Linear Regression in R
Hồi quy tuyến tính trên R
library(MASS)
library(ISLR)Hồi quy tuyến tính giản đơn
names(Boston)## [1] "crim" "zn" "indus" "chas" "nox" "rm" "age"
## [8] "dis" "rad" "tax" "ptratio" "black" "lstat" "medv"attach(Boston)## The following objects are masked from Boston (pos = 4):
##
## age, black, chas, crim, dis, indus, lstat, medv, nox, ptratio,
## rad, rm, tax, znlm.fit=lm(medv~lstat)
lm.fit##
## Call:
## lm(formula = medv ~ lstat)
##
## Coefficients:
## (Intercept) lstat
## 34.55 -0.95summary(lm.fit)##
## Call:
## lm(formula = medv ~ lstat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.17 -3.99 -1.32 2.03 24.50
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.5538 0.5626 61.4 <2e-16 ***
## lstat -0.9500 0.0387 -24.5 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.2 on 504 degrees of freedom
## Multiple R-squared: 0.544, Adjusted R-squared: 0.543
## F-statistic: 602 on 1 and 504 DF, p-value: <2e-16names(lm.fit)## [1] "coefficients" "residuals" "effects" "rank"
## [5] "fitted.values" "assign" "qr" "df.residual"
## [9] "xlevels" "call" "terms" "model"coef(lm.fit)## (Intercept) lstat
## 34.55 -0.95confint(lm.fit)## 2.5 % 97.5 %
## (Intercept) 33 35.66
## lstat -1 -0.87predict(lm.fit,data.frame(lstat=(c(5,10,15))), interval="confidence")## fit lwr upr
## 1 30 29 31
## 2 25 24 26
## 3 20 20 21predict(lm.fit,data.frame(lstat=(c(5,10,15))), interval="prediction")## fit lwr upr
## 1 30 17.6 42
## 2 25 12.8 37
## 3 20 8.1 33plot(lstat,medv)
abline(lm.fit)
abline(lm.fit,lwd=3)
abline(lm.fit,lwd=3,col="red")
plot(lstat,medv,col="red")
plot(lstat,medv,pch=20)
plot(lstat,medv,pch="+")
plot(1:20,1:20,pch=1:20)
par(mfrow=c(2,2))
plot(lm.fit)
plot(predict(lm.fit), residuals(lm.fit))
plot(predict(lm.fit), rstudent(lm.fit))
plot(hatvalues(lm.fit))
which.max(hatvalues(lm.fit))## 375
## 375
Hồi quy tuyến tính nhiều biến
lm.fit=lm(medv~lstat+age,data=Boston)
summary(lm.fit)##
## Call:
## lm(formula = medv ~ lstat + age, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.98 -3.98 -1.28 1.97 23.16
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.2228 0.7308 45.46 <2e-16 ***
## lstat -1.0321 0.0482 -21.42 <2e-16 ***
## age 0.0345 0.0122 2.83 0.0049 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.2 on 503 degrees of freedom
## Multiple R-squared: 0.551, Adjusted R-squared: 0.549
## F-statistic: 309 on 2 and 503 DF, p-value: <2e-16lm.fit=lm(medv~.,data=Boston)
summary(lm.fit)##
## Call:
## lm(formula = medv ~ ., data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.594 -2.730 -0.518 1.777 26.199
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.65e+01 5.10e+00 7.14 3.3e-12 ***
## crim -1.08e-01 3.29e-02 -3.29 0.00109 **
## zn 4.64e-02 1.37e-02 3.38 0.00078 ***
## indus 2.06e-02 6.15e-02 0.33 0.73829
## chas 2.69e+00 8.62e-01 3.12 0.00193 **
## nox -1.78e+01 3.82e+00 -4.65 4.2e-06 ***
## rm 3.81e+00 4.18e-01 9.12 < 2e-16 ***
## age 6.92e-04 1.32e-02 0.05 0.95823
## dis -1.48e+00 1.99e-01 -7.40 6.0e-13 ***
## rad 3.06e-01 6.63e-02 4.61 5.1e-06 ***
## tax -1.23e-02 3.76e-03 -3.28 0.00111 **
## ptratio -9.53e-01 1.31e-01 -7.28 1.3e-12 ***
## black 9.31e-03 2.69e-03 3.47 0.00057 ***
## lstat -5.25e-01 5.07e-02 -10.35 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.8 on 492 degrees of freedom
## Multiple R-squared: 0.741, Adjusted R-squared: 0.734
## F-statistic: 108 on 13 and 492 DF, p-value: <2e-16?summary.lm
summary(lm.fit)$r.sq## [1] 0.74summary(lm.fit)$sigma## [1] 4.7# install.packages("car")
library(car)
vif(lm.fit)## crim zn indus chas nox rm age dis rad
## 1.8 2.3 4.0 1.1 4.4 1.9 3.1 4.0 7.5
## tax ptratio black lstat
## 9.0 1.8 1.3 2.9lm.fit1=lm(medv~.-age,data=Boston)
summary(lm.fit1)##
## Call:
## lm(formula = medv ~ . - age, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.605 -2.731 -0.519 1.760 26.224
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 36.43693 5.08012 7.17 2.7e-12 ***
## crim -0.10801 0.03283 -3.29 0.00107 **
## zn 0.04633 0.01361 3.40 0.00072 ***
## indus 0.02056 0.06143 0.33 0.73799
## chas 2.68903 0.85960 3.13 0.00186 **
## nox -17.71354 3.67931 -4.81 2.0e-06 ***
## rm 3.81439 0.40848 9.34 < 2e-16 ***
## dis -1.47861 0.19061 -7.76 5.0e-14 ***
## rad 0.30579 0.06609 4.63 4.8e-06 ***
## tax -0.01233 0.00376 -3.28 0.00110 **
## ptratio -0.95221 0.13029 -7.31 1.1e-12 ***
## black 0.00932 0.00268 3.48 0.00054 ***
## lstat -0.52385 0.04763 -11.00 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.7 on 493 degrees of freedom
## Multiple R-squared: 0.741, Adjusted R-squared: 0.734
## F-statistic: 117 on 12 and 493 DF, p-value: <2e-16# lm.fit1=update(lm.fit, ~.-age)Tương tác giữa các biến
summary(lm(medv~lstat*age,data=Boston))##
## Call:
## lm(formula = medv ~ lstat * age, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.81 -4.04 -1.33 2.08 27.55
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 36.088536 1.469835 24.55 < 2e-16 ***
## lstat -1.392117 0.167456 -8.31 8.8e-16 ***
## age -0.000721 0.019879 -0.04 0.971
## lstat:age 0.004156 0.001852 2.24 0.025 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.2 on 502 degrees of freedom
## Multiple R-squared: 0.556, Adjusted R-squared: 0.553
## F-statistic: 209 on 3 and 502 DF, p-value: <2e-16Chuyển đổi phi tuyến các biến dự báo
lm.fit2=lm(medv~lstat+I(lstat^2))
summary(lm.fit2)##
## Call:
## lm(formula = medv ~ lstat + I(lstat^2))
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.28 -3.83 -0.53 2.31 25.41
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 42.86201 0.87208 49.1 <2e-16 ***
## lstat -2.33282 0.12380 -18.8 <2e-16 ***
## I(lstat^2) 0.04355 0.00375 11.6 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.5 on 503 degrees of freedom
## Multiple R-squared: 0.641, Adjusted R-squared: 0.639
## F-statistic: 449 on 2 and 503 DF, p-value: <2e-16lm.fit=lm(medv~lstat)
anova(lm.fit,lm.fit2)## Analysis of Variance Table
##
## Model 1: medv ~ lstat
## Model 2: medv ~ lstat + I(lstat^2)
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 504 19472
## 2 503 15347 1 4125 135 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1par(mfrow=c(2,2))
plot(lm.fit2)
lm.fit5=lm(medv~poly(lstat,5))
summary(lm.fit5)##
## Call:
## lm(formula = medv ~ poly(lstat, 5))
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.543 -3.104 -0.705 2.084 27.115
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 22.533 0.232 97.20 < 2e-16 ***
## poly(lstat, 5)1 -152.460 5.215 -29.24 < 2e-16 ***
## poly(lstat, 5)2 64.227 5.215 12.32 < 2e-16 ***
## poly(lstat, 5)3 -27.051 5.215 -5.19 3.1e-07 ***
## poly(lstat, 5)4 25.452 5.215 4.88 1.4e-06 ***
## poly(lstat, 5)5 -19.252 5.215 -3.69 0.00025 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.2 on 500 degrees of freedom
## Multiple R-squared: 0.682, Adjusted R-squared: 0.679
## F-statistic: 214 on 5 and 500 DF, p-value: <2e-16summary(lm(medv~log(rm),data=Boston))##
## Call:
## lm(formula = medv ~ log(rm), data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.49 -2.87 -0.10 2.84 39.82
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -76.49 5.03 -15.2 <2e-16 ***
## log(rm) 54.05 2.74 19.7 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.9 on 504 degrees of freedom
## Multiple R-squared: 0.436, Adjusted R-squared: 0.435
## F-statistic: 389 on 1 and 504 DF, p-value: <2e-16Các biến dự báo định tính
# fix(Carseats)
names(Carseats)## [1] "Sales" "CompPrice" "Income" "Advertising" "Population"
## [6] "Price" "ShelveLoc" "Age" "Education" "Urban"
## [11] "US"lm.fit=lm(Sales~.+Income:Advertising+Price:Age,data=Carseats)
summary(lm.fit)##
## Call:
## lm(formula = Sales ~ . + Income:Advertising + Price:Age, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.921 -0.750 0.018 0.675 3.341
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.575565 1.008747 6.52 2.2e-10 ***
## CompPrice 0.092937 0.004118 22.57 < 2e-16 ***
## Income 0.010894 0.002604 4.18 3.6e-05 ***
## Advertising 0.070246 0.022609 3.11 0.00203 **
## Population 0.000159 0.000368 0.43 0.66533
## Price -0.100806 0.007440 -13.55 < 2e-16 ***
## ShelveLocGood 4.848676 0.152838 31.72 < 2e-16 ***
## ShelveLocMedium 1.953262 0.125768 15.53 < 2e-16 ***
## Age -0.057947 0.015951 -3.63 0.00032 ***
## Education -0.020852 0.019613 -1.06 0.28836
## UrbanYes 0.140160 0.112402 1.25 0.21317
## USYes -0.157557 0.148923 -1.06 0.29073
## Income:Advertising 0.000751 0.000278 2.70 0.00729 **
## Price:Age 0.000107 0.000133 0.80 0.42381
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1 on 386 degrees of freedom
## Multiple R-squared: 0.876, Adjusted R-squared: 0.872
## F-statistic: 210 on 13 and 386 DF, p-value: <2e-16attach(Carseats)
contrasts(ShelveLoc)## Good Medium
## Bad 0 0
## Good 1 0
## Medium 0 1