crisp {crisp}

Convex Regression with Interpretable Sharp Partitions (CRISP).

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. 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 <http://jmlr.org/papers/volume17/15-344/15-344.pdf>.

Usage

crisp(y, X, q = NULL, lambda.min.ratio = 0.01, n.lambda = 50,
  lambda.seq = NULL, 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 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.

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 crisp, which can be summarized using summary, plotted using plot, and used to predict outcome values for new covariates using predict.

• M.hat.list: A list of length n.lambda giving M.hat for each value of lambda.seq.

• num.blocks: A vector of length n.lambda giving the number of blocks in M.hat for each value of lambda.seq.

• obj.vec: A vector of length n.lambda giving the value of the objective of Eqn (4) in the CRISP paper for each value of lambda.seq.

• Other elements: As specified by the user.

See Also

crispCV, plot, summary, predict

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 for a range of tuning parameters, i.e., lambda values
#lambda sequence is chosen automatically if not specified
crisp.out <- crisp(X = data$X, y = data$y)

## End(Not run)