Normal equations matrix

GREGPLUS in selecting pivots in the normal-equation matrix A for each constrained minimization of S 6). GREGPLUS does not judge a parameter estimable unless its test divisor exceeds ADTOL at pivoting time. 

A yVA)/ is the corresponding diagonal element of the inverse normal equation matrix,... 

The inverse of the normal equation matrix, (A WA)", may be used to evaluate the correlation coefficients (p ) among the pairs of free least squares variables (xi and xj) ... 

Cigrovski B., Lapain M., Petrovic S. (1984) Operating with a Sparse Normal Equations Matrix. In Milovanovic G. (ed) Numerical Methods and Approximaterion Theory, University of Nis, Nis, pp 67-72. 

The form of the symmetric matrix of coefficients in Eq. 3-20 for the normal equations of the quadratic is very regular, suggesting a simple expansion to higher-degree equations. The coefficient matrix for a cubic fitting equation is a 4 x 4... 

We have already seen the normal equations in matrix form. In the multivariate case, there are as many slope parameters as there are independent variables and there is one intercept. The simplest multivariate problem is that in which there are only two independent variables and the intercept is zero... 

The left side of the normal equations can be seen to be a product including X, its transpose, and m. Matrix multiplication shows that... 

Therefore, instead of modifying the normal equations, we propose a direct approach whereby the conditioning of matrix A can be significantly improved by using an appropriate section of the data so that most of the available sensitivity information is captured for the current parameter values. To be able to determine the proper section of the data where sensitivity infonnation is available, Kalo-gerakis and Lulls (1983b) introduced the Information Index for each parameter, defined as... 

The fact that the algorithm used by MATLAB does not return a normalized output matrix C can create problems when we do feedback calculations in Chapter 9. The easy solution is to rescale the model equations. The output equation can be written as... 

The recursive formulas that have been presented exhibit some advantages over classical batch processing. First, they avoid the inversion of the normal coefficient matrix, since we would usually process a few equations at a time. Obviously, when only one equation is involved each time, the inversion degenerates into computing the reciprocal of a scalar. Furthermore, these sequential relationships can also be used to isolate systematic errors that may be present in the data set, as will be shown in the next chapter. 

The normal equations, giving the estimates a, a2,..., ap of the model parameters, are thus written in matrix notation as... 

Experience has shown the covariance matrix 0rm to be conspicuously different when the same problem was treated by either the rs-method (without enforcing the first and second moment conditions) or any of the r0-derived methods. For the former method the errors of the coordinates were much less correlated. This, as well as the better condition number of the normal equation system, is no doubt a... 

The derivative of the normalized overlap matrix element, Ski, equation (48), presents no new challenges and it is a simple exercise to show... 

The particularization of the system of the normal equations (5.9) into an equivalent form of the relationship between the process variables (5.83), results in the system of equations (5.85). In matrix forms, the system can be represented by relation (5.86), and the matrix of the coefficients is given by relation (5.87). According to the inversion formula for a matrix, we obtain the elements for the inverse matrix of the matrix multiplication (XX ), where (X ) is the transpose matrix of the matrix of independent variables. 

The orthogonality of the planning matrix, results in an easier computation of the matrix of regression coefficients. In this case, the matrix of the coefficients of the normal equation system (X X) has a diagonal state with the same value N for all diagonal elements. As a consequence of the mentioned properties, the elements of the inverse matrix (X X) i have the values djj = 1/N, dj] = 0, j / k. 

Here the sum within the curly brackets denotes a square matrix with the indicated elements. The normal equations in either form give... 

Similarly, normal equations can be written more succinctly in matrix notation... 

Evaluating this equation for each k, and writing the result in matrix form, the normal equations are obtained ... 

To determine by least-squares the value of each coefficient requires we use eight simultaneous equations. In matrix notation the normal equations can be expressed as,... 

This is the suite of normal equations there is one for each parameter shift, Ap,. By accumulating terms, these equations can be given, in matrix form, from the following ... 

For N parameters the least-squares equations lead to an V x AT normal matrix. Because the restraints involve only near neighbour atoms, the matrix is sparse, with the majority of non-diagonal terms being zero and less than 1% of the elements nonzero [117]. For n atoms and m distance restraints the number of elements to be stored is 6n-l-9m. For example, with a small protein of 812 atoms and 2030 restraints (approximately 3 x the number of atoms), the number of elements is 23 142. For phosphorylase b with 6640 atoms there are 26 561 parameters and some 229451 nonzero elements on the normal matrix, which is still only 0.03% of the total matrix elements. In the restrained least-squares refinement (and many of the other refinement methods) the normal equations are solved by the conjugate-gradient algorithm [129]. 

The derivatives of the calculated data with respect to the force constants are then formed, and these are used to construct the normal equations from which corrections to the force constants are calculated in such a way as to minimize the sum of weighted squares of residuals. Because the relations between the data and the force constants are often very nonlinear, it is necessary to cycle this calculation until the changes in the force constants drop to zero when the calculation will have converged and the sum of weighted squares of errors will be minimized. The usual statistical formulas are then used to obtain the variance/covariance matrix in the derived best estimates of the force constants, and the estimated standard errors in the force constants are usually quoted along with their values. The whole procedure is referred to as a force constant refinement calculation. ... 

Rule 4 enables us to determine the coefficients of NBMOs without diagonalizing the Hiickel matrix.287 Start by assigning an arbitrary value of a to one of the atoms with nonvanishing coefficients. Then assign multiples or fractions of a to the other atoms in the same set, using rule 3 that the coefficients on atoms that are attached to an atom with Cnbmo, = 0 must add up to zero. Finally, the NBMO must be normalized (Equation 4.25), whereby the value of a is defined. 

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