Principal Component Analysis

Distribuzioni di probabilità multivariate in R

Surajit Ray

Professor, University of Glasgow

Principal Component Analysis (PCA) goals

Dimension reduction
Creating uncorrelated variables
Capturing variability in fewer dimensions

Algorithm

PC1 explains maximum variation in orange direction
PC2 uncorrelated to PC1 - explains maximum remaining variation in blue direction
PC3 uncorrelated to PC1 and PC2 - explains maximum remaining variation in green direction

princomp() function calculates PCs

Principal Component Analysis in R

Simplified format

princomp(x, cor = FALSE, scores = TRUE)

x: a numeric matrix or data frame

cor: use correlation matrix instead of covariance

scores: scores/projection of the data on principal components are produced

Principal Component Analysis of mtcars dataset

mtcars dataset relates to 11 variables on fuel consumption for 32 automobiles

head(mtcars,5)

                     mpg cyl  disp  hp drat    wt  qsec vs am gear carb
Mazda RX4           21.0   6 160.0 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag       21.0   6 160.0 110 3.90 2.875 17.02  0  1    4    4
Datsun 710          22.8   4 108.0  93 3.85 2.320 18.61  1  1    4    1
Hornet 4 Drive      21.4   6 258.0 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout   18.7   8 360.0 175 3.15 3.440 17.02  0  0    3    2

Selecting numeric columns from mtcars dataset

Exclude the vs and am variables - both binary
```
mtcars.sub <- mtcars[ , -c(8,9)] 
```
$$

Perform PCA

cars.pca <- princomp(mtcars.sub, cor = TRUE, scores = TRUE)

princomp function output

cars.pca

Standard deviations:
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9 
 2.378  1.443  0.710  0.515  0.428  0.352  0.324  0.242  0.149

summary(cars.pca)

Importance of components:
                       Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8  Comp.9
Standard deviation      2.378  1.443  0.710 0.5148 0.4280 0.3518 0.3241 0.2419 0.14896
Proportion of Variance  0.628  0.231  0.056 0.0294 0.0204 0.0138 0.0117 0.0065 0.00247
Cumulative Proportion   0.628  0.860  0.916 0.9453 0.9656 0.9794 0.9910 0.9975 1.00000

Let's apply principal component analysis!

Distribuzioni di probabilità multivariate in R