Log-Scaling rainfall data: effects on GPD Bayesian goodness of fit. - PowerPoint PPT Presentation

Motivation Model Estimation Model Checking Conclussions Log-Scaling rainfall data: effects on GPD Bayesian goodness of fit. M.I. Ortego J.J. Egozcue Departament de Matemàtica Aplicada III E.T.S. Enginyeria Camins Canals Ports Barcelona (Civil Engineering) Universitat Politècnica de Catalunya 4th conference on Extreme Value Analysis. Gothenburg, 15-19 August 2005 M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Model Estimation Model Checking Conclussions Outline Motivation 1 Rainfall data Problems with model adequacy p -values Model Estimation 2 Bayesian Generalized Pareto Estimation (BGPE) Priors and posteriors Model Checking 3 GPD goodness-of-fit Whole model Conclussions 4 M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions Vergel de Racons data. Vergel de Recons 350 mm daily-precipitation 300 250 200 150 100 50 0 1964 1969 1974 1979 1984 1989 1994 time (years) Main goals: • Finding suitable model. • Hazard analysis. • Occurrence probabilities; return periods. For reference, see Romero et al. (1998), [4] M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions The model Scale of the reference variable, precipitation: ⋄ is a positive variable: (0 mm rainfall is not rainfall!) ⋄ has a relative scale: 50 mm is double than 25mm daily rainfall, but 500mm and 525 mm daily rainfall is nearly the same! Logarithmic scale is needed!!! M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions The model Occurrence: Cramér-Lundberg model (Homogeneous Poisson process with intensity parameter λ ). Magnitude: Excesses over threshold described by a Generalized Pareto Distribution (GPD). Bayesian parameter estimation. M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions Is it a suitable model? Hazard Estimates: At high levels, great uncertainty of estimates due to scarcity of data. Estimates of typical hazard parameters (e.g. Return period) vary dramatically depending on the selected model: Distribution of log10(return period) Distribution of log10(return period) 4.0 7.00 3.0 log10(return period) log10(return period) 5.00 2.0 3.00 1.0 1.00 0.0 50.0 150.0 250.0 350.0 450.0 550.0 650.0 750.0 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 -1.0 -1.00 mm mm Quant0.05 Quant0.1 Quant0.25 Quant0.5 Quant0.75 Quant0.9 Quant0.95 Quant0.05 Quant0.1 Quant0.25 Quant0.5 Quant0.75 Quant0.9 Quant0.95 Return period of raw data Return period of log data C M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions Return period of raw data Distribution of log10(return period) 4.0 3.0 log10(return period) 2.0 1.0 0.0 50.0 150.0 250.0 350.0 450.0 550.0 650.0 750.0 -1.0 mm Quant0.05 Quant0.1 Quant0.25 Quant0.5 Quant0.75 Quant0.9 Quant0.95 Go back M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions Return period of log data Distribution of log10(return period) 7.00 log10(return period) 5.00 3.00 1.00 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 -1.00 mm Quant0.05 Quant0.1 Quant0.25 Quant0.5 Quant0.75 Quant0.9 Quant0.95 Go back M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions p -values Whole model checking (prior +likelihood +GPD) For reference, see Gelman et al. 1995, 1996, [6] Several ways of checking it. Goodness-of-fit checking GPD ( ξ, β ) goodness-of-fit assessing. Several ways of checking it M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions p -values A first approach: plug-in-p-value ( p plug ) p plug = P gpd ( ·| ❜ ξ, ❜ β ) [ t ( X ) ≥ t ( x obs )] , gpd ( x | ξ, β ) is replaced by gpd ( ·| � ξ, � β ) , where � ξ, � β is the maximum likelihood estimate of the parameters. M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions p -values Bayesian p -values: posterior predictive p-value ( p post ) Guttman (1967) and Rubin (1984) p post = P m post ( ·| x obs ) [ t ( X ) ≥ t ( x obs )] , where m post ( x | x obs ) is the posterior predictive distribution , � m post ( x | x obs ) = gpd ( x | ξ, β ) π ( ξ, β | x obs ) d ( ξ, β ) , and π ( ξ, β | x obs ) is the posterior density for ξ, β . M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions p -values discrepancy p-value ( p dis ) (Gelman et al., 1995) The test statistic t ( X ) is replaced by a discrepancy t ( X , ξ, β ) p dis = P m dis ( · ) [ t ( X , ξ, β ) ≥ t ( x obs , ξ, β )] , where m dis ( x , ξ, β | x obs ) is , m dis ( x , ξ, β | x obs ) = gpd ( x | ξ, β ) π ( ξ, β | x obs ) , and π ( ξ, β | x obs ) is the posterior density for ξ, β . M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Rainfall data Model Estimation Problems with model adequacy Model Checking p -values Conclussions p -values: pros and cons Desirable characteristics: • Uniform distribution. • Easy to compute. Other useful characteristics: • Known distribution of used statistic (even asymptotically). • Easiness of interpretation. Pros and cons: • plug-in p-value : Easy to compute. Uncertainty ignored. • posterior predictive p-value is not uniform. Easy to compute. • discrepancy p-value is not uniform. M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Model Estimation Bayesian Generalized Pareto Estimation (BGPE) Model Checking Priors and posteriors Conclussions Bayesian GP Estimation (BGPE) Three parameters to estimate in the model: Poisson rate, λ , of Poisson ( λ ) and ξ, β of the magnitude, modelled by GPD ( ξ, β ) : � � − 1 1 + ξ ξ GPD X ( x | ξ, β ) = 1 − β x A suitable joint prior distribution for λ, ξ, β is set. Prior distributions for λ and ξ, β are independent → the joint prior factorizes: π λ,ξ,β ( λ, ξ, β ) = π λ ( λ ) · π ξ,β ( ξ, β ) M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Model Estimation Bayesian Generalized Pareto Estimation (BGPE) Model Checking Priors and posteriors Conclussions The joint likelihood of parameters, L ( λ, ξ, β | x obs ) , splits into two terms: L ( λ, ξ, β | x obs ) = L ( λ | x obs ) · L ( ξ, β | x obs ) Finally, the Posterior distribution of λ, ξ, β , π λ,ξ,β ( λ, ξ, β | x obs ) , is obtained: π λ,ξ,β ( λ, ξ, β | x obs ) = L ( λ, ξ, β | x obs ) · π λ ( λ ) · π ξ,β ( ξ, β ) Attention is set to marginal posterior distribution of ξ, β : π ξ,β ( ξ, β | x obs ) = L ( ξ, β | x obs ) · π ξ,β ( ξ, β ) For reference, see Egozcue and Ramis (2001), [1], and Egozcue and Tolosana (2002), [2] . M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Model Estimation Bayesian Generalized Pareto Estimation (BGPE) Model Checking Priors and posteriors Conclussions Prior and posterior distributions : Raw data (I) Prior density Posterior density Something is lost!! M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Model Estimation Bayesian Generalized Pareto Estimation (BGPE) Model Checking Priors and posteriors Conclussions Prior and posterior distributions : Raw data (II) Prior density, ξ < 0 Posterior density, ξ < 0 M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Model Estimation Bayesian Generalized Pareto Estimation (BGPE) Model Checking Priors and posteriors Conclussions Prior and posterior distributions: log data Prior density Posterior density M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF

Motivation Model Estimation GPD goodness-of-fit Model Checking Whole model Conclussions p -values: Our alternative First approach � pp = ψ i p i , predictive KS p − value , where i p i = KSGOF ( ξ i , β i ) , for fixed ( ξ i , β i ) and ψ i = π ( ξ i , β i | x obs ) Our alternative   � n i = 1 ψ i Φ − 1 ( p i )   , 1 ≤ δ ≤ 2 , δ ≃ 1 . pp = Φ �� i ψ δ i p i = KSGOF ( ξ i , β i ) , for fixed ( ξ i , β i ) ; ψ i = π ( ξ i , β i | x obs ) M.I. Ortego, J.J.Egozcue Log-scalling rainfall data. Effects on GPD GOF