crisp (version 1.0.0)

crispCV: CRISP with Tuning Parameter Selection via Cross-Validation.

Description

This function implements CRISP, which considers the problem of predicting an outcome variable on the basis of two covariates, using an interpretable yet non-additive model. CRISP partitions the covariate space into blocks in a data-adaptive way, and fits a mean model within each block. Unlike other partitioning methods, CRISP is fit using a non-greedy approach by solving a convex optimization problem, resulting in low-variance fits. This function differs from the crisp function in that the tuning parameter, lambda, is automatically selected using K-fold cross-validation. More details are provided in Petersen, A., Simon, N., and Witten, D. (2016). Convex Regression with Interpretable Sharp Partitions. Journal of Machine Learning Research, 17(94): 1-31 .

Usage

crispCV(y, X, q = NULL, lambda.min.ratio = 0.01, n.lambda = 50, lambda.seq = NULL, fold = NULL, n.fold = NULL, seed = NULL, within1SE = FALSE, rho = 0.1, e_abs = 10^-4, e_rel = 10^-3, varyrho = TRUE, double.run = FALSE)

Arguments

y

An n-vector containing the response.

X

An n x 2 matrix with each column containing a covariate.

q

The desired granularity of the CRISP fit, M.hat, which will be a q by q matrix. M.hat is a mean matrix whose element M.hat[i,j] contains the mean for pairs of covariate values within a quantile range of the observed predictors X[,1] and X[,2]. For example, M.hat[1,2] represents the mean of the observations with the first covariate value less than the 1/q-quantile of X[,1], and the second covariate value between the 1/q- and 2/q-quantiles of X[,2]. If left NULL, then q=n is used when n<100, and="" q=100 is used when n>=100. We recommend using q<=100< code=""> as higher values take longer to fit and provide an unneeded amount of granularity.

lambda.min.ratio

The smallest value for lambda.seq, as a fraction of the maximum lambda value, which is the data-derived smallest value for which the fit is a constant value. The default is 0.01.

n.lambda

The number of lambda values to consider - the default is 50.

lambda.seq

A user-supplied sequence of positive lambda values to consider. The typical usage is to calculate lambda.seq using lambda.min.ratio and n.lambda, but providing lambda.seq overrides this. If provided, lambda.seq should be a decreasing sequence of values, since CRISP relies on warm starts for speed. Thus fitting the model for a whole sequence of lambda values is often faster than fitting for a single lambda value.

fold

User-supplied fold numbers for cross-validation. If supplied, fold should be an n-vector with entries in 1,...,K when doing K-fold cross-validation. The default is to choose fold using n.fold.

n.fold

The number of folds, K, to use for the K-fold cross-validation selection of the tuning parameter, lambda. The default is 10 - specification of fold overrides use of n.fold.

seed

An optional number used with set.seed() at the beginning of the function. This is only relevant if fold is not specified by the user.

within1SE

Logical value indicating how cross-validated tuning parameters should be chosen. If within1SE=TRUE, lambda is chosen to be the value corresponding to the most sparse model with cross-validation error within one standard error of the minimum cross-validation error. If within1SE=FALSE, lambda is chosen to be the value corresponding to the minimum cross-validation error.

rho

The penalty parameter for our ADMM algorithm. The default is 0.1.

e_abs, e_rel

Values used in the stopping criterion for our ADMM algorithm, and discussed in Appendix C.2 of the CRISP paper.

varyrho

Should rho be varied from iteration to iteration? This is discussed in Appendix C.3 of the CRISP paper.

double.run

The initial complete run of our ADMM algorithm will yield sparsity in z_1i and z_2i, but not necessarily exact equality of the rows and columns of M.hat. If double.run is TRUE, then the algorithm is run a second time to obtain M.hat with exact equality of the appropriate rows and columns. This issue is discussed further in Appendix C.4 of the CRISP paper.

Value

An object of class crispCV, which can be summarized using summary, plotted using plot, and used to predict outcome values for new covariates using predict.

• lambda.cv: Optimal lambda value chosen by K-fold cross-validation.
• index.cv: The index of the model corresponding to the chosen tuning parameter, lambda.cv. That is, lambda.cv=crisp.out$lambda.seq[index.cv].
• crisp.out: An object of class crisp returned by crisp.
• mean.cv.error: An m-vector containing cross-validation error where m is the length of lambda.seq. Note that mean.cv.error[i] contains the cross-validation error for the tuning parameter crisp.out$lambda.seq[i].
• se.cv.error: An m-vector containing cross-validation standard error where m is the length of lambda.seq. Note that se.cv.error[i] contains the standard error of the cross-validation error for the tuning parameter crisp.out$lambda.seq[i].
• Other elements: As specified by the user.

See Also

crisp, plot, summary, predict, plot.cvError

Examples

## Not run: 
# #See ?'crisp-package' for a full example of how to use this package
# 
# #generate data (using a very small 'n' for illustration purposes)
# set.seed(1)
# data <- sim.data(n = 15, scenario = 2)
# 
# #fit model and select lambda using 2-fold cross-validation
# #note: use larger 'n.fold' (e.g., 10) in practice
# crispCV.out <- crispCV(X = data$X, y = data$y, n.fold = 2)
# ## End(Not run)