变量选择——lasso、SCAD、MCP的实现（R语言）

本是古典何须时尚 2022-09-04 11:44 1299阅读 0赞

### 目录 ###

*  0引言
 *  1、glmnet
 *   *  1.1生成数据
     *  1.2 glmnet::cv.glmnet
 *  2、msaenet
 *  3、ncvreg
 *  4、结果对比
 *  5、总结
 *  6、纠错代码

# 0引言 #

自1996年lasso被提出以来，很多学者针对不同的模型提出有效的算法进行计算，例如多元线性线性模型、cox回归模型、广义线性模型等。也给出了很多选择惩罚参数的方式，比如cv、aic、bic。还有很多惩罚类型：lasso、适应性lasso、弹性网、SCAD、MCP。  
本文主要介绍下面三个包：`glmnet、ncvreg、msaenet`。先汇总每个包的主要函数、方法。如下表：

<table> 
 <thead> 
  <tr> 
   <th>函数</th> 
   <th>模型</th> 
   <th>惩罚参数</th> 
   <th>惩罚类型</th> 
   <th>包</th> 
   <th>是否弹性网</th> 
  </tr> 
 </thead> 
 <tbody> 
  <tr> 
   <td>glmnet</td> 
   <td>“gaussian”, “binomial”, “poisson”, “multinomial”, “cox”, “mgaussian”</td> 
   <td>—</td> 
   <td>lasso</td> 
   <td>glmnet</td> 
   <td>是</td> 
  </tr> 
  <tr> 
   <td>cv.glmnet</td> 
   <td>“gaussian”, “binomial”, “poisson”, “multinomial”, “cox”, “mgaussian”</td> 
   <td>CV</td> 
   <td>lasso</td> 
   <td>glmnet</td> 
   <td>是</td> 
  </tr> 
  <tr> 
   <td>aenet</td> 
   <td>“gaussian”, “binomial”, “poisson”, “cox”</td> 
   <td>“cv”, “ebic”, “bic”, “aic”</td> 
   <td>Adaptive lasso</td> 
   <td>msaenet</td> 
   <td>是</td> 
  </tr> 
  <tr> 
   <td>asnet</td> 
   <td>“gaussian”, “binomial”, “poisson”, “cox”</td> 
   <td>“cv”, “ebic”, “bic”, “aic”</td> 
   <td>SCAD</td> 
   <td>msaenet</td> 
   <td>是</td> 
  </tr> 
  <tr> 
   <td>amnet</td> 
   <td>“gaussian”, “binomial”, “poisson”, “cox”</td> 
   <td>“cv”, “ebic”, “bic”, “aic”</td> 
   <td>MCP</td> 
   <td>msaenet</td> 
   <td>是</td> 
  </tr> 
  <tr> 
   <td>ncvreg</td> 
   <td>“gaussian”, “binomial”, “poisson”</td> 
   <td>—</td> 
   <td>“MCP”, “SCAD”, “lasso”</td> 
   <td>ncvreg</td> 
   <td>否</td> 
  </tr> 
  <tr> 
   <td>cv.ncvreg</td> 
   <td>“gaussian”, “binomial”, “poisson”</td> 
   <td>CV</td> 
   <td>“MCP”, “SCAD”, “lasso”</td> 
   <td>ncvreg</td> 
   <td>否</td> 
  </tr> 
 </tbody> 
</table>

做lasso类变量选择的包有很多，上表只列出了三个我常用的。有知道更多包的大佬欢迎评论区留言或私信给我们安利，大家一块扩充知识。  
上表给出了包和函数主要的参数和模型，大部分熟悉R语言的可以很容易的安装实现自己想要的模型。但是对于初学者可能有些困难，为了大部分人快速入手上述包。下面生成合适的模型数据，以高斯的误差和lasso（或者Adaptive lasso）为例，使用CV选择作为选择惩罚因子的方法给出实现例子。

# 1、glmnet #

## 1.1生成数据 ##

本文使用的数据都是本节生成的，下面是生成代码：

> # R版本 3.6.3
    > Beta <- cbind(c(1,1,1,1,0,0)) # 真是的beta
    > n <- 500  # 样本量
    > set.seed(1234)  # 为了本文的复现，设置随机种子
    > x1 <- rnorm(n)
    > x2 <- rnorm(n)
    > x3 <- rnorm(n)
    > x4 <- rnorm(n)
    > x5 <- rnorm(n)
    > x6 <- rnorm(n)
    > x <- cbind(x1, x2, x3, x4, x5, x6)
    > y <- x %*% Beta + rnorm(n, 0, 0.5)
    > 
    > # 数据展示
    > head(x, 10)
                  x1          x2         x3         x4         x5         x6
     [1,] -1.2070657  0.98477997 -1.2053334  1.7940745 -0.9738186 -1.3662376
     [2,]  0.2774292 -1.22473788  0.3014667 -1.3645489 -0.0996312  0.5392093
     [3,]  1.0844412  0.70972622 -1.5391452 -0.7074400 -0.1107350 -1.3219320
     [4,] -2.3456977 -0.10921999  0.6353707 -0.5562843  1.1921946 -0.2812887
     [5,]  0.4291247  1.78260790  0.7029518 -0.3100811 -1.6558859 -2.1049469
     [6,]  0.5060559 -0.24344468 -1.9058829 -0.3761793 -1.0456433 -1.6176047
     [7,] -0.5747400 -1.52710702  0.9389214  0.4585260 -1.7402391 -0.7237319
     [8,] -0.5466319  0.49183437 -0.2244921 -1.2611491  0.5131208  0.3067410
     [9,] -0.5644520  0.35450366 -0.6738168 -0.5274652 -0.4459568  0.2255962
    [10,] -0.8900378 -0.01762635  0.4457874 -0.5568142 -1.8391938  0.9357160
    > head(y, 10)
                [,1]
     [1,]  0.1739191
     [2,] -1.7533630
     [3,] -0.2984157
     [4,] -1.4562544
     [5,]  3.4013056
     [6,] -2.1985494
     [7,] -1.2608381
     [8,] -1.2924795
     [9,] -2.0985074
    [10,] -0.7405859

## 1.2 glmnet::cv.glmnet ##

> (fit <- glmnet::cv.glmnet(x, y))
    
    Call:  glmnet::cv.glmnet(x = x, y = y) 
    
    Measure: Mean-Squared Error 
    
         Lambda Index Measure       SE Nonzero
    min 0.01019    52  0.2512 0.009655       5
    1se 0.05439    34  0.2605 0.010606       4
    > summary(fit)
               Length Class  Mode     
    lambda     57     -none- numeric  
    cvm        57     -none- numeric  
    cvsd       57     -none- numeric  
    cvup       57     -none- numeric  
    cvlo       57     -none- numeric  
    nzero      57     -none- numeric  
    call        3     -none- call     
    name        1     -none- character
    glmnet.fit 12     elnet  list     
    lambda.min  1     -none- numeric  
    lambda.1se  1     -none- numeric  
    index       2     -none- numeric  
    > str(fit)
    List of 12
     $ lambda    : num [1:57] 1.172 1.068 0.973 0.886 0.808 ...
     $ cvm       : num [1:57] 4.46 4.23 3.79 3.23 2.73 ...
     $ cvsd      : num [1:57] 0.369 0.376 0.364 0.323 0.272 ...
     $ cvup      : num [1:57] 4.83 4.6 4.16 3.56 3 ...
     $ cvlo      : num [1:57] 4.09 3.85 3.43 2.91 2.45 ...
     $ nzero     : Named int [1:57] 0 2 4 4 4 4 4 4 4 4 ...
      ..- attr(*, "names")= chr [1:57] "s0" "s1" "s2" "s3" ...
     $ call      : language glmnet::cv.glmnet(x = x, y = y)
     $ name      : Named chr "Mean-Squared Error"
      ..- attr(*, "names")= chr "mse"
     $ glmnet.fit:List of 12
      ..$ a0       : Named num [1:57] -0.000854 -0.000831 0.004037 0.005898 0.007593 ...
      .. ..- attr(*, "names")= chr [1:57] "s0" "s1" "s2" "s3" ...
      ..$ beta     :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
      .. .. ..@ i       : int [1:236] 0 1 0 1 2 3 0 1 2 3 ...
      .. .. ..@ p       : int [1:58] 0 0 2 6 10 14 18 22 26 30 ...
      .. .. ..@ Dim     : int [1:2] 6 57
      .. .. ..@ Dimnames:List of 2
      .. .. .. ..$ : chr [1:6] "x1" "x2" "x3" "x4" ...
      .. .. .. ..$ : chr [1:57] "s0" "s1" "s2" "s3" ...
      .. .. ..@ x       : num [1:236] 0.10041 0.00378 0.18545 0.09365 0.0017 ...
      .. .. ..@ factors : list()
      ..$ df       : int [1:57] 0 2 4 4 4 4 4 4 4 4 ...
      ..$ dim      : int [1:2] 6 57
      ..$ lambda   : num [1:57] 1.172 1.068 0.973 0.886 0.808 ...
      ..$ dev.ratio: num [1:57] 0 0.0537 0.1566 0.2905 0.4016 ...
      ..$ nulldev  : num 2235
      ..$ npasses  : int 206
      ..$ jerr     : int 0
      ..$ offset   : logi FALSE
      ..$ call     : language glmnet(x = x, y = y)
      ..$ nobs     : int 500
      ..- attr(*, "class")= chr [1:2] "elnet" "glmnet"
     $ lambda.min: num 0.0102
     $ lambda.1se: num 0.0544
     $ index     : int [1:2, 1] 52 34
      ..- attr(*, "dimnames")=List of 2
      .. ..$ : chr [1:2] "min" "1se"
      .. ..$ : chr "Lambda"
     - attr(*, "class")= chr "cv.glmnet"
    > coef(fit)
    7 x 1 sparse Matrix of class "dgCMatrix"
                         1
    (Intercept) 0.02380925
    x1          0.94151352
    x2          0.98011080
    x3          0.94789578
    x4          0.93311113
    x5          .         
    x6          .         
    >

# 2、msaenet #

> fit <- msaenet::aenet(x, y)
    > summary(fit)
                      Length Class     Mode   
    beta               6     dgCMatrix S4     
    model             12     elnet     list   
    beta.first         6     dgCMatrix S4     
    model.first       12     elnet     list   
    best.alpha.enet    1     -none-    numeric
    best.alpha.aenet   1     -none-    numeric
    best.lambda.enet   1     -none-    numeric
    best.lambda.aenet  1     -none-    numeric
    step.criterion     2     -none-    numeric
    adpen              6     -none-    numeric
    seed               1     -none-    numeric
    call               3     -none-    call   
    > str(fit)
    List of 12
     $ beta             :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
      .. ..@ i       : int [1:4] 0 1 2 3
      .. ..@ p       : int [1:2] 0 4
      .. ..@ Dim     : int [1:2] 6 1
      .. ..@ Dimnames:List of 2
      .. .. ..$ : chr [1:6] "x1" "x2" "x3" "x4" ...
      .. .. ..$ : chr "s0"
      .. ..@ x       : num [1:4] 0.98 1.026 0.997 0.978
      .. ..@ factors : list()
     $ model            :List of 12
      ..$ a0       : Named num 0.0248
      .. ..- attr(*, "names")= chr "s0"
      ..$ beta     :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
      .. .. ..@ i       : int [1:4] 0 1 2 3
      .. .. ..@ p       : int [1:2] 0 4
      .. .. ..@ Dim     : int [1:2] 6 1
      .. .. ..@ Dimnames:List of 2
      .. .. .. ..$ : chr [1:6] "x1" "x2" "x3" "x4" ...
      .. .. .. ..$ : chr "s0"
      .. .. ..@ x       : num [1:4] 0.98 1.026 0.997 0.978
      .. .. ..@ factors : list()
      ..$ df       : int 4
      ..$ dim      : int [1:2] 6 1
      ..$ lambda   : num 1.11e+13
      ..$ dev.ratio: num 0.945
      ..$ nulldev  : num 2235
      ..$ npasses  : int 5
      ..$ jerr     : int 0
      ..$ offset   : logi FALSE
      ..$ call     : language glmnet(x = x, y = y, family = family, alpha = best.alpha.aenet, lambda = best.lambda.aenet,      penalty.factor =| __truncated__
      ..$ nobs     : int 500
      ..- attr(*, "class")= chr [1:2] "elnet" "glmnet"
     $ beta.first       :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
      .. ..@ i       : int [1:5] 0 1 2 3 5
      .. ..@ p       : int [1:2] 0 5
      .. ..@ Dim     : int [1:2] 6 1
      .. ..@ Dimnames:List of 2
      .. .. ..$ : chr [1:6] "x1" "x2" "x3" "x4" ...
      .. .. ..$ : chr "s0"
      .. ..@ x       : num [1:5] 0.98 1.027 0.999 0.978 -0.017
      .. ..@ factors : list()
     $ model.first      :List of 12
      ..$ a0       : Named num 0.025
      .. ..- attr(*, "names")= chr "s0"
      ..$ beta     :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
      .. .. ..@ i       : int [1:5] 0 1 2 3 5
      .. .. ..@ p       : int [1:2] 0 5
      .. .. ..@ Dim     : int [1:2] 6 1
      .. .. ..@ Dimnames:List of 2
      .. .. .. ..$ : chr [1:6] "x1" "x2" "x3" "x4" ...
      .. .. .. ..$ : chr "s0"
      .. .. ..@ x       : num [1:5] 0.98 1.027 0.999 0.978 -0.017
      .. .. ..@ factors : list()
      ..$ df       : int 5
      ..$ dim      : int [1:2] 6 1
      ..$ lambda   : num 0.00686
      ..$ dev.ratio: num 0.945
      ..$ nulldev  : num 2235
      ..$ npasses  : int 5
      ..$ jerr     : int 0
      ..$ offset   : logi FALSE
      ..$ call     : language glmnet(x = x, y = y, family = family, alpha = best.alpha.enet, lambda = best.lambda.enet,      penalty.factor = p| __truncated__ ...
      ..$ nobs     : int 500
      ..- attr(*, "class")= chr [1:2] "elnet" "glmnet"
     $ best.alpha.enet  : num 0.85
     $ best.alpha.aenet : num 0.05
     $ best.lambda.enet : num 0.00686
     $ best.lambda.aenet: num 1.11e+13
     $ step.criterion   : num [1:2] 0.501 0.502
     $ adpen            : num [1:6, 1] 1.02 9.73e-01 1.00 1.02 4.50e+15 ...
      ..- attr(*, "dimnames")=List of 2
      .. ..$ : chr [1:6] "x1" "x2" "x3" "x4" ...
      .. ..$ : chr "s0"
     $ seed             : num 1001
     $ call             : language msaenet::aenet(x = x, y = y)
     - attr(*, "class")= chr [1:2] "msaenet" "msaenet.aenet"
    > coef(fit)
    [1] 0.9799217 1.0258625 0.9968319 0.9781661 0.0000000 0.0000000

# 3、ncvreg #

> fit <- ncvreg::cv.ncvreg(x, y)
    > summary(fit)
    MCP-penalized linear regression with n=500, p=6
    At minimum cross-validation error (lambda=0.1781):
    -------------------------------------------------
      Nonzero coefficients: 4
      Cross-validation error (deviance): 0.25
      R-squared: 0.94
      Signal-to-noise ratio: 16.69
      Scale estimate (sigma): 0.503
    > str(fit)
    List of 9
     $ cve       : num [1:100] 4.45 4.25 3.91 3.36 2.76 ...
     $ cvse      : num [1:100] 0.278 0.267 0.246 0.211 0.174 ...
     $ fold      : num [1:500] 2 2 5 1 1 2 8 9 7 4 ...
     $ lambda    : num [1:100] 1.172 1.093 1.019 0.95 0.886 ...
     $ fit       :List of 13
      ..$ beta          : num [1:7, 1:100] -0.000854 0 0 0 0 ...
      .. ..- attr(*, "dimnames")=List of 2
      .. .. ..$ : chr [1:7] "(Intercept)" "x1" "x2" "x3" ...
      .. .. ..$ : chr [1:100] "1.17175" "1.09278" "1.01913" "0.95044" ...
      ..$ iter          : int [1:100] 0 2 4 6 4 4 4 4 4 4 ...
      ..$ lambda        : num [1:100] 1.172 1.093 1.019 0.95 0.886 ...
      ..$ penalty       : chr "MCP"
      ..$ family        : chr "gaussian"
      ..$ gamma         : num 3
      ..$ alpha         : num 1
      ..$ convex.min    : NULL
      ..$ loss          : num [1:100] 2235 2103 1926 1642 1345 ...
      ..$ penalty.factor: num [1:6] 1 1 1 1 1 1
      ..$ n             : int 500
      ..$ X             : num [1:500, 1:6] -1.169 0.267 1.047 -2.271 0.413 ...
      .. ..- attr(*, "dimnames")=List of 2
      .. .. ..$ : NULL
      .. .. ..$ : chr [1:6] "x1" "x2" "x3" "x4" ...
      .. ..- attr(*, "center")= num [1:6] 0.00184 -0.05503 0.03168 -0.00266 0.05087 ...
      .. ..- attr(*, "scale")= num [1:6] 1.034 0.958 0.937 1.023 1.041 ...
      .. ..- attr(*, "nonsingular")= int [1:6] 1 2 3 4 5 6
      ..$ y             : num [1:500, 1] 0.175 -1.753 -0.298 -1.455 3.402 ...
      ..- attr(*, "class")= chr "ncvreg"
     $ min       : int 28
     $ lambda.min: num 0.178
     $ null.dev  : num 4.47
     $ Bias      : num 0.00686
     - attr(*, "class")= chr "cv.ncvreg"
    > coef(fit)
    (Intercept)          x1          x2          x3          x4          x5 
     0.02498015  0.98628670  1.03260613  1.00392827  0.98542619  0.00000000 
             x6 
     0.00000000

# 4、结果对比 #

> coef(fit1) # cv.glmnet
    7 x 1 sparse Matrix of class "dgCMatrix"
                        1
    (Intercept) 0.0235698
    x1          0.9323572
    x2          0.9693753
    x3          0.9364369
    x4          0.9224125
    x5          .        
    x6          .        
    > coef(fit2) # cv.ncvreg
    (Intercept)          x1          x2          x3          x4          x5 
     0.02483197  0.97635269  1.02273049  0.99368238  0.97404453  0.00000000 
             x6 
    -0.01177071 
    > coef(fit3) # aenet
    [1] 0.9804661 1.0261746 0.9968471 0.9786486 0.0000000 0.0000000

# 5、总结 #

希望可以帮助大家提高R水平。  
水平有限发现错误还望及时评论区指正，您的意见和批评是我不断前进的动力。

# 6、纠错代码 #

(fit1 <- glmnet::cv.glmnet(x, y, alpha = 1, family = "gaussian"))
    summary(fit1)
    str(fit1)
    coef(fit1) # cv.glmnet
    ?glmnet
    
    fit2 <- ncvreg::cv.ncvreg(x, y, family = "gaussian", penalty = "lasso")
    summary(fit2)
    str(fit2)
    coef(fit2) # cv.ncvreg
    
    
    
    fit3 <- msaenet::aenet(x, y, alphas = 1)
    summary(fit3)
    str(fit3)
    coef(fit3) # aenet