cv.trendfilter: Perform k-fold cross-validation to choose a trend filtering model

Description

This function performs k-fold cross-validation to choose the value of the regularization parameter lambda for a trend filtering problem, given the computed solution path. This function only applies to trend filtering objects with identity predictor matrix (no X passed).

Usage

cv.trendfilter(object, k = 5, mode = c("lambda", "df"),
               approx = FALSE, rtol = 1e-07, btol = 1e-07,
               verbose = FALSE)

Value

Returns and object of class "cv.trendfilter", a list with the following components:

err: a numeric vector of cross-validated errors.
se: a numeric vector of standard errors (standard deviations of the cross-validation error estimates).
mode: a character string indicating the mode, either "lambda" or "df".
lambda: if mode="lambda", the values of lambda at which the cross-validation errors in err were computed.
lambda.min: if mode="lambda", the value of lambda at which the cross-validation error is minimized.
lambda.1se: if mode="lambda", the value of lambda chosen by the one standard error rule (the largest value of lambda such that the cross-validation error is within one standard error of the minimum).
df: if mode="df", the degrees of freedom values at which the cross-validation errors in err were computed.
df.min: if mode="df", the degrees of freedom value at which the cross-validation error is minimized.
df.1se: if mode="df", the degrees of freedom value chosen by the one standard error rule (the smallest degrees of freedom value such that cross-validation error is within one standard error of the minimum).
i.min: the index of the model minimizing the cross-validation error.
i.1se: the index of the model chosen by the one standard error rule.
call: the matched call.

Arguments

object: the solution path object, of class "trendfilter", as returned by the trendfilter function.
k: an integer indicating the number of folds to split the data into. Must be between 2 and n-2 (n being the number of observations), default is 5. It is generally not a good idea to pass a value of k much larger than 10 (say, on the scale of n); see "Details" below.
mode: a character string, either "lambda" or "df". Specifying "lambda" means that the cross-validation error will be computed and reported at each value of lambda that appears as a knot in the solution path. Specifying "df" means that the cross-validation error will be computed and reported for every of degrees of freedom value (actually, estimate) incurred along the solution path. In the case that the same degrees of freedom value is visited multiple times, the model with the most regularization (smallest value of lambda) is considered. Default is "lambda".
approx: a logical variable indicating if the approximate solution path should be used (with no dual coordinates leaving the boundary). Default is FALSE.
rtol: a numeric variable giving the relative tolerance used in the calculation of the hitting and leaving times. A larger value is more conservative, and may cause the algorithm to miss some hitting or leaving events (do not change unless you know what you're getting into!). Defaultis 1e-7.
btol: similar to rtol but in absolute terms. If numerical instability is detected, first change rtol; then adjust btol if problems persist.
verbose: a logical variable indicating if progress should be reported after each knot in the path.

Details

For trend filtering (with an identity predictor matrix), the folds for k-fold cross-validation are chosen by placing every kth point into the same fold. (Here the points are implicitly ordered according to their underlying positions---either assumed to be evenly spaced, or explicitly passed through the pos argument.) The first and last points are not included in any fold and are always included in building the predictive model. As an example, with n=15 data points and k=4 folds, the points are assigned to folds in the following way: $$ x \; 1 \; 2 \; 3 \; 4 \; 1 \; 2 \; 3 \; 4 \; 1 \; 2 \; 3 \; 4 \; 1 \; x $$ where $x$ indicates no assignment. Therefore, the folds are not random and running cv.trendfilter twice will give the same result. In the calculation of the cross-validated error, the predicted value at a point is given by the average of the fits at this point's two neighbors (guaranteed to be in a different fold).

Running cross-validation in modes "lambda" and "df" often yields very similar results. The mode "df" simply gives an alternative parametrization for the sequence of cross-validated models and can be more convenient for some applications; if you are confused about its function, simply leave the mode equal to "lambda".

Examples

Run this code

# Constant trend filtering (the 1d fused lasso)
set.seed(0)
n = 50
beta0 = rep(sample(1:10,5),each=n/5)
y = beta0 + rnorm(n,sd=0.8)
a = fusedlasso1d(y)
plot(a)

# Choose lambda by 5-fold cross-validation
cv = cv.trendfilter(a)
plot(cv)
plot(a,lambda=cv$lambda.min,main="Minimal CV error")
plot(a,lambda=cv$lambda.1se,main="One standard error rule")

# \donttest{
# Cubic trend filtering
set.seed(0)
n = 100
beta0 = numeric(100)
beta0[1:40] = (1:40-20)^3
beta0[40:50] = -60*(40:50-50)^2 + 60*100+20^3
beta0[50:70] = -20*(50:70-50)^2 + 60*100+20^3
beta0[70:100] = -1/6*(70:100-110)^3 + -1/6*40^3 + 6000
beta0 = -beta0
beta0 = (beta0-min(beta0))*10/diff(range(beta0))
y = beta0 + rnorm(n)
a = trendfilter(y,ord=3,maxsteps=150)
plot(a,nlam=5)

# Choose lambda by 5-fold cross-validation
cv = cv.trendfilter(a)
plot(cv)
plot(a,lambda=cv$lambda.min,main="Minimal CV error")
plot(a,lambda=cv$lambda.1se,main="One standard error rule")
# }