### Description

Non-parametric adaptive regression method for diffusion map basis.

### Usage

adapreg.m(epsilon, D, y, mmax = min(50, length(y)), fold = NULL,
nfolds = 10, objfun = FALSE)


### Arguments

 epsilon diffusion map kernel parameter 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 objfun if the function is to be passed into an optimization routine (such as minimize()), then this needs to be set to TRUE, so that only the minimal CV risk is returned

### Details

Fits an adaptive regression model using the estimated diffusion map coordinates of a data set, while holding epsilon fixed and optimizing over m. The adaptive regression model is the expansion of the response function on the first m diffusion map basis functions.

For a given epsilon value, this routine finds the optimal m by minimizing the cross-validation risk (CV MSE) of the regression estimate. To optimize over (epsilon,m), use the function adapreg().

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) does 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 equals mmax, it is advised to increase mmax. cvrisk vector of CV risk estimates for model sizes from 1:mmax epsilon 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

If objfun is set to TRUE, then the returned value is the minimum cross-validation risk for the adaptive regression model for the given epsilon.

### References

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

diffuse(),adapreg()

### Examples

library(stats)
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))
# leave-one-out cross-validation:
print(paste("optimal model size:",AR$mopt,"; min. CV risk:", round(AR$mincvrisk,4)))
par(mfrow=c(2,1),mar=c(5,5,4,1))
plot(AR$cvrisks,typ='b',xlab="Model size",ylab="CV risk", cex.lab=1.5,cex.main=1.5,main="CV risk estimates") plot(y,AR$y.hat,ylab=expression(hat("y")),cex.lab=1.5,cex.main=1.5,
main="Predictions")
abline(0,1,col=2,lwd=2)

## swiss roll data
N=2000
t = (3*pi/2)*(1+2*runif(N));  height = runif(N);
X = cbind(t*cos(t), height, t*sin(t))
X = scale(X) + matrix(rnorm(N*3,0,0.05),N,3)
tcol = topo.colors(32)
colvec = floor((t-min(t))/(max(t)-min(t))*32); colvec[colvec==0] = 1
scatterplot3d(X,pch=18,color=tcol[colvec],xlab=expression("x"[1]),
ylab=expression("x"[2]),zlab=expression("x"[3]),cex.lab=1.5,
main="Swiss Roll, Noise = 0.05",cex.main=1.5,xlim=c(-2,2),
ylim=c(-2,2),zlim=c(-2,2),col.axis="gray")

D = as.matrix(dist(X))
# 10-fold cross-validation:
print(paste("optimal model size:",AR$mopt,"; min. CV risk:", round(AR$mincvrisk,4)))
plot(AR$cvrisks,typ='b',xlab="Model size",ylab="CV risk", cex.lab=1.5,cex.main=1.5,main="CV risk estimates") plot(t,AR$y.hat,ylab=expression(hat("t")),cex.lab=1.5,cex.main=1.5,