## Deriving meaning from principal components analysis

I'm going to illustrate how you can use principal components analysis to find underlying trends in your data. If you want to reproduce what I'm doing, I've put the R code on github. I'm using an example dataset to illustrate how PCA can be used. This data contains five different morphological measurements from 200 crabs. These measurements are:

`**FL**`

- Frontal lobe size`**RW**`

- Rear width`**CL**`

- Carapace length`**CW**`

- Carapace width`**BD**`

- Body depth

The first rows of these data can be seen as follows:

```
R> library(MASS)
R> data(crabs)
R> head(crabs)
sp sex index FL RW CL CW BD
1 B M 1 8.1 6.7 16.1 19.0 7.0
2 B M 2 8.8 7.7 18.1 20.8 7.4
3 B M 3 9.2 7.8 19.0 22.4 7.7
4 B M 4 9.6 7.9 20.1 23.1 8.2
5 B M 5 9.8 8.0 20.3 23.0 8.2
6 B M 6 10.8 9.0 23.0 26.5 9.8
```

Each row contains the species of the crab (Blue/Orange), the crab sex (Male/Female), the row index and the five measured morphological characteristics in millimetres.

I'm going to outline how you might apply PCA to analysing multivariate data such as this. If you are interested in learning how PCA is calculated there is a video by Andrew Ng. In addition the machine learning course at coursera also covers the application of PCA.

PCA describes the underlying structure of your data where component relates to variation. The first component describes the greatest degree of variation in the data, the second component the next largest component and so forth. Furthermore each component describes variation orthogonal to previous components. This may sound esoteric, PCA is however very useful for exploratory data analysis without understanding the algorithm itself. If you begin using PCA regularly, taking the time to understand the algorithm will however help you understand the results of using it.

I'll begin by using R to calculate the PCA on the crab data. I'll then attach the first three components to the original crab data so that they can be compared.

```
# Perform PCA on the data
# retx returns the principal component weights for each crab
R> crab.pca <- prcomp(crabs[,4:8],retx=TRUE)
# Append the components for each crab to the original data
R> crabs$PC1 <- crab.pca$x[,1]
R> crabs$PC2 <- crab.pca$x[,2]
R> crabs$PC3 <- crab.pca$x[,3]
R> head(crabs)
sp sex index FL RW CL CW BD PC1 PC2 PC3
1 Blue Male 1 8.1 6.7 16.1 19.0 7.0 -26.46 -0.5765 -0.61157
2 Blue Male 2 8.8 7.7 18.1 20.8 7.4 -23.56 -0.3364 -0.23739
3 Blue Male 3 9.2 7.8 19.0 22.4 7.7 -21.74 -0.7119 0.06550
4 Blue Male 4 9.6 7.9 20.1 23.1 8.2 -20.34 -0.8358 -0.21830
5 Blue Male 5 9.8 8.0 20.3 23.0 8.2 -20.21 -0.6955 -0.36252
6 Blue Male 6 10.8 9.0 23.0 26.5 9.8 -15.34 -0.7950 -0.08414
```

The column `PC1`

is the first component, and so forth. Each row shows the value
each crab is given in each component e.g. the weight for the first crab in the
first component is -26.46. The weight for the second crab in the second
component is -0.3364. In addition to each of the crabs (observations) having PCA
weights, the variables also have weights. These can be observed in the
`rotations`

part of the returned PCA object.

```
R> head(crab.pca$rotation)
PC1 PC2 PC3 PC4 PC5
FL 0.2890 0.3233 -0.5072 0.7343 0.1249
RW 0.1973 0.8647 0.4141 -0.1483 -0.1409
CL 0.5994 -0.1982 -0.1753 -0.1436 -0.7417
CW 0.6617 -0.2880 0.4914 0.1256 0.4712
BD 0.2837 0.1598 -0.5469 -0.6344 0.4387
```

A common application for PCA is to discriminate your data into partitions based
on the underlying structure in the data. I can use the components to spread and
separate the data. PCA discriminating components should have both positive and
negative values. If you look at all the value for the first component `PC1`

you
will see that they are all negative and suggest there is little discriminatory
power for this component.

Looking to the second component, this have both positive and negative values and therefore this second component can be used to discriminate the data. To visualise this effect we'll use the two values furthest apart - the most negative and the most positive values. In this case it's rear width (RW) 0.87, and carapace width (CW) -0.29. Plotting these two characteristics we get the following graph.

<%= image(amzn('/principal_components_analysis/second_component_dotplot.png'),'') %>

Here you can see that data forms a noisy V shape. Imagine that you drew a line through each arm of the V, you would get two sets of data. Therefore, we might assume that there are two different distributions of the ratio of carapace to rear width. We can test this by plotting the density of the crabs according to this ratio

<%= image(amzn('/principal_components_analysis/second_component_density.png'),'') %>

There are two peaks, with some overlap. So the biological meaning of the second component is shows up the difference in ratios of rear and carapace widths. What this means, I'll examine later .

Looking at the third component, again this is discriminating, the most extreme values are carapace width and body depth. Plotting these, we get this figure.

<%= image(amzn('/principal_components_analysis/third_component_dotplot.png'),'') %>

This time we have two vaguely parallel lines, so again there appears to be different ratio of the two characteristics. A density plot illustrates the distribution of this ratio.

<%= image(amzn('/principal_components_analysis/third_component_density.png'),'') %>

Two more overlapping but obvious distributions, so it appears that the third component highlights a discriminating effect based on the ratio of body depth to carapace width. As I wrote, PCA can be used to discriminate based on the underlying trends in the data. I can therefore plot the distribution of crabs based on these two derived ratios, which produces this figure.

<%= image(amzn('/principal_components_analysis/morphology.png'),'') %>

I coloured the crabs based on their species and used different point types to further highlight gender. Looking at this plot, I think you can see a biological relationship from the second and third principal components.

The x-axis, which we derived from the second component, appears to be related to crab gender, as sex appears roughly separated based on this ratio. The male crabs seem to have a larger carapace to rear width compared with that of the female crabs. On the y-axis, the ratio derived from the third component, appears to separate crab species. The orange crabs have a larger body depth to their carapace width, compared with their blue counterparts.

So far, I haven't so far plotted the component plots - which is often the first thing people do after performing PCA. So here they are, the first/second components.

<%= image(amzn('/principal_components_analysis/first_components.png'),'') %>

And the second and third components.

<%= image(amzn('/principal_components_analysis/second_components.png'),'') %>

You can see that the first and second component plot doesn't discriminate the crabs very well. This why it is important to also look at the component weights. The second/third component plot discriminates the data similarly to the morphology ratios plot, however the axis that would split gender and species are rotated.

As a small test of how these ratios discriminate the data I created two generalised linear binomial models to predict either crab species or gender. Each model was generated one of two ways, the first using simple additive terms and the second using the additive terms plus the ratio of the terms indicated by the principal components analysis. I then bootstrapped the crab data 100 times to test how accurately each model predicted either species or gender. The R code for this is available on github also.

<%= image(amzn('/principal_components_analysis/accuracy.png'),'') %>

This plot shows how effectively the models predicted each response variable. You can see that for species including the ratio had little effect. The model predicting species however performs slightly better when the ratio term is included. This is not a terribly scientific example but goes to show how the results of PCA might be applied.

So here you have it, I hope this has illustrated how the data produced from PCA relates to your original data, and how you can begin to interpret it. Here's the message I've tried to convey:

- The first two components are not always the most useful.
- Look at the components (columns) with the positive and negative weights.
- Look at you original data in terms of the most positive and negative values for these components.