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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 highjump shot run200m longjump javelin run800m

hurdles 1.00 0.81 0.65 0.77 0.91 0.01 0.78

highjump 0.81 1.00 0.44 0.49 0.78 0.00 0.59

shot 0.65 0.44 1.00 0.68 0.74 0.27 0.42

run200m 0.77 0.49 0.68 1.00 0.82 0.33 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

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)

Principal Components Proportion of Sample VariancesFirst two Principal Componentsbiplot(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.

The correlation between Y1 and the scoring looks very strong.

cor(heptathlon$score, heptathlon_pca$x[,1])

[1] -0.9910978

How well does it fit the Scoring?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)!

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