GENERALIZED LINEAR MODELING APPROACH TO STOCHASTIC WEATHER GENERATORS

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1 GENERALIZED LINEAR MODELING APPROACH TO STOCHASTIC WEATHER GENERATORS Rick Katz Institute for Study of Society and Environment National Center for Atmospheric Research Boulder, CO USA Joint work with Eva Furrer (NCAR/IMAGE) Talk: Paper (Climate Research, 2007):

2 QUOTE Jean-Baptiste Lamarck (French biologist, ~ 1800)... l efficacité de toute influence que l atmosphère reçoit, est constamment en raison de l état préexistant des choses dans cette atmosphère...

3 OUTLINE (1) Stochastic Weather Generators (2) Generalized Linear Models (GLMs) (3) GLM Weather Generator (4) Application to Daily Weather at Pergamino (5) Extensions (6) Resources

4 (1) Stochastic Weather Generators Historical Perspective -- Precipitation occurrence (Quetelet 1850s; Gabriel & Neumann 1962) Markov chain model (Two-state, First-order) -- Chain-dependent process (Todorovic & Woolhiser 1975; Katz 1977) Retain Markov chain model for precipitation occurrence Precipitation intensity Conditionally independent & identically distributed (e. g., Gamma distribution)

5 Richardson Model (e. g., Richardson 1981; WGEN) -- Precipitation as chain-dependent process -- Minimum & maximum temperature Bivariate first-order autoregressive [AR(1)] process -- Dependence between precipitation & temperatures Conditional means of minimum & maximum temperature depend on whether or not precipitation occurs

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7 Current Issues -- Parameter estimation (Software) -- Covariates (Downscaling) -- Extremes (Improved treatment) -- Uncertainty analysis -- Multisite

8 Alternative Approach (Resampling) -- Use resampling scheme (such as bootstrap ) to draw new weather data from observations -- Advantages Nonparametric (Capable of reproducing any desired statistic) -- Disadvantages Synthetic weather looks too much like observations Unrealistic properties for extremes Difficult to incorporate predictors ( Covariates ) Not amenable to uncertainty analysis

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11 (2) Generalized Linear Models Statistical Framework -- Multiple Regression Analysis (Linear model or LM) Response Y has normal distribution with conditional mean expressed as linear function of covariates X 1, X 2,..., X p E(Y X 1, X 2,..., X p ) = β 0 + β 1 X β p X p Var(Y X 1, X 2,..., X p ) constant Parameter estimation by ordinary least squares (Equivalent to maximum likelihood estimation)

12 -- GLMs Response Y can have non-normal distribution Skewed distributions (e. g., Gamma) Discrete distributions (e. g., Binomial) Mean of distribution E(Y X 1, X 2,..., X p ) linear function of covariates (Or nonlinear transformation of mean, such as logarithm, to preserve range) Var(Y X 1, X 2,..., X p ) no longer necessarily constant Estimate parameters by iterative weighted least squares (Equivalent to maximum likelihood estimation)

13 Application to Weather Variables -- Stern & Coe (1984) Daily precipitation with marked wet season (e. g., tropics) Fit entire model via GLM Markov chain model for daily precipitation occurrence: J t = 1 if tth day wet, J t = 0 if tth day dry Transition probs. P ij = Pr{J t = j J t 1 = i }, 0 < P ij < 1; i, j = 0, 1 Logistic model for annual cycle (T 365 days) ln[p i1 / (1 P i1 )] = β i0 + β i1 cos(2πt /T) + β i2 sin(2πt /T), i = 0, 1

14 Gamma distribution for daily precipitation intensity: Shape parameter α > 0 Scale parameter β > 0 Mean μ = α β > 0 Logarithmic model for annual cycle ln μ = γ 0 + γ 1 cos(2πt /T) + γ 2 sin(2πt /T) Really model for ln β Constraining coefficient of variation 1/α 1/2 (or α) to be constant

15 -- Chandler et al. (2002, 2005) More recently: Revisited Stern & Coe approach Geophysical covariates (e.g., NAO) Other weather variables (wind speed, temperature) Multisites (Spatial dependence of daily weather) -- Software R open source statistical programming language: Function glm Family of distributions (e. g., Binomial, Gamma) Link function (e. g., logistic, logarithm) -- Model selection: Use Likelihood Ratio Test, AIC, or BIC

16 (3) GLM Weather Generator Precipitation -- Essentially as in Stern & Coe (1984) -- Probability of precipitation occurrence p t = Pr{J t = 1} Apply logistic transformation: ln[p t / (1 p t )] Use ENSO index (monthly mean) as covariate: Z t

17 ln[p t / (1 p t )] = β 0 + β 1 J t 1 (Markov dependence)

18 ln[p t / (1 p t )] = β 0 + β 1 J t 1 (Markov dependence) + β 2 cos(2πt /T) + β 3 sin(2πt /T) (Single annual cycle)

19 ln[p t / (1 p t )] = β 0 + β 1 J t 1 (Markov dependence) + β 2 cos(2πt /T) + β 3 sin(2πt /T) (Single annual cycle) + β 4 Z t (Single ENSO effect)

20 ln[p t / (1 p t )] = β 0 + β 1 J t 1 (Markov dependence) + β 2 cos(2πt /T) + β 3 sin(2πt /T) (Single annual cycle) + β 4 Z t (Single ENSO effect) + β 5 J t 1 cos(2πt /T) + β 6 J t 1 sin(2πt /T) (Two annual cycles)

21 ln[p t / (1 p t )] = β 0 + β 1 J t 1 (Markov dependence) + β 2 cos(2πt /T) + β 3 sin(2πt /T) (Single annual cycle) + β 4 Z t (Single ENSO effect) + β 5 J t 1 cos(2πt /T) + β 6 J t 1 sin(2πt /T) (Two annual cycles) + β 7 J t 1 Z t (Different ENSO effect)

22 -- Precipitation intensity Gamma distribution: Annual cycle in mean (log transform) Dependence of mean (log transform) on ENSO index ln μ = γ 0 (Single gamma dist.) + γ 1 cos(2πt /T) + γ 2 sin(2πt /T) (Annual cycle) + γ 3 Z t (ENSO effect)

23 Daily Minimum X t & Maximum Y t Temperature -- Coupled univariate models Essentially equivalent to bivariate AR(1) X t = α 0 + α 1 J t + α 2 X t 1 + α 3 Y t 1 + α 4 cos(2πt /T) + α 5 sin(2πt /T) + α 6 Z t + ε t (X) Y t = λ 0 + λ 1 J t + λ 2 X t + λ 3 Y t 1 + λ 4 cos(2πt /T) + λ 5 sin(2πt /T) + λ 6 Z t + ε t (Y) Error terms with normal distributions & Corr[ε t (X), ε t (Y)] = 0 Parameter estimation: Could use R function lm instead of glm

24 (4) Application to Daily Weather at Pergamino Pampas Region of Argentina -- Marked wet season Southern Hemisphere summer -- ENSO teleconnections Wetter than normal during El Niño events Smaller temperature range during El Niño events (i. e., higher minimum & lower maximum temperature)

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26 Transition probability p 11 (t) Day of the year ENSO

27 Transition probability p 01 (t) Day of the year ENSO

28 Mean maximum temperature Day of the year ENSO

29 Mean minimum temperature Day of the year ENSO

30 (5) Extensions Weather Variability -- Annual cycles GLM imposes possibly unrealistic constraints (e. g., assumes variance & autocorrelation of minimum & maximum temperature constant) Test for winter / summer differences (via GLM with interaction terms)

31 Extremes -- More realistic treatment High precipitation amounts (Evidence of apparent heavy or Pareto upper tail) Issue of parsimony (Not parsimonious to directly add Pareto tail) Stretched exponential (or Weibull) distribution Pre-asymptotic extreme value theory (Asymptotic light tail, but pre-asymptotic heavy tail)

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36 Coupled modeling approach -- Not feasible with more than two variables (Besides precipitation) -- Alternative Multivariate AR(1) process with covariates

37 (6) Resources R (Open source statistical programming language) -- GLM Weather Generator -- Statistics of Weather and Climate Extremes -- Extremes Toolkit (extremes) --