More General Need different response curves for each predictor Need more complex responses

Generalized Additive Models

Spline Curves Knots Bell-shaped Irwin-Hall spline

Spline Curves in R Wrap predictors in a spline function: –s(predictor) Use “gamma” parameter to set the number of knots –Controls over-fitting –1.4 is recommended In R: –TheModel=gam(Height~s(AnnualPrecip), data=TheData,gamma=1.4)

Reading Read Hastie and Tibshirani when you have “time” –“All considered, it is conceivable that in a minor way, nonparametric regression might, like linear regression, become an object treasured for both its artistic merit as well as usefulness” L. Breiman, 1977 Read Martinez-Rincon and Jensen for next time

Which Approach? GAM Kernel Smoother Income Age Hastie and Tibshirani 1986, Generalized Additive Models Z-axis shows the proportion of families with a telephone at home

GAM Plots in R Modeled Response Curve 95% CI Sample point “Grass” FIA Doug-Fir height data vs. BioClim Annual Precipitation “Partial” = 1 Covariate

Brown Shrimp in GOM Data from SeaMap and NOAA

Gamma=1.4 Explained Deviance: 59%, AIC=57807 Data from FIA and BioClim

Gamma=10 Explained Deviance: 59%, AIC=57961 Data from FIA and BioClim

Gamma=20 Explained Deviance: 57%, AIC=58081 Data from FIA and BioClim

Gamma=20 Explained Deviance: 51%, AIC=58796 Data from FIA and BioClim

Gamma=0.1 Explained Deviance: 59%, AIC=57811 Data from FIA and BioClim

GAM Model Runs LayersGammaExplained Deviance AIC All All All Best All

Best Model? Best 3 predictors, gamma=20 Data from FIA and BioClim

Blue Crab Distribution Model

Blue Crab vs. Salinity Jensen et. al. 2005, Winter distribution of blue crab Callinectes sapidus in Chesapeake Bay: application and cross- validation of a two-stage generalized additive model

Response Curves (partial) GAMs BRTs

GAMs vs. BRTs Martinez-Rincon 2012, Comparative performance of generalized additive models and boosted regression trees for statistical modeling of incidental catch of wahoo (Acanthocybium solandri) in the Mexican tuna purse-seine fishery “Results indicate little difference between the performance of GAM and BRT models”

Gamma in GAMs

Anderson We are not trying to model the data; instead, we are trying to model the information in the data. The goal is to recover the information that applies more generally to the process, not just to the particular data set. If we were merely trying to model the data well, we could fit high order Fourier series terms or polynomial terms until the fit is perfect. Data contain both information and noise; fitting the data perfectly would include modeling the noise and this is counter to our science objective.

Additional Resources Generalized Additive Models: an introduction with R –Copyrighted book –Includes: Linear models GLMs GAMs Examples in R Some matrix algebra

Additional Resources Geospatial Analysis with GAMs: – 1/handouts/C3-Guszcza.pdf Disease mapping using GAMs (workshop): – -mapWorkshop Mapping population based studies: – healthgeographics.com/content/5/1/26