# Principal Components Analysis - how to get the contribution (%) of each parameter to a Prin.Comp.?

I want to know to what degree a measurement/parameter contributes to one of the calculated principal components.

A real-world description:

1. i've got five climatic parameters to the geographic distribution of a species
2. i performed a PCA with these five parameters
3. 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?

```R> class(pca\$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
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.

```R> load <- with(pca, unclass(loadings))
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
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

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)
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
```