Assignment 2
2025-06-21
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Q2. Carefully explain the diferences between the KNN classifer and KNN regression methods.
KNN classifer assigns the value to the point by evaluating in function of the neighbors. The point assumes the value with the higher probabilty making the values of KNN cl;assifier qulitative. KNN regression methods assigns the value that has the greater probability tp the point. Averages the value of the k-neighbors and assigns the value to the average point making values quantitative.
library(MASS)auto <- read.csv("Auto.csv", na.strings = "?")
auto <- na.omit(auto)
str(auto)## 'data.frame': 392 obs. of 9 variables:
## $ mpg : num 18 15 18 16 17 15 14 14 14 15 ...
## $ cylinders : int 8 8 8 8 8 8 8 8 8 8 ...
## $ displacement: num 307 350 318 304 302 429 454 440 455 390 ...
## $ horsepower : int 130 165 150 150 140 198 220 215 225 190 ...
## $ weight : int 3504 3693 3436 3433 3449 4341 4354 4312 4425 3850 ...
## $ acceleration: num 12 11.5 11 12 10.5 10 9 8.5 10 8.5 ...
## $ year : int 70 70 70 70 70 70 70 70 70 70 ...
## $ origin : int 1 1 1 1 1 1 1 1 1 1 ...
## $ name : chr "chevrolet chevelle malibu" "buick skylark 320" "plymouth satellite" "amc rebel sst" ...
## - attr(*, "na.action")= 'omit' Named int [1:5] 33 127 331 337 355
## ..- attr(*, "names")= chr [1:5] "33" "127" "331" "337" ...plot(auto)
Auto_new <- auto[,-c(9)]
cor(Auto_new)## mpg cylinders displacement horsepower weight
## mpg 1.0000000 -0.7776175 -0.8051269 -0.7784268 -0.8322442
## cylinders -0.7776175 1.0000000 0.9508233 0.8429834 0.8975273
## displacement -0.8051269 0.9508233 1.0000000 0.8972570 0.9329944
## horsepower -0.7784268 0.8429834 0.8972570 1.0000000 0.8645377
## weight -0.8322442 0.8975273 0.9329944 0.8645377 1.0000000
## acceleration 0.4233285 -0.5046834 -0.5438005 -0.6891955 -0.4168392
## year 0.5805410 -0.3456474 -0.3698552 -0.4163615 -0.3091199
## origin 0.5652088 -0.5689316 -0.6145351 -0.4551715 -0.5850054
## acceleration year origin
## mpg 0.4233285 0.5805410 0.5652088
## cylinders -0.5046834 -0.3456474 -0.5689316
## displacement -0.5438005 -0.3698552 -0.6145351
## horsepower -0.6891955 -0.4163615 -0.4551715
## weight -0.4168392 -0.3091199 -0.5850054
## acceleration 1.0000000 0.2903161 0.2127458
## year 0.2903161 1.0000000 0.1815277
## origin 0.2127458 0.1815277 1.0000000lm_fit<- lm(mpg ~ ., data = Auto_new)
summary(lm_fit)##
## Call:
## lm(formula = mpg ~ ., data = Auto_new)
##
## 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- Is there a relationship between the predictors and the response?
- F-STATISTIC : 252.4 AND P-VALUE < 2.2e-16 provide evidence that there is a relationship between at least one predictor and the response
- Which predictors appear to have a statistically significant relationship to the response?
- Displacement, Year, Weight and origin
- What does the coefficient for the year variable suggest?
- there is a positive relationship between mpg and year. Each year representa a 0.750773 increase of mpg
par (mfrow = c(2, 2))
plot(lm_fit)
summary(lm(mpg ~ .:., data = Auto_new))##
## Call:
## lm(formula = mpg ~ .:., data = Auto_new)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.6303 -1.4481 0.0596 1.2739 11.1386
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.548e+01 5.314e+01 0.668 0.50475
## cylinders 6.989e+00 8.248e+00 0.847 0.39738
## displacement -4.785e-01 1.894e-01 -2.527 0.01192 *
## horsepower 5.034e-01 3.470e-01 1.451 0.14769
## weight 4.133e-03 1.759e-02 0.235 0.81442
## acceleration -5.859e+00 2.174e+00 -2.696 0.00735 **
## year 6.974e-01 6.097e-01 1.144 0.25340
## origin -2.090e+01 7.097e+00 -2.944 0.00345 **
## cylinders:displacement -3.383e-03 6.455e-03 -0.524 0.60051
## cylinders:horsepower 1.161e-02 2.420e-02 0.480 0.63157
## cylinders:weight 3.575e-04 8.955e-04 0.399 0.69000
## cylinders:acceleration 2.779e-01 1.664e-01 1.670 0.09584 .
## cylinders:year -1.741e-01 9.714e-02 -1.793 0.07389 .
## cylinders:origin 4.022e-01 4.926e-01 0.816 0.41482
## displacement:horsepower -8.491e-05 2.885e-04 -0.294 0.76867
## displacement:weight 2.472e-05 1.470e-05 1.682 0.09342 .
## displacement:acceleration -3.479e-03 3.342e-03 -1.041 0.29853
## displacement:year 5.934e-03 2.391e-03 2.482 0.01352 *
## displacement:origin 2.398e-02 1.947e-02 1.232 0.21875
## horsepower:weight -1.968e-05 2.924e-05 -0.673 0.50124
## horsepower:acceleration -7.213e-03 3.719e-03 -1.939 0.05325 .
## horsepower:year -5.838e-03 3.938e-03 -1.482 0.13916
## horsepower:origin 2.233e-03 2.930e-02 0.076 0.93931
## weight:acceleration 2.346e-04 2.289e-04 1.025 0.30596
## weight:year -2.245e-04 2.127e-04 -1.056 0.29182
## weight:origin -5.789e-04 1.591e-03 -0.364 0.71623
## acceleration:year 5.562e-02 2.558e-02 2.174 0.03033 *
## acceleration:origin 4.583e-01 1.567e-01 2.926 0.00365 **
## year:origin 1.393e-01 7.399e-02 1.882 0.06062 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.695 on 363 degrees of freedom
## Multiple R-squared: 0.8893, Adjusted R-squared: 0.8808
## F-statistic: 104.2 on 28 and 363 DF, p-value: < 2.2e-16summary(lm(mpg ~ .*., data = Auto_new))##
## Call:
## lm(formula = mpg ~ . * ., data = Auto_new)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.6303 -1.4481 0.0596 1.2739 11.1386
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.548e+01 5.314e+01 0.668 0.50475
## cylinders 6.989e+00 8.248e+00 0.847 0.39738
## displacement -4.785e-01 1.894e-01 -2.527 0.01192 *
## horsepower 5.034e-01 3.470e-01 1.451 0.14769
## weight 4.133e-03 1.759e-02 0.235 0.81442
## acceleration -5.859e+00 2.174e+00 -2.696 0.00735 **
## year 6.974e-01 6.097e-01 1.144 0.25340
## origin -2.090e+01 7.097e+00 -2.944 0.00345 **
## cylinders:displacement -3.383e-03 6.455e-03 -0.524 0.60051
## cylinders:horsepower 1.161e-02 2.420e-02 0.480 0.63157
## cylinders:weight 3.575e-04 8.955e-04 0.399 0.69000
## cylinders:acceleration 2.779e-01 1.664e-01 1.670 0.09584 .
## cylinders:year -1.741e-01 9.714e-02 -1.793 0.07389 .
## cylinders:origin 4.022e-01 4.926e-01 0.816 0.41482
## displacement:horsepower -8.491e-05 2.885e-04 -0.294 0.76867
## displacement:weight 2.472e-05 1.470e-05 1.682 0.09342 .
## displacement:acceleration -3.479e-03 3.342e-03 -1.041 0.29853
## displacement:year 5.934e-03 2.391e-03 2.482 0.01352 *
## displacement:origin 2.398e-02 1.947e-02 1.232 0.21875
## horsepower:weight -1.968e-05 2.924e-05 -0.673 0.50124
## horsepower:acceleration -7.213e-03 3.719e-03 -1.939 0.05325 .
## horsepower:year -5.838e-03 3.938e-03 -1.482 0.13916
## horsepower:origin 2.233e-03 2.930e-02 0.076 0.93931
## weight:acceleration 2.346e-04 2.289e-04 1.025 0.30596
## weight:year -2.245e-04 2.127e-04 -1.056 0.29182
## weight:origin -5.789e-04 1.591e-03 -0.364 0.71623
## acceleration:year 5.562e-02 2.558e-02 2.174 0.03033 *
## acceleration:origin 4.583e-01 1.567e-01 2.926 0.00365 **
## year:origin 1.393e-01 7.399e-02 1.882 0.06062 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.695 on 363 degrees of freedom
## Multiple R-squared: 0.8893, Adjusted R-squared: 0.8808
## F-statistic: 104.2 on 28 and 363 DF, p-value: < 2.2e-16summary(lm(mpg ~ acceleration+I(acceleration^2), data = Auto_new))##
## Call:
## lm(formula = mpg ~ acceleration + I(acceleration^2), data = Auto_new)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18.0877 -5.5700 -0.8524 4.3827 22.9813
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -15.26045 7.79899 -1.957 0.051095 .
## acceleration 3.79787 0.98283 3.864 0.000131 ***
## I(acceleration^2) -0.08156 0.03056 -2.669 0.007934 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.025 on 389 degrees of freedom
## Multiple R-squared: 0.194, Adjusted R-squared: 0.1898
## F-statistic: 46.8 on 2 and 389 DF, p-value: < 2.2e-16summary(lm(mpg ~ acceleration+I(sqrt(acceleration)), data = Auto_new))##
## Call:
## lm(formula = mpg ~ acceleration + I(sqrt(acceleration)), data = Auto_new)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.9559 -5.5979 -0.8015 4.6222 22.8777
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -72.877 29.051 -2.509 0.01253 *
## acceleration -3.845 1.885 -2.040 0.04203 *
## I(sqrt(acceleration)) 39.750 14.823 2.682 0.00764 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.025 on 389 degrees of freedom
## Multiple R-squared: 0.1941, Adjusted R-squared: 0.19
## F-statistic: 46.85 on 2 and 389 DF, p-value: < 2.2e-16data("Carseats", package = "ISLR")lm_sales <- lm(Sales ~ Price + Urban + US, Carseats)
summary(lm_sales)##
## 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- Provide an interpretation of each coefficient in the model. Be careful—some of the variables in the model are qualitative!
- The data set contains sales of child car seats, measured in thousands. Price is a continuous variable, while Urban and Us are categorical. - The Price coefficient (-0.054459) indicates that for every unit of price increased, sales decreased by 54 units. The P-value shows evidence of a strong relationship between Price and Sales. - The Urban variable can take 2 values: Yes(1) or NO(0) to indicate if the store is in an Urban location. Its coefficient (-0.021916) indicates that whenever a store is in a urban location, sales decrease by 21 units. Its p-value, however, shows that there is not enough evidence to indicate a relation between Urban and Price. - The US variable can take 2 values: Yes or No, to indicate if the store is in the US. Its coefficient (1.200573) indicates that for sales increase by 1200 units whenever a store is located in the US. The P-value shows evidence of a relation between Sales and US.
- Write out the model in equation form, being careful to handle the qualitative variables properly.
- Sales = 13.043469 - 0.054459(Price) - 0.021916(Urban)[1 if the Store is in Urban area; 0 otherwise] + 1.200573(US)[1 if the store is in the US; 0 otherwise]
d.For which of the predictors can you reject the null hypothesis H0 : βj = 0? - The null hypothesis can be rejected for the Price and US variables, since their p-values is 2e-16 and 4.86e-06, respectively
lm_sales_new <- lm(Sales ~ Price + US, Carseats)
summary(lm_sales_new)##
## 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- How well do the models in (a) and (e) fit the data?
- Model a has an Adjusted R-squared of:0.2335.Model b has an Adjusted R-squared of: 0.2354. Both models explain around 23% of the variance, which confirms that dropping the Urban variable from the set didn’t affect the fit of the model to the data.
confint(lm_sales_new, level = 0.95)## 2.5 % 97.5 %
## (Intercept) 11.79032020 14.27126531
## Price -0.06475984 -0.04419543
## USYes 0.69151957 1.70776632par (mfrow = c(3, 2))
plot(hatvalues(lm_sales_new))
plot(lm_sales_new)
a. Recall that the coefficient estimate βˆ for the linear regression of
Y onto X without an intercept is given by (3.38). Under what
circumstance is the coefficient estimate for the regression of X onto Y
the same as the coefficient estimate for the regression of Y onto X? -
After equating the respective quotients to calculate the coefficients
for both x and y, respectively, we can see that the estimates are equal
for both x and y when the summation of x squared is equal to the
summation of y squared.
set.seed(99)
x <- rnorm(100)
y= 3*x+rnorm(100)
df <- data.frame(x, y)
fit1 <- lm(y ~ x + 0, data = df)
fit2 <- lm(x ~ y + 0, data = df)
summary(fit1)$coefficients## Estimate Std. Error t value Pr(>|t|)
## x 3.087646 0.1182444 26.1124 3.415688e-46summary(fit2)$coefficients## Estimate Std. Error t value Pr(>|t|)
## y 0.2828097 0.01083047 26.1124 3.415688e-46set.seed(89)
x1 <- rnorm(100)
y1= x1
df1 <- data.frame(x1, y1)
fit2_1 <- lm(y1 ~ x1 + 0, data = df1)
fit2_2 <- lm(x1 ~ y1 + 0, data = df1)
summary(fit2_1)$coefficients## Warning in summary.lm(fit2_1): essentially perfect fit: summary may be
## unreliable## Estimate Std. Error t value Pr(>|t|)
## x1 1 3.450101e-17 2.898466e+16 0summary(fit2_2)$coefficients## Warning in summary.lm(fit2_2): essentially perfect fit: summary may be
## unreliable## Estimate Std. Error t value Pr(>|t|)
## y1 1 3.450101e-17 2.898466e+16 0