Singular value decomposition method

Hull et al. have developed the Latent Semantic Structure Indexing (LaSSI) approach to perform similarity search in low-dimensional chemical space. To reduce the dimension of initial chemical space, the singular value decomposition method is applied for the descriptor-molecule matrix. Ranking molecules by similarity to a query molecule was performed in the reduced space... 

For a better understanding of how the regularization parameter or damping factor" a works, let us apply a singular value decomposition method to matrix A ... 

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 scores and loadings of a data set can be determined by the singular value decomposition method (8), which decomposes the m x n matrix X according to the following relationship ... 

Figure B2.1.8 Dynamic absorption trace obtained with the dye IR144 in methanol, showing oscillations arising from coherent wavepacket motion (a) transient observed at 775 nm (b) frequency analysis of the oscillations obtained using a linear prediction, singular-value-decomposition method.

Broughton 1991) and (Oppe, 2001a) they developed another technique, the singular value decomposition method , in comparing road safety trends between ditferent countries. This technique investigates the similarities and dissimilarities between... 

Singular value decomposition methods can be used to solve this minimization problem. The final transformation matrix, T, has the form ... 

The linear equations are usually solved with the singular value decomposition method [9,13]. After the calculation of the weighing matrix A, clear representations of the separated fluorophores can be created (Figs. 2 and 3). 

Hendler R W and Shrager R I 1994 Deconvolutions based on singular value decomposition and the pseudoinverse—a guide for beginners J. Blochem. Blophys. Methods 28 1-33... 

Widely used methods of data transformation are Fast Fourier and Wavelet Transformations or Singular Value Decomposition... 

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 ... 

Here the pair-force fj (r, r -) is unknown, so a model pair-force fij(r , rj, p, P2 pm) is chosen, which depends linearly upon m unknown parameters p, p2 - Pm- Consequently, the set of Eq. (8-2) is a system of linear equations with m unknowns p, P2 - - Pm- The system (8-2) can be solved using the singular value decomposition (SVD) method if n > m (over-determined system), and the resulting solution will be unique in a least squares sense. If m > n, more equations from later snapshots along the MD trajectory should be added to the current set so that the number of equations is greater than the number of unknowns. Mathematically, n = qN > m where q is the number of MD snapshots used to generate the system of equations. 

Clearly, the total number of unknowns that need to be determined is m = a + +. .. + z and a solution set for parameters p, P2 pm is determined using the singular value decomposition or any other suitable method. The mean pair-force corresponding to the potential of mean force can be obtained in a systematic manner by averaging a number of sets of solutions for parameters p, P2 Pm obtained along the atomistic MD trajectory in which the phase space is sampled extensively. 

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. 

There are also forms of nonlinear PCR and PLS where the linear PCR or PLS factors are subjected to a nonlinear transformation during singular value decomposition the nonlinear transformation function can be varied with the nonlinearity expected within the data. These forms of PCR/PLS utilize a polynomial inner relation as spline fit functions or neural networks. References for these methods are found in [7], A mathematical description of the nonlinear decomposition steps in PLS is found in [8],... 

Calculation of eigenvectors requires an iterative procedure. The traditional method for the calculation of eigenvectors is Jacobi rotation (Section 3.6.2). Another method—easy to program—is the NIPALS algorithm (Section 3.6.4). In most software products, singular value decomposition (SVD), see Sections A.2.7 and 3.6.3, is applied. The example in Figure A.2.7 can be performed in R as follows ... 

Selected entries from Methods in Enzymology [vol, page(s)] Overview, 226, 119, 147 absorption apparatus, 226, 131 apparatus, 226, 152 detectors, 226, 126 detector systems, 226, 125 excitation source, 226, 121 global analysis, 226, 146, 155 kinetic applications, 226, 134 heme proteins, 226, 142 multiphoton effects, 226, 141 nanosecond time-resolved recombination, 226, 141 quantum yields, 226, 139 singular value decomposition, 226, 146, 155 spectral dynamics, 226, 136 time delay generators, 226, 130. 

Matrix methods such as singular value decomposition and others based on pseudoinverses have been described in the digital image processing literature. A book chapter by Andrews (1975) and texts by Pratt (1978), Andrews and Hunt (1977), and Hall (1979) treat this subject and give further references. These works also contain abundant references and considerable... 

We note that the values X /2 are called the singular values of the matrix A and they can be determined directly from A, without forming ATA. The corresponding numerical method called singular value decomposition is relatively complex but somewhat more accurate then the procedure described here, for details see (ref. 11). 

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). 

Bos et al. [94] compared the performance of ANNs for modelling the Cr-Ni-Fe system in quantitative XRF spectroscopy with the classical Rasberry-Heinrich model and a previously published method applying the linear learning machine in combination with singular value decomposition. They studied whether ANNs were able to model nonlinear relationships, and also their ability to handle non-ideal and noisy data. They used more than 100 steel samples with large variations in composition to calibrate the model. ANNs were found to be robust and to perform better than the other methods. 

The multiple linear regression (MLR) method was historically the first and, until now, the most popular method used for building QSPR models. In MLR, a property is represented as a weighted linear combination of descriptor values F=ATX, where F is a column vector of property to be predicted, X is a matrix of descriptor values, and A is a column vector of adjustable coefficients calculated as A = (XTX) XTY. The latter equation can be applied only if the matrix XTX can be inverted, which requires linear independence of the descriptors ( multicollinearity problem ). If this is not the case, special techniques (e.g., singular value decomposition (SVD)26) should be applied. 

Henry, E. R., and Hofrichter, J. (1992). Singular value decomposition—Application to analysis of experimental-data. Methods Enzymol. 210, 129-192. 

In mathematical system theory, the subject of model reduction has been studied for about 30 years. The focus is on model reduction of linear systems, in particular methods based on singular value decomposition. One of the best known of these methods is balanced truncation. It is used extensively for various engineering purposes, such as electronic chip design and the reduction of models of aerospace structures. This method does not require the type of a priori information about the system mentioned above. Only recently has it been tried out on biochemical systems [105, 106]. 

---