What data are we going to use?
"Cars93" is a list of cars on
 |
| Photo: www.caranddriver.com |
The data frame contains 93 rows and 27 columns.
More about the data:
Cars were selected at random from among 1993 passenger car models that were listed in both the Consumer Reports issue and the PACE Buying Guide. Pickup trucks and Sport/Utility vehicles were eliminated due to incomplete information in the ConsumerManufacturersource. Duplicate models (e.g., Dodge Shadow and Plymouth Sundance) were listed at most once. Reports
Manufacturer.
Model
Model.
Type
Type: a factor with levels "Small", "Sporty", "Compact", "Midsize", "Large" and "Van".
Min
Minimum Price (in \$1,000): price for a basic version.
Price
Midrange Price (in \$1,000): average of Min
Max
Maximum Price (in \$1,000): price for “a premium version”.
MPG
City MPG (miles per US gallon by EPA rating).
MPG
Highway MPG.
Air Bags standard. Factor: none, driver only, or driver & passenger.
Drive train type: rear wheel, front wheel or 4WD; (factor).
Cylinders
Number of cylinders (missing for Mazda RX-7, which has a rotary engine).
Engine size (
Horsepower
Horsepower (maximum).
RPM
RPM (revs per minute at maximum horsepower).
Rev
Engine revolutions per mile (in highest gear).
Man
Is a manual transmission version available? (
Fuel
Fuel tank capacity (US gallons).
Passengers
Passenger capacity (persons)
Length
Length (inches).
Wheelbase
Wheelbase (inches).
Width
Width (inches).
Turn
U-turn space (feet).
Rear
Rear seat room (inches) (missing
Luggage
Luggage capacity (cubic feet) (missing for vans).
Weight
Weight (pounds).
Origin
Of non-USA or USA company origins? (
Make
Combination of Manufacturer and Model (character).
Goal:
R Code:
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| > library(MASS) | |
| > data("Cars93") | |
| > attach(Cars93) | |
| > model <- lm(Weight~MPG.city+EngineSize+Horsepower+Passengers) | |
| > summary(model) | |
| Call: | |
| lm(formula = Weight ~ MPG.city + EngineSize + Horsepower + Passengers) | |
| Residuals: | |
| Min 1Q Median 3Q Max | |
| -370.02 -131.60 10.57 108.59 530.42 | |
| Coefficients: | |
| Estimate Std. Error t value Pr(>|t|) | |
| (Intercept) 1764.4733 269.9261 6.537 3.97e-09 *** | |
| MPG.city -28.9017 5.7062 -5.065 2.23e-06 *** | |
| EngineSize 147.6183 33.1217 4.457 2.44e-05 *** | |
| Horsepower 4.0593 0.6773 5.994 4.41e-08 *** | |
| Passengers 192.1325 24.4279 7.865 8.72e-12 *** | |
| --- | |
| Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 | |
| Residual standard error: 185.9 on 88 degrees of freedom | |
| Multiple R-squared: 0.905, Adjusted R-squared: 0.9007 | |
| F-statistic: 209.6 on 4 and 88 DF, p-value: < 2.2e-16 | |
| > attributes(model) | |
| $names | |
| [1] "coefficients" "residuals" "effects" "rank" "fitted.values" "assign" | |
| [7] "qr" "df.residual" "xlevels" "call" "terms" "model" | |
| $class | |
| [1] "lm" | |
| > #2 ways to extract an attribute | |
| > model$residuals | |
| 1 2 3 4 5 6 7 8 | |
| -131.603519 70.863050 116.373195 -74.660985 381.842685 -172.710463 -149.160791 78.066640 | |
| 9 10 11 12 13 14 15 16 | |
| 67.971697 -370.023995 -204.236201 -283.872897 11.127103 104.744171 113.387845 -125.194971 | |
| 17 18 19 20 21 22 23 24 | |
| -147.545379 55.893820 -336.613541 67.557578 -182.740661 146.913169 -211.866518 -92.668774 | |
| 25 26 27 28 29 30 31 32 | |
| -86.403312 67.657431 -5.305004 131.592684 -186.866518 -234.581267 -239.688287 -311.636456 | |
| 33 34 35 36 37 38 39 40 | |
| -128.510096 187.089005 108.585239 27.676250 195.647707 102.657543 120.596725 207.524359 | |
| 41 42 43 44 45 46 47 48 | |
| 36.632804 395.395322 307.579942 -92.214601 -238.360353 -91.439108 -76.924467 -26.565166 | |
| 49 50 51 52 53 54 55 56 | |
| 111.275695 146.037575 58.628476 126.472239 61.096788 -159.705629 -38.457027 73.788676 | |
| 57 58 59 60 61 62 63 64 | |
| 10.574073 -94.328498 -4.242766 -60.380141 304.749653 -186.866518 262.268187 -24.695125 | |
| 65 66 67 68 69 70 71 72 | |
| 55.331135 426.124044 -10.537597 -90.203358 58.323718 -125.194971 -149.160791 132.570314 | |
| 73 74 75 76 77 78 79 80 | |
| 176.374295 -227.152615 104.744171 -39.758925 -124.160791 -250.397483 -46.400689 -7.715720 | |
| 81 82 83 84 85 86 87 88 | |
| -143.640259 172.138413 83.110398 -299.568788 266.777960 88.236721 281.367186 -164.974384 | |
| 89 90 91 92 93 | |
| 530.422351 27.621635 -338.653293 64.521438 61.657591 | |
| > residuals(model) | |
| 1 2 3 4 5 6 7 8 | |
| -131.603519 70.863050 116.373195 -74.660985 381.842685 -172.710463 -149.160791 78.066640 | |
| 9 10 11 12 13 14 15 16 | |
| 67.971697 -370.023995 -204.236201 -283.872897 11.127103 104.744171 113.387845 -125.194971 | |
| 17 18 19 20 21 22 23 24 | |
| -147.545379 55.893820 -336.613541 67.557578 -182.740661 146.913169 -211.866518 -92.668774 | |
| 25 26 27 28 29 30 31 32 | |
| -86.403312 67.657431 -5.305004 131.592684 -186.866518 -234.581267 -239.688287 -311.636456 | |
| 33 34 35 36 37 38 39 40 | |
| -128.510096 187.089005 108.585239 27.676250 195.647707 102.657543 120.596725 207.524359 | |
| 41 42 43 44 45 46 47 48 | |
| 36.632804 395.395322 307.579942 -92.214601 -238.360353 -91.439108 -76.924467 -26.565166 | |
| 49 50 51 52 53 54 55 56 | |
| 111.275695 146.037575 58.628476 126.472239 61.096788 -159.705629 -38.457027 73.788676 | |
| 57 58 59 60 61 62 63 64 | |
| 10.574073 -94.328498 -4.242766 -60.380141 304.749653 -186.866518 262.268187 -24.695125 | |
| 65 66 67 68 69 70 71 72 | |
| 55.331135 426.124044 -10.537597 -90.203358 58.323718 -125.194971 -149.160791 132.570314 | |
| 73 74 75 76 77 78 79 80 | |
| 176.374295 -227.152615 104.744171 -39.758925 -124.160791 -250.397483 -46.400689 -7.715720 | |
| 81 82 83 84 85 86 87 88 | |
| -143.640259 172.138413 83.110398 -299.568788 266.777960 88.236721 281.367186 -164.974384 | |
| 89 90 91 92 93 | |
| 530.422351 27.621635 -338.653293 64.521438 61.657591 | |
| > plot(residuals(model)) | |
| > plot(Weight, model$fitted, type="p", col="blue") | |
| > abline(c(1,1)) | |
| > #Index Cars93, round any decimal value in the data, partition 60 as Train | |
| > trainindex <- sample(1:nrow(Cars93), size=round(nrow(Cars93)*.60)) | |
| > train <- Cars93[trainindex,] | |
| > test <- Cars93[-trainindex,] | |
| > attach(train) | |
| > model <- lm(Weight~MPG.city+EngineSize+Horsepower+Passengers) | |
| > summary(model) | |
| Call: | |
| lm(formula = Weight ~ MPG.city + EngineSize + Horsepower + Passengers) | |
| Residuals: | |
| Min 1Q Median 3Q Max | |
| -427.74 -137.13 -10.58 121.31 474.01 | |
| Coefficients: | |
| Estimate Std. Error t value Pr(>|t|) | |
| (Intercept) 1459.043 353.166 4.131 0.000134 *** | |
| MPG.city -19.732 7.286 -2.708 0.009185 ** | |
| EngineSize 201.967 50.050 4.035 0.000183 *** | |
| Horsepower 4.098 1.058 3.873 0.000308 *** | |
| Passengers 182.523 35.668 5.117 4.75e-06 *** | |
| --- | |
| Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 | |
| Residual standard error: 192.6 on 51 degrees of freedom | |
| Multiple R-squared: 0.9003, Adjusted R-squared: 0.8925 | |
| F-statistic: 115.1 on 4 and 51 DF, p-value: < 2.2e-16 | |
| > predict(model) | |
| 1 2 3 4 5 6 7 8 9 10 11 | |
| 2743.276 2889.000 2573.372 3314.338 4047.745 2733.637 2835.799 2479.407 4127.354 3988.672 2795.499 | |
| 12 13 14 15 16 17 18 19 20 21 22 | |
| 2809.266 3845.693 3337.968 2468.260 2203.848 3589.101 2773.481 4167.904 3460.915 3034.933 2479.407 | |
| 23 24 25 26 27 28 29 30 31 32 33 | |
| 3156.623 2604.810 2391.323 1708.818 4193.964 3247.412 2997.536 2610.730 3136.930 3428.463 3218.895 | |
| 34 35 36 37 38 39 40 41 42 43 44 | |
| 3380.547 2997.830 3431.796 3643.438 2910.374 3453.052 2590.788 3532.352 2475.868 2098.178 2289.152 | |
| 45 46 47 48 49 50 51 52 53 54 55 | |
| 2081.347 2815.641 2974.892 2801.829 2434.326 3015.200 3449.667 3150.233 2914.642 3625.985 3054.540 | |
| 56 | |
| 3128.944 | |
| > model | |
| Call: | |
| lm(formula = Weight ~ MPG.city + EngineSize + Horsepower + Passengers) | |
| Coefficients: | |
| (Intercept) MPG.city EngineSize Horsepower Passengers | |
| 1459.043 -19.732 201.967 4.098 182.523 | |
| > #this is the same as the Fitted values of the model | |
| > attributes(model) | |
| $names | |
| [1] "coefficients" "residuals" "effects" "rank" "fitted.values" "assign" | |
| [7] "qr" "df.residual" "xlevels" "call" "terms" "model" | |
| $class | |
| [1] "lm" | |
| > fitted.values(model) | |
| 1 2 3 4 5 6 7 8 9 10 11 | |
| 2743.276 2889.000 2573.372 3314.338 4047.745 2733.637 2835.799 2479.407 4127.354 3988.672 2795.499 | |
| 12 13 14 15 16 17 18 19 20 21 22 | |
| 2809.266 3845.693 3337.968 2468.260 2203.848 3589.101 2773.481 4167.904 3460.915 3034.933 2479.407 | |
| 23 24 25 26 27 28 29 30 31 32 33 | |
| 3156.623 2604.810 2391.323 1708.818 4193.964 3247.412 2997.536 2610.730 3136.930 3428.463 3218.895 | |
| 34 35 36 37 38 39 40 41 42 43 44 | |
| 3380.547 2997.830 3431.796 3643.438 2910.374 3453.052 2590.788 3532.352 2475.868 2098.178 2289.152 | |
| 45 46 47 48 49 50 51 52 53 54 55 | |
| 2081.347 2815.641 2974.892 2801.829 2434.326 3015.200 3449.667 3150.233 2914.642 3625.985 3054.540 | |
| 56 | |
| 3128.944 | |
| > fitted(model) | |
| 1 2 3 4 5 6 7 8 9 10 11 | |
| 2743.276 2889.000 2573.372 3314.338 4047.745 2733.637 2835.799 2479.407 4127.354 3988.672 2795.499 | |
| 12 13 14 15 16 17 18 19 20 21 22 | |
| 2809.266 3845.693 3337.968 2468.260 2203.848 3589.101 2773.481 4167.904 3460.915 3034.933 2479.407 | |
| 23 24 25 26 27 28 29 30 31 32 33 | |
| 3156.623 2604.810 2391.323 1708.818 4193.964 3247.412 2997.536 2610.730 3136.930 3428.463 3218.895 | |
| 34 35 36 37 38 39 40 41 42 43 44 | |
| 3380.547 2997.830 3431.796 3643.438 2910.374 3453.052 2590.788 3532.352 2475.868 2098.178 2289.152 | |
| 45 46 47 48 49 50 51 52 53 54 55 | |
| 2081.347 2815.641 2974.892 2801.829 2434.326 3015.200 3449.667 3150.233 2914.642 3625.985 3054.540 | |
| 56 | |
| 3128.944 | |
| > #Input "new" data | |
| > test.predict <- predict(model, newdata=test) | |
| > #Check predictions | |
| > test.actual <- Cars93$Weight[-trainindex] | |
| > #To get errors, just subtract the two | |
| > errors <- test.actual-test.predict | |
| > #Plot to see our erros(residuals) | |
| > #Our model was off by as much as 500 pounds | |
| > plot(errors) | |
| > #Calculate Mean Squared Error to see how well model performed | |
| >#RMSE - same units as response variable. The lower, the better | |
| >#RMSE is a good measure of how accurately the model predicts the response, | |
| >#and is the most important criterion for fit if the main purpose of the model is prediction | |
| > testerrors <- sqrt(mean(errors^2)) | |
| > testerrors | |
| [1] 188.7558 | |
| > #Get MSE for original training data as well | |
| > trainerrors <- sqrt(mean(model$residuals^2)) | |
| > trainerrors | |
| [1] 183.8388 |
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