---
tags: ISLR
---

# ISLR hw5

Range: 「課本第三章習題：第7題、第9題、第10題、第12題、第四章習題：第2題」

## Q7
pass

## Q9

library(ISLR)
data(Auto)

### a

pairs(Auto)


### b

# exclude column name
cor(Auto[, -9])

#                     mpg  cylinders displacement horsepower
# mpg           1.0000000 -0.7776175   -0.8051269 -0.7784268
# cylinders    -0.7776175  1.0000000    0.9508233  0.8429834
# displacement -0.8051269  0.9508233    1.0000000  0.8972570
# horsepower   -0.7784268  0.8429834    0.8972570  1.0000000
# weight       -0.8322442  0.8975273    0.9329944  0.8645377
# acceleration  0.4233285 -0.5046834   -0.5438005 -0.6891955
# year          0.5805410 -0.3456474   -0.3698552 -0.4163615
# origin        0.5652088 -0.5689316   -0.6145351 -0.4551715
#                  weight acceleration       year     origin
# mpg          -0.8322442    0.4233285  0.5805410  0.5652088
# cylinders     0.8975273   -0.5046834 -0.3456474 -0.5689316
# displacement  0.9329944   -0.5438005 -0.3698552 -0.6145351
# horsepower    0.8645377   -0.6891955 -0.4163615 -0.4551715
# weight        1.0000000   -0.4168392 -0.3091199 -0.5850054
# acceleration -0.4168392    1.0000000  0.2903161  0.2127458
# year         -0.3091199    0.2903161  1.0000000  0.1815277
# origin       -0.5850054    0.2127458  0.1815277  1.0000000

### c

mpg_lm <-lm(mpg ~ . - name, data = Auto)
summary(mpg_lm)

# Call:
# lm(formula = mpg ~ . - name, data = Auto)

# Residuals:
#     Min      1Q  Median      3Q     Max 
# -9.5903 -2.1565 -0.1169  1.8690 13.0604 

# Coefficients:
#                Estimate Std. Error t value Pr(>|t|)    
# (Intercept)  -17.218435   4.644294  -3.707  0.00024 ***
# cylinders     -0.493376   0.323282  -1.526  0.12780    
# displacement   0.019896   0.007515   2.647  0.00844 ** 
# horsepower    -0.016951   0.013787  -1.230  0.21963    
# weight        -0.006474   0.000652  -9.929  < 2e-16 ***
# acceleration   0.080576   0.098845   0.815  0.41548    
# year           0.750773   0.050973  14.729  < 2e-16 ***
# origin         1.426141   0.278136   5.127 4.67e-07 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

# Residual standard error: 3.328 on 384 degrees of freedom
# Multiple R-squared:  0.8215,	Adjusted R-squared:  0.8182 
# F-statistic: 252.4 on 7 and 384 DF,  p-value: < 2.2e-16

#### i
Yes, p < 0.05

#### ii
displacement, weight, year and origin have stronger confidence have relation in statistic

#### iii
This lm suggest 0.751 for year.

### d

There are some outliers through the residual plot.
Leverge plot show a value have very high leverge 14.

### e
From last week's cor plot

displacement:cylinders and displacement:weight have quiet high correlation.
mpg:weight have the highest negative in correlation.
We check this 3 interaction in lm

cor_lm <-lm(mpg ~ displacement*cylinders + displacement*weight + mpg*weight, data = Auto)
summary(cor_lm)

# Call:
# lm(formula = mpg ~ displacement * cylinders + displacement * 
#     weight + mpg * weight, data = Auto)

# Residuals:
#     Min      1Q  Median      3Q     Max 
# -3.8503 -0.6281 -0.0573  0.5452  4.7186 

# Coefficients:
#                          Estimate Std. Error t value Pr(>|t|)    
# (Intercept)             3.162e+01  6.493e-01  48.696  < 2e-16 ***
# displacement           -7.004e-02  4.342e-03 -16.129  < 2e-16 ***
# cylinders               1.283e+00  1.996e-01   6.425 3.89e-10 ***
# weight                 -1.280e-02  3.481e-04 -36.786  < 2e-16 ***
# displacement:cylinders -2.823e-03  8.910e-04  -3.169  0.00165 ** 
# displacement:weight     2.463e-05  1.302e-06  18.921  < 2e-16 ***
# mpg:weight              3.691e-04  5.060e-06  72.945  < 2e-16 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

# Residual standard error: 1.067 on 385 degrees of freedom
# Multiple R-squared:  0.9816,	Adjusted R-squared:  0.9813 
# F-statistic:  3422 on 6 and 385 DF,  p-value: < 2.2e-16

Interaction part, displacement:weight and mpg:weight have stronger confidence have relation with y, displacement:cylinders's confidence is smaller.

### f

Apply four kinds of transformation into weight, show plot with mpg


log and sqrt have better shape in my opinion, they are more likely to linear relation.

## Q10

library(ISLR)
data(Carseats)

### a

carseat_lm <- lm(Sales ~ Price + Urban + US, data=Carseats)
summary(carseat_lm)

# Call:
# lm(formula = Sales ~ Price + Urban + US, data = Carseats)

# Residuals:
#     Min      1Q  Median      3Q     Max 
# -6.9206 -1.6220 -0.0564  1.5786  7.0581 

# Coefficients:
#              Estimate Std. Error t value Pr(>|t|)    
# (Intercept) 13.043469   0.651012  20.036  < 2e-16 ***
# Price       -0.054459   0.005242 -10.389  < 2e-16 ***
# UrbanYes    -0.021916   0.271650  -0.081    0.936    
# USYes        1.200573   0.259042   4.635 4.86e-06 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

# Residual standard error: 2.472 on 396 degrees of freedom
# Multiple R-squared:  0.2393,	Adjusted R-squared:  0.2335 
# F-statistic: 41.52 on 3 and 396 DF,  p-value: < 2.2e-16

### b
Price, there are some relation because have low p value.
Higher Price will cause lower Sales.

UrbanYes, there are no relation because p value is so high.

USYes, have relation.
If the store located in US, Sales will be higher about 1.2

### c
$Sales = 13.043 - 0.054 * Price + 1.201 * USYes + b$
If the shop located in US, USYes will be 1, otherwise, will be 0.

### d
We can reject null hypothesis in Price and USYes, cuz their p-value is low(< 0.05)

### e

evidence_lm <- lm(Sales ~ Price + US, data=Carseats)
summary(evidence_lm)

# Call:
# lm(formula = Sales ~ Price + US, data = Carseats)

# Residuals:
#     Min      1Q  Median      3Q     Max 
# -6.9269 -1.6286 -0.0574  1.5766  7.0515 

# Coefficients:
#             Estimate Std. Error t value Pr(>|t|)    
# (Intercept) 13.03079    0.63098  20.652  < 2e-16 ***
# Price       -0.05448    0.00523 -10.416  < 2e-16 ***
# USYes        1.19964    0.25846   4.641 4.71e-06 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

# Residual standard error: 2.469 on 397 degrees of freedom
# Multiple R-squared:  0.2393,	Adjusted R-squared:  0.2354 
# F-statistic: 62.43 on 2 and 397 DF,  p-value: < 2.2e-16

### f
They are almost the same.
f just be a little better than a

### g

confint(evidence_lm)

#                   2.5 %      97.5 %
# (Intercept) 11.79032020 14.27126531
# Price       -0.06475984 -0.04419543
# USYes        0.69151957  1.70776632

### h

par(mfrow =c(2, 2))
plot(evidence_lm)


From leverge, there are some outliers.

## Q12
### a

$\beta_x = \sum_{i=1}^{n} x_i y_i / \sum_{i'=1}^{n} x_{i'} ^ 2$
$\beta_y = \sum_{i=1}^{n} x_i y_i / \sum_{i'=1}^{n} y_{i'} ^ 2$
$\beta_y = \beta_x$
$\sum_{i=1}^{n} x_i y_i / \sum_{i'=1}^{n} x_{i'} ^ 2 = \sum_{i=1}^{n} x_i y_i / \sum_{i'=1}^{n} y_{i'} ^ 2$
$\sum_{i'=1}^{n} x_{i'} ^ 2 = \sum_{i'=1}^{n} y_{i'} ^ 2$
When sum of X^2 equal to sum of Y^2

### b

x <- rnorm(100)
y <- x ^2
yx_lm <- lm(y ~x +0)
xy_lm <- lm(x ~y +0)

summary(yx_lm)
# Call:
# lm(formula = y ~ x + 0)

# Residuals:
#     Min      1Q  Median      3Q     Max 
# -0.0053  0.1184  0.4420  1.3340 10.6699 

# Coefficients:
#   Estimate Std. Error t value Pr(>|t|)
# x   0.1456     0.1828   0.797    0.428

# Residual standard error: 1.886 on 99 degrees of freedom
# Multiple R-squared:  0.00637,	Adjusted R-squared:  -0.003666 
# F-statistic: 0.6347 on 1 and 99 DF,  p-value: 0.4275

summary(xy_lm)
# Call:
# lm(formula = x ~ y + 0)

# Residuals:
#      Min       1Q   Median       3Q      Max 
# -2.81406 -0.57183  0.07921  0.74082  2.85206 

# Coefficients:
#   Estimate Std. Error t value Pr(>|t|)
# y  0.04375    0.05491   0.797    0.428

# Residual standard error: 1.034 on 99 degrees of freedom
# Multiple R-squared:  0.00637,	Adjusted R-squared:  -0.003666 
# F-statistic: 0.6347 on 1 and 99 DF,  p-value: 0.4275

### c

x <- rnorm(100)
y <- -rnorm(100)
# sum(X^2) should be very close sum(Y^2)
sum(x^2)
# [1] 86.55331
sum(y^2)
# [1] 87.43572
yx_lm <- lm(y ~x +0)
xy_lm <- lm(x ~y +0)

summary(yx_lm)
# Call:
# lm(formula = y ~ x + 0)

# Residuals:
#     Min      1Q  Median      3Q     Max 
# -2.6074 -0.5135  0.1752  0.7049  2.0976 

# Coefficients:
#   Estimate Std. Error t value Pr(>|t|)
# x   0.0339     0.1010   0.336    0.738

# Residual standard error: 0.9392 on 99 degrees of freedom
# Multiple R-squared:  0.001138,	Adjusted R-squared:  -0.008952 
# F-statistic: 0.1128 on 1 and 99 DF,  p-value: 0.7377

summary(xy_lm)
# Call:
# lm(formula = x ~ y + 0)

# Residuals:
#      Min       1Q   Median       3Q      Max 
# -2.90954 -0.69906 -0.09423  0.51060  2.00552 

# Coefficients:
#   Estimate Std. Error t value Pr(>|t|)
# y  0.03356    0.09994   0.336    0.738

# Residual standard error: 0.9345 on 99 degrees of freedom
# Multiple R-squared:  0.001138,	Adjusted R-squared:  -0.008952 
# F-statistic: 0.1128 on 1 and 99 DF,  p-value: 0.7377

Same.

## ch4 Q2
pass