Weighting matrix

To determine the vibrational motions of the system, the eigenvalues and eigenvectors of a mass-weighted matrix of the second derivatives of potential function has to be calculated. Using the standard normal mode procedure, the secular equation... 

State weighting matrix %Control weighting matrix... 

Feed-back models can be constructed and trained. In a constructed model, the weight matrix is created by adding the output product of every input pattern vector with itself or with an associated input. After construction, a partial or inaccurate input pattern can be presented to the network and, after a time, the network converges to one of the original input patterns. Hopfield and BAM are two well-known constructed feed-back models. 

Theorem 5 [goles87a] If the synaptic-weight matrix A is symmetric, and the number of sites in the lattice is finite, then the orbits of the generalized threshold rule (equation 5.121) are either fixed points or cycles of period two. 

It is important to have a symmetric synaptic weight matrix, since even a small deviation away from this symmetry can often make the net unstable and therefore unable to settle into a required set of patterns. While it is true that a symmetric weight set w,... 

Step 1 For a given set of patterns Pi,P2,..., Vn, where the pattern is encoded by the neuronal values, .., define the synaptic-weight matrix... 

In this case we minimize a weighted SSE with constant weights, i.e., the user-supplied weighting matrix is kept the same for all experiments, Qj=Q for all... 

Of course, it is not at all clear how one should select the weighting matrices Q i=l,...,N, even for cases where a constant weighting matrix Q is used. Practical guidelines for the selection of Q can be derived from Maximum Likelihood (ML) considerations. 

The above choice has the computational disadvantage that the weights are a function of the unknown parameters. If the magnitude of the errors is not excessive, we could use the measurements in the weighting matrix with equally good results, namely... 

SELECTION OF WEIGHTING MATRIX Q IN LEAST SQUARES ESTIMATION... 

If however the measurements of a response variable change over several orders of magnitude, it is better to use the non-constant diagonal weighting matrix Qj given below... 

The unknown parameter vector k is obtained by minimizing the corresponding least squares objective function where the weighting matrix Q, is chosen based on the statistical characteristics of the error term e, as already discussed in Chapter 2. 

The above expressions for the CO l (k ) and of are valid, if the statistically correct choice of the weighting matrix Q, (i=1,...,N) is used in the formulation of the problem. Namely, if the errors in the response variables (e, i=l,...,N) are normally distributed with zero mean and covariance matrix,... 

---