Generalized Singular Value Decomposition And Its Applications In Model Analysis

For a variety of processes we can observe and register their characteristics, making up a sequence of measurement vectors or matrices (rectangular in general). Our goal is to extract some model dependent information using the available information. Such approaches are typical in technology (for a neat chemistry example, see [7,9]) and model analysis like parameter identification of linear stochastic dynamic systems. Since a stochastic nature of financial and economic data is evident, we can extend this data analysis technique to a number of new applications.If we are successful, some kind of adaptive filter can be further constructed (similar to the classic Kalman's one, for example). Inspired with formal model parameters, we can apply this filter to process financial data like stock information to predict and verify how close is a mathematical model to a real-time data.Namely, when provided with a set measurements represented by matrices Ai ∈ Mm,n(ℝ), we have to estimate a problem dependent characteristic matrices \[ \min_{r,P,B_1,\ldots,B_k,Q}\sum_{i=1}^k \|A_i-PB_iQ^T\|_F^2, \] with P,Q being orthonormal matrices, Bi ∈ Mr(ℝ), r ≤ min{m,n}. Formulated as above, the problem is usually called a generalized singular value decomposition (GSVD) problem and could be solved numerically [1, 2]. These matrices provide some basic information applicable for higher level automated problem solver or human interpretation.