This notebook will examine some characterisitics of the "housing-header.txt" dataset, a dataset of housing data in Boston neighborhoods. For reference:

1. CRIM: per capita crime rate by town
2. ZN: proportion of residential land zoned for lots over 25,000 sq.ft .
3. INDUS: proportion of non retail business acres per town
4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
5. NOX: nitric oxides concentration (parts per 10 million)
6. RM: average number of rooms per dwelling
7. AGE: proportion of owner occupied units built prior to 1940
8. DIS: weighted distances to five Boston employment centers
9. RAD: index of accessibility to radial highways
10. TAX: full value property tax rate per 10,000
11. PTRATIO: pupil teacher ratio by town
12. B: 1000(Bk 0.63)^2 where Bk is the proportion of blacks by town
13. LSTAT: percent lower status of the population
14. MEDV: Median value of owner occupied homes in 1000's

First we must load the dataset:

housing<-read.table("housing.header.txt", header=TRUE, sep=",")
colnames(housing)<-c("Crim", "Zn", "Indus", "Chas", "Nox", "Rm", "Age", "Dis", "Rad", "Tax", "Ptratio", "B", "Lstat", "Medv")
head(housing)

Crim	Zn	Indus	Chas	Nox	Rm	Age	Dis	Rad	Tax	Ptratio	B	Lstat	Medv
0.00632	18	2.31	0	0.538	6.575	65.2	4.0900	1	296	15.3	396.90	4.98	24.0
0.02731	0	7.07	0	0.469	6.421	78.9	4.9671	2	242	17.8	396.90	9.14	21.6
0.02729	0	7.07	0	0.469	7.185	61.1	4.9671	2	242	17.8	392.83	4.03	34.7
0.03237	0	2.18	0	0.458	6.998	45.8	6.0622	3	222	18.7	394.63	2.94	33.4
0.06905	0	2.18	0	0.458	7.147	54.2	6.0622	3	222	18.7	396.90	5.33	36.2
0.02985	0	2.18	0	0.458	6.430	58.7	6.0622	3	222	18.7	394.12	5.21	28.7

We then plot all samples with respect to the crime rate:

attach(housing)
plot(Crim)

We then continue with our example using crime rate: plotting its density as well as a histogram:

par(mfrow=c(1,2))
hist(Crim)
title ("Histogram of Crim")
plot(density(Crim, na.rm=TRUE))

We then plot crime rate, number of rooms, age of the home, and property tax rate with respect to median house value, in order to examine the correlation:

par(mfrow=c(1,4))
plot(Crim, Medv)
title("Crim-Medv")
plot(Rm, Medv)
title("Rm-Medv")
plot(Age, Medv)
title("Age-Medv")
plot(Tax, Medv)
title("Tax-Medv")

From the scatterplots, we can observe that:

• Crim and Medv appear to have a negative correlation
• Rm and Medv appear to have a positive correlation
• Age and Medv appear to have a negative correlation
• Tax and Medv appear to have a negative correlation

We can then report the pairwise correlation between every two variables using a level plot:

cv<-cor(housing)
library(lattice)
levelplot(cv)

Observe the level plot above. The following variables are negatively correlated with Medv:

• Lstat
• Ptratio
• Tax
• Rad
• Age
• Nox
• Indus
• Crim

The following variables have little or no correlation with Medv:

• B
• Dis
• Chas
• Zn

The following variables are positively correlated with Medv:

• Medv itself (correlation of 1 expected)
• Rm

We also demonstrate this same correlation with scatterplots:

par(mfrow=c(2, 7))
plot(Crim, Medv)
title("Crim-Medv")
plot(Zn, Medv)
title("Zn-Medv")
plot(Indus,Medv)
title("Indus-Medv")
plot(Chas, Medv)
title("Chas-Medv")
plot(Nox,Medv)
title("Nox-Medv")
plot(Rm, Medv)
title("Rm-Medv")
plot(Age, Medv)
title("Age-Medv")
plot(Dis, Medv)
title("Dis-Medv")
plot(Rad, Medv)
title("Rad-Medv")
plot(Tax, Medv)
title("Tax-Medv")
plot(Ptratio, Medv)
title("Ptratio-Medv")
plot(B, Medv)
title("B-Medv")
plot(Lstat, Medv)
title("Lstat-Medv")

The attributes that are posiitvely correlated with Medv will have a generally increasing trend on the scatterplot (x increases as y increases). The attributes that are independent of Medv will have no easily observable trend. The attributes that are negatvely correlated with Medv will have a generally decreasing trend on the scatterplot (x decreases as y increases)