Determinant criterion

If the variances in Equation 2.25 are totally unknown, the ML parameter estimates can only be obtained by the determinant criterion presented later in this chapter. 

It is worthwhile noting that Box and Draper (1965) arrived at the same determinant criterion following a Bayesian argument and assuming that is unknown and that the prior distribution of the parameters is noninformative. 

The determinant criterion is very powerful and it should be used to refine the parameter estimates obtained with least squares estimation if our assumptions about the covariance matrix are suspect. 

The F factor is varied until L ls equal to 2N, where N is the order of the system. For N = 1, the SISO case, we get the familiar -f2 dB maximum closedloop log modulus criterion. For a 2 x 2 system, a +4 dB value of is used for a 3 x 3, -1-6 dB and so forth. This empirically determined criterion has been tested on a large number of cases and gives reasonable performance, which is a little on the conservative side. 

Unfortunately, there are some technical difficulties associated with the determinant criterion (ref. 28). Minimizing the determinant (3.66) is not a trivial task. In addition, the method obviously does not apply if det[ V(p) ] is zero or nearly zero for all parameter values. This is the case if there exist affine linear relationships among the responses y, y2,. .., yny, as we discussed in Section 1.8.7. To overcome this problem the principal component analysis of the observations is applied before the estimation step. 

Example 5.6 Comparison of the determinant criterion with least squares... 

Because of the use of determinants it is also called the determinant criterion, for which test the user also needs special table values or subprograms for the calculation of the respective / -approximation. 

In Table 3 several conditions for application of the various criteria are summarized. Special treatment will be required when some observations of response variables are missing, such as when an analysis was only partially successful [9]. It is noted that the determinant criterion is not a sum of squares criterion, so the minimization routines based on the Hessian approximation are not applicable and the other techniques have to be applied. 

The first two criteria should be considered a minimum requirement and should always be reported otherwise, the racemization date should be considered meaningless. The second criterion is by far the most important. The d/l amino acid ratio for a particular bone can generally be imequivocally determined (criterion 1). However, the conversion of this ratio into an age estimate can be done only if a suitable calibration value has been determined for the site or locality where the bone was found. The reliability of the date increases with the number of criteria which are fulfilled. In the future, it should eventually be possible to satisfy an additional criterion and report an average racemization age for a sample deduced using several different amino acids. 

The determining criterion to differentiate major projects from small projects is the degree of complexity, not necessarily the project cost. Occasionally, an expenditure of tens of millions of dollars could be simple enough that it might be better handled using the small project approach to conserve spending, rather than following the conventional approach used in major projects. 

Choosing the basis candidates in order of their current test divisors Di — AiijAn,re selects those parameters for which the normal equations are best conditioned. This order of selection may sometimes exclude a parameter that is important on other grounds. Such a parameter might be made estimable by providing additional experiments (preferably selected by the determinant criterion in Section 6.6), or by using an ADTOL value smaller than the excluded test divisor Di, or by rewriting the model in a better form. 

The determinant criterion and the trace criterion are well regarded, and both are provided as options in GREGPLUS. Example C.l applies the full determinant criterion to a three-parameter model. 

A determinant criterion is used to obtain least-squares estimates of model parameters. This entails minimizing the determinant of the matrix of cross products of the various residuals. The maximum likehhood estimates of the model parameters are thus obtained without knowledge of the variance-covariance matrix. The residuals e, , and correspond to the difference between predicted and actual values of the dependent variables at the different values of the Mth independent variable (m = to to u = tn), for the ith, 7th, and kth experiments (A, B, and C), respectively. It is possible to constmct an error covariance matrix with elements v,y ... 

If no replicate experiments were carried out and the error covariance matrix is completely unknown, the determinant criterion of Box and Draper [1965] can be used ... 

It is not simply a matter of measuring viscosity it is the visual aspect of the wine that is the determining criterion for characterizing this problem. For example, a wine with a viscosity of 1.637 centistoke (cst) and a polysaccharide concentration of 95 mg/1 is not ropy, as opposed to a wine with a viscosity of 1.615 and a polysaccharide concentration of 300 mg/1. Many other medium factors contribute to wine viscosity, notably the alcoholic content. 

A window-size of zero gives no smoothing at all, whereas any number larger than half the number of data points will render the data virtually featureless. Therefore the smoothing window should be optimized according to a pre-determined criterion. When the undisturbed... 

Cost considerations can be a key factor in decisions to implement countermeasures. Cost-benefit analysis is a cost analysis methodology used to justify safety expenditure. The cost-benefit analysis is a comparative assessment of the benefits from your safety measure and the costs to perform it, in relation to the financial impact resulting from potential loss to the asset. The safety manager should consider costs for all phases of the safety project. Costs may be one-time capital costs or recurring costs. The determining criterion in a cost-benefit analysis is that the safety benefit exceeds the cost. The amount that the benefit exceeds the cost is based on the safety manager s determination in consultation with organizational leadership. 

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