Principal Component Analysis
Olympic Heptathlon
Ch. 13
Principal components
• The Principal components method
summarizes data by finding the major
correlations in linear combinations of the
obervations.
– Little information lost in process, usually
– Major application: Correlated variables are
transformed into uncorrelated variables
Olympic Heptathlon Data
• 7 events: Hurdles, Highjump, Shot,
run200m, longjump, javelin, run800m
– The scores for these events are all on
different scales
– A relatively high number could be good or bad
depending on the event
• 25 Olympic competitors
R Commands
• Reorder the scores so that a high number
means a good score
– heptathlon$hurdles <max(heptathlon$hurdles) –
heptathlon$hurdles
• Hurdles, Run200m, Run800m requires reordering
Basic plot to look at data
• R Commands
– score <- which(colnames(heptathlon) ==
“score”)
• “which” searches the column names of the
heptathlon data.frame for “score” and stores it in a
variable “score” above
– plot(heptathlon[, -score])
• Scatterplot matrix, excluding the score column
Interpretation
• The data looks correlated except for the
javelin event.
– The book speculates the javelin is a ‘technical’
event, whereas the others are all ‘power’
events
To get numerical correlation values:
• round(cor(heptathlon[, -score]), 2)
• The cor(data.frame) function finds the actual
correlation values
• The cor(data.frame) function is in agreement with
this interpretation
hurdles
hurdles
1.00
highjump 0.81
shot
0.65
run200m 0.77
longjump 0.91
javelin
0.01
run800m 0.78
highjump
0.81
1.00
0.44
0.49
0.78
0.00
0.59
shot run200m
0.65 0.77
0.44 0.49
1.00 0.68
0.68 1.00
0.74 0.82
0.27 0.33
0.42 0.62
longjump
0.91
0.78
0.74
0.82
1.00
0.07
0.70
javelin
0.01
0.00
0.27
0.33
0.07
1.00
-0.02
run800m
0.78
0.59
0.42
0.62
0.70
-0.02
1.00
Running a Principal Component
analysis
• heptathlon_pca <- prcomp(heptathlon[, -score],
scale = TRUE)
• print(heptathlon_pca)
Standard deviations:
[1] 2.1119364 1.0928497 0.7218131 0.6761411 0.4952441 0.2701029 0.2213617
Rotation:
PC1
PC2
PC3
PC4
PC5
PC6
PC7
hurdles -0.4528710 0.15792058 -0.04514996 0.02653873 -0.09494792 -0.78334101 0.38024707
highjump -0.3771992 0.24807386 -0.36777902 0.67999172 0.01879888 0.09939981 -0.43393114
shot -0.3630725 -0.28940743 0.67618919 0.12431725 0.51165201 -0.05085983 -0.21762491
run200m -0.4078950 -0.26038545 0.08359211 -0.36106580 -0.64983404 0.02495639 -0.45338483
longjump -0.4562318 0.05587394 0.13931653 0.11129249 -0.18429810 0.59020972 0.61206388
javelin -0.0754090 -0.84169212 -0.47156016 0.12079924 0.13510669 -0.02724076 0.17294667
run800m -0.3749594 0.22448984 -0.39585671 -0.60341130 0.50432116 0.15555520 -0.09830963
•a1 <- heptathlon_pca$rotation[, 1]
•a1
•This shows the coefficients for the first principal
component y1
•Y1 is the linear combination of observations that
maximizes the sample variance as a portion of the
overall sample variance.
•Y2 is the linear combination that maximizes out of the
remaining portion of sample variance, with the added
constraint of being uncorrelated with Y1
• > a1<-heptathlon_pca$rotation[,1]
• > a1
hurdles
highjump
shot
run200m
longjump
javelin
run800m
-0.4528710 -0.3771992 -0.3630725 -0.4078950 -0.4562318 -0.0754090 -0.3749594
Interpretation
• 200m and long jump is the most important
factor
• Javelin result is less important
Data Analysis using the first
principal component
• center <- heptathlon_pca$center
– This is the center or mean of the variables, it can also be a flag
in the prcomp() function that sets the center at 0.
• scale <- heptathlon_pca$scale
– This is also a flag in the prcomp() function that can scale the
variables to fit between 0 and 1, as it is, its just storing the
current scale.
– hm <- as.matrix(heptathlon[, -score])
• This coerces the data.frame heptathlon into a matrix and excludes
score
– drop(scale(hm, center = center, scale = scale) %*%
heptathlon_pca$rotation[, 1])
• rescales the raw heptathlon data to the Principal component scale
• performs matrix multiplication on the coefficients of the linear
combination for the first principal component (Y1)
• Drop() prints the resulting matrix
Joyner-Kersee (USA)
John (GDR)
Behmer (GDR) Sablovskaite
(URS)
-4.121447626
-2.882185935
-2.649633766
-1.343351210
Choubenkova (URS)
Schulz (GDR)
Fleming (AUS)
Greiner (USA)
-1.359025696
-1.043847471
-1.100385639
-0.923173639
Lajbnerova (CZE)
Bouraga (URS)
Wijnsma (HOL) Dimitrova (BUL)
-0.530250689
-0.759819024
-0.556268302
-1.186453832
Scheider (SWI)
Braun (FRG) Ruotsalainen (FIN)
Yuping (CHN)
0.015461226
0.003774223
0.090747709
-0.137225440
Hagger (GB)
Brown (USA)
Mulliner (GB) Hautenauve (BEL)
0.171128651
0.519252646
1.125481833
1.085697646
Kytola (FIN)
Geremias (BRA)
Hui-Ing (TAI)
Jeong-Mi (KOR)
1.447055499
2.014029620
2.880298635
2.970118607
Launa (PNG)
6.270021972
An easier way
• predict(heptathlon_pca)[, 1]
– Accomplishes the same thing as the previous
set of commands
Principal Components Proportion of
Sample Variances
• The first component
contributes the vast
majority of total
sample variance
– Just looking at the first
two (uncorrelated!)
principal components
will account for most of
the overall sample
variance (~81%)
• plot(heptathlon_pca)
First two Principal Components
biplot(heptathlon_pca,col=c("gray","black"))
Interpretation
• The Olympians with the highest score
seem to be at the bottom left of the graph,
while
• The javelin event seems to give the scores
a more fine variation and award the
competitors a slight edge.
How well does it fit the Scoring?
• The correlation
between Y1 and the
scoring looks very
strong.
• cor(heptathlon$score,
heptathlon_pca$x[,1])
– [1] -0.9910978
Homework! (Ch.13)
• Use the “meteo” data on page 225 and create
scatterplots to check for correlation (don’t recode/reorder
anything, and remember not to include columns in the
analysis that don’t belong!
– Is there correlation? Don’t have R calculate the numerical values
unless you really want to
• Run PCA using the long way or the shorter “predict”
command (remember not to include the unneccesary
column!)
• Create a biplot, but use colors other than gray and black!
• Create a scatterplot like on page 224 of the 1st principle
component and the yield
– What is the numerical value of the correlation?
• Don’t forget to copy and paste your commands into word
and print it out for me (and include the scatterplot)!