xtune: Tuning feature-specific shrinkage parameters of penalized regression models based on external information

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📗 Introduction

Motivation

In standard regularized regression (Lasso, Ridge, and Elastic-net), a single penalty parameter \(\lambda\) applied equally to all regression coefficients to control the amount of regularization in the model.

Better prediction accuracy may be achieved by allowing a different amount of shrinkage. Ideally, we want to give a small penalty to important features and a large penalty to unimportant features. We guide the penalized regression model with external data \(Z\) that are potentially informative for the importance/effect size of coefficients and allow feature-specific shrinkage modeled as a log-linear function of the external data.

The objective function of feature-specific shrinkage integrating external information is:

\min_f \sum_{i = 1}^n V(f(x_i), y_i) + \textcolor{red}{\lambda} R(f)
\textcolor{red}{\lambda = e^{Z \cdot \alpha}}

where \(V\) represents loss function, \(\lambda\) is the penalty/tuning parameter, and \(R(f)\) is the regularization/penalty term. Specifically, we use Elastic-net type of penalty:

\[R(f) = \left[\sum_{k = 1}^K\bigg((1-c)||\beta_k||_2^2/2 + c||\beta_k||_1 \bigg) \right]\]

when \(c = 1, 0\) or any value between 0 to 1, the model is equivalent to LASSO, Ridge, and Elastic-net, respectively.

The idea of external data is that it provides us information on the importance/effect size of regression coefficients. It could be any nominal or quantitative feature-specific information, such as the grouping of predictors, prior knowledge of biological importance, external p-values, function annotations, etc. Each column of \(Z\) is a variable for features in design matrix \(X\). \(Z\) is of dimension \(p \times q\), where \(p\) is the number of features and \(q\) is the number of variables in \(Z\).

Tuning multiple penalty parameters

Penalized regression fitting consists of two phases: (1) learning the tuning parameter(s) (2) estimating the regression coefficients giving the tuning parameter(s). Phase (1) is the key to achieve good performance. Cross-validation is widely used to tune a single penalty parameter, but it is computationally infeasible to tune more than three penalty parameters. We propose an Empirical Bayes approach to estimate the multiple tuning parameters. The individual penalties are interpreted as variance terms of the priors (exponential prior for Elastic-net) in a random effect formulation of penalized regressions. A majorization-minimization algorithm is employed for implementation. Once the tuning parameters \(\lambda\) s are estimated, and therefore the penalties are known, phase (2) - estimating the regression coefficients is done using glmnet.

Data structure examples

Suppose we want to predict a person’s weight loss using his/her weekly dietary intake. Our external information Z could incorporate information about the levels of relevant food constituents in the dietary items.

Primary data X and Y: predicting an individual’s weight loss by his/her weekly dietary items intake

External information Z: the nutrition facts about each dietary item

📙 Installation

xtune can be installed from Github using the following command:

# install.packages("devtools")

library(devtools)
devtools::install_github("JingxuanH/xtune", 
                         build_vignettes = TRUE)

library(xtune)

✍ Citation

citation("xtune")
#> Warning in citation("xtune"): no date field in DESCRIPTION file of package
#> 'xtune'
#> Warning in citation("xtune"): could not determine year for 'xtune' from package
#> DESCRIPTION file
#> 
#> To cite package 'xtune' in publications use:
#> 
#>   Jingxuan He and Chubing Zeng (NA). xtune: Regularized Regression with
#>   Feature-specific Penalties Integrating External Information. R
#>   package version 0.99.0.
#> 
#> A BibTeX entry for LaTeX users is
#> 
#>   @Manual{,
#>     title = {xtune: Regularized Regression with Feature-specific Penalties Integrating External Information},
#>     author = {Jingxuan He and Chubing Zeng},
#>     note = {R package version 0.99.0},
#>   }

Feel free to contact hejingxu@usc.edu if you have any questions.

📘 Examples

To show some examples on how to use this package, we simulated an example of data that contains 100 observations, 200 predictors, and a continuous outcome. The external information Z contains 4 columns, each column is indicator variable (can be viewed as the grouping of predictors).

library(xtune)

## load the example data
data(example)

The data looks like:

example$X[1:3,1:5]
#>               Predictor_1 Predictor_2 Predictor_3 Predictor_4 Predictor_5
#> Observation_1  -0.7667960   0.9212806   2.0149030  0.79004563  -1.4244699
#> Observation_2  -0.8164583  -0.3144157  -0.2253684  0.08712746  -1.0296026
#> Observation_3  -0.1415352   0.6623149  -1.0398456  1.87611212   0.7340254
example$Z[1:5,]
#>             External_variable_1 External_variable_2 External_variable_3
#> Predictor_1                   1                   0                   0
#> Predictor_2                   1                   0                   0
#> Predictor_3                   0                   1                   0
#> Predictor_4                   0                   1                   0
#> Predictor_5                   0                   0                   1
#>             External_variable_4
#> Predictor_1                   0
#> Predictor_2                   0
#> Predictor_3                   0
#> Predictor_4                   0
#> Predictor_5                   0

xtune() is the core function to fit the integrated penalized regression model. At a minimum, you need to specify the predictor matrix X, outcome variable Y. If an external information matrix Z is provided, the function will incorporate Z to allow differential shrinkage based on Z. The estimated tuning parameters are returned in $penalty.vector.

If you do not provide external information Z, the function will perform empirical Bayes tuning to choose the single penalty parameter in penalized regression, as an alternative to cross-validation. You could compare the tuning parameter chosen by empirical Bayes tuning to that choose by cross-validation (see also cv.glmnet). The default penalty applied to the predictors is the Elastic-net penalty.

If you provide an identify matrix as external information Z to xtune(), the function will estimate a separate tuning parameter (_j) for each regression coefficient (_j).

xtune.fit <- xtune(example$X,example$Y,example$Z, family = "linear")
#> Z provided, start estimating individual tuning parameters 
#> Start estimating alpha:
#> #-----------------Inner loop Iteration 1 Done-----------------#
#> #-----------------Inner loop Iteration 2 Done-----------------#
#> #-----------------Inner loop Iteration 3 Done-----------------#
#> #-----------------Inner loop Iteration 4 Done-----------------#
#> #-----------------Inner loop Iteration 5 Done-----------------#
#> #-----------------Inner loop Iteration 6 Done-----------------#
#> #-----------------Inner loop Iteration 7 Done-----------------#
#> Difference between alpha_old and alpha_new: 5.693111 
#> Start estimating alpha:
#> #-----------------Inner loop Iteration 1 Done-----------------#
#> #-----------------Inner loop Iteration 2 Done-----------------#
#> #-----------------Inner loop Iteration 3 Done-----------------#
#> #-----------------Inner loop Iteration 4 Done-----------------#
#> #-----------------Inner loop Iteration 5 Done-----------------#
#> #-----------------Inner loop Iteration 6 Done-----------------#
#> #-----------------Inner loop Iteration 7 Done-----------------#
#> Difference between alpha_old and alpha_new: 2.281027 
#> ...
#> Done!

To view the penalty parameters estimated by xtune()

xtune.fit$penalty.vector[1:5]
#> [1] 0.005381686 0.005381686 0.015186170 0.015186170 0.052245694

The coef and predict functions can be used to extract beta coefficient estimates and predict response on new data.

coef_xtune(xtune.fit)[1:5]
#> [1]  0.07943206  2.08369420 -1.95701978  0.86767453 -1.31125776
predict_xtune(xtune.fit, example$X)[1:5]
#> Observation_1 Observation_2 Observation_3 Observation_4 Observation_5 
#>     -2.573221     -2.915913     -5.589512      2.196957      1.684783

More details and examples are also described in the vignettes to further illustrate the usage and syntax of this package.