Homework_3
Contreras, Eric
2024-10-06
Load the data and the packages:
library(ggplot2)Exploratory Data Analysis (EDA)
Read the data document on the data
?mtcarsExamine data and variable types
#Sampling the first few rows of data for familiarization
head(mtcars)## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1#Review the structure of the data
str(mtcars)## 'data.frame': 32 obs. of 11 variables:
## $ mpg : num 21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
## $ cyl : num 6 6 4 6 8 6 8 4 4 6 ...
## $ disp: num 160 160 108 258 360 ...
## $ hp : num 110 110 93 110 175 105 245 62 95 123 ...
## $ drat: num 3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
## $ wt : num 2.62 2.88 2.32 3.21 3.44 ...
## $ qsec: num 16.5 17 18.6 19.4 17 ...
## $ vs : num 0 0 1 1 0 1 0 1 1 1 ...
## $ am : num 1 1 1 0 0 0 0 0 0 0 ...
## $ gear: num 4 4 4 3 3 3 3 4 4 4 ...
## $ carb: num 4 4 1 1 2 1 4 2 2 4 ...Summary statistics
summary(mtcars)## mpg cyl disp hp
## Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
## Median :19.20 Median :6.000 Median :196.3 Median :123.0
## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
## drat wt qsec vs
## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
## Median :3.695 Median :3.325 Median :17.71 Median :0.0000
## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :0.0000 Median :4.000 Median :2.000
## Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :1.0000 Max. :5.000 Max. :8.000table(mtcars$cyl)##
## 4 6 8
## 11 7 14Visualization Of the Data
ggplot(mtcars, aes(x = mpg, y = wt)) +
geom_point() +
labs(title = "MPG vs Vehicle Weight", x = "MPG", y = "WT (1000 lbs)")
Anaylsis
This visualization shows that the heavier the vehicle the less miles it will be able to travel per gallon.
ggplot(mtcars, aes(x = mpg, y = hp)) +
geom_point() +
labs(title = "MPG vs Horsepower", x = "MPG", y = "Horsepower")
Anaylsis
The visual indicates there is a increase in MPG as the weight of the vehicle drops.
ggplot(mtcars, aes(x = wt, y = hp)) +
geom_point() +
labs(title = "Horsepower required by Weight", x = "WT (1000 lbs)", y = "Horsepower")
Anaylsis
After comparing how horsepower and weight effect the MPG, a third relationship can be seen between the horsepower required depending on the weight on the vehicle. With the three visual present, one can start to determine the ideal range of motor requirements to achieve the best mileage.
corr_matrix <- cor(mtcars)
print(corr_matrix)## mpg cyl disp hp drat wt
## mpg 1.0000000 -0.8521620 -0.8475514 -0.7761684 0.68117191 -0.8676594
## cyl -0.8521620 1.0000000 0.9020329 0.8324475 -0.69993811 0.7824958
## disp -0.8475514 0.9020329 1.0000000 0.7909486 -0.71021393 0.8879799
## hp -0.7761684 0.8324475 0.7909486 1.0000000 -0.44875912 0.6587479
## drat 0.6811719 -0.6999381 -0.7102139 -0.4487591 1.00000000 -0.7124406
## wt -0.8676594 0.7824958 0.8879799 0.6587479 -0.71244065 1.0000000
## qsec 0.4186840 -0.5912421 -0.4336979 -0.7082234 0.09120476 -0.1747159
## vs 0.6640389 -0.8108118 -0.7104159 -0.7230967 0.44027846 -0.5549157
## am 0.5998324 -0.5226070 -0.5912270 -0.2432043 0.71271113 -0.6924953
## gear 0.4802848 -0.4926866 -0.5555692 -0.1257043 0.69961013 -0.5832870
## carb -0.5509251 0.5269883 0.3949769 0.7498125 -0.09078980 0.4276059
## qsec vs am gear carb
## mpg 0.41868403 0.6640389 0.59983243 0.4802848 -0.55092507
## cyl -0.59124207 -0.8108118 -0.52260705 -0.4926866 0.52698829
## disp -0.43369788 -0.7104159 -0.59122704 -0.5555692 0.39497686
## hp -0.70822339 -0.7230967 -0.24320426 -0.1257043 0.74981247
## drat 0.09120476 0.4402785 0.71271113 0.6996101 -0.09078980
## wt -0.17471588 -0.5549157 -0.69249526 -0.5832870 0.42760594
## qsec 1.00000000 0.7445354 -0.22986086 -0.2126822 -0.65624923
## vs 0.74453544 1.0000000 0.16834512 0.2060233 -0.56960714
## am -0.22986086 0.1683451 1.00000000 0.7940588 0.05753435
## gear -0.21268223 0.2060233 0.79405876 1.0000000 0.27407284
## carb -0.65624923 -0.5696071 0.05753435 0.2740728 1.00000000library(corrplot)## corrplot 0.94 loadedcorrplot(corr_matrix, method="circle", type="upper", order="hclust",
tl.col="black", tl.srt=45)
Anaylsis
From the correlation visual above, we can confirm that for mpg, weight, horsepower, cylinders, and displacement hold the strongest relationships.
Data Preprocessing
The Motor Trend Car Road Tests data set is a well-studied and cleaned data set that does not require much pre-processing but a check can always be performed.
Check whether there is missing Value for each column
colSums(is.na(mtcars))## mpg cyl disp hp drat wt qsec vs am gear carb
## 0 0 0 0 0 0 0 0 0 0 0Outliers
boxplot(mtcars, las=2, cex.axis=0.6)
Outliers: disp and hp both show outliers, indicated by points outside the whiskers. These outliers represent values that are notably higher or lower than the typical range.
Skewness: disp and hp distributions are somewhat skewed, with larger spreads and asymmetry in their box plots, suggesting potential non-normality in these variables.
# Truncate data based on a specific threshold (80 in this case)
mtcars_truncated <- mtcars[mtcars$mpg <= 80, ]
dim(mtcars_truncated)## [1] 32 11# Calculate the 1st and 99th percentiles for 'cyl'
lower_bound_cyl <- quantile(mtcars_truncated$cyl, 0.01, na.rm = TRUE)
upper_bound_cyl <- quantile(mtcars_truncated$cyl, 0.99, na.rm = TRUE)
mtcars_truncated$cyl[mtcars_truncated$cyl < lower_bound_cyl] <- lower_bound_cyl
mtcars_truncated$cyl[mtcars_truncated$cyl > upper_bound_cyl] <- upper_bound_cylsummary(mtcars_truncated[,c("mpg","cyl")])## mpg cyl
## Min. :10.40 Min. :4.000
## 1st Qu.:15.43 1st Qu.:4.000
## Median :19.20 Median :6.000
## Mean :20.09 Mean :6.188
## 3rd Qu.:22.80 3rd Qu.:8.000
## Max. :33.90 Max. :8.000summary(mtcars_truncated$mpg)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 10.40 15.43 19.20 20.09 22.80 33.90Simple regression
mtcars_lm <- lm(mpg ~disp, data = mtcars)
summary(mtcars_lm)##
## Call:
## lm(formula = mpg ~ disp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.8922 -2.2022 -0.9631 1.6272 7.2305
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.599855 1.229720 24.070 < 2e-16 ***
## disp -0.041215 0.004712 -8.747 9.38e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.251 on 30 degrees of freedom
## Multiple R-squared: 0.7183, Adjusted R-squared: 0.709
## F-statistic: 76.51 on 1 and 30 DF, p-value: 9.38e-10The coefficient for disp is -0.041215, which indicates that for every unit increase in disp (displacement), the mpg (miles per gallon) is expected to decrease by 0.041215, holding all else constant. This negative relationship is highly significant, as indicated by the low p-value (9.38e-10).
Linear regression modal
When using linear regression, the following assumptions are made:
1. Relationship between the independent and dependent variables is linear.
2. The residuals (errors) are independent of each other, with no autocorrelation.
3. The residuals are normally distributed.
4. In cases of multiple regression, the independent variables should not be highly correlated with each other.Evaluate the model via Mean Squared Error (MSE) for a fitted model:
lm_mse <- mean((mtcars_lm$fitted.values - mtcars$mpg)^2)
print(paste("Mean Squared Error for Linear Model:", round(lm_mse, 2)))## [1] "Mean Squared Error for Linear Model: 9.91"Linear regression model with interaction terms
model <- lm(mpg ~ disp * hp + wt * qsec + cyl * am, data = mtcars)
summary(model)##
## Call:
## lm(formula = mpg ~ disp * hp + wt * qsec + cyl * am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.5462 -1.4358 -0.6963 1.5130 3.2800
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 21.4289697 41.2342873 0.520 0.608
## disp -0.0295298 0.0319580 -0.924 0.366
## hp -0.0665118 0.0412461 -1.613 0.121
## wt -2.2081431 13.4093655 -0.165 0.871
## qsec 0.9229689 2.1373401 0.432 0.670
## cyl 0.3772102 0.9917047 0.380 0.707
## am 4.9354769 4.3727586 1.129 0.271
## disp:hp 0.0001912 0.0001473 1.299 0.208
## wt:qsec -0.0870827 0.7262635 -0.120 0.906
## cyl:am -0.5917330 0.6892465 -0.859 0.400
##
## Residual standard error: 2.345 on 22 degrees of freedom
## Multiple R-squared: 0.8925, Adjusted R-squared: 0.8486
## F-statistic: 20.3 on 9 and 22 DF, p-value: 1.108e-08Anaylsis
The inclusion of interaction terms has led to a multiple R-squared of 0.8925, indicating that approximately 89.25% of the variance in mpg is explained by the model with these interaction terms. This is a relatively high R-squared value, suggesting a good model fit. However, the p-values indicate that most terms, including interactions, are not statistically significant, so while R-squared is high, the individual predictor contributions are not strong.