Singular Value Decomposition Algebra

An important theorem of matrix algebra, called singular value decomposition (SVD), states that any nxp table X can be written as the matrix product of three terms U, A and V ... 

It can be proved that the decomposition is always possible and that the solution is unique (except for the algebraic signs of the columns of U and V) [3]. Singular value decomposition of a rectangular table is an extension of the classical work of Eckart and Young [4] on the decomposition of matrices. The decomposition of X into U, V and A is illustrated below using a 4x3 data table which has been adapted from V an Borm [5]. (A similar example has been used for the introduction to PC A in Chapter 17.)... 

In the previous section we have developed principal components analysis (PCA) from the fundamental theorem of singular value decomposition (SVD). In particular we have shown by means of eq. (31.1) how an nxp rectangular data matrix X can be decomposed into an nxr orthonormal matrix of row-latent vectors U, a pxr orthonormal matrix of column-latent vectors V and an rxr diagonal matrix of latent values A. Now we focus on the geometrical interpretation of this algebraic decomposition. 

In Chapters 21-23 and in this chapter, we have described the most basic calculations for MLR, PCR, and PLS. To reiterate, our intention is to demonstrate these basic computations for each mathematical method presently, and then to delve into greater detail as the chapters progress consider these articles linear algebra bytes. For this chapter we will illustrate the basic calculation and mathematical relationships of different matrices for the calculations of Singular Value Decomposition or SVD. 

We will describe the PCA method following the treatment of Ressler et al. (2000) but with the above notation. PCA can be derived from the singular-value decomposition theorem from linear algebra, which says that any rectangular matrix can be decomposed as follows... 

The second approach, the more commonly apphed singular value decomposition (SVD) is an algebraic reformulation of the convolution integrals of (5.11), rewritten as ... 

The terms factor analysis, principal components analysis, and singular value decomposition (SVD) are used by spectroscopists to describe the fitting of a two-way array of data with a general bilinear model. We will use the term factor analysis in this sense, although this term has a somewhat different meaning in statistics. SVD is a specific algebraic procedure, discussed by Henry and Hofrichter and briefly later in this chapter, whose use alone is often not the best way to fit a general bilinear model. 

The singular value decomposition (SVD) method, and the similar principal component analysis method, are powerful computational tools for parametric sensitivity analysis of the collective effects of a group of model parameters on a group of simulated properties. The SVD method is based on an elegant theorem of linear algebra. The theorem states that one can represent an w X n matrix M by a product of three matrices ... 

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