Embed Size (px)
Citation preview
More General
• Need different response curves for each predictor
• Need more complex responses
Generalized Additive Models
• Adds functions to linearize each predictor variable
• = )• Functions can be parametric or non-
parametric: Including splines• Makes GAMS:
– Very general– Prone to over-fitting
𝑓 (𝑥 )={14
(𝑥+2)3−2≤ 𝑥≤−1
14
(3|𝑥|3−6 𝑥2+4 )−1≤𝑥 ≤114
(2− 𝑥 )31≤𝑥 ≤2
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
IncomeIncome AgeAge
Hastie and Tibshirani 1986, Generalized Additive Models
Z-axis shows the proportion of families with a telephone at home
GAM Plots in RModeled Response Curve
95% CI
Sample point “Grass”
FIA Doug-Fir height data vs. BioClim Annual Precipitation
“Partial” = 1 Covariate
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 RunsLayers Gamma Explained
DevianceAIC
All 6 1.4 59 57807
All 6 10 58 57961
All 6 20 57 58081
Best 3 20 51 58796
All 6 0.1 59 57811
Best Model?
Best 3 predictors, gamma=20 Data from FIA and BioClim
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
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”
• = degrees of freedom• Degrees of freedom =
• – number of estimated parameters • gam() chooses smoothing parameters to
minimize:
• Note: The reason the effect of gamma reverses itself at large values is that becomes larger than
∑ ( �̂�−𝑦 𝑖)2
(𝑛−𝑔𝑎𝑚𝑎∗𝑥)2
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:– http://www.casact.org/education/annual/
2011/handouts/C3-Guszcza.pdf• Disease mapping using GAMs
(workshop):– http://www.cireeh.org/pmwiki.php/Main/
Gam-mapWorkshop• Mapping population based studies:
– http://www.ij-healthgeographics.com/content/5/1/26
