Generalized singular value decomposition

We assume that Z is a transformed nxp contingency table (e.g. by means of row-, column- or double-closure) with associated metrics defined by W and W. Generalized SVD of Z is defined by means of  

It can be proved that the above generalized SVD of the matrix Z can be derived from the usual SVD of the matrix  

The results of applying these operations to the double-closed data in Table 32.6 are shown in Table 32.7. The analysis yielded two latent vectors with associated singular values of 0.567 and 0.433. 

Usual singular vectors extracted from the transformed Z in Table 32.6, i.e.  

In Table 32.7 we observe a contrast (in the sense of difference) along the first row-singular vector u, between Clonazepam (0.750) and Lorazepam (-0.619). Similarly we observe a contrast along the first column-singular vector v, between epilepsy (0.762) and anxiety (-0.644). If we combine these two observations then we find that the first singular vector (expressed by both u, and v,) is dominated by the positive correspondence between Clonazepam and epilepsy and between Lorazepam and anxiety. Equivalently, the observations lead to a negative correspondence between Clonazepam and anxiety, and between Lorazepam and epilepsy. In a similar way we can interpret the second singular vector (expressed by both U2 and V2) in terms of positive correspondences between Triazolam and sleep and between Diazepam and anxiety. 

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Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

The decomposition in eqn (3.30) is general for PCR, PLS and other regression methods. These methods differ in the criterion (and the algorithm) used for calculating P and, hence, they characterise the samples by different scores T. In PCR, T and P are found from the PCA of the data matrix R. Both the NIPALS algorithm [3] and the singular-value decomposition (SVD) (much used, see Appendix) of R can be used to obtain the T and P used in PCA/PCR. In PLS, other algorithms are used to obtain T and P (see Chapter 4). 

Generally, the five independent components of the alignment tensor A can be derived by mathematical methods like the singular value decomposition (SVD)23 as long as a minimum set of five RDCs has been measured in which no two internuclear vectors for the RDCs are oriented parallel to each other and no more than three RDC vectors lie in a plane. Any further measured RDC directly... 

The paper describes the application of singular-value decomposition method for resolution estimation and resolution enhancement in near-field optics with focus at scanning near-field microscopy. Analysis of general properties of singular functions and associated singular values is presetted. 

The analysis of a series of chiroptical spectra and recovery of systematic trends in a given set can be carried out in several ways. In the past, the results strongly depended on the spectroscopist s personal experience actually, this was the least objective part of the circular dichroism application. Nowadays, we can rely on general procedures of statistical data treatment like singular value decomposition, factor analysis (especially its first part, analysis of the correlation matrix and the projection of the experimental spectra onto the space of orthogonal components), cluster analysis and the use of neural networks. This field has been pioneered by Pancoska and Keiderling [72-76], and also by Johnson [77] when analyzing the chiroptical properties of biopolymers. 

A whole spectrum of statistical techniques have been applied to the analysis of DNA microarray data [26-28]. These include clustering analysis (hierarchical, K-means, self-organizing maps), dimension reduction (singular value decomposition, principal component analysis, multidimensional scaling, or correspondence analysis), and supervised classification (support vector machines, artificial neural networks, discriminant methods, or between-group analysis) methods. More recently, a number of Bayesian and other probabilistic approaches have been employed in the analysis of DNA microarray data [11], Generally, the first phase of microarray data analysis is exploratory data analysis. 

The PARAFAC model is introduced here by generalizing the singular value decomposition. A two-way model of a matrix X (7 x J), with typical elements xy, based on a singular value decomposition truncated to R components reads in summation notation... 

To solve for Ck, the generalized eigenvalue problem is used with the singular-value decomposition technique. The results of the problem indicate both the pure component response patterns x and y and the ratio of concentrations of the pure components to the standard response concentration. 

Singular value decomposition (SVD) analysis on spectro-photometric data obtained from an oxygen atom transfer (OAT) reaction involving a molybdoenzyme model system is reported. Specifically, the rate of solvolysis reaction of a phosphoryl intermediate complex has been compared with independent measurements. The SVD derived reaction rates are consistent with other measurements. This generalized approach is applicable in examining other bioinorganic reactions, and data processing. 

In general one can decide how many independent decay components are needed to describe the observed kinetics by closely inspecting the residuals (difference between measured and fitted curves). The singular value decomposition (SVD) method is a statistical tool to determine the maximum number of components that can be extracted with confidence from the experimental traces. 

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. 

To solve equations (7-9), an approach with direct solution of the system (8) at each step of the iterative process by the pseudo- (or generalized) inversion method is used. It is based on singular value decomposition (SVD). It is well-known that the SSVDC procedure in the Linpack library is used to calculate SVD [12]. Paper [13] presents a standard SVD procedure in Fortran-IV used in the present paper. The current MATLAB system versions have a built-in function svd(A) implementing this decomposition for an arbitrary nxm matrix A. The calculation scheme of the SVD procedure is in decomposing the matrix (9) at each step of the iterative process into the product of three matrices. 

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