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README.md

SparseStep R Package

SparseStep is an R package for sparse regularized regression and provides an alternative to methods such as best subset selection, elastic net, lasso, and lars. The SparseStep method is introduced in the following paper:

SparseStep: Approximating the Counting Norm for Sparse Regularization by G.J.J. van den Burg, P.J.F. Groenen, and A. Alfons (Arxiv preprint arXiv:1701.06967 [stat.ME], 2017).

This R package can be easily installed by running install.packages('sparsestep') in R. If you use the package in your work, please cite the above reference using, for instance, the following BibTeX entry:

@article{vandenburg2017sparsestep,
  title = {{SparseStep}: Approximating the Counting Norm for Sparse Regularization},
  author = {{Van den Burg}, G. J. J. and Groenen, P. J. F. and Alfons, A.},
  journal = {arXiv preprint arXiv:1701.06967},
  year = {2017}
}

Introduction

The SparseStep method solves the regression problem regularized with the l_0 norm. Since the l_0 term is highly non-convex and therefore difficult to optimize, this non-convexity is introduced gradually in SparseStep during optimization. As in other regularized regression methods such as ridge regression and lasso, a regularization parameter lambda can be specified to control the amount of regularization. The choice of regularization parameter affects how many non-zero variables remain in the final model.

We will give a quick guide to SparseStep using the Prostate dataset from the book Elements of Statistical Learning.

We will show a few examples of running SparseStep on the Prostate dataset from the lasso2 package. First we load the data and create a data matrix and outcome vector:

> prostate <- read.table("http://statweb.stanford.edu/~tibs/ElemStatLearn/datasets/prostate.data")
> X <- prostate[prostate$train == T, c(-1, -10)]
> X <- as.matrix(X)
> y <- prostate[prostate$train == T, 1]
> y <- as.vector(y)

The easiest way to fit a SparseStep model is to use the path.sparsestep function. This estimates the entire path of solutions for the SparseStep model for different values of the regularization parameter using a golden section search algorithm.

> path <- path.sparsestep(X, y)
Found maximum value of lambda: 2^( 7 )
Found minimum value of lambda: 2^( -3 )
Running search in interval [ -3 , 7 ] ...
Running search in interval [ -3 , 2 ] ...
Running search in interval [ -3 , -0.5 ] ...
Running search in interval [ -3 , -1.75 ] ...
Running search in interval [ -0.5 , 2 ] ...
Running search in interval [ -0.5 , 0.75 ] ...
Running search in interval [ 0.125 , 0.75 ] ...
Running search in interval [ 2 , 7 ] ...

> plot(path, col=1:nrow(path$beta))     # col specifies colors to matplot
> legend('topleft', legend=rownames(path$beta), lty=1, col=1:nrow(path$beta))

In the resulting plot we can see the coefficients of the features that are included in the model at different values of lambda:

SparseStep regression on Prostate dataset

The coefficients of the model can be obtained using coef(path), which returns a sparse matrix:

> coef(path)
9 x 9 sparse Matrix of class "dgCMatrix"
                   s0           s1           s2          s3           s4         s5        s6       s7
Intercept  1.31349155  1.313491553  1.313491553  1.31349155  1.313491553 1.31349155 1.3134916 1.313492
lweight   -0.11336968 -0.113485291  .            .           .           .          .         .
age        0.02010188  0.020182049  0.018605327  0.01491472  0.018704172 0.01623212 .         .
lbph      -0.05698125 -0.059026246 -0.069116923  .           .           .          .         .
svi        0.03511645  .            .            .           .           .          .         .
lcp        0.41845469  0.423398063  0.420516410  0.43806447  0.433449263 0.38174743 0.3887863 .
gleason    0.22438690  0.222333394  0.236944796  0.23503609  .           .          .         .
pgg45     -0.00911273 -0.009084031 -0.008949463 -0.00853420 -0.004328518 .          .         .
lpsa       0.57545508  0.580111724  0.561063637  0.53017309  0.528953966 0.51473225 0.5336907 0.754266
                s8
Intercept 1.313492
lweight   .
age       .
lbph      .
svi       .
lcp       .
gleason   .
pgg45     .
lpsa      .

Note that the final model included in coef(beta) is a intercept-only model, which is generally not very useful. Predicting out-of-sample data can be done easily using the predict function.

By default SparseStep centers the regressors and outcome variable y and normalizes the regressors X to ensure that the regularization is applied evenly among them and the intercept is not penalized. If you prefer to use a constant term in the regression and penalize this as well, you'll have to transform the input data and disable the intercept:

> Z <- cbind(constant=1, X)
> path <- path.sparsestep(Z, y, intercept=F)
...
> plot(path, col=1:nrow(path$beta))
> legend('bottomright', legend=rownames(path$beta), lty=1, col=1:nrow(path$beta))

Note that since we add the constant through the data matrix it is subject to regularization and therefore sparsity:

SparseStep regression on Prostate dataset (with constant)

For more information and examples, please see the documentation included with the package. In particular, the following pages are good places to start:

> ?'sparsestep-package'
> ?sparsestep
> ?path.sparsestep

Reference

If you use SparseStep in any of your projects, please cite the paper using the information available through the R command:

citation('sparsestep')

or use the following BibTeX code:

@article{van2017sparsestep,
  title = {{SparseStep}: Approximating the Counting Norm for Sparse Regularization},
  author = {Gerrit J.J. {van den Burg} and Patrick J.F. Groenen and Andreas Alfons},
  journal = {arXiv preprint arXiv:1701.06967},
  archiveprefix = {arXiv},
  year = {2017},
  eprint = {1701.06967},
  url = {https://arxiv.org/abs/1701.06967},
  primaryclass = {stat.ME},
  keywords = {Statistics - Methodology, 62J05, 62J07},
}

Notes

This package is licensed under GPLv3. Please see the LICENSE file for more information. If you have any questions or comments about this package, please open an issue on GitHub (don't hesitate, you're helping to make this project better for everyone!). If you prefer to use email, please write to gertjanvandenburg at gmail dot com.