Lab 1
Akshay Kumar
02/03/2025
# Load necessary libraries
library(ISLR2)## Warning: package 'ISLR2' was built under R version 4.4.3library(ggplot2)9. This exercise involves the Auto data set studied in the lab. We have to make sure that the missing values have been removed from the data.
# Load the Auto dataset and remove missing values
data(Auto)
Auto <- na.omit(Auto)(a) Which of the predictors are quantitative, and which are qualitative?
sapply(Auto, class)## mpg cylinders displacement horsepower weight acceleration
## "numeric" "integer" "numeric" "integer" "integer" "numeric"
## year origin name
## "integer" "integer" "factor"# (a) Identify quantitative and qualitative predictors
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 : Factor w/ 304 levels "amc ambassador brougham",..: 49 36 231 14 161 141 54 223 241 2 ...
## - attr(*, "na.action")= 'omit' Named int [1:5] 33 127 331 337 355
## ..- attr(*, "names")= chr [1:5] "33" "127" "331" "337" ...# (b) Range of quantitative predictors
sapply(Auto[sapply(Auto, is.numeric)], range)## mpg cylinders displacement horsepower weight acceleration year origin
## [1,] 9.0 3 68 46 1613 8.0 70 1
## [2,] 46.6 8 455 230 5140 24.8 82 3(b) Range of Quantitative Predictors
sapply(Auto[sapply(Auto, is.numeric)], range)## mpg cylinders displacement horsepower weight acceleration year origin
## [1,] 9.0 3 68 46 1613 8.0 70 1
## [2,] 46.6 8 455 230 5140 24.8 82 3(c) Mean and Standard Deviation of Quantitative Predictors
sapply(Auto[sapply(Auto, is.numeric)], function(x) c(mean = mean(x), sd = sd(x)))## mpg cylinders displacement horsepower weight acceleration
## mean 23.445918 5.471939 194.412 104.46939 2977.5842 15.541327
## sd 7.805007 1.705783 104.644 38.49116 849.4026 2.758864
## year origin
## mean 75.979592 1.5765306
## sd 3.683737 0.8055182(d) Now remove the 10th through 85th observations. What is the range, mean, and standard deviation of each predictor in the subset of the data that remains?
Auto_subset <- Auto[-c(10:85),]
sapply(Auto_subset[sapply(Auto_subset, is.numeric)], function(x) c(range = diff(range(x)), mean = mean(x), sd = sd(x)))## mpg cylinders displacement horsepower weight acceleration
## range 35.600000 5.000000 387.00000 184.00000 3348.0000 16.300000
## mean 24.404430 5.373418 187.24051 100.72152 2935.9715 15.726899
## sd 7.867283 1.654179 99.67837 35.70885 811.3002 2.693721
## year origin
## range 12.000000 2.000000
## mean 77.145570 1.601266
## sd 3.106217 0.819910This part creates a subset of the data, excluding observations 10-85, and calculates range, mean, and standard deviation for numeric variables.
(e) Using the full data set, investigate the predictors graphically, using scatterplots or other tools of your choice. Create some plots highlighting the relationships among the predictors. Comment on your findings.
# Scatterplot matrix
pairs(Auto[,c("mpg","cylinders","displacement","horsepower","weight","acceleration","year")])
Based on the pairs plot:
Cylinder count doesn’t linearly affect MPG. 4-cylinder vehicles tend to be most fuel-efficient, while 3-cylinder models are least efficient.
MPG decreases as displacement, horsepower, and weight increase, showing a negative correlation.
Newer car models generally demonstrate improved fuel efficiency compared to older ones.
These observations highlight the complex relationships between various vehicle characteristics and fuel economy.
(f) Suppose that we wish to predict gas mileage (mpg) on the basis of the other variables. Do your plots suggest that any of the other variables might be useful in predicting mpg? Justify your answer.
# Scatterplot of mpg vs horsepower with smoothed trend line
ggplot(Auto, aes(x=horsepower, y=mpg)) +
geom_point() +
geom_smooth() +
labs(title="MPG vs Horsepower",
x="Horsepower",
y="Miles per Gallon")## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
# Boxplot of mpg by number of cylinders
ggplot(Auto, aes(x=as.factor(cylinders), y=mpg)) +
geom_boxplot() +
labs(title="MPG by Number of Cylinders",
x="Number of Cylinders",
y="Miles per Gallon")
# Scatterplot of mpg vs weight, colored by origin
ggplot(Auto, aes(x=weight, y=mpg, color=origin)) +
geom_point() +
labs(title="MPG vs Weight by Origin",
x="Weight",
y="Miles per Gallon")
# Faceted scatterplots of mpg vs displacement for each origin
ggplot(Auto, aes(x=displacement, y=mpg)) +
geom_point() +
geom_smooth(method="lm") +
facet_wrap(~origin) +
labs(title="MPG vs Displacement by Origin",
x="Displacement",
y="Miles per Gallon")## `geom_smooth()` using formula = 'y ~ x'
- The scatter plots show that weight, horsepower, and displacement have strong negative correlations with mpg.
- Cars with higher weight, horsepower, and displacement tend to have lower mpg, suggesting these predictors could be useful in predicting gas mileage.
Exercise 10
(a) Loading and Examining the Boston Dataset
library(ISLR2)
data(Boston)
dim(Boston)## [1] 506 13str(Boston)## 'data.frame': 506 obs. of 13 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...The Boston Housing Data set is a classic dataset frequently used in statistics and machine learning for regression tasks. It consists of 506 rows, where each row represents a suburb of Boston, and 14 columns, which include various features (predictors) as well as the target variable (housing prices). Here’s an explanation of each column:
| Feature | Description |
|---|---|
| crim | Crime rate per person by town. |
| zn | Percentage of residential land zoned for large lots (greater than 25,000 sq. ft.). |
| indus | Percentage of land in a town used for non-retail business purposes. |
| chas | A binary variable indicating whether the area is near the Charles River (1 = yes, 0 = no). |
| nox | Concentration of nitrogen oxides (measured in parts per 10 million). |
| rm | Average number of rooms per house. |
| age | Percentage of houses occupied by owners that were built before 1940. |
| dis | Weighted average distance to major employment hubs in Boston. |
| rad | Measure of how easily accessible radial highways are. |
| tax | Property tax rate per $10,000 of property value. |
| ptratio | Ratio of students to teachers in town schools. |
| b | Proportion of residents of African American descent. |
| lstat | Percentage of the population considered lower-income or lower-status. |
| medv | Median value of homes occupied by their owners (in $1,000s). |
(b) Make some pairwise scatter plots of the predictors in this data set. Describe your findings.
pairs(Boston[, 1:6])
Variables like RM (average number of rooms per dwelling) and LSTAT (percentage of lower status population) show strong correlations with MEDV (median home value). RM positively correlates with MEDV, while LSTAT has a negative correlation.
Predictors such as ZN (proportion of residential land zoned for lots over 25,000 sq.ft.) and CHAS (proximity to Charles River) exhibit weak or no clear correlation with MEDV.
(c) Are any of the predictors associated with percapita crime rate? If so, explain the relationship
cor(Boston$crim, Boston)## crim zn indus chas nox rm age
## [1,] 1 -0.2004692 0.4065834 -0.05589158 0.4209717 -0.2192467 0.3527343
## dis rad tax ptratio lstat medv
## [1,] -0.3796701 0.6255051 0.5827643 0.2899456 0.4556215 -0.3883046There is a strong relationship between the crime rate and the accessibility of radial highways, with higher crime rates observed in areas with better highway access. Additionally, crime rate tends to increase with a higher proportion of non-retail business land in a town and higher property tax rates.
On the other hand, there is an inverse relationship between the crime rate and the distance to employment centers towns located farther from employment hubs generally experience lower crime rates.
(d) Do any of the suburbs of Boston appear to have particularly high crime rates? Tax rates? Pupil-teacher ratios? Comment on the range of each predictor.
summary(Boston[, c("crim", "tax", "ptratio")])## crim tax ptratio
## Min. : 0.00632 Min. :187.0 Min. :12.60
## 1st Qu.: 0.08205 1st Qu.:279.0 1st Qu.:17.40
## Median : 0.25651 Median :330.0 Median :19.05
## Mean : 3.61352 Mean :408.2 Mean :18.46
## 3rd Qu.: 3.67708 3rd Qu.:666.0 3rd Qu.:20.20
## Max. :88.97620 Max. :711.0 Max. :22.00Crime Rates: The crime rates in the data set span a broad range, from as low as 0.00632 to as high as 88.9762. This indicates a significant disparity, with the majority of suburbs having minimal crime, while a few experience exceptionally high levels.
Tax Rates: Property tax rates vary considerably, ranging from 187 to 711. This highlights that some towns have relatively modest tax burdens, whereas others impose substantially higher taxes.
Pupil-Teacher Ratios: The pupil-teacher ratio falls within a narrower range of 12.60 to 22.00 compared to crime or tax rates. However, this still reflects some differences in the availability of educational resources across the suburbs.
(e) How many of the suburbs in this data set bound the Charles river?
sum(Boston$chas == 1)## [1] 35(f) What is the median pupil-teacher ratio among the towns in this data set?
median(Boston$ptratio)## [1] 19.05(g) Which suburb of Boston has lowest median value of owner-occupied homes? What are the values of the other predictors for that suburb, and how do those values compare to the overall ranges for those predictors? Comment on your findings.
lowest_medv <- which.min(Boston$medv)
Boston[lowest_medv,]## crim zn indus chas nox rm age dis rad tax ptratio lstat medv
## 399 38.3518 0 18.1 0 0.693 5.453 100 1.4896 24 666 20.2 30.59 5# Range of the predictors
smry=data.frame(sapply(Boston, function(x) c(min = min(x), max = max(x),avg=mean(x))))
smry## crim zn indus chas nox rm age
## min 0.006320 0.00000 0.46000 0.00000000 0.3850000 3.561000 2.9000
## max 88.976200 100.00000 27.74000 1.00000000 0.8710000 8.780000 100.0000
## avg 3.613524 11.36364 11.13678 0.06916996 0.5546951 6.284634 68.5749
## dis rad tax ptratio lstat medv
## min 1.129600 1.000000 187.0000 12.60000 1.73000 5.00000
## max 12.126500 24.000000 711.0000 22.00000 37.97000 50.00000
## avg 3.795043 9.549407 408.2372 18.45553 12.65306 22.53281The suburb with the lowest median home value exhibits distinct characteristics when compared to other suburbs in the dataset. The crime rate in this area is significantly higher than the average across all suburbs, and there is no residential zoning for large lots in the vicinity.
The proportion of land used for non-retail businesses in this suburb exceeds the average seen in other areas. Additionally, it does not border the Charles River, and nitrogen oxide levels are extremely elevated, indicating poor air quality.
This suburb is located far from employment centers and is the most distant from radial highways among all suburbs. The population here has a high concentration of lower-income residents, and the majority of the community consists of individuals of African American descent.
(h) In this data set, how many of the suburbs average more than seven rooms per dwelling? More than eight rooms per dwelling? Comment on the suburbs that average more than eight rooms per dwelling.
sum(Boston$rm > 7)## [1] 64sum(Boston$rm > 8)## [1] 13mt8=data.frame(sapply(Boston[Boston$rm>8,], function(x) Avg=mean(x)))
mt8## sapply.Boston.Boston.rm...8.....function.x..Avg...mean.x..
## crim 0.7187954
## zn 13.6153846
## indus 7.0784615
## chas 0.1538462
## nox 0.5392385
## rm 8.3485385
## age 71.5384615
## dis 3.4301923
## rad 7.4615385
## tax 325.0769231
## ptratio 16.3615385
## lstat 4.3100000
## medv 44.2000000There are 64 suburbs with homes averaging more than 7 rooms, and 13 suburbs where houses have over 8 rooms. The suburbs with homes exceeding 8 rooms generally exhibit distinct features:
Low Crime Rates: These areas experience significantly lower crime compared to other suburbs.
Residential Zoning: They feature a high proportion of land designated for large residential lots.
Proximity to Charles River: Many of these suburbs are situated near the Charles River.
Nitrogen Oxide Levels: Air pollution levels are moderate in these regions.
Older Homes: The majority of houses in these suburbs are older compared to those in other areas.
Accessibility: These suburbs are closer to employment hubs and radial highways, making them more conveniently located.
Tax Rates and Education: Property tax rates align closely with Boston’s average, and pupil-teacher ratios indicate a favorable educational environment.
Demographics: The population consists predominantly of African American residents, with a very small percentage of lower-income individuals.
Housing Values: Median home values in these suburbs are nearly double the average across all suburbs and approach the highest values in the data set.