Recover original data from first principal component

[Guest]

Hello,

I would like to know if you can help me with the following issue:

I performed the PCA on bank's credit default swaps spreads and I predicted the first principal component scores (that in the standardize form). I would like to know how I can recover the original unit of measure (in basis points) using only the first principal component? The objective is to use the first principal component as a measure of bank's credit risk on my regression.
Thank you very much.

Ana Vasconcelos

[Clyde Schechter]

I don't think this is even possible. PCA can be used to create components from variables that have incompatible dimensions. The component scores are calculated from standardized values of the constituent variables (hence dimensionless), and dimensionless coefficients are used to calculate the appropriate linear combination. So the component scores are also, inherently, dimensionless. Even if all of the constituent variables are denominated in basis points, unless they all have the same standard deviation in your sample (which would be truly astonishing), they will be scaled differently during standardization, and in general the coefficients applied to them will be different, so there will be no canonical way of rescaling from the dimensionless results back to the common unit of the constituent variables.

[Clyde Schechter]

Statistically, the first component in a PCA analysis is just the linear combination of the standardized constituent variables with the maximum variance, subject to the constraint that the sum of squares of the coefficients is 1. In general, the first component can be thought of as a single variable which comes as close as possible (which may not be particularly close) to expressing all of the variation seen in all of the constituent variables. Principal components are often used in regression analyses in order to 1) reduce the number of variables in the model by selecting only some of the components, and 2) assure that the model variables are independent of each other.

That probably doesn't help you very much. It certainly won't help you explain it to an audience of non-statisticians. Interpreting a component in terms of the content of the subject matter domain you are working in requires subject-matter expertise as much as it does statistical knowledge (actually, more). As I know nothing about spreads and swaps and the like, I cannot offer you any advice. There are a number of people on this forum who do know about these things, and perhaps one will chime in.

Failing that, if you can find a colleague who works in your field and also knows statistics well enough that he/she understands principal components analysis, that person would be ideally positioned to help you interpret the component in a meaningful way. If you yourself came up with the idea of applying principal components analysis to this data, then you probably had some reason in mind, and revisiting that thinking will be helpful. If you are implementing calculations suggested by somebody else, that person might be your best resource.