So, Durgaryan and Pashchenko (2001), Pashchenko (2001) consider the mutual Shannon information of model “output” and system ”output” as an identification criterion to derive the required model. Such a criterion, which has been referred as the information one, is to be maximized, and the model “output” is just considered as the maximization argument. The approach proposed by Durgaryan and Pashchenko (2001), Pashchenko (2001) thus cannot be considered as a constructive one, because it initially is based either on a requirement that the joint distribution density of the model “output” and system “output” is to be preliminary known (what is nonsense, in entity), or the above “outputs” are able to be observed. But this, the second, way is not applicable because the problem is just to derive the model, and hence its “output” can naturally not be observed. As to the first way, it also can not be considered as acceptable, because it requires such an amount of a priori knowledge under which the identification problem already is to loose its sense: the joint distribution of model and system “outputs” is a final result of many factors (system and model structure, statistical properties of “inputs”, etc.). However, just postulating a concrete kind of the joint distribution density of the “outputs” of model and system has been used as a basis for analytical inferences of Durgaryan and Pashchenko (2001), Pashchenko (2001). Specifically, Durgaryan and Pashchenko (2001), Pashchenko (2001) assume the joint distribution of the model and system “outputs” to be the Gaussian one, having known parameters, what gives rice the initial identification problem to the problem of maximizing the correlation coefficient of the “outputs” of model and system. From a substantial point of view, the assumption that the joint distribution of “outputs” of the model and system to be Gaussian is equivalent to that, for instance, if there would be proposed a new method of matrix inversion followed by an assumption that the matrix subject to inversion to be the diagonal one. One also should be noted that the assumption the joint distribution to be Gaussian is always not valid, for instance, under identification of the identity transformer. In fact, let the “input” X have the standard Gaussian distribution, i.e. , the system “output” ; the model “output” ; the joint distribution of the model and system “outputs” is of the form:

.

Hence, the joint distribution density of the model and system “outputs” is not Gaussian.

As to those seldom cases, when the assumption that the joint distribution density is Gaussian is valid (if the property is implied by the system and model structure, probabilistic properties of the input signal, etc.) reasonability of such is approach is quite questionable since, for the case, it is enough to apply ordinary least squared criterion (for the joint Gaussian distribution, the maximal correlation is well known to be linear and to coincide with the ordinary one).

A constructive approach to identification by means of the information criterion for a rather general case might be derived on basis of using a linear model (with parameters subject to identification) approximating an initial system under study driven by Gaussian “inputs”. For the case, an approximation of the joint distribution density of the system and model “outputs” may be based on using the criterion of minimum of relative entropy, determining the conditional distribution of the model “output” with respect to the system “output” (within this, additional conditions imposed on moments of such a conditional density may be taken into account). In turn, the system “output” distribution is estimated via sampled data.

More over, Pashchenko (2001, Section 7.4.1 (pages 456-457)) has introduced a number of definitions relating to the entropy. These are:

• dynamic entropy

;

• generalized dynamic entropy

;

• total entropy

;

• maximal entropy

.

In the definitions, l_Y “causes a reference mark on a scale of entropies” (Pashchenko 2001, p. 454), and B is a nonlinear transformation. The elements By form the set of all states {BY} which is the result of acting of arbitrary transformations B on the initial random value Y. Within such a framework, it is noted also that and . In turn, Pashchenko (2001) states that the results of Durgaryan and Pashchenko (2001), Pashchenko (2001) are valid both for the conventional entropy and the above considered generalized one. However, Pashchenko (2201) provides no details concerning such issues as existence of the values , , , , as well as a definition of the measure m (B). At the same time, simple examples show that, for instance, the maximal entropy introduced does not meet the condition of finiteness. Namely, let the random value Y has the standard Gaussian distribution density, and the nonlinear transformation B of this random value be chosen in the form

.

For the case, the plot of the subintegral function in (entering the sign “minus” under the integral) is of the kind presented at figure 1 (the all figures are presented at the end of the paper), and is practically a direct line which is parallel to the abscissa axis and situated at distance from it. Obviously, that for the case is equal to infinity.

For the cases of , , and the corresponding plots are presented at figures 2 to 4.

It is not also difficult to provide the corresponding example for any of known distributions, as well as to prove to be equal to infinity for the general case of distribution. In turn, taking into account the opinion of Pashchenko (2001) that the above mention inequalities and hold, one should conclude the results of Pashchenko (2001), relating to the general entropy, and, correspondingly, the general information, to be at premium.

Fig. 1. Toward infiniteness of as .

Fig. 2. Toward infiniteness of as .

Fig. 3. Toward infiniteness of as .

Fig. 4. Toward infiniteness of as .