What is the meaning of these principal components?

Question

I have a matrix of data. I computed the principal components of my matrix using SVD (code shown below):

subtract mean...then

$$[U,S,V] = SVD({\rm matrix})$$

for $V$, which is the principal components, I obtain the following values:

$$ V = \begin{matrix} \text{Noise} &-0.2344 & 0.9548 & -0.0170 & 0.0947 & 0.1551 \\ \text{Size} &-0.9643 & -0.2296 & 0.0853 & 0.0666 & -0.0753 \\ \text{Speed} &0.0890 & 0.0479 & 0.9869 & 0.0770 & -0.0993 \\ \text{Electric} &0.0079 & -0.1823 & 0.0658 & 0.4101 & 0.8912 \\ \text{Lorry} &-0.0847 & 0.0045 & 0.1187 & -0.9014 & 0.4077 \\ \end{matrix} $$

How do I interpret these data and how am I suppose to know if any of these correlate?

Answer 1

The principal components don't correlate. Per definition. They are chosen to be orthogonal.

The data suggest that speed is not correlated with anything else, as it is basically identical to the 3rd PC. However, if that is your question, you should look at the correlation coefficient table, not the PC vectors. Note that the PC vectors don't contain sufficient information about correlations between the variables since you also need eigenvalues.

Interpretation: It looks like a loco can be understood in terms of these three concepts: 1) Size 2) Noiseness adjusted for size 3) Speed

However, you need to look at the eigenvalues also. Do the three first components account for the bulk of the variance in the data?

Answer 2

There are a couple of things you could do with this information. One is to just use the principal components (PCs) directly for further analysis. But another is that you could actually use the original data, and you just know which ones to use based on this.

The first column is your first (and most important) PC. That PC is primarily made up of the size (-.9643). Your next column is the second most important, and fortunately, that's also dominated by a single feature - noise (0.9548). Your third PC is almost completely speed (0.9869).

So you could take either the PCs or those features and see how much of the variation is explained by those alone. To do that, you need to make a scree plot (which you can find tutorials on).