主成分分析の続き

最近主成分分析を再び学んでいます。
I am not sure exactly what types of functions this applies to. It seems to have applications to finding the principal components of an image in computer vision or compression. It also has applications to financial portfolio analysis and risk management.
An example in a risk management text defines the portfolio variance in terms of the eigenvectors of the portfolio's covariance matrix. The portfolio variance is W^T(V)W. However, I am wondering whether such an analysis is possible when the quantity we are interested in assessing is not a linear combination of variables. For example, the present value of a bond might be PV = P/*1,which is a function of interest rate and time to maturity. Can this type of function be written as a product of matrices? Wikipedia defines the first principal component as
この相関を1つの変数が作った偽相関と仮定し、数学的に算出する。それを第一主成分と呼ぶ。
or the combining of several factors' correlations into one synthetic variable.
It seems that Principal Component Analysis can be carried out whenever you begin with a matrix of data from which a covariance matrix can be derived. Thus, in the case of portfolio variance, the components (eigenvectors of covariance matrix) actually indicate the directions of variance of the points of returns of the underlying assets at different times. However, because of the formula for portfolio variance and the fact that the eigenvectors span the space of the weight vectors, the weights can be written as a linear combination of the principal components. This enables the W^T(V)W expression to be written in terms of eigenvectors multiplied by the coefficients used for the principal components in the weight vectors. Thus, the weights of assets in the portfolio indicate how much that portfolio will vary from the average return, it seems.

*1:1+r)^(T