How to apply PCA results on a Future Dataset?

Question

I have a fundamental question regarding the applications of the results of PCA:

If we have already performed a successful PCA on a dataset of, say, real estate prices of a certain region over the last 20 years, then we've figured out the Eigenvalues and Eigenvectors that maximize the explanatory power of a mathematically descriptive system, while simultaneously lowering its complexity.

If we're now 2 years into the future and possess new real estate prices, how can we judge if certain real estate is under- or overvalued with these PCA results from the initial dataset?

Do we have to apply a new PCA on a dataset that contains the initial 20 years as well as the 2 future years of data in order to make such an assessment?

This is the initial dataset:

Here are the eigenvectors and values:

Here's the result that I get when I înclude all 22 years of data, plot the first column of the PC matrix and compare it against the standardized price of a real estate A:

I've generated eigenvector and eigenvalue tables with the scikit tools and then I calculated the dot product between the standardized price matrix and the eigenvector matrix to get a principal components matrix. The blue line reprsents the first column of this PC matrix and the orange line is the standardized Real Estate A price.

Is this how it's supposed to be used? You always need to include the latest data so as to see the most accurate discrepancies between modeled PCA value and the real current real estate value?

Answer 1

There's one thing you haven't mentioned, which is how you will use the PCA results to judge if something is undervalued. Like you said, PCA reduces dimensionality, and will tell you the directions in which the data has most variance. However, in order to use this to judge new properties, you need some classifier. Reducing dimensions is not enough.

But to answer your question, assuming you have a classifier, you seem to be asking if the PCA results on the first 20 years will be similar to the PCA results on the entire 22 years. This really depends on the data. One thing you can do to empirically check this is to just run both of these PCA's and compare them. I would expect that the initial PCA will generalize well to the next two years, but that's just a guess. It would be much better to get your answers from the data.

Answer 2

we've figured out the Eigenvalues and Eigenvectors that maximize the explanatory power of a mathematically descriptive system, while simultaneously lowering its complexity

You might have reduced dimensionality with PCA/PCR, but you probably haven't reduced the complexity in terms of the original predictor variables. Each principal component might include contributions from all of those original predictors.

For predictions on new data, it's best to move from the retained principal components back to corresponding regression coefficients expressed in terms of the original predictors. That's done, for example, by the R pls package; see Section 8 of the vignette. Those coefficients won't be the same as what you would have gotten with ordinary regression, as they only contain predictor contributions included in the retained principal components.

Moving back to the predictor space from principal components removes potential problems in mapping between principal components of your original and your new data sets. In terms of the original predictor variables, whether to continue with the first model or update isn't any different for PCA/PCR than it is for other modeling approaches.