Generalized Additive Models Keith D. Holler September 19, 2005 Keith D. Holler September 19, 2005

GLM’s – The Challenge What to do with continuous variables? –Eg. Age, credit score, amount of insurance Options –Categorize – but how? Equal volume, Tree, judgment –Appendix H, “A Practioner’s Guide to GLMs” by Duncan et al –Treat as polynomial The Weierstrass Approximation Theorem Eg Mileage (2 miles)^4 = 16 (25 miles)^4 = 390,625 –Look at categorical estimates, transform, rerun Newage variable = age^3 if age < 20 + age^2 if age < 80 + minimum (age, 80) All forms must be decided BEFORE model is run Obviously, no clear winner!

Modelers Aspiration

Generalized Additive Models - GAMS GLMs are special case of GAMs Eg LN(E[PP]) = Intercept + f1(age) + f2(gender) + f3(symbol) + f4(marital) The functions f1,f2,f3,f4 can be anything –GLM - Categorical, polynomial, transforms –Non-parametric functional smoothers –Decision trees Balance degrees of freedom, amount of data, and functional form better

Smoothers – Partial List Locally weighted running line smoother (LOESS) Regression splines Cubic smoothing splines Monotonic splines B-splines Kernel smoothers Running medians, means, lines GLM – categories or polynomials Decision Trees Many can be extended to multiple dimensions

GAM – Keys Backfitting allows reduction of dimension –Residual Z = LN(E[PP]) – intercept – f1(age) – f2(gender) – f4(marital) –Fit Z = f3(symbol) –Now a 2-dimensional problem “Y vs X” Data drives the shape –Not determined apriori –Use of cross validation to find smoothing parameter “Local” – many of the smoothers use only data points close to the point being predicted, instead of all.

Example – SAS Code proc gam data=all; class gender marital2; model clclmonz = param(gender marital2) spline(age2,df=4) spline(symbol,df=3) / dist=Poisson; output out=estall p; run;

Example – Degrees of Freedom

Smoothing Spline Error Criteria ∑ {Y i – g(t i ) } ² + λ ∫ { g” (t)} ² dt –λ is smoothing parameter –Reference: Nonparametric Regression and Generalized Linear Models, Green and Silverman

Example – Cross Validation proc gam data=all; class gender marital2; model clclmonz = param(gender marital2) spline(age2) spline(symbol) / method=GCV dist=Poisson; output out=estGCV p; run; Results in degrees of freedom of 17 and 14.

Miscellaneous Parameter Estimates – 1 for each value SPLUS References –SAS Proc Gam –Generalized Additive Models, Hastie and Tibshirani

Q & A Keith D. Holler PhD, FCAS, ASA, ARM Personal Lines Research Department St. Paul Travelers k d (860) 277 – 4808 Research paper in progress for Ratemaking call