Singular vector decomposition

From these results we can now define the singular vector decomposition of the 4x2 data matrix X ... 

Note that the algebraic signs of the columns in U and V are arbitrary as they have been computed independently. In the above illustration, we have chosen the signs such as to be in agreement with the theoretical result. This problem does not occur in practical situations, when appropriate algorithms are used for singular vector decomposition. 

By analogy with singular vector decomposition (SVD) and ordinary PCA, one can also define the TuckerS model in an extended matrix notation ... 

From this point on, the analysis is identical to that of CFA. Briefly, this involves a generalized singular vector decomposition (SVD) of Z using the metrics W and Wp, such that ... 

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... 

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 ... 

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. 

We now have both the data matrix A and the concentration vector c required to calculate PLS S VD. Both A and c are necessary to calculate the special case of PLS singular value decomposition (PLSSVD). The operation performed in PLSSVD is sometimes referred to as the PLS form of eigenanalysis, or factor analysis. If we perform PLSSVD on the A matrix and the c vector, 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. 

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.

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. 

The first equation is the well-known Singular Value Decomposition. In the context of PCR the eigenvectors U form the basis for the column vectors of Y. The second equation in (5.72) attempts to also represent the column vector q of qualities in the same space U. If both representations are good then PCR works well, resulting in accurate predictions. A potential drawback of PCR is the fact that U is defined solely by Y. Even if there is good reasoning for a relationship between q and U, as indicated in the derivation of equation (5.60), it is somehow accidental. ... 

Self-organizing map Singular value decomposition Support vector machine... 

Figure 4.11 A singular value decomposition of the preconditioned in situ spectroscopic data showing the 1st, 4th, 5th, 7th significant vectors and the 713th vector. The marked extrema are those which were used to recover the organometallic pure component spectra as well as alkene and aldehyde by STEM. Atmospheric moisture and CO2, hexane, and dissolved CO were removed from the experimental data during preconditioning. (C. Li, E. Widjaja, M. Garland,/ Am. Chem. Soc., 2003, 725, 5540-5548.)...

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. 

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... 

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,... 

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