Week 6 Homework
Question 9.1
Using the same crime data set as in Question 8.2, apply Principal Component Analysis and then create a
regression model using the first few principal components. Specify your new model in
...
Week 6 Homework
Question 9.1
Using the same crime data set as in Question 8.2, apply Principal Component Analysis and then create a
regression model using the first few principal components. Specify your new model in terms of the original
variables (not the principal components), and compare its quality to that of your solution to Question 8.2.
You can use the R function prcomp for PCA. Note that to first scale the data, you can include scale. =
TRUE to scale as part of the PCA function. Don’t forget that, to make a prediction for the new city, you’ll
need to unscale the coefficients (i.e., do the scaling calculation in reverse!
require("knitr")
## Loading required package: knitr
opts_knit$set(root.dir = "~/Desktop/GT OMSA/ISYE 6501/Wk6")
Setting up the environment
rm(list=ls())
set.seed(1)
library(MASS)
library(reshape2)
library(ggplot2)
library(Hmisc)
## Loading required package: lattice
## Loading required package: survival
## Loading required package: Formula
##
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:base':
##
## format.pval, units
library(dplyr)
##
## Attaching package: 'dplyr'
## The following objects are masked from 'package:Hmisc':
##
## src, summarize
## The following object is masked from 'package:MASS':
##
## select
## The following objects are masked from 'package:stats':
##
## filter, lag
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
1
library(DAAG)
##
## Attaching package: 'DAAG'
## The following object is masked from 'package:survival':
##
## lung
## The following object is masked from 'package:MASS':
##
## hills
crime <- read.table("uscrime.txt", header = TRUE)
head(crime)
## M So Ed Po1 Po2 LF M.F Pop NW U1 U2 Wealth Ineq
## 1 15.1 1 9.1 5.8 5.6 0.510 95.0 33 30.1 0.108 4.1 3940 26.1
## 2 14.3 0 11.3 10.3 9.5 0.583 101.2 13 10.2 0.096 3.6 5570 19.4
## 3 14.2 1 8.9 4.5 4.4 0.533 96.9 18 21.9 0.094 3.3 3180 25.0
## 4 13.6 0 12.1 14.9 14.1 0.577 99.4 157 8.0 0.102 3.9 6730 16.7
## 5 14.1 0 12.1 10.9 10.1 0.591 98.5 18 3.0 0.091 2.0 5780 17.4
## 6 12.1 0 11.0 11.8 11.5 0.547 96.4 25 4.4 0.084 2.9 6890 12.6
## Prob Time Crime
## 1 0.084602 26.2011 791
## 2 0.029599 25.2999 1635
## 3 0.083401 24.3006 578
## 4 0.015801 29.9012 1969
## 5 0.041399 21.2998 1234
## 6 0.034201 20.9995 682
Reading in and viewing the data
crime <- read.table("uscrime.txt", header = TRUE)
head(crime)
## M So Ed Po1 Po2 LF M.F Pop NW U1 U2 Wealth Ineq
## 1 15.1 1 9.1 5.8 5.6 0.510 95.0 33 30.1 0.108 4.1 3940 26.1
## 2 14.3 0 11.3 10.3 9.5 0.583 101.2 13 10.2 0.096 3.6 5570 19.4
## 3 14.2 1 8.9 4.5 4.4 0.533 96.9 18 21.9 0.094 3.3 3180 25.0
## 4 13.6 0 12.1 14.9 14.1 0.577 99.4 157 8.0 0.102 3.9 6730 16.7
## 5 14.1 0 12.1 10.9 10.1 0.591 98.5 18 3.0 0.091 2.0 5780 17.4
## 6 12.1 0 11.0 11.8 11.5 0.547 96.4 25 4.4 0.084 2.9 6890 12.6
## Prob Time Crime
## 1 0.084602 26.2011 791
## 2 0.029599 25.2999 1635
## 3 0.083401 24.3006 578
## 4 0.015801 29.9012 1969
## 5 0.041399 21.2998 1234
## 6 0.034201 20.9995 682
Variable “So” is binary, as this doesnt make sense in a PCA model i am removing it.
crime1 <- crime[-2]
head(crime1)
## M Ed Po1 Po2 LF M.F Pop NW U1 U2 Wealth Ineq Prob
## 1 15.1 9.1 5.8 5.6 0.510 95.0 33 30.1 0.108 4.1 3940 26.1 0.084602
2
## 2 14.3 11.3 10.3 9.5 0.583 101.2 13 10.2 0.096 3.6 5570 19.4 0.029599
## 3 14.2 8.9 4.5 4.4 0.533 96.9 18 21.9 0.094 3.3 3180 25.0 0.083401
## 4 13.6 12.1 14.9 14.1 0.577 99.4 157 8.0 0.102 3.9 6730 16.7 0.015801
## 5 14.1 12.1 10.9 10.1 0.591 98.5 18 3.0 0.091 2.0 5780 17.4 0.041399
## 6 12.1 11.0 11.8 11.5 0.547 96.4 25 4.4 0.084 2.9 6890 12.6 0.034201
## Time Crime
## 1 26.2011 791
## 2 25.2999 1635
## 3 24.3006 578
## 4 29.9012 1969
## 5 21.2998 1234
## 6 20.9995 682
Running the PCA model based on the crime data
pca <- prcomp(crime1[,1:15], scale = TRUE)
Summarizing and plotting the PCA
summary(pca)
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 2.3802 1.6756 1.4202 1.16749 1.03667 0.74864 0.5988
## Proportion of Variance 0.3777 0.1872 0.1345 0.09087 0.07165 0.03736 0.0239
## Cumulative Proportion 0.3777 0.5649 0.6993 0.79020 0.86185 0.89921 0.9231
## PC8 PC9 PC10 PC11 PC12 PC13
## Standard deviation 0.55069 0.48478 0.44375 0.42652 0.32674 0.26644
## Proportion of Variance 0.02022 0.01567 0.01313 0.01213 0.00712 0.00473
## Cumulative Proportion 0.94334 0.95900 0.97213 0.98426 0.99138 0.99611
## PC14 PC15
## Standard deviation 0.2324 0.06595
## Proportion of Variance 0.0036 0.00029
## Cumulative Proportion 0.9997 1.00000
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