Suppose that I have a series of $M$ time-observations of $N$ "quantities" $z_1(t_1),...,z_1(t_M)$, ..., $z_N(t_1),...,z_N(t_M)$. I want to estimate the values of $z_1(t_{M+1}),...,z_N(t_{M+1})$. This a problem of interest, for example, in stock asset prediction. I want to use Principal Component Analysis (PCA) performed by a Singular Value Decomposition (SVD).
My questions:
- What is the physical meaning of the first singular vector (namely, that one corresponding to the largest singular value)? What is the physical meaning of the remaining singular vectors? I understand that the singular vectors provide uncorrelated linear combinations of the above random variables.
- What is the physical meaning of the singular values? As long as I know, they are related to the variances associated to the singular vectors. But then, does it mean that the most probable value of the quantities at time $t_{M+1}$ is due to the least singular values (least variances)?
- How do I use the singular values and vectors of PCA to predict the value of the quantitites at time $t_{M+1}$? Could you let me understand the idea behind?
Thank you very much in advance.