### Introduction

### Methods

### Penalized regression model

*n*observations, and the data consist of

*n*×1 including the response variable

*y*and a

*n*×

*p*matrix

*X*of predictors. Assume that

*y*and X have been centered. The objective function of Lasso is

*λ*is a tuning parameter, ||∙||

_{1}stands for the vector

*l*-norm and ||∙||

_{1}_{2}stands for the vector

*l*-norm. This penalty shrinks any coefficients contributing to the minimization problem to 0 [1].

_{2}*G*groups. Let

*X*be a matrix for the predictors of the g

_{g}^{th}group with the corresponding coefficient vector

*β*. The objective function of group Lasso is

_{g}_{2}is the Euclidean norm [4,9]. Group Lasso shrinks all

*β*values in irrelevant groups to 0. When

*λ*=0, this criterion is equivalent to Lasso.

*α*∈[0,1] and

*β*=(

*β*

_{1},…,

*β*) [5,9].

_{G}*α*is a convex combination of Lasso and group Lasso. This criterion is equivalent to group Lasso if

*α*=0, and to Lasso if

*α*=1.

### MP-Lasso charts

### Results

### Implementation

*λ*values that minimize CV error. The group vector represents the group structure of variables. Group names should be in character type or integer type. The group vector should be identical to the one used when the CV object is created. MP-Lasso chart supports three methods: Lasso, group Lasso, and sparse group Lasso. Table 1 summarizes the functions of the developed package for MP-Lasso charts and related packages to obtain input data.

*α*is set to 1 in the cv.glmnet function.

*λ*to use for each method, and it can take two values (“min” and “1se”), with “min” as default. The “min” option chooses the

*λ*value that minimizes the CV loss. The “1se” chooses the largest

*λ*with a CV error not 1 standard error further from the minimum CV loss. The “1se” option chooses fewer variables. The sort.type option determines which numeric feature represents the coefficients of variables in each group. Two choices are available for the sort.type option, “max” and “mean.” The “max” option uses the maximum absolute coefficient in each group as the feature of the group, while the “mean” option uses the mean of the absolute coefficients in each group. The last option is the max.shown option. When the number of chosen group is large, the chart can be too crowded with segments, making the chart difficult to interpret. By choosing max.shown, the user can decide the maximum number of segments shown on the chart. In the resulting MP-Lasso chart, interactive features are used. Moving the cursor over a point in the inner level displays information about that variable. For Lasso, moving the mouse over the sector area displays the corresponding information.

### Real data analysis

*λ*value that yields the minimum 10-fold CV loss, and each group is represented by maximum absolute coefficients. For Lasso analysis, we set

*max.shown*=30 for better representation. A summary of the results is presented in Table 2.