Matrix singularity

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... 

The matrix cannot be solved in this form because for each row the sum of Xa or values is equal to one. These linear dependences make the matrix singular and therefore an infinite number of solutions is possible. To eliminate singularities, the calculated... 

To obtain initial estimates, an Emax model was fit to the data set in a na ive-pooled manner, which does not take into account the within-subject correlations and assumes each observation comes from a unique individual. The final estimates from this nonlinear model, 84% maximal inhibition and 0.6 ng/mL as the IC50, were used as the initial values in the nonlinear mixed effects model. The additive variance component and between-subject variability (BSV) on Emax was modeled using an additive error models with initial values equal to 10%. BSV in IC50 was modeled using an exponential error model with an initial estimate of 10%. The model minimized successfully with R-matrix singularity and an objective function value (OFV) of 648.217. The standard deviation (square root of the variance component) associated with IC50 was 6.66E-5 ng/mL and was the likely source of the... 

Note All models with using Laplacian estimation. The symbol R denotes R-matrix singularity. —denote that the parameter was not included in the model. 

All models were fit using FOCE-I. Data are reported as estimate (standard error of the estimate).R denotes model minimized successfully but with R-matrix singularity. (A) denotes additive residual error term. (E) denotes exponential residual error term. 

Next, a 2-compartment model (ADVAN3 TRANS4) was then fit to the data. Initial values were taken from the literature (Winslade et al., 1987) 5 L/h for CL, 17 L for VI, 1 L/h for intercompartmental clearance to the peripheral compartment (Q2), and 94 L for the peripheral compartment (V2). BSV was set to 70% for CL and 32% for all remaining pharmacokinetic variance terms. The exponential component of the residual error was set to 23% while the additive component was set to 1 mg/L. Optimization minimized successfully with an OFV of —6.280 (Table 9.4). R-matrix singularity was observed which indicated that the model was overparameterized. The 2-compartment fit was a significant improvement in the goodness of fit compared to the... 

The resulting modeling minimized successfully, again with R-matrix singularity, and an OFV of... 

The first model resulted in successful minimization but with R-matrix singularity and had an OFV of —133.282, a difference of 1.31 (p = 0.2522), which was not considered a significant improvement in the model. The second model resulted in successful minimization with no singularities and had an OFV of -134.073, a difference of 2.101 (p = 0.1472), which was also not considered a significant improvement over the model presented in Eq. (9.14). Hence, this concluded model development with the model given by Eq. (9.14) being considered the final covariate model with the final parameter estimates given in Table 9.15 (all data). 

Contains the number of the compartment that is too small. IERROR = 5 when this occurs. Toggle for matrix singular, usually used as part of calling sequence for MATINV. Maximum number of constraints allowed Maximum number of compartments allowed Maximum number of columns allowed Maximum value for constraints plus compartments... 

KE 1 I Toggle for matrix singular, usually used as part of calling sequence for MATINV. 

Write out M mass balance or equilibrium constant equations (j = 1,2,..., M). These should be written so the lowest activities are in the denominator to reduce the chances of matrix singularity. 

The closure constraint has to be taken into account also in modelling results of mixture experiments. The closure means that the columns of the model matrix are linearly dependent making the matrix singular. One way to overcome this problem is to make the model using only N-1 variables, because we need to know only the values of N-1 variables, and the value of the N th variable is one minus the sum of the others. However, this may make the interpretation of the model coefficients quite difficult. Another alternative is to use the so-called Scheffe polynomials, i.e. polynomials without the intercept and the quadratic terms. It can be shown that Scheffe polynomials of N variables represent the same model as an ordinary polynomial of N-1 variables, naturally with different values for the polynomial coefficients. For example the quadratic polynomial of two... 

Fisher, A. T. M. B. Underwood, 1995. Calibration of an X-ray diffraction method to determine relative mineral abundances in bulk powders using matrix singular value decomposition a test from the Barbados accretionary complex. In Proceedings Ocean Drilling Program, Initial Reports 156 29-37. 

Matrix singularity can also occur when the elements in two or more rows of a matrix are linearly dependent. For example, if we multiply the elements of the second row of matrix [Ai by... 

There are some limitations to the K-matrix approach. To obtain the matrix of concentration information, C must be inverted during the operations. This inversion demands that C be nonsingular requiring that there are no linear relationships between its component rows and columns. However, if rows or columns of the matrix have a linear relationship to each other, the determinant will be zero, the matrix singular, and noninvertable. This problem can be avoided by paying careful attention to the makeup of the calibrating standards. 

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