS Anderson-Darling statistic (AD) 2i - ( ) ˆ ( ) Gives more weight to observations in the tails of the distribution as compared to Cramér-von Mises (CvM) and Kolmogorov-Smirnov (KS) statistics. Ratio of maximized likelihood statistic (RML) Most widely investigated method. { } ˆ ( ) ( ( ) ) n n i n+-i i= A = A X = -n - lnf X +ln - F X n ( ) T = T X n ( ˆ n θ0 ) ( ˆ n θ ) LX; = ln LX; 4

5 DISCRIMINATION STATISTICS Transformation to normality followed by the application of Shapiro-Wilk GoF statistic () - Z= φ FX ˆ ( ) i ( i ) 2 n n S= vz i ( ) / Z i ( i) -Z i= i= Transformation to normality followed by the application of the Probability plot correlation coefficient statistic () Z W = Φ * i = Φ * i ( Fˆ ( Xi ) ) ( p ) i 5

6 DISCRIMINATION STATISTICS The statistic is then calculated as follows: R * * * * ( Z( ) Z )( Wi W i ) * * * * ( Z( i) Z ) ( Wi W ) * = Note: When the two frequency models have the same number of unknown parameters, applying RML is equivalent to using AIC or BIC.

7 PARAMETER ESTIMATION METHODS In practice, the parameters of the model are unknown, so they need to be estimated from the data. We considered three parameter estimation methods: Maximum likelihood (ML) Moments (MOM) Probability weighted moments (PWM). 7

8 DISCRIMINATION STUDIES The discriminations considered are between: GP and KAP models Gumbel and some alternative frequency models Model pairs belonging to the group {GEV, P, GLO): GEV vs P, GEV vs GLO and P vs GLO. 8

9 GP AND KAP 00 AD Test Statistic PCS (%)_GP*shape parameter_gp 2 4 PCS (%)_KAP*shape parameter_kap 00 RML Test Statistic PCS (%)_GP*shape parameter_gp 2 4 PCS (%)_KAP*shape parameter_kap 00 Test Statistic PCS (%)_GP*shape parameter_gp 2 4 PCS (%)_KAP*shape parameter_kap n n n Fig. PCS(%) using the AD statistic when GP is the true sampled distribution (left) and when KAP is the true sampled distribution (right). Fig. 2 PCS (%) using the RML statistic when GP is the true sampled distribution (left) and when KAP is the true sampled distribution (right). Fig. PCS(%) using the statistic when GP is the true sampled distribution (left) and when KAP is the true sampled distribution (right).

10 GP AND KAP Application with Eight Hydrological Datasets Table. Selecting between the GP and KAP distribution for fitting POT flood data at eight hydrometric stations Station n GP Estimates (shape; scale) KAP Estimates (shape; scale) AD statistics (a GP ; a KAP ) statistics (s GP ; s KAP ) 0AQ00 5 (-0.25;.8) (.5; 8.8) (0.77; 0.2) (0.8; 0.4) KAP is selected 0BL002 4 (0.0;.72) (2.77; 2.8) (.0; 0.8) (0.; 0.82) KAP is selected 02FC002 5 (0.;.5) (2.; 0.4) (0.40;.08) (0.2; 0.8) 0BJ (-0.04; 40.8) (2.; 8.) (0.40; 0.) (0.8; 0.7) 0AF (0.2; 28.4) (4.00; 27.04) (0.; 0.24) (0.85; 0.0 ) KAP is selected 04CA002 2 (0.0; 57.4) (2.74; 5.2) (.00;.5) (0.2; 0.5) RML statistics (t GP ; t KAP ) (-.88;.88) KAP is selected (-2.0; 2.0) KAP is selected (.0; -.0) (.05; -.05) (-0.; 0.) KAP is selected (.00; -.00) 0 02LB (-0.4; 4) (.2;.) (0.2; 0.4) (0.77; 0.72) 0BJ00 5 (0.; 7.28) (.; 5.5) (0.27; 0.4) (0.85; 0.87) KAP is selected (0.8; -0.8) (0.52; -0.52)

11 Gumbel and some alternative frequency models The alternative models are: The normal: N ( ) The logistic: LOG μ,σ ( ) Two student s t models μ,σ to which a location parameter was added: STU ( ) Three models μ;ν from the -parameter gamma family: GAM ( ) Four models from the GEV family: ( μ,σ ) GEV μ,σ

12 Gumbel and some alternative frequency models Boxplot of PCS.mean Boxplot of PCS.abs.diff n = 0 A RML n = n = n = 0 B RML n = n = PCS.mean (%) 00 0 n = 0 n = n = 00 0 PCS.abs.diff (%) 0 n = 0 n = n = RML Discrimination statistic RML RML Discrimination statistic RML 2

13 GEV, P and GLO Discrimination between GEV and GLO DS DS PCS_mean (GEV_GLO) Increase in population skew ---> Theta_input vector # ( to 8) A PCS_abs.diff (GEV_GLO) Increase in population skew ---> Theta_input vector # ( to 8) B Panel variable: Sample size n Panel variable: Sample size n Fig. 5 PCS means for comparing and Fig. PCS absolute difference for comparing and

14 GEV, P and GLO Discrimination between P and GLO DS TN SW DS PCS_mean (P_GLO) PCS_abs.diff (P_GLO) 0 00 Increase in population skew ---> Increase in population skew ---> 2 Panel variable: Sample size n 5 8 Theta_input vector # ( to 8) 4A Panel variable: Sample size n 5 8 Theta_input vector # ( to 8) 4B Fig. 7 PCS means for comparing and 4 Fig. 8 PCS absolute difference for comparing and

15 GEV, P and GLO Discrimination between GEV and P DS DS PCS_mean (GEV_P) Panel variable: Sample size n Increase in Population skew ---> 5 8 Theta_input vector # ( to 8) A PCS_abs.diff (GEV_P) 00 0 Increase in population skew ---> Theta_input vector # ( to 8) Panel variable: Sample size n B Fig. PCS means for comparing and 5 Fig. 0 PCS absolute difference for comparing and

16 GEV, P and GLO Table 2. Use of the statistic s and the statistic r * to choose between the GEV, GLO and P models for the 8 data series ID Sample size s.gev s.p s.glo r*.gev r*.p r*.glo Chosen.Model # GEV # GEV # P # P # * # P # P # P # GLO # P # GLO # * # GEV # NA NA 0.88 GEV # P # GLO # NA NA GEV # NA NA GEV

17 CONCLUSIONS To discriminate between the KAP and GP models, use of the AD statistic leads to bias for one model over the other. The use of RML for discriminating between three-parameter distributions led to some serious numerical problems. The and statistics proved to be the most advantageous among those considered, they would be recommendable in practice for this reason. We found a difficulty in discriminating between the P and GEV models. 7