Vector Generalized Additive Models and applications to extreme value analysis Olivier Mestre (1,2) (1) Météo-France, Ecole Nationale de la Météorologie, Toulouse, France (2) Université Paul Sabatier, LSP, Toulouse, France Based on previous studies realized in collaboration with : Stéphane Hallegatte (CIRED, Météo-France) Sébastien Denvil (LMD)

SMOOTHER « Smoother=tool for summarizing the trend of a response measurement Y as a function of predictors » (Hastie & Tibshirani) estimate of the trend that is less variable than Y itself  Smoothing matrix S Y*=S  Y The equivalent degrees of freedom (df) of the smoother S is the trace of S. Allows compare with parametric models.  Pointwise standard error bands COV(Y*)=V=S t S  ²given an estimation of  ², this allows approximate confidence intervals (values : ±2  square root of the diagonal of V)

SCATTERPLOT SMOOTHING EXAMPLE  Data: wind farm production vs numerical windspeed forecasts

SMOOTHING  Problems raised by smoothers How to average the response values in each neighborhood? How large to take the neighborhoods?  Tradeoff between bias and variance of Y*

SMOOTHING: POLYNOMIAL (parametric)  Linear and cubic parametric least squares fits: MODEL DRIVEN APPROACHES

SMOOTHING: BIN SMOOTHER  In this example, optimum intervals are determined by means of a regression tree

SMOOTHING: RUNNING LINE  Running line

KERNEL SMOOTHER  Watson-Nadaraya

SMOOTHING: LOESS  The smooth at the target point is the fit of a locally-weighted linear fit (tricube weight)

CUBIC SMOOTHING SPLINES  This smoother is the solution of the following optimization problem: among all functions f(x) with two continuous derivatives, choose the one that minimizes the penalized sum of squares Closeness to the datapenalization of the curvature of f It can be shown that the unique solution to this problem is a natural cubic spline with knots at the unique values x i Parameter can be set by means of cross-validation

CUBIC SMOOTHING SPLINES  Cubic smoothing splines with equivalent df=5 and 10

Additive models  Gaussian Linear Model:IE[Y]=  o +  1 X 1 +  2 X 2  Gaussian Additive model:IE[Y]=S 1 (X 1 )+S 2 (X 2 ) S 1, S 2 smooth functions of predictors X 1, X 2, usually LOESS, SPLINE Estimation of S 1, S 2 : « Backfitting Algorithm »  PRINCIPLE OF THE BACKFITTING ALGORITHM Y=S 1 (X 1 )+e  estimation S 1 * Y-S 1 *(X 1 )=S 2 (X 2 )+e  estimation S 2 * Y-S 2 *(X 2 )=S 1 (X 1 )+e  estimation S 1 ** Y-S 1 **(X 1 )=S 2 (X 2 )+e  estimation S 2 ** Y-S 2 **(X 2 )=S 1 (X 1 )+e  estimation S 1 *** Etc… until convergence

Additive models  Additive models One efficient way to perform non-linear regression, but…  Crucial point ADAPTED WHEN ONLY FEW PREDICTORS 2, 3 predictors at most

Additive models  Philosophy DATA DRIVEN APPROACHES RATHER THAN MODEL DRIVEN APPROACH USEFUL AS EXPLORATORY TOOLS  Approximate inference tests are possible, but full inferences are better assessed by means of parametric models

Generalized Additive models (GAM)  Extension to non-normal dependant variables  Generalized additive models: additive modelling of the natural parameter of exponential family laws (Poisson, Binomial, Gamma, Gauss…). g[µ]=  =S 1 (X 1 )+S 2 (X 2 )  Vector Generalized Additive Models (VGAM): one step beyond…

Example 1 Annual umber and maximum integrated intensity (PDI) of hurricane tracks over the North Atlantic

Number of Hurricanes  Number of Hurricanes in North Atlantic ~ Poisson distribution

Factors influencing the number of hurricanes  GAM applied to number of hurricanes (YEAR,SST,SOI,NAO)

GAM model  Log( )=  o +S 1 (SST)+S 2 (SOI)

PARAMETRIC model  “broken stick model” (with continuity constraint) in SOI, revealed by GAM analysis  log( )=  o+  SOI (1) SOI+  SST SST SOI<K =  o+  SOI (1) SOI+  SOI (2) (SOI-K)+  SST SST SOI  K  The best fit obtained for SOI value K=1 log-likelihood= , to be compared with (linearity) standard deviance test allows reject linearity (p value=0.02)  Expectation of the hurricane number is then straightforwardly computed as a function of SOI and SST

EXPECTATION OF HURRICANE NUMBERS

OBSERVED vs EXPECTED: r=0.6