tutoring2
mtcars %>% head
## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1
y <- mtcars$mpg
x <- mtcars$disp
plot(y,x)
lm(y ~ x) %>% summary()
##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.8922 -2.2022 -0.9631 1.6272 7.2305
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.599855 1.229720 24.070 < 2e-16 ***
## x -0.041215 0.004712 -8.747 9.38e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.251 on 30 degrees of freedom
## Multiple R-squared: 0.7183, Adjusted R-squared: 0.709
## F-statistic: 76.51 on 1 and 30 DF, p-value: 9.38e-10
lm(y ~ poly(x,10)) %>% summary()
##
## Call:
## lm(formula = y ~ poly(x, 10))
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.1828 -1.7280 -0.1159 1.0308 3.6278
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20.09062 0.38020 52.842 < 2e-16 ***
## poly(x, 10)1 -28.44097 2.15075 -13.224 1.19e-11 ***
## poly(x, 10)2 9.15235 2.15075 4.255 0.000353 ***
## poly(x, 10)3 -9.74460 2.15075 -4.531 0.000183 ***
## poly(x, 10)4 -0.01148 2.15075 -0.005 0.995792
## poly(x, 10)5 -4.50020 2.15075 -2.092 0.048731 *
## poly(x, 10)6 -0.47751 2.15075 -0.222 0.826443
## poly(x, 10)7 3.06010 2.15075 1.423 0.169478
## poly(x, 10)8 3.23277 2.15075 1.503 0.147706
## poly(x, 10)9 0.11928 2.15075 0.055 0.956296
## poly(x, 10)10 -0.99337 2.15075 -0.462 0.648923
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.151 on 21 degrees of freedom
## Multiple R-squared: 0.9137, Adjusted R-squared: 0.8727
## F-statistic: 22.24 on 10 and 21 DF, p-value: 5.663e-09
lm(y ~ poly(x,19)) %>% summary()
##
## Call:
## lm(formula = y ~ poly(x, 19))
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.0007 -0.9331 0.0001 0.5718 3.4166
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20.09062 0.39167 51.294 1.98e-15 ***
## poly(x, 19)1 -28.44097 2.21564 -12.836 2.27e-08 ***
## poly(x, 19)2 9.15235 2.21564 4.131 0.001394 **
## poly(x, 19)3 -9.74460 2.21564 -4.398 0.000868 ***
## poly(x, 19)4 -0.01148 2.21564 -0.005 0.995951
## poly(x, 19)5 -4.50020 2.21564 -2.031 0.065002 .
## poly(x, 19)6 -0.47751 2.21564 -0.216 0.832984
## poly(x, 19)7 3.06010 2.21564 1.381 0.192415
## poly(x, 19)8 3.23277 2.21564 1.459 0.170222
## poly(x, 19)9 0.11928 2.21564 0.054 0.957951
## poly(x, 19)10 -0.99337 2.21564 -0.448 0.661893
## poly(x, 19)11 0.54355 2.21564 0.245 0.810352
## poly(x, 19)12 1.34286 2.21564 0.606 0.555750
## poly(x, 19)13 -4.50100 2.21564 -2.031 0.064960 .
## poly(x, 19)14 2.15493 2.21564 0.973 0.349950
## poly(x, 19)15 2.08077 2.21564 0.939 0.366185
## poly(x, 19)16 2.12840 2.21564 0.961 0.355698
## poly(x, 19)17 -0.97140 2.21564 -0.438 0.668863
## poly(x, 19)18 -1.01995 2.21564 -0.460 0.653501
## poly(x, 19)19 0.62153 2.21564 0.281 0.783858
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.216 on 12 degrees of freedom
## Multiple R-squared: 0.9477, Adjusted R-squared: 0.8649
## F-statistic: 11.44 on 19 and 12 DF, p-value: 5.419e-05
这里可以看到把 x 的 1到10次方都加入后，R^2 会越来越高。 如果 x 放到无穷，那么 R^2 = 100%，相当于影响关系和噪音都抓取了。
但是这其实使用的是泰勒公式展开，有兴趣你可以去查下。
另外，这种处理方法，很容易让这个 x 的组合变量是过拟合。
所以一般来说，大家考虑使用 交叉项，也就是 x1 和 x2 等的乘数，而不是这种暴力的提高x的次方。
更多地，你可以参考 James et al. (2013, 115)
James, Gareth, Daniela Witten, Trevor Hastie, and Robert Tibshirani.
2013. *An Introduction to Statistical Learning with Applications in R*.
8th ed. Springer New York.