V. I. Krasinsky COMPARISON OF OBJECTS BY THE DEGREE OF FUZZYNESS OF NOMINAL CHARACTERS

V. I. Krasinsky COMPARISON OF OBJECTS BY THE DEGREE OF FUZZYNESS OF NOMINAL CHARACTERS

For classification and recognition of botanical objects the nominal (qualitative) characters such as types of leaves, fruits, etc. are often used. A subsequent analyses by statistical methods is impeded by the multivaluedness of objects (by fuzzyness of the initial information) as an every character of a majority of objects is described by not one but a list of possible values. Besides it, the values of nominal characters could not be compared, that is the focal subsets in the sense of Dempster-Shafer theory [1,2] could not be defined.

To overcome these difficulties in the building up of a recognition program of the Siberian plants families a theory of possibilities [3,4] has been applied. It allowed to derive a function of membership (FM) of objects to the fuzzy sets (FS), the semantics of which is defined as "reliability of diagnosis by the multivalued character". A number of fuzzy sets is equal to number of characters.

For each character the sum of units in a binary incidence matrix t_ik defined by the set "object – value of character" is considered as "a mass of diagnosis unreliability". In this case a distribution of this mass over nominal character values corresponds to the relative weights of this character values m_k (k – is an index of the character value). FM function of objects to the fuzzy set V(X) is calculated as:

_V(x_i) = _k t_ik m_k , where x_i is object; i=1, ... , n.

For normalization of FS a following technique is applied: to the initial objects an additional simulated one is introduced which has the incidence to the all values of all characters. For this object it always true _V(x_n+1)=1. As the result a usually needed scaling of values of each FM by its maximum value is not necessary which is important for multidimensional analyses of objects.

The supplement V(x) to FS means "reliability of diagnosis" of object by the character value: _F(x_i)=1–_V(x_i). Through the decreasing of _F(x_i) values variational rows of objects–families for each character have been obtained, that is a transformation of nominal listed characters of objects into a strong numerical scale of intervals (comparison of objects by reliability of diagnosis) has been performed.

Implemented by the author a computational program of recognition of 102 families of Siberian plants operates by four numerical and seven nominal multivalued characters and is based on a superposition of all fuzzy sets.

1. 
   Dempster Upper and lower probabilities induced by a multivalued mapping. // Ann. Math. Statistics. 1967. 38. Pp. 325–339.
   
2. 
   G. Shafer A Mathematical Theory of Evidence. Princeton Univ. Press. 1976.
   
3. 
   Dubois P., Prade H. A set-theoretic view of belief functions. Logical operations and approximations by fuzzy sets. // Int. J. of General Systems. 1986. 12(3). Pp. 193-226.
   
4. 
   Dubois, H. Prade Theorie des possibilites. Applications a la representation des connaissances en informatique. Paris Milan Barcelone Mexico: MASSON. 1988.
   

A.N. Tselykh DECISION MAKING ON THE BASIS OF FUZZY SEMANTIC NETWORK

In the report the questions of construction models of decision-making based on use fuzzy semantic network with different representation of the initial data are considered.

The fuzzy semantic network is represents as a fuzzy hyper-graph including two types of tops: tops-attributes and tops-conclusions. Tops-attributes match factors with the most essential influence on decision making. Tops-conclusions match various meanings of decisions. In a common case the edges of hyper-graph represent n-measured relations, where n = 1,2,3,... , determined on the set of tops-attributes and characterizing associative links between attributes or steady groups of attributes. Weight coefficients describing the strength of semantic perception of attributes or the strength of links between the attribute (group of attributes) and the conclusion are attributed to hyper-graph edges. The manager-expert builds the set of attributes, their meanings, links between attributes and also links between attributes and conclusions.

The field of knowledge of an semantic network formally is written: K = (X, Y, R), where X - tops-attributes, Y - tops-conclusions, R - edges representing n–measured links between tops xX and yY. To each edge rR the measure of strength of connection (r)[0,1] between the appropriate tops is correlated. If the edge r of the hyper-graph K includes more than two tops from X, the measure (r,y) characterizes the strength of connection of the given subset of tops from X included in an edge r with top-conclusion y  Y.

The inquiry for the search of the appropriate conclusion in a common case represents the fuzzy subset of attributes given by meanings of belonging degrees describing an initial situation.

The logic conclusion adds up to the definition of a way with the maximal estimation between top appropriate to an initial situation and all tops-conclusions of the hyper-graph. This way is formed by the affiliation of top appropriate to an initial situation to the hyper-graph specifying the field of knowledge. Thus affiliated top links with tops-attributes in a field of knowledge by oriented edges with weights appropriate to the belonging degrees of the given attribute in the description of an initial situation (search image). The degree of suitability of some conclusion to the sent inquiry is determined by an estimation of a way received as a result of a logic conclusion.

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