Criterion functions

More correctly, the regression problem involves means instead of averages in (1). Furthermore, when the criterion function is quadratic, the general (usually nonlinear) optimal solution is given by y = [p u ], i.e., the conditional mean of y given the observation u . 

Table 10.2 Values of the criterion function Fij calculated for five different fibre coatings...

System Identification Techniques. In system identification, the (nonlinear) resi pnses of the outputs of a system to the input signals are approximated by a linear model. The parameters in this linear model are determined by minimizing a criterion function that is based on some difference between the input-output data and the responses predictedv by the model. Several model structures can be chosen and depending on this structure, different criteria can be used (l ,IX) System identification is mainly used as a technique to determine models from measured input-output data of processes, but can also be used to determine compact models for complex physical models The input-output data is then obtained from simulations of the physical model. 

With the system embedded in the economic environment, all the energies and materials of interest are evaluated according to their economic potentials (costs). The criterion function is a cost measure. A typical function rated per unit time is... 

To obtain a distribution throughout the system of the items involved in the criterion function... 

Both modes of thermoeconomic analysis, accounting and optimization, are illustrated in example 2, considering a cost criterion function. The case of an energy criterion function may be included as a special case. 

When H is block-diagonal, as in Table 7.2, Oj is preferably calculated according to Eq. (7.1-16). Corresponding tests of goodness of fit can then be made for the individual response blocks b via the criterion functions... 

The optimal fuzzy n-partition and its representation could be obtained as a minimum of criterion function. To minimize the objective function an iterative method is at hand. This method is based on the successive minimization of the functions J(P, X where partition P is fixed, and J, L where the representation L is fixed. 

Using this criterion function we can obtain the membership degrees thus ... 

The best basis algorithm seeks a basis in the WPT which optimizes some criterion function. Thus, the best basis algorithm is a task-specific algorithm in that the particular basis is dependent upon the role for which it will be used. For example, a basis chosen for compressing data may be quite different from a basis that might be used for classifying or calibrating data, since different criterion functions would be optimized. The wavelet packet coefficients which are resultant of the best basis, may then be used for some specific task such as compression or classification for instance. 

The adaptive wavelet algorithm outlined in Section 6 can be used for a variety of situations, and its goal is reflected by the particular criterion which is to be optimized. In this chapter, we apply the filter coefficients produced from the adaptive wavelet algorithm for discriminant analysis. It was stated earlier that the dimensionality is reduced by selecting some band(jg,xg) of wavelet coefficients from the discrete wavelet transform. It then follows that the criterion function will be based on the same coefficients i.e. Xl " (xg). 

If the filter coefficients are to be used for discriminatory purposes, then the criterion function should strive to reflect differences among classes. In this section three suitable discriminant criterion functions are described. These discriminant criterion functions are Wilk s lambda (3a), entropy (3e), and the cross-validated quadratic probability measure (3cvqpm)-... 

The denominator is a normalization constant. The numerator is simply the sum of squares of the wavelet coefficients from either the DWT or WPT which occur in the same position of the wavelet trees, where the DWT or WPT has been performed for objects belonging to the same class. The discriminatory criterion function is then... 

The cross-validated quadratic probability measure (CVQPM) (see Chapter 12 for more details) assesses the trustworthiness of the class predictions made by the discriminant model. The CVQPM ranges from 0 to 1. Ideally, larger values of the QPM are preferred, since this implies the classes can be differentiated with a higher degree of certainty. The CVQPM criterion function based on a band of coefficients x[ ) (t) would be defined as follows... 

Here the classification rates of the individual bands at initialization and at completion of the algorithm are shown. Note that the same starting parameters for V, U and un have been used for the implementation involving the different modelling criteria, hence the same classification results occur at initialization for each of the criterion functions 3a, 3e, and 3cvqpm- The shading indicates which band optimization was based upon. 

Table 3. The percentage of correctly classified spectra, using the coefficients X (t) for T = 0,..., 3 at initialization and at termination of the adaptive wavelet algorithm. The discriminant criterion functions were Wilk s Lambda, symmetric entropy and the CVQPM.

One reason why the CVQPM, maybe outperforming the other criterion functions could be due to the fact that optimization and hence classification is based on scaling coefficients. So that a fair comparison could be made, the optimization routine using the Wilk s Lambda, and symmetric criterion functions was repeated, this time forcing optimization over the scaling band. These results are summarized in Table 4. 

Optimization over the scaling band did improve the results slightly for the Wilk s Lambda and symmetric entropy criterion, but these criterion functions were not able to improve upon the results previously obtained with the CVQPM criterion function. 

The correct classification rate (CCR) or misclassification rate (MCR) are perhaps the most favoured assessment criteria in discriminant analysis. Their widespread popularity is obviously due to their ease in interpretation and implementation. Other assessment criteria are based on probability measures. Unlike correct classification rates which provide a discrete measure of assignment accuracy, probability based criteria provide a more continuous measure and reflect the degree of certainty with which assignments have been made. In this chapter we present results in terms of correct classification rates, for their ease in interpretation, but use a probability based criterion function in the construction of the filter coefficients (see Section 2.3). Whilst we speak of correct classification rates, misclassification rates (MCR == 1 - CCR) would equally suffice. The correct classification rate is typically formulated as the ratio of correctly classified objects with the total... 

Classification criterion functions for the adaptive wavelet algorithm... 

The discriminant criterion function implemented by the adaptive wavelet algorithm is the CVQPM criterion function discussed in Section 1.4. The adaptive wavelet algorithm is applied using several settings of the m, q and jo... 

A suitable criterion function for regression analysis should reflect how well the response values are predicted. In the adaptive wavelet algorithm, the criterion function considered for regression is based on the PRESS statistic and is then converted to a leave-one-out cross-validated R-squared measure as follows... 

The cross-validated R-squared criterion function is defined as... 

The most conceivable difference between the AWA when applied for regression (as opposed to classification) is the criterion function which is implemented. Here, the cross-validated R-squared criterion which is based on the PRESS statistic, is the regression criterion function which is implemented... 

Not all linear programming relaxations are close approximations of the discrete problem of interest. Because very complex criterion functions and constraints can be modeled by the hnear part of an ILP, however, it is often the case that an optimal solution to the relaxation (ILP) provides a sound starting point for construction of a good feasible solution to the discrete problem. Such constructions may loosely be termed rounding. ... 

It is possible to assume that kinetic and hydrodynamic methods of reactors analysis and design are well advanced at present. Methods of computer simulation and modelling are widely used. So, we can say that if we know processes kinetic and hydrodynamic parameters and fundamental particularities of reactor functioning we can calculate all process characteristics and its stmcture, we also can predict effectiveness of apparatus operation and consumer properties of chemical production. Meanwhile criterion function development for calculation and organization of novel processes and optimization of present productions require overcoming of a big number of problems in production practice. This principle is satisfactory enough for processes with low or medium reactions rates, when creation of isothermal conditions in apparatus is easy. In this case it is easy to calculate and reproduce in working conditions all characteristics of chemical process and to control the last ones. 

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