Collinear singularities

Highly correlating (collinear) variables make the covariance matrix singular, and consequently the inverse cannot be calculated. This has important consequences on the applicability of several methods. Data from chemistry often contain collinear variables, for instance the concentrations of similar elements, or IR absorbances at neighboring wavelengths. Therefore, chemometrics prefers methods that do not need the inverse of the covariance matrix, as for instance PCA, and PLS regression. The covariance matrix becomes singular if... 

In order for this matrix to be singular, the three points must be non-collinear. Otherwise an infinite number of planes exist that pass through the given line. It is difficult... 

We define the order of the singular values as a > a2 > 31. The planar and collinear configurations give a3 0 and a2 a3 = 0, respectively. Furthermore, we let the sign of a3 specify the permutational isomers of the cluster [14]. That is, if (det Ws) = psl (ps2 x ps3) > 0, which is the case for isomer (A) in Fig. 12, fl3 >0. Otherwise, a3 < 0. Eigenvectors ea(a = 1,2,3) coincide with the principal axes of instantaneous moment of inertia tensor of the four-body system. We thereby refer to the principal-axis frame as a body frame. On the other hand, the triplet of axes (u1,u2,u3) or an SO(3) matrix U constitutes an internal frame. Rotation of the internal frame in a three-dimensional space, which is the democratic rotation in the four-body system, is parameterized by three... 

Collinearity may also be caused by poor scaling and/or near singularity of the xTx matrix. If the collinearity is due to scaling, then one simple way to remove the collinearity is by centering. Centering creates a new variable x using... 

CMB Resolution A final issue that may complicate the application of the CMB on ambient data sets is existence of two. sources with similar fingerprints or, more generally, a source whose profile is a linear combination of other source profiles. This is called the collinearity problem. If this is the ca.se then the matrix [A WA] u.sed in (24.15) has two columns that are almost similar, or a linear combination of. several others. This matrix from a mathematical point of view is close to singular and the result of its inversion is extremely. sensitive to small errors. Often, if this is the case, the results of CMB are large positive and negative. source contributions. The simplest solution to this problem is identification of the "offending sources and elimination of one of them. Physically, becau.se the source.s are too... 

Equation [9] highlights the most frequent problem in multiple linear regression, namely, the matrix whose inverse must be calculated, (C C), may suffer from collinearity or singularity problems. The matrix may not possess... 

In case the columns of matrix are hnearly dependent (A is collinear), one can use singular value decomposition to calculate the pseudo inverse. A matrix (w, ) is factored into ... 

All variances and covariances of a multivariate data set can be arranged in a symmetric matrix, with the variances located in the main diagonal. This covariance matrix C plays an important role in multivariate statistics it describes the dispersion of multivariate data. Highly correlating (collinear) features - a frequent situation in chemical data - make C singular and then the inversion of C (which is necessary in some methods)... 

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