Performing pca using r

Question

I am trying to perform pca with R.

I have the following data matrix:

         V2  V3  V4 V5   V6
2430   0 168 290 45 1715
552928  188  94 105 60 3374
55267    0   0 465  0 3040
27787   0   0   0  0 3380
938270   0  56  56  0 2039
249165   0   0 332  0 2548
31009   0   0   0  0 2690
314986   0   0   0  0 2897
5001    0   0   0  0 3453
28915   0 262 175  0 2452
5261    0   0 351  0 3114
74412   0 109  54  0 2565
16007   0   0 407  0 1730
6614   0  71 179  0 2403
419    0   0   0  0 2825

with 15 variables and 5 samples.

I tried the following code (which uses the transpose of my data matrix):

fit <- prcomp(t(dt))
summary(fit) # print variance accounted for
loadings(fit) # pc loadings
plot(fit,type="lines") # scree plot
fit$scores # the principal components
biplot(fit)

which returns:

> summary(fit) # print variance accounted for
Importance of components:
                             PC1       PC2       PC3      PC4       PC5
Standard deviation     4651.1348 298.09026 126.79032 41.03270 3.474e-13
Proportion of Variance    0.9951   0.00409   0.00074  0.00008 0.000e+00
Cumulative Proportion     0.9951   0.99918   0.99992  1.00000 1.000e+00


loadings(fit) # pc loadings
NULL
> plot(fit,type="lines") # scree plot
> fit$scores # the principal components
NULL

I then tried with the original data matrix (not transposed):

fit <- prcomp(dt)
summary(fit) # print variance accounted for
loadings(fit) # pc loadings
plot(fit,type="lines") # scree plot
fit$scores # the principal components
biplot(fit)

Importance of components:
                            PC1       PC2      PC3      PC4     PC5
Standard deviation     562.2600 156.13452 75.59006 43.63721 9.21936
Proportion of Variance   0.9079   0.07001  0.01641  0.00547 0.00024
Cumulative Proportion    0.9079   0.97788  0.99429  0.99976 1.00000

> loadings(fit) # pc loadings
NULL
> plot(fit,type="lines") # scree plot
> fit$scores # the principal components
NULL
> biplot(fit)

In both cases, I have 5 principal components that explain 100% of the variability. However, since I have 15 variables, shouldn't 100% of variability be explained by 15 variables?

Answer 1

The number of principal components can never exceed the number of samples. Perhaps too simply put, since you only have 5 samples you only need 5 variables to explain the variability.