Linear Regression
yongchan
2015년 6월 22일
Linear Regression
#########################################
######################################### Linear regression
######################################### by Housing Values in Suburbs of Boston
#install.packages("MASS")
library(MASS)
plot(medv~lstat,Boston)
fit1=lm(medv~lstat,data=Boston) #simple linear regression
fit1##
## Call:
## lm(formula = medv ~ lstat, data = Boston)
##
## Coefficients:
## (Intercept) lstat
## 34.55 -0.95summary(fit1)##
## Call:
## lm(formula = medv ~ lstat, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.168 -3.990 -1.318 2.034 24.500
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.55384 0.56263 61.41 <2e-16 ***
## lstat -0.95005 0.03873 -24.53 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.216 on 504 degrees of freedom
## Multiple R-squared: 0.5441, Adjusted R-squared: 0.5432
## F-statistic: 601.6 on 1 and 504 DF, p-value: < 2.2e-16abline(fit1,col="red")
fit2=lm(medv~lstat+age,data=Boston) #multiple linear regression
summary(fit2)##
## Call:
## lm(formula = medv ~ lstat + age, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.981 -3.978 -1.283 1.968 23.158
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.22276 0.73085 45.458 < 2e-16 ***
## lstat -1.03207 0.04819 -21.416 < 2e-16 ***
## age 0.03454 0.01223 2.826 0.00491 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.173 on 503 degrees of freedom
## Multiple R-squared: 0.5513, Adjusted R-squared: 0.5495
## F-statistic: 309 on 2 and 503 DF, p-value: < 2.2e-16fit3=lm(medv~.,Boston) #multiple linear regression(all variables)
summary(fit3)##
## Call:
## lm(formula = medv ~ ., data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.595 -2.730 -0.518 1.777 26.199
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.646e+01 5.103e+00 7.144 3.28e-12 ***
## crim -1.080e-01 3.286e-02 -3.287 0.001087 **
## zn 4.642e-02 1.373e-02 3.382 0.000778 ***
## indus 2.056e-02 6.150e-02 0.334 0.738288
## chas 2.687e+00 8.616e-01 3.118 0.001925 **
## nox -1.777e+01 3.820e+00 -4.651 4.25e-06 ***
## rm 3.810e+00 4.179e-01 9.116 < 2e-16 ***
## age 6.922e-04 1.321e-02 0.052 0.958229
## dis -1.476e+00 1.995e-01 -7.398 6.01e-13 ***
## rad 3.060e-01 6.635e-02 4.613 5.07e-06 ***
## tax -1.233e-02 3.760e-03 -3.280 0.001112 **
## ptratio -9.527e-01 1.308e-01 -7.283 1.31e-12 ***
## black 9.312e-03 2.686e-03 3.467 0.000573 ***
## lstat -5.248e-01 5.072e-02 -10.347 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.745 on 492 degrees of freedom
## Multiple R-squared: 0.7406, Adjusted R-squared: 0.7338
## F-statistic: 108.1 on 13 and 492 DF, p-value: < 2.2e-16par(mfrow=c(2,2)) #Resdiual analysis for fit3
plot(fit3)
#########################################
######################################### Advanced Linear regression
######################################### by Housing Values in Suburbs of Boston
#install.packages("car")
library(car)
#VIF
vif(fit3)## crim zn indus chas nox rm age dis
## 1.792192 2.298758 3.991596 1.073995 4.393720 1.933744 3.100826 3.955945
## rad tax ptratio black lstat
## 7.484496 9.008554 1.799084 1.348521 2.941491# condition number
X=model.matrix(fit3)[,-1]
sqrt(eigen(cor(X))$values[1]/eigen(cor(X))$values)## [1] 1.000000 2.067539 2.220496 2.672899 2.709089 3.052821 3.382963
## [8] 3.932943 4.703521 5.274395 5.739120 6.015702 9.822003# AIC
step(fit3,direction="backward")## Start: AIC=1589.64
## medv ~ crim + zn + indus + chas + nox + rm + age + dis + rad +
## tax + ptratio + black + lstat
##
## Df Sum of Sq RSS AIC
## - age 1 0.06 11079 1587.7
## - indus 1 2.52 11081 1587.8
## <none> 11079 1589.6
## - chas 1 218.97 11298 1597.5
## - tax 1 242.26 11321 1598.6
## - crim 1 243.22 11322 1598.6
## - zn 1 257.49 11336 1599.3
## - black 1 270.63 11349 1599.8
## - rad 1 479.15 11558 1609.1
## - nox 1 487.16 11566 1609.4
## - ptratio 1 1194.23 12273 1639.4
## - dis 1 1232.41 12311 1641.0
## - rm 1 1871.32 12950 1666.6
## - lstat 1 2410.84 13490 1687.3
##
## Step: AIC=1587.65
## medv ~ crim + zn + indus + chas + nox + rm + dis + rad + tax +
## ptratio + black + lstat
##
## Df Sum of Sq RSS AIC
## - indus 1 2.52 11081 1585.8
## <none> 11079 1587.7
## - chas 1 219.91 11299 1595.6
## - tax 1 242.24 11321 1596.6
## - crim 1 243.20 11322 1596.6
## - zn 1 260.32 11339 1597.4
## - black 1 272.26 11351 1597.9
## - rad 1 481.09 11560 1607.2
## - nox 1 520.87 11600 1608.9
## - ptratio 1 1200.23 12279 1637.7
## - dis 1 1352.26 12431 1643.9
## - rm 1 1959.55 13038 1668.0
## - lstat 1 2718.88 13798 1696.7
##
## Step: AIC=1585.76
## medv ~ crim + zn + chas + nox + rm + dis + rad + tax + ptratio +
## black + lstat
##
## Df Sum of Sq RSS AIC
## <none> 11081 1585.8
## - chas 1 227.21 11309 1594.0
## - crim 1 245.37 11327 1594.8
## - zn 1 257.82 11339 1595.4
## - black 1 270.82 11352 1596.0
## - tax 1 273.62 11355 1596.1
## - rad 1 500.92 11582 1606.1
## - nox 1 541.91 11623 1607.9
## - ptratio 1 1206.45 12288 1636.0
## - dis 1 1448.94 12530 1645.9
## - rm 1 1963.66 13045 1666.3
## - lstat 1 2723.48 13805 1695.0##
## Call:
## lm(formula = medv ~ crim + zn + chas + nox + rm + dis + rad +
## tax + ptratio + black + lstat, data = Boston)
##
## Coefficients:
## (Intercept) crim zn chas nox
## 36.341145 -0.108413 0.045845 2.718716 -17.376023
## rm dis rad tax ptratio
## 3.801579 -1.492711 0.299608 -0.011778 -0.946525
## black lstat
## 0.009291 -0.522553