1. PCA Example Air pollution in 41 cities in the USA.
R data “USairpollution”
Variables:
SO2: SO2 content of air in micrograms per cubic meter
temp: average annual temperature in degrees Fahrenheit
manu: number of manufacturing enterprises employing 20 or more workers
popul: population size (1970 census) in thousands
wind: average annual wind speed in miles per hour
precip: average annual precipitation in inches
predays: average number of days with precipitation per year

2. PCA Example Read the US air pollution data:
>USairpollution_data=read.csv("E:/Multivariate_analysis/Data/USairpollution.csv",header=T)
Remove the first two columns (City, SO2) and leave only the variables related to human ecology (popul, manu) and four to climate (temp, wind, precip, predays).
>USairpollution=USairpollution_data[,-c(1,2)]
Transform temperature to negative values. This way, high values for all six variables
indicate unfriendly environment.
> USairpollution$negtemp=USairpollution$temp*(-1)
> USairpollution$temp=NULL
We need to extract the principal components from the correlation matrix
rather than covariance matrix, since the variables are on very different scales

3. PCA Example Correlation matrix for Usairpollution:
>round(cor(USairpollution),2)
manu popul wind precip predays negtemp
manu
popul
wind
precip
predays
negtemp
High correlations between popul and manu.

4. PCA Example
We construct a scatter plot of the six variables and we include the histograms for
each variable on the main diagonal. For this we need the package “HSAUR2”
>library(HSAUR2)
Build the function “panel.hist” to insert the histograms on the main diagonal.
>panel.hist<-function(x, ...){
usr<-par("usr");on.exit(par(usr))
par(usr=c(usr[1:2],0,1.5)) # set the plotting coordinates
h=hist(x,plot=FALSE)
breaks<-h$breaks;nB<-length(breaks)
y<-h$counts;y<-y/max(y)
rect(breaks[-nB],0,breaks[-1],y,col="grey",...) }
Plot the scatterplot matrix:
>pairs(USairpollution,diag.panel=panel.hist,pch=".",cex=1.5)

5. PCA Example
Outliers

6. PCA Example Extract the principal components:
> USairpollution_PCA=princomp(USairpollution,cor=TRUE)
> summary(USairpollution_PCA,loadings=TRUE)
Importance of components:
Comp Comp Comp Comp Comp.5
Standard deviation
Proportion of Variance
Cumulative Proportion
Comp.6
Standard deviation
Proportion of Variance
Cumulative Proportion
Loadings:
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6
manu
popul
wind
precip
Predays
negtemp

7. PCA Example Eigenvectors and eigenvalues
> eigen(cor(USairpollution))
$values
[1]
$vectors
[,1] [,2] [,3] [,4] [,5] [,6]
[1,]
[2,]
[3,]
[4,]
[5,]
[6,]

8. PCA Example
The first three components accounts for almost 85% of the variance of the original
variables and have eigenvalues greater than one.
Equations of the first three principal components:
The first component is an indicator of quality of life with high values
showing poor environment, the second component is concerned with city’s
rainfall with high values for precip and predays, and the third component is a
contrast between precip and negtemp.

9. PCA Example
Scatter plot matrix of the first three principal components:
> library(MVA)
> pairs(USairpollution_PCA$scores[,1:3],xlim=c(-6,4),ylim=c(-6,4),
panel=function(x,y,...){
text(x,y,abbreviate(USairpollution_data[,1]),
cex=0.75)
bvbox(cbind(x,y),add=TRUE) })

10. PCA Example
The scatter plots indicate that Chicago and possible Phoenix and Philadelphia
are outliers.

11. PCA Example
We want to determine which of the climate and human ecology variables are the best predictors of the degree of air pollution in a city as measured by the SO2 in the air. We use multiple regression to address this problem.
Scatter plots of the 6 principal components vs. SO2 concentration:
par(mfrow=c(2,3))
out=sapply(1:6,function(i){
plot(USairpollution_data$SO2,USairpollution_PCA$scores[,i],
xlab=paste("PC",i,sep=""),
ylab="SO2 concentration")})
The first principal component is the most predictive of sulfur dioxide concentration
(p-value =2.28e-07).

12. PCA Example

13. PCA Example
> usair_reg=lm(SO2~USairpollution_PCA$scores,data=USairpollution_data)
> summary(usair_reg)
Call:
lm(formula = SO2 ~ USairpollution_PCA$scores, data = USairpollution_data)
Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-15 ***
USairpollution_PCA$scoresComp e-07 ***
USairpollution_PCA$scoresComp
USairpollution_PCA$scoresComp
USairpollution_PCA$scoresComp **
USairpollution_PCA$scoresComp *
USairpollution_PCA$scoresComp **
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: on 34 degrees of freedom
Multiple R-squared: , Adjusted R-squared:
F-statistic: on 6 and 34 DF, p-value: 5.419e-07