Matrix Singular Value Decomposition

Fisher, A. T. M. B. Underwood, 1995. Calibration of an X-ray diffraction method to determine relative mineral abundances in bulk powders using matrix singular value decomposition a test from the Barbados accretionary complex. In Proceedings Ocean Drilling Program, Initial Reports 156 29-37. 

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... 

It may look weird to treat the Singular Value Decomposition SVD technique as a tool for data transformation, simply because SVD is the same as PCA. However, if we recall how PCR (Principal Component Regression) works, then we are really allowed to handle SVD in the way mentioned above. Indeed, what we do with PCR is, first of all, to transform the initial data matrix X in the way described by Eqs. (10) and (11). 

The scaled data matrix D is decomposed using singular value decomposition (see Bonvin and Rippin (1990), Hamer (1989), Golub and van Loan (1983)) into matrices with one containing stoichiometric information which can be processed into acceptable stoichiometry. The decomposition can be easily done by any available software packages (e.g. Dongarra etal. (1979), Press et ai, (1989)). Upon decomposing one obtains ... 

Singular value decomposition (SVD) of a rectangular matrix X is a method which yields at the same time a diagonal matrix of singular values A and the two matrices of singular vectors U and V such that ... 

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 ... 

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. 

This is just the same matrix discussed with singular value decomposition. For data analysis, though, it is necessary to adjust the columns to have zero mean by subtracting from each entry in the column the average of the column entries. Once this is done, the loadings are the v, and satisfy... 

We now have the data necessary to calculate the singular value decomposition (SVD) for matrix A. The operation performed in SVD is sometimes referred to as eigenanal-ysis, principal components analysis, or factor analysis. If we perform SVD on the A matrix, the result is three matrices, termed the left singular values (LSV) matrix or the V matrix the singular values matrix (SVM) or the S matrix and the right singular values matrix (RSV) or the V matrix. 

Given any data matrix A of arbitrary size (as rows x columns) the matrix A can be written or defined using the computation of Singular Value Decomposition [6-8] as... 

In practice, the transformation matrix M can be found numerically from the singular value decomposition of X. The rank of X is equal to the number of non-zero singular values, and the transformation matrix corresponds to the transpose of die premultiplier orthogonal matrix. This process is illustrated below for die non-elementary reaction case. 

In order to find a linear transformation matrix to simplify the scalar transport equation, we will make use of the singular value decomposition (SVD) of Y ... 

Figure 5.2. The composition vector c can be partitioned by a linear transformation into two parts c,., a reacting-scalar vector of length /VT and cc, a conserved-scalar vector of length N. The linear transformation is independent of x and t, and is found from the singular value decomposition of the reaction coefficient matrix Y.

A symmetric matrix A, can usually be factored using the common-dimension expansion of the matrix product (Section 2.1.3). This is known as the singular value decomposition (SVD) of the matrix A. Let A, and u, be a pair of associated eigenvalues and eigenvectors. Then equation (2.3.9) can be rewritten, using equation (2.1.21)... 

Matlab is a matrix oriented language that is just about perfect for most data analysis tasks. Those readers who already know Matlab will agree with that statement. Those who have not used Matlab so far, will be amazed by the ease with which rather sophisticated programs can be developed. This strength of Matlab is a weak point in Excel. While Excel does include matrix operations, they are clumsy and probably for this reason, not well known and used. An additional shortcoming of Excel is the lack of functions for Factor Analysis or the Singular Value Decomposition. Nevertheless, Excel is very powerful and allows the analysis of fairly complex data. 

In the standard equation for multiwavelength spectrophotometric investigations, based on Beer-Lambert s law, the matrix Y is written as the product of the matrices C and A. According to the Singular Value Decomposition (SVD), Y can also be decomposed into the product of three matrices... 

L contains normalised rows while T is weighted by the matrix S. This, however, is somewhat ambiguous as the decomposition of the transposed, Y1, is equally possible and then the score and loading matrices are simply exchanged. For this reason, we do not use the expressions scores and loadings. The Singular Value Decomposition maintains some kind of symmetry between the decompositions of Y and Yl. 

The Singular Value Decomposition of a matrix Y into the product USV is full of rich and powerful information. The model-free analyses we discussed so far are based on the examination of the matrices of eigenvectors U and V. Evolving Factor Analysis, EFA, is primarily based on the analysis of the matrix S of singular values. 

The basic principle of EFA is very simple. Instead of subjecting the complete matrix Y to the Singular Value Decomposition, specific sub-matrices of Y are analysed. In the original EFA, these sub-matrices are formed by the first i spectra of Y where i increases from 1 to the total number of spectra, ns. The appearance of a new compound during the acquisition of the data is indicated by the emergence of a new significant singular value. 

The use of singular value decomposition (SVD), introduced into chemical engineering by Moore and Downs Proc. JACC, paper WP-7C, 1981) can give some guidance in the question of what variables to control. They used SVD to select the best tray temperatures. SVD involves expressing the matrix of plant transfer function steady state gains as the product of three matrices a V matrix, a diagonal Z matrix, and a matrix. 

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