### Description

Non-parametric adaptive regression method for diffusion map basis.

### Usage

adapreg(D, y, mmax = min(50, length(y)), fold = NULL, nfolds = 10,
nrep = 5)


### Arguments

 D n-by-n pairwise distance matrix for a data set with n points, or alternatively output from the dist() function y vector of responses to model mmax maximum model size to consider fold vector of integers of size n specifying the k-fold cross-validation allocation. Default does nfolds-fold CV by sample(1:nfolds,length(y),replace=T) nfolds number of folds to do CV. If fold is supplied, nfolds is ignored nrep number of times optimization algorithm is run (with random initializations). Higher nrep allows algorithm to avoid getting stuck in local minima

### Details

Fits an adaptive regression model leaving as free parameters both the diffusion map localness parameter, epsilon, and the size of the regression model, m. The adaptive regression model is the expansion of the response function on the first m diffusion map basis functions.

This routine searches for the optimal (epsilon,m) by minimizing the cross-validation risk (CV MSE) of the regression estimate. The function uses optimize() to search over an appropriate range of epsilon and calls the function adapreg.m() to find the optimal m for each epsilon.

Default uses 10-fold cross-validation to choose the optimal model size. User may also supply a vector of fold allocations. For instance, sample(1:10,length(y),replace=T) does 10-fold CV while 1:length(y) performs leave-one-out CV.

### Value

The returned value is a list with components

 mincvrisk minimum cross-validation risk for the adaptive regression model for the given epsilon mopt size of the optimal regression model. If mopt == mmax, it is advised to increase mmax. epsopt optimal value of epsilon used in diffusion map construction y.hat predictions of the response, y-hat, for the optimal model coeff coefficients of the optimal model

### References

Richards, J. W., Freeman, P. E., Lee, A. B., and Schafer, C. M., (2009), ApJ, 691, 32

diffuse(),adapreg.m()

### Examples

library(scatterplot3d)
## trig function on circle
t=seq(-pi,pi,.01)
x=cbind(cos(t),sin(t))
y = cos(3*t) + rnorm(length(t),0,.1)
tcol = topo.colors(32)
colvec = floor((y-min(y))/(max(y)-min(y))*32); colvec[colvec==0] = 1
scatterplot3d(x[,1],x[,2],y,color=tcol[colvec],pch=20,
main="Cosine function supported on circle",angle=55,
cex.main=2,col.axis="gray",cex.symbols=2,cex.lab=2,
xlab=expression("x"[1]),ylab=expression("x"[2]),zlab="y")

D = as.matrix(dist(x))
# do 10-fold cross-validation to optimize (epsilon, m):
print(paste("optimal model size:",AR$mopt,"; optimal epsilon:", round(AR$epsopt,4),"; min. CV risk:",round(AR$mincvrisk,5))) plot(y,AR$y.hat,ylab=expression(hat("y")),cex.lab=1.5,cex.main=1.5,