SVD singular value decomposition

The SVD method is based on the following theorem of linear algebra, whose proof is beyond our scope. 

Theorem 88 (SVD Theorem) Any N x L) matrix A whose number of rows is greater or equal to its number of columns can be represented as the product of an N X L column-orthogonal matrix U, an L x L) diagonal matrix Q with positive or zero elements  

Qr+i = 0. In other words, the matrix A of a rank r has r non-zero singular values. 

If the matrix A is a square N x N) matrix, then U, Q and V are all square matrices of the same size. In this case we can easily calculate the inverse matrix  

Note that if U is an A x L colunui-orthogonal matrix, then there exists an N X N — L) matrix U such that matrix U composed of the columns of the matrices U and U is orthogonal  

A brief survey of SVD, references and a FORTRAN program are given by Forsythe et al. [127]. 

The quantities of are called the singular values of A and the columns of U and V are called the left and right singular vectors. If A is symmetric, positive-semidefinite, the eigenvalues and the singular values of A are equal if A is not symmetric, they are not. 

If a set of independent vectors is multiplied by an orthogonal matrix, the resulting set is still independent. Thus, the ranks of A and 2 are the same. Consequently, the rank of a matrix is the number of non-zero singular values. 

One mathematical procedure that can be used in the context of PCA is singular value decomposition (SVD). SVD decomposes a matrix A created from m compounds each having n descriptor components by 

We can discard the principal components with small variance and make a reduced matrix S  

For a square, symmetric matrix X, singular value decomposition is equivalent to diagonalization, or solution of the eigenvalue problem. 

PCA is equivalent to finding the SVD of X and can be used to obtain the regression vector estimate w from 

Eliminating the identity matrices VV and UU that do not change the results, we can reduce this equation to 

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Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

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). 

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. 

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. 

In order to accelerate the minimization of Eq. (2.7.12), the data and the kernel can be compressed to a smaller number of variables using singular value decompositions (SVD) of K,... 

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 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. 

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 ... 

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. 

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 utility of singular value decomposition (SVD) for the determination of the order tensor stems from the formation of the M-P inverse, which is straightforward based on the SVD of a matrix. All matrices can be factored into a product of three matrices via SVD,92... 

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 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. 

The loading and scores for PCA can be generated by singular value decomposition (SVD). Instead of expressing the matrix containing the mixture spectra, A, as a product of two matrices as in Equation (4.4), SVD expresses it as a product of three matrices... 

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