Adaptive Regression
Non-parametric adaptive regression method for diffusion map basis.
adapreg(D, y, mmax = min(50, length(y)), fold = NULL, nfolds = 10, nrep = 5)
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 |
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.
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 |
Richards, J. W., Freeman, P. E., Lee, A. B., and Schafer, C. M., (2009), ApJ, 691, 32
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):
AR = adapreg(D,y, mmax=5,nfolds=2,nrep=2)
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,
main="Predictions")
abline(0,1,col=2,lwd=2)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.