Singular matrix problem

As stated earlier, LDA requires that the variance-covariance matrices of the classes being considered can be pooled. This is only so when these matrices can be considered to be equal, in the same way that variances can only be pooled, when they are considered equal (see Section 2.1.4.4). Equal variance-covariance means that the 95% confidence ellipsoids have an equal volume (variance) and orientation in space (covariance). Figure 33.10 illustrates situations of unequal variance or covariance. Clearly, Fig. 33.1 displays unequal variance-covariance, so that one must expect that QDA gives better classification, as is indeed the case (Fig. 33.2). When the number of objects is smaller than the number of variables m, the variance-covariance matrix is singular. Clearly, this problem is more severe for QDA (which requires m < n ) than for LDA, where the variance-covariance matrix is pooled and therefore the number of objects N is the sum of all objects... 

The latest iteration matrix is singular. Some of the equations or variables may be redundant. If so, you need to review the problem formulation. 

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

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... 

If no laminae have failed, the load must be determined at which the first lamina fails (so-called first-ply failure), that is, violates the lamina failure criterion. In the process of this determination, the laminae stresses must be found as a function of the unknown magnitude of loads first in the laminate coordinates and then in the principal material directions. The proportions of load (i.e., the ratios of to Ny, to My,/ etc.) are, of course, specified at the beginning of the analysik The loaa parameter is increased until some individual lamina fails. The properties, of the failed lamina are then degraded in one of two ways (1) totally to zero if the fibers in the lamina fail or (2) to fiber-direction properties if the failure is by cracking parallel to the fibers (matrix failure). Actually, because of the matrix manipulations involved in the analysis, the failed lamina properties must not be zero, but rather effectively zero values in order to avoid a singular matrix that could not be inverted in the structural analysis problem. The laminate strains are calculated from the known load and the stiffnesses prior to failure of a lamina. The laminate deformations just after failure of a lamina are discussed later. 

If two or more of the unknown parameters are highly correlated, or one of the parameters does not have a measurable effect on the response variables, matrix A may become singular or near-singular. In such a case we have a so called ill-posed problem and matrix A is ill-conditioned. 

It is a common problem to solve a set of homogeneous equations of the form Ax = 0. If the matrix is non-singular the only solutions are the trivial ones, x = x2 = = xn = 0. It follows that the set of homogeneous equations has non-trivial solutions only if A = 0. This means that the matrix has no inverse and a new strategy is required in order to get a solution. 

Note that for high-order modes, cos (A() = cosh (I AC) and isin( A AC) =-sinh(pA AC) -coshfyA/ l AC), and as a result, the matrix A(AC) becomes singular. An alternative approach to solve this problem, the immittance matrix method, will be discussed in the next section, another one, the scattering matrix method, is described in ". ... 

Thus, with a nearly zero eigenvalue of the covariance matrix of the independent variables the estimates tend to be inflated and the results are meaningless. Therefore, in nearly singular estimation problems reducing the mean square... 

There are some special cases that require modification of the L matrix. A problem arises as a pure species situation is approached, because all Xk except one approach zero, and this causes the L matrix to become singular. Therefore, for the purpose of forming L in a computational setting, we do not allow a pure species situation to occur. A small number <5 (insignificant compared to any mass fraction of interest) is added to each mole fraction, to prevent such an occurance. 

If A is not a square matrix and we command A b in MATLAB, then the SVD is invoked and finds the least squares solution to the minimization problem min., Ax — b. A slight variant that uses only the QR factorization mentioned in subsection (F) for a singular but square system matrix A Rn,n is used inside our modified boundary value solver bvp4cf singhouseqr. m in Chapter 5 in order to deal successfully with singular Jacobian matrices inside its embedded Newton iteration. 

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