模拟假数据集
set.seed(0820) n <- 50 p <- 25 beta <- rep(0,p) beta\[1:5\] <- 1:5/5 X <- matrix(rnorm(n*p),n,p) X <- scale(X) Xb <- X%*%beta Y <- X%*%beta+rnorm(n) Y <- Y-mean(Y) plot(cor(X,Y),xlab="j",ylab="Cor(Y,X_j)",main="Sample correlations",cex=2)
标准最小二乘法
summary(ols) ## ## Call: ## ## Residuals: ## Min 1Q Median 3Q Max ## -2.54540 -0.38971 -0.00738 0.49058 1.90900 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -3.914e-16 1.648e-01 0.000 1.00000 ## X1 6.020e-01 2.097e-01 2.871 0.00841 ** ## X2 5.924e-01 2.013e-01 2.944 0.00709 ** ## X3 -1.106e-01 2.290e-01 -0.483 0.63363 ## X4 1.117e+00 2.058e-01 5.427 1.42e-05 *** ## X5 1.234e+00 2.190e-01 5.633 8.46e-06 *** ## X6 -3.225e-01 2.322e-01 -1.389 0.17755 ## X7 -1.954e-01 2.150e-01 -0.909 0.37231 ## X8 1.466e-01 2.532e-01 0.579 0.56803 ## X9 4.678e-02 2.353e-01 0.199 0.84409 ## X10 -2.779e-01 2.151e-01 -1.292 0.20864 ## X11 -7.308e-02 2.553e-01 -0.286 0.77717 ## X12 -4.424e-02 2.642e-01 -0.167 0.86839 ## X13 -1.078e-01 2.101e-01 -0.513 0.61270 ## X14 3.000e-01 2.263e-01 1.326 0.19743 ## X15 2.396e-01 2.480e-01 0.966 0.34365 ## X16 -1.178e-01 2.285e-01 -0.515 0.61100 ## X17 -2.409e-01 2.280e-01 -1.057 0.30104 ## X18 -3.191e-01 2.396e-01 -1.332 0.19551 ## X19 -1.207e-01 2.372e-01 -0.509 0.61553 ## X20 1.721e-01 2.179e-01 0.790 0.43733 ## X21 -1.677e-02 2.144e-01 -0.078 0.93831 ## X22 3.706e-02 2.145e-01 0.173 0.86426 ## X23 3.233e-02 2.108e-01 0.153 0.87938 ## X24 -1.541e-01 2.343e-01 -0.658 0.51691 ## X25 -1.970e-01 1.965e-01 -1.002 0.32622 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 1.166 on 24 degrees of freedom ## Multiple R-squared: 0.8416, Adjusted R-squared: 0.6767 ## F-statistic: 5.102 on 25 and 24 DF, p-value: 7.861e-05
LASSO
`````` plot(lasso)
使用BIC选择路径上的最佳点
## df MSE bic ## Intercept 1 4.1174138 74.67329 ## 2 3.8224639 74.86881 ## 3 1.9171062 44.27691 ## 4 1.9136899 48.09976 ## 5 1.5118875 40.22806 ## 6 1.3016560 36.65400 ## 7 1.2693779 39.31051 ## 8 1.2124870 40.92986 ## 9 1.1814011 43.54326 ## 10 1.1728179 47.09070 ## 11 1.1016346 47.87201 ## 12 1.0050559 47.19643 ## 13 0.9867377 50.18875 ## 14 0.9636054 52.91465 ## 15 0.8686856 51.64164 ## 16 0.7777734 50.02637 ## 17 0.7700763 53.44111 ## 18 0.7663544 57.11089 ## 19 0.7510361 60.01336 ## 20 0.7451598 63.53263 ## 19 0.7196873 57.88151 ## 20 0.7149486 61.46323 ## 21 0.7141592 65.32002 ## 20 0.7051259 60.77152 ## 21 0.6875391 63.42065 ## 22 0.6764241 66.51776 ## 23 0.6739037 70.24313 ## 24 0.6570954 72.89225 ## 25 0.6564105 76.75213 ## 26 0.6520870 80.33373
结果
# beta真值 ## \[1\] 0.2 0.4 0.6 0.8 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ## \[18\] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 #使用最小二乘预测beta round(beta_ols,3) ## X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 ## 0.602 0.592 -0.111 1.117 1.234 -0.323 -0.195 0.147 0.047 -0.278 ## X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 ## -0.073 -0.044 -0.108 0.300 0.240 -0.118 -0.241 -0.319 -0.121 0.172 ## X21 X22 X23 X24 X25 ## -0.017 0.037 0.032 -0.154 -0.197 # LASSO预测beta round(beta_lasso,3) ## \[1\] 0.238 0.238 0.000 0.900 0.786 0.000 0.000 0.000 0.000 0.000 ## \[11\] 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 ## \[21\] 0.000 0.000 0.000 0.000 -0.075 #MSE - OLS mean((beta-beta_ols)^2) ## \[1\] 0.06204978 #MSE - LASSO mean((beta-beta_lasso)^2) ## \[1\] 0.01795647
