Applications of the diagonal representation IV
Principal component analysis is a general method based on diagonalization of the variance matrix. We consider it in a financial context. The variance matrix measures riskiness of the portfolio. We want to see which stocks contribute most to the portfolio risk. The surprise is that the answer is given not in terms of the vector of returns but in terms of its linear transformation.
8. Principal component analysis (PCA)
With such a matrix, instead of we can consider its transformation for which
We know that has variances on the main diagonal. It follows that for all Variance is a measure of riskiness. Thus, the transformed variables are put in the order of declining risk. What follows is the realization of this idea using sample data.
In a sampling context, all population means should be replaced by their sample counterparts. Let be a vector of observations on at time These observations are put side by side into a matrix where is the number of moments in time. The population mean is estimated by the sample mean
The variance matrix is estimated by
where is a vector of ones. It is this matrix that is diagonalized:
In general, the eigenvalues in are not ordered. Ordering them and at the same time changing places of the rows of correspondingly we get a new orthogonal matrix (this requires a small proof) such that the eigenvalues in will be ordered. There is a lot more to say about the method and its applications.