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P. Neittaanmäki, T. Rossi, K. Majava, and O. Pironneau (eds.)
COMPARATIVE IDENTIFICATION OF LINEAR MODEL OF MULTIFACTOR CHOICE Eduard G. Petrov , Atageldy O. Ovezgeldyev and Konstantin E. Petrov
National University of Radio Electronics, Lenin Avenue, 14, 61166, Kharkov, Ukraine
National University of the Internal Affairs, 50-letiya SSSR Avenue, 27, 61080, Kharkov, Ukraine
Key words: Comparative Identification, Model of Mental Activity, Multifactor Choice Model, Preference Matrix. Abstract. The analysis of phenomenological model of intellectual activity of the person is conducted at decision-making. The problem of identification of a linear additive function of usefulness is formulated on the basis of the decision, received by an individual. The mathematical model of solution of this task is offered, which one is based on ideas of comparative identification.
Eduard G. Petrov, Atageldy O. Ovezgeldyev and Konstantin E. Petrov.
INTRODUCTION
A retrospective analysis of scientific and technical progress shows that all previous stages
have been aimed at multiplying man's physical power and creating new materials. From this point of view the current stage of scientific and technical progress is unique. Since the creation of the computer, has for the first time in its history humanity obtained a tool for increasing its intellectual ability. Nowadays, more and more complicated mental functions are being passed to the machines. The process started with computerizing procedures with well-known algorithm; it is now fast developing towards automation of rather sophisticated mental processes, such as situation recognition, semantic analysis of information, gaining new knowledge, taking decisions, etc. As a whole, it is the problem of creating artificial intelligence. A central task of this problem is to construct a system of formal models for man's mental activity.
A PHENOMENOLOGICAL MODEL OF MAN'S MENTAL ACTIVITY
Informally, a general model of mental activity can be described as follows. Using his
sensitive system, an individual creates his own informational image of some situation in the context of the environment, and then analyzing the information obtained, realizes some behavior, in the broad sense of the word. This can be any kind of activity, e.g. drawing a logical inference, obtaining new knowledge, etc. Let us denote by Θ the set of external stimuli onto man's sensory system in a concrete situation and by J its informational image
Here t is the current time; Ψ - the individual operator of informational mapping; τ - the
delay time, since the sensory system does not react at once. The individual behavior is described by the equation
where Π is the individual behavior operator. It is reasonable to distinguish the two models:
that of unconscious (reflex) behavior when rigid algorithms are realized (like the mass center stabilization in walking) and that of conscious behavior based on the intelligent processing of information. In what follows, we shall consider only models of the latter type. Transition to the formal models is connected with the identification of the operators Ψ
PECULIARITIES OF THE IDENTIFICATION TASK
The synthesis of the formal models are connected with solving tasks of structural
identification, i.e. finding the form of the operator establishing the relation between the input and output, and tasks of the parametrical i.e. quantitative identification. Classical Identification Theory involves solving the above-mentioned tasks, and using the experimental
Eduard G. Petrov, Atageldy O. Ovezgeldyev and Konstantin E. Petrov.
information on the quantitative values of the input stimuli parameters and on the reaction of the system under study. Unfortunately, in many cases of the identification of intellectual activity models one is unable to quantitatively measure the individual reaction, though it is possible to register the type of behavior and its qualitative characteristics. In this case two approaches are possible. The first is to make the experimentee analyze his behavior introspectively, i.e. to compose and formalize a model of some mental process. The tools to implement this approach are questionnaires, interviews with the experimentees, expert estimates, etc. The peculiarity of this approach is the necessity of training the experimentee, subjectivism and possibility of conscious or unconscious, distortion of information, correlation of the results with the poll procedure, ability for the investigator to influence the results. Nevertheless, in many cases this approach is fruitful especially in solving structural identification problems.
In an attempt to overcome the subjectivism in the above techniques, a second approach has
been developed. It consist of carrying out active and passive experiments with the aim of registering the type and other characteristics of the experimentee's behavior, and the further use of this information for identification. Here the classical identification theory becomes useless, and one has to create an alternative theoretical basis, which can be constructed using the theory of comparative identification [1].
Neither approach is antagonistic, they both complement each other and have their own
problems. In what follows we shall consider only the second approach.
POSING THE PROBLEM OF COMPARATIVE IDENTIFICATION
Let A be a set of situations, and element being characterized by n parameters that can be
objectively quantitatively measured. Each situation causes the individual to behave in a certain way, i.e. there exists a mapping onto the set of different ways of behavior
Then F is an operator and B = F(A) is a model of mental activity. The elements of the set B ("behaviors" of the individual) cannot be measured quantitatively, but can be analyzed quantitatively.
In what follows we shall assume that the set B satisfies two conditions: firstly it does not
contain incomparable elements, which is ensured by the correct way of fixing the set A, and secondly, there exists the partial order relations, i.e. the reflexivity and transitivity axioms are
satisfied. Then, as a result of analyzing the set B, one can construct the factor set '
uniting into one class all the elements u, v ∈ B such that u ≥ v and simultaneously, v ≥ u
which means that u = v. It can happen that several or all classes of '
element. As is known, the partial order on B ensures the order on '
the assumptions we have made, it is possible to order '
elements (the smallest and the largest) and to indicate the previous element to any except the smallest. Then, using the information obtained, one must identify the operator F.
Eduard G. Petrov, Atageldy O. Ovezgeldyev and Konstantin E. Petrov.
PROCEDURE FOR OBTAINING INITIAL INFORMATION
A pair of situations x, y ∈ A are presented to the experimentee. Reacting to this input, the
experimentee behaves in the manner u = F(x), v = F(y), u, v ∈ B. Then the analysis of these manners is carried out. Formally, the analysis can be described as a system of comparators, the first of which realizes the predicate of the form
B (i.e. each pair of elements from B is comparable). A
superposition of predicates (4) form the predicate
A and values (1, 0). It is easy to show that the predicates
are reflexive, transitive and symmetric, so they are the predicates of equality and
equivalence, respectively. On applying the predicates
A whose elements are the classes of equivalent
behaviors and situations, respectively. Here in both sets, several or all elements can contain only one element.
The next stage of the analysis is to rearrange the elements of
complete ordering. To this end, a pair of behaviors u and v, for which
second comparator, realizing the predicate
A superposition of predicates (6) form the predicate
The predicates E and D are transitive and comparable (u < v and u > v cannot hold at
the same time) and, by virtue of this, are strict ordering predicates, establishing the relation of
The experimentee can serve as the above-described comparators by analyzing his
perceptions; if the experimentee's reactions have observable expression, then an observer can play the part of the comparators.
Depending on the peculiarities of the mental process under investigation, on the aim of
modeling, on the opportunities to carry out a passive or an active experiment, it can happen that the experimentee will be shown more than two situations. It is assumed in this case that the analysis procedure and the final result are similar to those described above and consists in establishing binary relations of elements of the sets A and B. The information thus obtained
Eduard G. Petrov, Atageldy O. Ovezgeldyev and Konstantin E. Petrov.
serves as initial for the problem of identification of the operator F. For each pair of the equivalent situations x and x (5) implies that
while for the situations that fulfill the preference relation (7) implies that
The total number of relations of the form (8) and (9) depend on the cardinality of the set A
and on its structure, that is the number of the equivalence classes and the number of elements in these classes. In a particular case, the set A may contain either one equivalence class of all comparable elements, or several linearly ordered equivalence classes comprising one or more elements, or consists of one-element classes only, i.e. to be a completely ordered set. The first case is described by equalities (8), the second - by a composition of equalities (8) and inequalities (9), the third - the inequalities (9).
The complete information on the structure of the set A and B can, as a rule, be obtained in
an active specially planned experiment with a specially trained experimentee or a group. If such an experiment is impossible or undesirable, then a passive experiment is carried out, which consists of observing an untrained experimentee under natural conditions. As a rule, it is impossible to obtain sufficient information on the structure of the sets A and B in the latter case. For example, if a choice process is investigated, then as a result of a passive experiment one can obtain results on most preferable but not all feasible alternatives.
The number of equations needed to solve a concrete identification problem depends on the
process under study, on the dimensionality of the operator F, on the experimental conditions, etc. One should bear in mind that relations of the form (8) are more informative than those of the form (9), since the former enables one, in principle, to unambiguously determine the unknown variables, whereas (9) bound a certain range of solutions, and to choose a solution from this range is a task thus needing additional information.
A TECHNIQUE FOR SOLVING THE COMPARATIVE IDENTIFICATION PROBLEM
The operator F which we have to identify can be chosen among the models of psycho-
physical processes of reflecting the environment (the operator Ψi ), such as visual, audio or tactile perceptions, or among the models of mental processing information (the operator Πi ), such as comprehension, recognition, classification, choice, etc.
A generalized behavior model, independent of a concrete mental process, can be written as
where x is n-dimensional quantitatively measured input (a situation);
, the m-dimensional vector of the quantitative features (parameters) of
the model. The subscript “M” shows that the quantity relates to a model (not real) process. In
Eduard G. Petrov, Atageldy O. Ovezgeldyev and Konstantin E. Petrov.
the classical identification theory the output U can be quantitatively measured, and the identification problem consists in finding
which have been obtained with the same input x. In the case of the comparative identification of mental processes, as has been shown above, the experimental data enables one to separate the equivalence classes and the preference relation from the set of behaviors (output u) and the set of situations (input x). Under these conditions the identification problem consists in finding
such that they do not contradict (8) and (9) following from the
equivalence and preference relations. This problem as it is posed, completely coincides with the identification of utility functions, i.e. from the models describing the preference in the numerical form.
The solution of the comparative identification problem, as in the classical case, requires
(the structural identification) and the values of the
(the parametric identification). There exist two approaches to identify
first one consists in synthesizing an operator, which describes as accurately as possible the actual physical, physiological, biochemical and other processes in man's brain. The second approach consists in constructing a simplest operator
the real system but which simulates the reactions that coincide with the real ones to within the required accuracy (equivalence in reactions). The latter approach is in better agreement with the problem of formalizing mental processes with the aim of constructing artificial intelligence, as well as it better agree with the ideas of the utility theory. Independently of an approach accepted, the structural identification consists in an iteration procedure to formulate a hypothesis for the structure of
using the additional information, the quantitative
and checking how well the model agrees with the
is given, the procedure of determining q
the form of the information obtained in the experiment. Let q
as a result of a comparative experiment, k > m equations of the form (8) have been obtained, then m compatible equations are used for learning, in order to determine the parameters q
while the remaining equations are used for checking, which makes it possible to objectively evaluate the model. If k < m or k = 0, but there are compatible relations of the form (9) bounding some range of solutions, then all the equations of the form (8) are used for decreasing the number of the unknown variables by substitution, after which the remaining unknown variables are found as the Chebyshev approximation, i.e. as a point equidistant from the boundaries of the constrained range.
In the framework of this general approach and accounting for peculiarities of the mental
process studied, the form of the operator
, the available and additional experimental data,
Eduard G. Petrov, Atageldy O. Ovezgeldyev and Konstantin E. Petrov.
the aims of the modeling process, it is reasonable to develop problem-oriented procedures which are always more efficient in concrete situations.
In what follows we shall describe several procedures of this kind for solving concrete
problems of the identification of models of mental activity.
IDENTIFICATION OF MULTIFACTOR CHOICE MODEL
Informally, the process of the multifactor choice can be described as follows. A decision
taker (DT) considers N > 1 alternatives, each being characterized by n natural numbers. DT prefers one alternative from of the displayed set. The problem is to identify the model for this mental process of choice.
The theoretical base here is the utility theory whose subject mater is to express numerically
the preference relations. The main assumption of this theory is the condition that each alternative may be related to a number U (utility) such that for any two alternatives x and y one is preferable with respect to the other only if the utility of the first exceeds the utility of the second, i.e.
The function U satisfying (12) is called the utility function. For the utility function we shall take the linear form
where p are the utility functions of the factors characterizing the alternatives and are the
µ weight coefficients (DT preferences). Then the formal process of choosing the most
Relationships (13) and (14) determine the structure of the model of choice. The problem of
the parameter identification consists in finding the matrix of the DT's preferences M = µi
The difficulty in solving this problem lies in the fact that a quantitative measurement of the
utility function values of different alternatives U is impossible, so we shall resort to the
comparative identification technique. We assume that a passive experiment is carried out, i.e. that a current situation needing a choice is considered.
Let the quantitative characteristics of N situations, which a DT must analyze, be described
Eduard G. Petrov, Atageldy O. Ovezgeldyev and Konstantin E. Petrov.
Then the utility of each of the alternatives for the DT is given by the matrix
As a result of a passive experiment, let it be established that the DT chose alternative c. It follows from (14) that
θ µ + θ µ + . + θ µ ≤ θ µ + θ µ + + θ
The characteristics θvi for each alternative are given to the DT in the natural form. Each of
them has a different physical meaning and range; so, for the sake convenience, we shall reduce them to one range and a dimensionless form, using the formula
θ means the utility of the local indexes of alternatives. Then (18) will become
Besides, we shall assume that we are interested in the relative "weights" of the
Eduard G. Petrov, Atageldy O. Ovezgeldyev and Konstantin E. Petrov.
The problem is in finding the preference matrix M, taking account of restrictions (20), (21)
and (22). Equation (21) is that of a plane in the n-dimensional space where restrictions (20) and (22) make a convex polyhedron, which is the set of feasible solutions Ω . The set is empty if restrictions (20) and (22) are incompatible. For choosing one solution (the matrix M
values) from Ω one need some criterion. We recommend the Chebyshev point M ={
1 n , that is take as a solution, which is positioned at the maximum (by the modulus)
For the case considered problem (23) is an LP problem, and its solution does not present
any principal difficulties. Here the quantity L characterizes the size of the domain Ω ; therefore the accuracy of finding the matrix M. Besides, the system of restrictions is compatible only if L ≥ 0.
The accuracy of the identification can be refined if an active experiment is possible.
Otherwise, the accuracy is increased by repetitive experiments with the consequent averaging of the estimates obtained.
An investigation into the method described, particularly an estimate of how adequate and
accurate the model is, the conditions of the restriction compatibility, and the results of practical application in the identification of the model hiring personnel was described detail in [2].
CONCLUSIONS
Comparative identification opens broad opportunities for identifying models of various
mental processes, including pattern recognition, decision taking, comprehension and so forth. However, the theory of comparative identification is still in development now, and much should be done to develop the underlying general theory, and to create problem-oriented methods and algorithms for solving practical problems.
REFERENCES
[1]. Yu.P. Shabanov-Kushnarenko. Theory of intelligence. Mathematical Tools. Vyshcha
[2]. A.O. Ovezgeldyev, E.G. Petrov, K.E. Petrov. Synthesis and Identification of Models of Multifactor Estimation and Optimizations. Naukova dumka, Kyiv, 2002.

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