I want to know to what degree a measurement/parameter contributes to one of the calculated principal components.
A real-world description:
- i've got five climatic parameters to the geographic distribution of a species
- i performed a PCA with these five parameters
- the plot of the PC1 vs. PC2 shows an interesting pattern
Question: How do I get the percentage of contribution (of each parameter) to each PC?
What I expect: PC1 is composed to 30% of parameter1, to 50% of parameter2, to 20% of parameter3, 0% of parameter4 and 0% of parameter5. PC2 is composed...
An example with 5 dummy-parameters:
a <- rnorm(10, 50, 20)
b <- seq(10, 100, 10)
c <- seq(88, 10, -8)
d <- rep(seq(3, 16, 3), 2)
e <- rnorm(10, 61, 27)
my_table <- data.frame(a, b, c, d, e)
pca <- princomp(my_table, cor=T)
biplot(pca) # same: plot(pca$scores[,1], pca$scores[,2])
pca
summary(pca)
Where is my information hidden?
You want the $loadings component of the returned object:
R> class(pca$loadings)
[1] "loadings"
R> pca$loadings
Loadings:
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
a -0.198 0.713 -0.671
b 0.600 0.334 -0.170 0.707
c -0.600 -0.334 0.170 0.707
d 0.439 -0.880 -0.180
e 0.221 0.701 0.678
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
SS loadings 1.0 1.0 1.0 1.0 1.0
Proportion Var 0.2 0.2 0.2 0.2 0.2
Cumulative Var 0.2 0.4 0.6 0.8 1.0
Note that this has a special print() method which suppresses printing of small loadings.
If you want this as a relative contribution then sum up the loadings per column and express each loading as a proportion of the column (loading) sum, taking care to use the absolute values to account for negative loadings.
R> load <- with(pca, unclass(loadings))
R> load
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
a -0.1980087 0.712680378 0.04606100 -0.6713848 0.000000e+00
b 0.5997346 -0.014945831 0.33353047 -0.1698602 7.071068e-01
c -0.5997346 0.014945831 -0.33353047 0.1698602 7.071068e-01
d 0.4389388 0.009625746 -0.88032515 -0.1796321 5.273559e-16
e 0.2208215 0.701104321 -0.02051507 0.6776944 -1.110223e-16
This final step then yields the proportional contribution to the each principal component
R> aload <- abs(load) ## save absolute values
R> sweep(aload, 2, colSums(aload), "/")
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
a 0.09624979 0.490386943 0.02853908 0.35933068 0.000000e+00
b 0.29152414 0.010284050 0.20665322 0.09091055 5.000000e-01
c 0.29152414 0.010284050 0.20665322 0.09091055 5.000000e-01
d 0.21336314 0.006623362 0.54544349 0.09614059 3.728970e-16
e 0.10733880 0.482421595 0.01271100 0.36270762 7.850462e-17
R> colSums(sweep(aload, 2, colSums(aload), "/"))
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
1 1 1 1 1
If using the preferred prcomp() then the relevant loadings are in the $rotation component:
R> pca2 <- prcomp(my_table, scale = TRUE)
R> pca2$rotation
PC1 PC2 PC3 PC4 PC5
a -0.1980087 0.712680378 -0.04606100 -0.6713848 0.000000e+00
b 0.5997346 -0.014945831 -0.33353047 -0.1698602 -7.071068e-01
c -0.5997346 0.014945831 0.33353047 0.1698602 -7.071068e-01
d 0.4389388 0.009625746 0.88032515 -0.1796321 -3.386180e-15
e 0.2208215 0.701104321 0.02051507 0.6776944 5.551115e-17
And the relevant incantation is now:
R> aload <- abs(pca2$rotation)
R> sweep(aload, 2, colSums(aload), "/")
PC1 PC2 PC3 PC4 PC5
a 0.09624979 0.490386943 0.02853908 0.35933068 0.000000e+00
b 0.29152414 0.010284050 0.20665322 0.09091055 5.000000e-01
c 0.29152414 0.010284050 0.20665322 0.09091055 5.000000e-01
d 0.21336314 0.006623362 0.54544349 0.09614059 2.394391e-15
e 0.10733880 0.482421595 0.01271100 0.36270762 3.925231e-17
来源：https://stackoverflow.com/questions/12760108/principal-components-analysis-how-to-get-the-contribution-of-each-paramete