VGAM (version 0.7-1)

vgam: Fitting Vector Generalized Additive Models

Description

Fit a vector generalized additive model (VGAM). This is a large class of models that includes generalized additive models (GAMs) and vector generalized linear models (VGLMs) as special cases.

Usage

vgam(formula, family, data = list(), weights = NULL, subset = NULL, 
     na.action = na.fail, etastart = NULL, mustart = NULL, 
     coefstart = NULL, control = vgam.control(...), offset = NULL, 
     method = "vgam.fit", model = FALSE, x.arg = TRUE, y.arg = TRUE, 
     contrasts = NULL, constraints = NULL, 
     extra = list(), qr.arg = FALSE, smart = TRUE, ...)

Value

• An object of class "vgam" (see vgam-class for further information).

Details

A vector generalized additive model (VGAM) is loosely defined as a statistical model that is a function of $M$ additive predictors. The central formula is given by $$\eta_j = \sum_{k=1}^p f_{(j)k}(x_k)$$ where $x_k$ is the $k$th explanatory variable (almost always $x_1=1$ for the intercept term), and $f_{(j)k}$ are smooth functions of $x_k$ that are estimated by smoothers. The first term in the summation is just the intercept. Currently only one type of smoother is implemented and this is called a vector (cubic smoothing spline) smoother. Here, $j=1,\ldots,M$ where $M$ is finite. If all the functions are constrained to be linear then the resulting model is a vector generalized linear model (VGLM). VGLMs are best fitted with vglm.

Vector (cubic smoothing spline) smoothers are represented by s() (see s). Local regression via lo() is not supported. The results of vgam will differ from the S-PLUS and Rgam function (in the gam Rpackage) because vgam uses a different knot selection algorithm. In general, fewer knots are chosen because the computation becomes expensive when the number of additive predictors $M$ is large.

The underlying algorithm of VGAMs is iteratively reweighted least squares (IRLS) and modified vector backfitting using vector splines. B-splines are used as the basis functions for the vector (smoothing) splines. vgam.fit is the function that actually does the work. The smoothing code is based on F. O'Sullivan's BART code. A closely related methodology based on VGAMs called constrained additive ordination (CAO) first forms a linear combination of the explanatory variables (called latent variables) and then fits a GAM to these. This is implemented in the function cao for a very limited choice of family functions.

References

Yee, T. W. and Wild, C. J. (1996) Vector generalized additive models. Journal of the Royal Statistical Society, Series B, Methodological, 58, 481--493.

http://www.stat.auckland.ac.nz/~yee

See Also

vgam.control, vgam-class, vglmff-class, plotvgam, vglm, s, vsmooth.spline, cao.

Examples

# Nonparametric proportional odds model 
data(pneumo)
pneumo = transform(pneumo, let=log(exposure.time))
vgam(cbind(normal,mild,severe) ~ s(let), cumulative(par=TRUE), pneumo)

# Nonparametric logistic regression 
data(hunua) 
fit = vgam(agaaus ~ s(altitude), binomialff, hunua)
plot(fit, se=TRUE)

# Fit two species simultaneously 
fit2 = vgam(cbind(agaaus, kniexc) ~ s(altitude), binomialff(mv=TRUE), hunua)
coef(fit2, mat=TRUE)   # Not really interpretable 
plot(fit2, se=TRUE, overlay=TRUE, lcol=1:2, scol=1:2)
attach(hunua)
o = order(altitude)
matplot(altitude[o], fitted(fit2)[o,], type="l", lwd=2, las=1,
    xlab="Altitude (m)", ylab="Probability of presence",
    main="Two plant species' response curves", ylim=c(0,.8))
rug(altitude)
detach(hunua)