Chapter 3 Homework
Charles Ponthieux
2022-09-23
Problem 2
KNN Classifier – this identifies the closest items to K to determine the intended item. In the case of K=3, you would find information about 3 closest neighbors to determine the information.
KNN Regression Methods – This is the method in which we model as opposed to linear regression
Problem 9
Auto <- read.table("~/1 Data/Auto.data", stringsAsFactors = T, na.strings = "?", header = T)
attach(Auto)
par(mfrow = c(2,2))
summary(Auto)## mpg cylinders displacement horsepower weight
## Min. : 9.00 Min. :3.000 Min. : 68.0 Min. : 46.0 Min. :1613
## 1st Qu.:17.50 1st Qu.:4.000 1st Qu.:104.0 1st Qu.: 75.0 1st Qu.:2223
## Median :23.00 Median :4.000 Median :146.0 Median : 93.5 Median :2800
## Mean :23.52 Mean :5.458 Mean :193.5 Mean :104.5 Mean :2970
## 3rd Qu.:29.00 3rd Qu.:8.000 3rd Qu.:262.0 3rd Qu.:126.0 3rd Qu.:3609
## Max. :46.60 Max. :8.000 Max. :455.0 Max. :230.0 Max. :5140
## NA's :5
## acceleration year origin name
## Min. : 8.00 Min. :70.00 Min. :1.000 ford pinto : 6
## 1st Qu.:13.80 1st Qu.:73.00 1st Qu.:1.000 amc matador : 5
## Median :15.50 Median :76.00 Median :1.000 ford maverick : 5
## Mean :15.56 Mean :75.99 Mean :1.574 toyota corolla: 5
## 3rd Qu.:17.10 3rd Qu.:79.00 3rd Qu.:2.000 amc gremlin : 4
## Max. :24.80 Max. :82.00 Max. :3.000 amc hornet : 4
## (Other) :368pairs(Auto)
(b)
cor(Auto[,-9], use = "pairwise.complete.obs")## mpg cylinders displacement horsepower weight
## mpg 1.0000000 -0.7762599 -0.8044430 -0.7784268 -0.8317389
## cylinders -0.7762599 1.0000000 0.9509199 0.8429834 0.8970169
## displacement -0.8044430 0.9509199 1.0000000 0.8972570 0.9331044
## horsepower -0.7784268 0.8429834 0.8972570 1.0000000 0.8645377
## weight -0.8317389 0.8970169 0.9331044 0.8645377 1.0000000
## acceleration 0.4222974 -0.5040606 -0.5441618 -0.6891955 -0.4195023
## year 0.5814695 -0.3467172 -0.3698041 -0.4163615 -0.3079004
## origin 0.5636979 -0.5649716 -0.6106643 -0.4551715 -0.5812652
## acceleration year origin
## mpg 0.4222974 0.5814695 0.5636979
## cylinders -0.5040606 -0.3467172 -0.5649716
## displacement -0.5441618 -0.3698041 -0.6106643
## horsepower -0.6891955 -0.4163615 -0.4551715
## weight -0.4195023 -0.3079004 -0.5812652
## acceleration 1.0000000 0.2829009 0.2100836
## year 0.2829009 1.0000000 0.1843141
## origin 0.2100836 0.1843141 1.0000000lm.fit=lm(mpg~.,data = Auto[,-9])
summary(lm.fit)##
## Call:
## lm(formula = mpg ~ ., data = Auto[, -9])
##
## 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
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8215, Adjusted R-squared: 0.8182
## F-statistic: 252.4 on 7 and 384 DF, p-value: < 2.2e-16i.There are relationships between the predictors and response. ii.Displacement, Acceleration, Year, and origin all have a significant relationship iii.That newer cars have increased MPG. (d)
plot(lm.fit)
Residuals vs fitted start off very close to the abline, but start to
balloon out toward the right. This also shows a non-linear abline
Normal Q-Q appears to be very linear and normally distributed
Outliers appear to be very inconsistent here at first, but they all fall within the standard deviation, so no issues here.
Residuals vs Leverage plot shows that even with observation 14 removed, it doesn’t appear it would change the model drastically.
interact.fit1 = lm(mpg~.-name + horsepower*displacement, data=Auto)
interact.fit2 = lm(mpg~.-name + horsepower*origin, data=Auto)
interact.fit3 = lm(mpg~.-name + displacement*origin, data=Auto)
interact.fit4 = lm(mpg~.-name + weight*origin, data=Auto)
interact.fit5 = lm(mpg~.-name + weight*displacement, data=Auto)
interact.fit6 = lm(mpg~.-name + year*origin, data=Auto)
interact.fit7 = lm(mpg~.-name + acceleration*origin, data=Auto)
interact.fit8 = lm(mpg~.-name + acceleration*weight, data=Auto)
interact.fit9 = lm(mpg~.-name + acceleration*year, data=Auto)
interact.fit10 = lm(mpg~.-name + acceleration*displacement, data=Auto)
interact.fit11 = lm(mpg~.-name + acceleration*horsepower, data=Auto)
summary(interact.fit1)##
## Call:
## lm(formula = mpg ~ . - name + horsepower * displacement, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.7010 -1.6009 -0.0967 1.4119 12.6734
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.894e+00 4.302e+00 -0.440 0.66007
## cylinders 6.466e-01 3.017e-01 2.143 0.03275 *
## displacement -7.487e-02 1.092e-02 -6.859 2.80e-11 ***
## horsepower -1.975e-01 2.052e-02 -9.624 < 2e-16 ***
## weight -3.147e-03 6.475e-04 -4.861 1.71e-06 ***
## acceleration -2.131e-01 9.062e-02 -2.351 0.01921 *
## year 7.379e-01 4.463e-02 16.534 < 2e-16 ***
## origin 6.891e-01 2.527e-01 2.727 0.00668 **
## displacement:horsepower 5.236e-04 4.813e-05 10.878 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.912 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8636, Adjusted R-squared: 0.8608
## F-statistic: 303.1 on 8 and 383 DF, p-value: < 2.2e-16summary(interact.fit2)##
## Call:
## lm(formula = mpg ~ . - name + horsepower * origin, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.277 -1.875 -0.225 1.570 12.080
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.196e+01 4.396e+00 -4.996 8.94e-07 ***
## cylinders -5.275e-01 3.028e-01 -1.742 0.0823 .
## displacement -1.486e-03 7.607e-03 -0.195 0.8452
## horsepower 8.173e-02 1.856e-02 4.404 1.38e-05 ***
## weight -4.710e-03 6.555e-04 -7.186 3.52e-12 ***
## acceleration -1.124e-01 9.617e-02 -1.168 0.2434
## year 7.327e-01 4.780e-02 15.328 < 2e-16 ***
## origin 7.695e+00 8.858e-01 8.687 < 2e-16 ***
## horsepower:origin -7.955e-02 1.074e-02 -7.405 8.44e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.116 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8438, Adjusted R-squared: 0.8406
## F-statistic: 258.7 on 8 and 383 DF, p-value: < 2.2e-16summary(interact.fit3)##
## Call:
## lm(formula = mpg ~ . - name + displacement * origin, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.2312 -2.0927 -0.1295 1.7287 12.2761
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.140e+01 4.827e+00 -4.434 1.21e-05 ***
## cylinders -4.731e-01 3.204e-01 -1.477 0.140577
## displacement 3.802e-02 9.768e-03 3.892 0.000117 ***
## horsepower -1.523e-02 1.367e-02 -1.114 0.265920
## weight -5.940e-03 6.723e-04 -8.835 < 2e-16 ***
## acceleration 4.578e-02 9.868e-02 0.464 0.642939
## year 7.682e-01 5.087e-02 15.103 < 2e-16 ***
## origin 3.810e+00 8.762e-01 4.349 1.76e-05 ***
## displacement:origin -2.262e-02 7.891e-03 -2.866 0.004383 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.297 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8252, Adjusted R-squared: 0.8216
## F-statistic: 226.1 on 8 and 383 DF, p-value: < 2.2e-16summary(interact.fit4)##
## Call:
## lm(formula = mpg ~ . - name + weight * origin, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.6056 -2.0253 -0.1315 1.6295 12.5217
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.482e+01 4.891e+00 -5.075 6.06e-07 ***
## cylinders -5.799e-01 3.171e-01 -1.829 0.0682 .
## displacement 8.833e-03 7.810e-03 1.131 0.2588
## horsepower -1.068e-02 1.358e-02 -0.787 0.4321
## weight -2.782e-03 1.084e-03 -2.566 0.0107 *
## acceleration 3.766e-02 9.729e-02 0.387 0.6989
## year 7.632e-01 4.998e-02 15.270 < 2e-16 ***
## origin 6.633e+00 1.265e+00 5.244 2.60e-07 ***
## weight:origin -2.309e-03 5.477e-04 -4.215 3.11e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.257 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8294, Adjusted R-squared: 0.8258
## F-statistic: 232.7 on 8 and 383 DF, p-value: < 2.2e-16summary(interact.fit5)##
## Call:
## lm(formula = mpg ~ . - name + weight * displacement, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.9027 -1.8092 -0.0946 1.5549 12.1687
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.389e+00 4.301e+00 -1.253 0.2109
## cylinders 1.175e-01 2.943e-01 0.399 0.6899
## displacement -6.837e-02 1.104e-02 -6.193 1.52e-09 ***
## horsepower -3.280e-02 1.238e-02 -2.649 0.0084 **
## weight -1.064e-02 7.136e-04 -14.915 < 2e-16 ***
## acceleration 6.724e-02 8.805e-02 0.764 0.4455
## year 7.852e-01 4.553e-02 17.246 < 2e-16 ***
## origin 5.610e-01 2.622e-01 2.139 0.0331 *
## displacement:weight 2.269e-05 2.257e-06 10.054 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.964 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8588, Adjusted R-squared: 0.8558
## F-statistic: 291.1 on 8 and 383 DF, p-value: < 2.2e-16summary(interact.fit6)##
## Call:
## lm(formula = mpg ~ . - name + year * origin, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.6072 -2.0439 -0.0596 1.7121 12.3368
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.492e+00 9.044e+00 0.939 0.348353
## cylinders -5.042e-01 3.192e-01 -1.579 0.115082
## displacement 1.567e-02 7.530e-03 2.081 0.038060 *
## horsepower -1.399e-02 1.364e-02 -1.025 0.305786
## weight -6.352e-03 6.449e-04 -9.851 < 2e-16 ***
## acceleration 9.185e-02 9.766e-02 0.941 0.347546
## year 4.189e-01 1.125e-01 3.723 0.000226 ***
## origin -1.405e+01 4.699e+00 -2.989 0.002978 **
## year:origin 1.989e-01 6.030e-02 3.298 0.001064 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.286 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8264, Adjusted R-squared: 0.8228
## F-statistic: 227.9 on 8 and 383 DF, p-value: < 2.2e-16summary(interact.fit7)##
## Call:
## lm(formula = mpg ~ . - name + acceleration * origin, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.4106 -1.8805 -0.2471 1.7891 11.9680
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.3327273 5.0570077 -0.066 0.9476
## cylinders -0.5881258 0.3063127 -1.920 0.0556 .
## displacement 0.0086251 0.0073062 1.181 0.2385
## horsepower -0.0250843 0.0131049 -1.914 0.0564 .
## weight -0.0052351 0.0006439 -8.131 5.98e-15 ***
## acceleration -1.0340600 0.1896960 -5.451 8.98e-08 ***
## year 0.7623813 0.0482774 15.792 < 2e-16 ***
## origin -9.3089774 1.6109675 -5.779 1.56e-08 ***
## acceleration:origin 0.6546959 0.0969263 6.755 5.34e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.15 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8405, Adjusted R-squared: 0.8371
## F-statistic: 252.2 on 8 and 383 DF, p-value: < 2.2e-16summary(interact.fit8)##
## Call:
## lm(formula = mpg ~ . - name + acceleration * weight, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.247 -2.048 -0.045 1.619 12.193
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.364e+01 5.811e+00 -7.511 4.18e-13 ***
## cylinders -2.141e-01 3.078e-01 -0.696 0.487117
## displacement 3.138e-03 7.495e-03 0.419 0.675622
## horsepower -4.141e-02 1.348e-02 -3.071 0.002287 **
## weight 4.027e-03 1.636e-03 2.462 0.014268 *
## acceleration 1.629e+00 2.422e-01 6.726 6.36e-11 ***
## year 7.821e-01 4.833e-02 16.184 < 2e-16 ***
## origin 1.033e+00 2.686e-01 3.846 0.000141 ***
## weight:acceleration -5.826e-04 8.408e-05 -6.928 1.81e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.141 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8414, Adjusted R-squared: 0.838
## F-statistic: 253.9 on 8 and 383 DF, p-value: < 2.2e-16summary(interact.fit9)##
## Call:
## lm(formula = mpg ~ . - name + acceleration * year, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.2277 -2.1428 -0.0839 1.8216 12.2997
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 97.6199866 20.8291407 4.687 3.86e-06 ***
## cylinders -0.2163880 0.3148719 -0.687 0.49236
## displacement 0.0087776 0.0074936 1.171 0.24219
## horsepower -0.0269212 0.0133812 -2.012 0.04493 *
## weight -0.0058738 0.0006363 -9.232 < 2e-16 ***
## acceleration -7.1818103 1.2900806 -5.567 4.88e-08 ***
## year -0.7461986 0.2696927 -2.767 0.00593 **
## origin 1.2630333 0.2691459 4.693 3.76e-06 ***
## acceleration:year 0.0945502 0.0167501 5.645 3.22e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.202 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8352, Adjusted R-squared: 0.8317
## F-statistic: 242.6 on 8 and 383 DF, p-value: < 2.2e-16summary(interact.fit10)##
## Call:
## lm(formula = mpg ~ . - name + acceleration * displacement, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.129 -1.899 -0.135 1.755 12.119
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.005e+01 4.737e+00 -6.343 6.36e-10 ***
## cylinders 2.136e-03 3.125e-01 0.007 0.994550
## displacement 7.022e-02 1.005e-02 6.989 1.24e-11 ***
## horsepower -5.515e-02 1.407e-02 -3.920 0.000105 ***
## weight -4.211e-03 6.929e-04 -6.077 2.96e-09 ***
## acceleration 7.530e-01 1.332e-01 5.653 3.09e-08 ***
## year 7.722e-01 4.811e-02 16.051 < 2e-16 ***
## origin 1.057e+00 2.671e-01 3.958 9.01e-05 ***
## displacement:acceleration -4.855e-03 6.879e-04 -7.058 7.99e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.134 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.842, Adjusted R-squared: 0.8387
## F-statistic: 255.2 on 8 and 383 DF, p-value: < 2.2e-16summary(interact.fit11)##
## Call:
## lm(formula = mpg ~ . - name + acceleration * horsepower, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.0329 -1.8177 -0.1183 1.7247 12.4870
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -32.499820 4.923380 -6.601 1.36e-10 ***
## cylinders 0.083489 0.316913 0.263 0.792350
## displacement -0.007649 0.008161 -0.937 0.349244
## horsepower 0.127188 0.024746 5.140 4.40e-07 ***
## weight -0.003976 0.000716 -5.552 5.27e-08 ***
## acceleration 0.983282 0.161513 6.088 2.78e-09 ***
## year 0.755919 0.048179 15.690 < 2e-16 ***
## origin 1.035733 0.268962 3.851 0.000138 ***
## horsepower:acceleration -0.012139 0.001772 -6.851 2.93e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.145 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.841, Adjusted R-squared: 0.8376
## F-statistic: 253.2 on 8 and 383 DF, p-value: < 2.2e-16Yes, displacement and horse as well as displacement and weight power have a T stat value.
summary(lm(mpg ~ . -name + log(acceleration), data=Auto))##
## Call:
## lm(formula = mpg ~ . - name + log(acceleration), data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.7931 -2.0052 -0.1279 1.9299 13.1085
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.552e+01 1.479e+01 3.077 0.00224 **
## cylinders -2.796e-01 3.193e-01 -0.876 0.38172
## displacement 8.042e-03 7.805e-03 1.030 0.30344
## horsepower -3.434e-02 1.401e-02 -2.450 0.01473 *
## weight -5.343e-03 6.854e-04 -7.795 6.15e-14 ***
## acceleration 2.167e+00 4.782e-01 4.532 7.82e-06 ***
## year 7.560e-01 4.978e-02 15.186 < 2e-16 ***
## origin 1.329e+00 2.724e-01 4.877 1.58e-06 ***
## log(acceleration) -3.513e+01 7.886e+00 -4.455 1.10e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.249 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8303, Adjusted R-squared: 0.8267
## F-statistic: 234.2 on 8 and 383 DF, p-value: < 2.2e-16summary(lm(mpg ~ . -name + sqrt(displacement), data=Auto))##
## Call:
## lm(formula = mpg ~ . - name + sqrt(displacement), data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.6874 -1.8084 -0.0046 1.6599 11.7939
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.3425501 5.4548602 2.446 0.014895 *
## cylinders 0.1791258 0.3043002 0.589 0.556444
## displacement 0.1628037 0.0174800 9.314 < 2e-16 ***
## horsepower -0.0479788 0.0130456 -3.678 0.000269 ***
## weight -0.0044272 0.0006375 -6.944 1.64e-11 ***
## acceleration -0.0160378 0.0907715 -0.177 0.859851
## year 0.7781542 0.0465748 16.708 < 2e-16 ***
## origin 0.4130678 0.2780263 1.486 0.138177
## sqrt(displacement) -4.7444871 0.5338893 -8.887 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.034 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.852, Adjusted R-squared: 0.8489
## F-statistic: 275.6 on 8 and 383 DF, p-value: < 2.2e-16summary(lm(mpg ~ . -name + I(horsepower^2), data=Auto))##
## Call:
## lm(formula = mpg ~ . - name + I(horsepower^2), data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.5497 -1.7311 -0.2236 1.5877 11.9955
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.3236564 4.6247696 0.286 0.774872
## cylinders 0.3489063 0.3048310 1.145 0.253094
## displacement -0.0075649 0.0073733 -1.026 0.305550
## horsepower -0.3194633 0.0343447 -9.302 < 2e-16 ***
## weight -0.0032712 0.0006787 -4.820 2.07e-06 ***
## acceleration -0.3305981 0.0991849 -3.333 0.000942 ***
## year 0.7353414 0.0459918 15.989 < 2e-16 ***
## origin 1.0144130 0.2545545 3.985 8.08e-05 ***
## I(horsepower^2) 0.0010060 0.0001065 9.449 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.001 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8552, Adjusted R-squared: 0.8522
## F-statistic: 282.8 on 8 and 383 DF, p-value: < 2.2e-16summary(lm(mpg ~ . -name + log(weight), data=Auto))##
## Call:
## lm(formula = mpg ~ . - name + log(weight), data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.6516 -1.6398 -0.1671 1.5973 12.7247
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 269.474171 31.136919 8.654 < 2e-16 ***
## cylinders -0.498204 0.292415 -1.704 0.08924 .
## displacement 0.013527 0.006832 1.980 0.04843 *
## horsepower -0.022137 0.012483 -1.773 0.07696 .
## weight 0.007657 0.001631 4.694 3.73e-06 ***
## acceleration 0.045763 0.089486 0.511 0.60936
## year 0.797808 0.046383 17.200 < 2e-16 ***
## origin 0.719552 0.262819 2.738 0.00647 **
## log(weight) -41.320927 4.446725 -9.292 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.01 on 383 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.8543, Adjusted R-squared: 0.8513
## F-statistic: 280.8 on 8 and 383 DF, p-value: < 2.2e-16The weight of the car vs log(weight) has a drastic impact.
Problem 10
Carseats<- read.csv("~/1 Data/Carseats.csv", stringsAsFactors = T, header = T, na.strings = "?")
View(Carseats)
par(mfrow = c(2,2))
summary(Carseats)## Sales CompPrice Income Advertising
## Min. : 0.000 Min. : 77 Min. : 21.00 Min. : 0.000
## 1st Qu.: 5.390 1st Qu.:115 1st Qu.: 42.75 1st Qu.: 0.000
## Median : 7.490 Median :125 Median : 69.00 Median : 5.000
## Mean : 7.496 Mean :125 Mean : 68.66 Mean : 6.635
## 3rd Qu.: 9.320 3rd Qu.:135 3rd Qu.: 91.00 3rd Qu.:12.000
## Max. :16.270 Max. :175 Max. :120.00 Max. :29.000
## Population Price ShelveLoc Age Education
## Min. : 10.0 Min. : 24.0 Bad : 96 Min. :25.00 Min. :10.0
## 1st Qu.:139.0 1st Qu.:100.0 Good : 85 1st Qu.:39.75 1st Qu.:12.0
## Median :272.0 Median :117.0 Medium:219 Median :54.50 Median :14.0
## Mean :264.8 Mean :115.8 Mean :53.32 Mean :13.9
## 3rd Qu.:398.5 3rd Qu.:131.0 3rd Qu.:66.00 3rd Qu.:16.0
## Max. :509.0 Max. :191.0 Max. :80.00 Max. :18.0
## Urban US
## No :118 No :142
## Yes:282 Yes:258
##
##
##
## attach(Carseats)lm.fit = lm(Sales ~ Price+Urban+US, data= Carseats)
summary(lm.fit)##
## 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-16Price – It appears that there is an inverse relationship between price and sales. I.E. if price goes up, sales go down, but if sales go up, price goes down.
UrbanYes – There does not seem to be a strong relationship here.
USYes – There seems to be a relationship to sales and whether the store is in the U.S. The Estimate shows there are likely to be ~1,200 more units sold for U.S. stores (1.200573*1000)
Sales = 13.04-.05Price-.02UrbanYes+1.20USYes
Price and US
lm.fit2 = lm(Sales ~ Price + US, data = Carseats)
summary(lm.fit2)##
## 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)The models seem to be very close, neither fitting much better than the other. (g)
confint(lm.fit2)## 2.5 % 97.5 %
## (Intercept) 11.79032020 14.27126531
## Price -0.06475984 -0.04419543
## USYes 0.69151957 1.70776632(h)There do not appear to be any outliers or high leverage observations
plot(lm.fit2)
Problem 12 (a)Pass (b)
x<-1:100
sum(x^2)## [1] 338350y <- 2 * x + rnorm(100, sd = 0.1)
sum(y^2)## [1] 1353395fit.y <- lm(y ~ x + 0)
fit.x <- lm(x ~ y + 0)
summary(fit.y)##
## Call:
## lm(formula = y ~ x + 0)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.227533 -0.070909 0.002441 0.078863 0.203463
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## x 1.9999958 0.0001679 11909 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09769 on 99 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.418e+08 on 1 and 99 DF, p-value: < 2.2e-16summary(fit.x)##
## Call:
## lm(formula = x ~ y + 0)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.101710 -0.039425 -0.001187 0.035474 0.113785
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## y 5.000e-01 4.199e-05 11909 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04885 on 99 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.418e+08 on 1 and 99 DF, p-value: < 2.2e-16x <- 1:100
y <- 1:100
fit.y <- lm(y ~ x + 0)
fit.x <- lm(x ~ x + 0)## Warning in model.matrix.default(mt, mf, contrasts): the response appeared on the
## right-hand side and was dropped## Warning in model.matrix.default(mt, mf, contrasts): problem with term 1 in
## model.matrix: no columns are assignedsummary(fit.y)## Warning in summary.lm(fit.y): essentially perfect fit: summary may be unreliable##
## Call:
## lm(formula = y ~ x + 0)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.082e-13 -2.094e-15 2.900e-17 2.218e-15 1.294e-14
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## x 1.000e+00 5.379e-17 1.859e+16 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.129e-14 on 99 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 3.457e+32 on 1 and 99 DF, p-value: < 2.2e-16summary(fit.x)##
## Call:
## lm(formula = x ~ x + 0)
##
## Residuals:
## Min 1Q Median 3Q Max
## 1.00 25.75 50.50 75.25 100.00
##
## No Coefficients
##
## Residual standard error: 58.17 on 100 degrees of freedom