The Singular Value Decomposition, SVD

The Singular Value Decomposition, SVD, has superseded earlier algorithms that perform Factor Analysis, e.g. the NIPALS or vector iteration algorithms. SVD is one of the most stable, robust and powerful algorithms existing in the world of numerical computing. It is clearly the only algorithm that should be used for any calculation in the realm of Factor Analysis. 

According to the SVD any matrix Y can be decomposed into the product of three matrices 

The dimensions are as follows Y is an mxn matrix where m n, U is an mxn matrix as well, while S and V are nxn matrices. Matlab delivers the above economy sized dimensions only if the following command is used  

The important special properties of the three product matrices U, S and V are the following S is a diagonal matrix, containing the so-called singular values in descending order. Note that the singular values of real matrices are always positive and real. U and V are orthonormal matrices, which means they are comprised of orthonormal vectors. In matrix notation  

T is often called the score matrix and L the loadings matrix. The relationship between decompositions (5.1) and (5.3) is 

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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). 

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. 

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. 

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 ... 

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)... 

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... 

The controllability analysis was conducted in two parts. The theoretical control properties of the three schemes were first predicted through the use of the singular value decomposition (SVD) technique, and then closed-loop dynamic simulations were conducted to analyze the control behavior of each system and to compare those results with the theoretical predictions provided by SVD. 

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). 

The singular-value decomposition (SVD) [25] decomposes a matrix X as a product of three matrices ... 

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 singular-value decomposition (SVD) is a computational method for simultaneously calculating the complete set of column-mode eigenvectors, row-mode eigenvectors, and singular values of any real data matrix. These eigenvectors and singular values can be used to build a principal component model of a data set. 

To conduct SFA in practice, the singular-value decomposition (SVD, see Chapter 4) of the two subwindows yields a basis of orthogonal vectors spanning the (A,B) subspace, called ej, and another basis for the (B,C) subspace, called fj. The spectrum of B, sB, can be obtained from these two sets of basis vectors as shown in Equation 11.9,... 

PCA is simple in Matlab. The singular value decomposition (SVD) algorithm is employed, but this should normally give equivalent results to NIPALS except diat all the PCs are calculated at once. One difference is that die scores and loadings are bodi normalised, so that for SVD... 

The singular value decomposition (SVD) of the LLS matrix A can be used to both detect and remedy rank deficiency. The SVD writes the matrix A as... 

One of the basic and most important tools of modem numerical analysis is the Singular value decomposition (SVD). One important use of the SVD is in the study of the theoretical control properties in chemical process. One definition of SVD is ... 

SVD-Index. This is a commercial indexing program belonging to the TOPAS suite from Bruker AXS. The reciprocal-lattice relationship defined in Equation (8) is solved via the iterative use of the Singular Value Decomposition (SVD) approach. This method is recommended for cases in which there are more equations than variables. ... 

There are a variety of methods used to obtain the loading and scores matrix in Eq. (15). Perhaps, the most common methods employed are non-linear iterative partial least squares (NIPALS), and the singular value decomposition (SVD). Being an iterative method, NIPALS allows the user to calculate a minimum number of factors, whereas the SVD is more accurate and robust, but in most implementations provides all the factors, thus can be slow with large data sets. During SVD the data matrix can be expressed as... 

Principal Components Analysis (PCA) is a multivariable statistical technique that can extract the strong correlations of a data set through a set of empirical orthogonal functions. Its historic origins may be traced back to the works of Beltrami in Italy (1873) and Jordan in Prance (1874) who independently formulated the singular value decomposition (SVD) of a square matrix. However, the first practical application of PCA may be attributed to Pearson s work in biology [226] following which it became a standard multivariate statistical technique [3, 121, 126, 128]. 

Suppose that a matrix X (I x J) is available. Assuming that J < /, the singular value decomposition (SVD) of X is... 

The osculating orbit (24) is an ellipse centered at v = 0, or u = F0/ljq. A more natural choice of orbital elements than vq, v 0 are four geometric parameters of the ellipse (24). We suggest to use the singular-value decomposition (SVD)... 

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. 

A decomposition algorithm is one that is inherently noniterative and that will yield the parameters of the model if no noise is present in the data. The singular value decomposition (SVD) is a well-known decomposition for bilinear models. There are decompositions for the PARAFAC, Tucker2, and Tucker3 trilinear models. 

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 ... 

A little more expensive [n (m + I7nl3) flops and 2mn space versus n (m—nl3) and mn in the Householder transformation] but completely stable algorithm relies on computing the singular value decomposition (SVD) of A. Unlike Householder s transformation, that algorithm always computes the least-squares solution of the minimum 2-norm. The SVD of an m x n matrix A is the factorization A = ITEV, where U and V are two square orthogonal matrices (of sizes mxm and nxn, respectively), U U = Im, y V = In, and where the m x n matrix S... 

We solve Eq. (2) for a using the singular value decomposition (SVD) [25]. However, the range of boundary pressure actuations is limited by contact angle saturation. Using experimental data on the saturation characteristics for the... 

Figure 4.4 Principal component analysis of a 6x3 matrix (a) the six samples in the original space of three measured variables, (b) the new axes (principal components PCi and PC2) obtained from the singular-value decomposition (SVD) of the 6x3 matrix and (c) representation of the six samples in the space of the principal components. Notice how the three original variables are correlated (the higher xi and X2 are, the higher is X3). Notice also how using only the coordinates (scores) of the samples on these two principal components, the relative position of the samples in the initial variable space is captured. This is possible because the original variables are correlated. Principal component regression (PCR) uses the scores on these two new variables (the two principal components) instead of the three originally measured variables.

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