Eigenvalue prior

Practical experience has shown that (i) if we have a relatively large number of data points, the prior has an insignificant effect on the parameter estimates (ii) if the parameter estimation problem is ill-posed, use of "prior" information has a stabilizing effect. As seen from Equation 8.48, all the eigenvalues of matrix A are increased by the addition of positive terms in its diagonal. It acts almost like Mar-quadt s modification as far as convergence characteristics are concerned. 

All possible single assembly perturbations s to the reference LP are executed individually prior to the start of an optimization, and the changes in flux A( ), eigenvalue AX and NEM coupling corrections ad are found using the exact NEM neutronics solver and stored for later use, as in equations (4)-(6). 

Hopefully, just the first few principal components contain the majority of the variance in X. The outcome of PCA is to take a set of p-variables and reduce it to a set of q-variables (q < p) that contain most of the information within X. It turns out that the solution to PCA is to simply computes the eigenvalues and eigenvectors of X. If the variances of the columns of X are of different scale it is recommended that X be standardized prior to per-... 

One way to try to alleviate the problem of correlated descriptors is to perform a principal components analysis (see Section 9.13). Those principal components which explain (say) 90% of the variance may be retained for the subsequent calculations Alternatively, those principal components for which the associated eigenvalue exceeds unity may be chosen, or the principal components may be selected using more complex approaches based on cross-validation (see Section 12.12.3). It may be important to scale the descriptors (e.g. using autoscaling) prior to calculating the principal components. However, unless each principal component is largely associated with any particular descriptor it can be difficult to interpret the physical meaning of any subsequent results. ... 

From prior experience with the one-dimensional quantum mechanical harmonic oscillator [4], we know that its eigenfunctions and eigenvalues are... 

Consider the nondegenerate case, that each eigenfunction of the operator A has a distinct eigenvalue. Since the set of eigenfunctions is assumed to be a complete set, the wave function immediately prior to a measurement of A can be represented as a linear combination of eigenfunctions of A, as in Eq. (16.3-34) ... 

We now make a set of many measurements of A, ensuring somehow that the system is in the same state prior to each measurement. Each outcome will be an eigenvalue of A. Let the fraction that results in the value aj be equal to pj. By Eq. (16.4-33),... 

This theorem shows that every eigenfunction of a Hermitian operator is orthogonal to every other eigenfunction with a different eigenvalue. Furthermore, it is possible to prove that eigenfunctions of a Hermitian operator can be constructed to be orthogonal even if some have like eigenvalues. This is a corollary to the prior theorem which we consider without a detailed proof. 

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