Interpreting PCA plots

[Paul Evans]

All,

I often use the stylo package as a classroom demo to reproduce the classic Mosteller and Wallace attribution of the disputed numbers from The Federalist Papers. The question comes up how to interpret the scale on the axes (-8 to 4 on PC1, -5 to 5 on PC2 in this example). I took a look at the documentation for princomp, which I believe is generating this plot, but wasn't able to find an answer.

Thanks,

Paul Evans

​

[Maciej Eder]

Hi Paul,

good question. In short, the principal components are linear combinations of the original variables, meaning that the numbers you get on the plot, are summed up contributions of particular dimensions (i.e. words or other variables). These 'contributions', referred to as 'loadings', are computed using some linear algebra. Now, if the original variables have different units, then the final scores in a plot will not really give any meaning (say, if you take human height, heartbeat, blood pressure, and age as your variables). However, if your variables have the same unit, then the principal component will have that same unit as well. This is the case of word frequencies. In your plot, one can see the word frequencies scaled (or z-scored), because you've chosen using the correlation matrix. This means that your units are standard deviations. If a given word (say, "federal") is used 1 st.dev. more frequently than in the entire corpus, it will get 1 on your scale. This is a reasonably good explanation of it:

    https://stats.stackexchange.com/questions/80534/what-are-the-units-in-this-pca-biplot

My favourite introduction to PCA (and many other topics), however, is that one:

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning with applications in R. New York: Springer.

I hope this helps!

All the best,

Maciej

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