Two new Fuzzy Models Using Fuzzy Cognitive Maps Model and Kosko Hamming Distance

Transcription

1 Ultra Scientist Vol (1)B, 43- (01) Two new Fuzzy Models Using Fuzzy ognitie Maps Model and Kosko Hamming Distance K THULUKKANAM 1 and RVASUKI (Acceptance Date th April, 01) Abstract In this paper for the first time two new fuzzy models iz Merged Fuzzy ognitie Maps (MFMs) models and Specially Merged Linked Fuzzy ognitie Maps (SMLFMs) are introduced To compare the experts opinion a new techniques called Kosko Hamming distance and Kosko Hamming weight are introduced Key words : Merged graphs, Fuzzy ognitie Maps, Merged matrices, Merged Fuzzy ognitie Maps (MFMs), Specially Merged Linked Fuzzy ognitie Maps (SMLFMs), Kosko Hamming Distance 1 Introduction This paper has six sections The first section is introductory in nature In the second section the concept of merged graphs and merged matrices are recalled from 11 For the notion of Fuzzy ognitie Maps model and its application to social problems please refer 1-11 In section three the new notion of Merged FM is introduced and illustrated by an example Section four introduces the new Specially Merged Linked FM model Section fie introduces the new technique of comparing the iews of the experts using the Kosko Hamming distance and Kosko Hamming weight The final section gies the conclusion obtained from this research Merged Graphs and their Merged Matrices : In this section the definition and properties of merged graphs and their merged matrices are recalled This is illustrated by examples For more about merged graphs please refer 11 Definition 1: Let G 1 = {V 1, E 1 } and G = {V, E } be two graphs with V i ertex set and E i edge set of G i ; i = 1, We can merge one or more ertices of G 1 with G and or one or more edges of G 1 with G which are common to G 1 and G The resultant graph got so by merging common ertices

2 44 K Thulukkanam, et al (and or) common edges is a graph defined as the merged graph 1 We will illustrate this definition by an example G = Example 1: Let G 1 be the directed graph gien in Figure Figure Let M be the matrix associated with the directed graph G gien in Figure G 1 = 3 1 Figure 1 M = The matrix related with G 1 gien in Figure 1 be M 1, M 1 is as follows: M 1 = Let G be the merged graph of the directed graphs G 1 and G gien in Figure 3 got by merging the edge 1 and the ertices 1 and The merged graph G is depicted in Figure 3 in the following: G = 3 1 Let G be the directed graph which is gien in Figure 4 8 Figure 3

3 Two new Fuzzy Models---and Kosko Hamming Distance 4 Let the matrix M denote the merged matrices; M 1 and M and is the matrix associated with the merged directed graph G which is gien in Figure 3 M = Let M 1 be the matrix associated with the graph G 1, M 1 = Let G be the directed graph gien in Figure G = 3 4 This is the way two graphs are merged and the merging is unique and their related matrices are merged and matrix so obtained is defined as the merged matrix Next the notion of specially merged linked graph is described by an example Example : Let G 1, G and G 3 be the three directed graphs gien in the following figures 4, and G 1 = Figure 4 Let M be the matrix associated with the graph G, M = 3 4 Figure Let G 3 be the directed graph gien in Figure

4 4 K Thulukkanam, et al 1 G 3 = G = 3 Figure Let M 3 be the matrix associated with the graph G 3, 10 M 3 = We see the graphs G 1 and G 3 gien in Figures 4 and can not be merged as they do not hae a common ertex or edge Howeer graphs G 1 and G hae 3 to be the common ertex So G 1 and G can be merged uniquely only in one way Likewise graphs G and G 3 hae and as common ertices so they can be merged in a unique way Thus all the three graphs G 1, G and G 3 can be merged in a unique way into a single graph G gien in Figure Let M denote the specially linked merged matrix of the matrices M 1, M and M 3 which is also the matrix associated with the merged linked graph G M = Figure

5 Two new Fuzzy Models---and Kosko Hamming Distance 4 This sort of merging more than two graphs under certain constraints is known as the specially merged linked graphs In the next section how these concepts are used in the construction of merged FMs is described 3 Definition of New Merged FMs and their properties : In this section the notion of Merged Fuzzy ognitie Maps (MFMs) is introduced Merged FMs are built using the concept of merged graphs and the related merged matrices and described by an example Definition 31: Let = { 1,, n } be the n nodes associated with a real world problem Suppose t experts want to work with this problem using FMs but only using some selected nodes from the set of nodes Let the directed graphs gien by the t experts be G 1, G,, G t such that the ertex set of the graph G i with G j is non empty for i j; G i Gj ; 1 i, j t Then we can merge some graphs say k of them, k t from G 1,, G t so that the ertices of all these graphs gie all the nodes of the set Let G be the merged graph and the FMs associated with G will be known as the Merged Fuzzy ognitie Maps (MFMs) and the connection matrix associated with G will be known as the merged connection matrix of the MFMs or the merged dynamical system of the FMs We will first illustrate this situation by an example Example 31: Let us suppose we hae = { 1,, 3,, 1 } to be the set of nodes/attributes associated with a problem Let fie experts work with the problem using FMs and the nodes from the set Suppose the first expert wants to work with the set of nodes gien by X 1 where X 1 = { 1,, 3, 4, 8 } and the second expert wishes to work with the set of nodes gien by X, where X = { 3,,, 1 } The third expert works with the set of nodes gien by the subset X 3, where X 3 = { 1,, 10, 11 } Let the forth expert work with the nodes,, 1, 10, 1 gien by the set X 4 = {,, 1, 10, 1 } The fifth expert works with X = {,, 10,,, } Now we can get the merged FMs in two ways Taking the nodes X 1 X X 3 X 4 so that we get the merged graph G of the graphs G 1, G, G 3 and G 4 or X 1 X X 3 X that is we get the merged graph G' of the graphs G 1, G, G 3 and G So we get two integrated merged FMs model to work with Let us consider the directed graphs gien by the fie experts Let G 1 be the directed graph gien in Figure 31 by the first expert using the set of attributes X 1

6 48 K Thulukkanam, et al 1 Let M be the matrix associated with the graph G G 1 = M = Figure 31 Let M 1 be the matrix associated with the graph G 1 Let G 3 be the directed graph gien in Figure 33 gien by the third expert using the nodes X 3 = { 1,, 10, 11 }; M 1 = G 3 = Let G be the directed graph gien by the second expert using the attributes X = { 3,,, 1 } gien in Figure 3 G = 3 1 Figure 3 Let M 3 be the matrix associated with the graph G 3 M 3 = Figure Let G 4 be the directed graph proided

7 Two new Fuzzy Models---and Kosko Hamming Distance 4 by the fourth expert using the set of nodes X 4 = {,, 1, 1, 10 } gien in Figure 34 G 4 = 1 Let M 4 be the matrix associated with the graph G 4 M 4 = Figure Using X = {,, 10,,, }, let G be the directed associated with the fifth expert gien in Figure Let M be the matrix associated with the graph G M = Let M 1, M,, M be the matrices associated with the graphs G 1, G,, G respectiely Let the merged matrix of the matrices be M which is also the merged connection matrix M of the specially linked merged graph G gien in Figure 3 Now to get the integrated merged FMs we hae to merge the graphs G 1,G, G 3 and G 4 or G 1, G, G 3 and G So we get in total using merged FMs two integrated merged FMs using all the 1 attributes The merged graph G of the experts 1 to 4 is gien in Figure G = Figure 3 Figure 3

8 0 K Thulukkanam, et al Using this merged connection matrices M 1, M, M 3 and M 4 of graphs G 1, G, G 3 and G 4 respectiely we can study the integrated merged dynamical system of the integrated MFMs M = Now using the experts 1,, 3, and we get the merged graph G' of the four directed graphs of the FM gien in Figure Figure 3 Let the merged connection matrix of the matrices M 1, M, M 3 and M of the merged graph G' of Figure 33 be M' which is as follows: M = Thus we can find the merged connection matrix M' of the FM and study the problem The adantages of using this new merged FMs models are; 1 Experts can choose any number of attributes from the gien set of attributes so that they can be free to do work with the problem with their choice When the number of attributes is a small number; working is easy and apt 3 While getting merged model by combining all the experts opinion (so no expert is left out), eeryone is gien the same degree of importance 4 By combining them differently using different sets of experts we get seeral merged FMs for the same problem The alues in the connection merged

9 Two new Fuzzy Models---and Kosko Hamming Distance 1 integrated matrices need not be thresholded for they take only alues from the set {1, 1, 0} 4 Definition and Description of Specially Merged Linked FMs : The Specially Merged Linked Fuzzy ognitie Maps (SMLFMs) model are defined and described in the following Definition 41: Let = { 1,, n } be the n attributes of a problem with which t experts E 1, E,, E t work using FMs model Each of the experts work with X i set of attributes from the n attribute set ; that is X i, X i the subset of Now the sets X i s 1 i t are so formed such that X i X i+1, X i X j =, for j = 3, 4, t, i = 1,,, t 1 That is X 1 X, X X 3 but X X i =, for i = 4,,,, t; similarly is X 3 X 4 but X 3 X i =, for i =,,,t and so on Thus we see een when G 1, G,, G t are the directed graphs gien by the experts E 1, E,, E t using the set of nodes X 1, X,, X t respectiely we see we cannot merge any G i with G j G i can be merged with G j proided j = i +1 or i 1 (i 1) Thus we defined this special type of merging as specially merged linked graphs Let M 1, M,, M n be the n matrices associated with the directed graphs G 1, G,, G n respectiely Let M be the specially merged linked matrix associated with the specially merged linked graph We call the FMs associated with these specially merged linked graphs are defined as Specially Merged Linked Fuzzy ognitie Maps (SMLFMs) model We will illustrate this situation by an example Example 41: Let = { 1,,, 11 } be the collection of nodes associated with a problem Let three experts E 1, E and E 3 work with the same problem using the FMs model choosing a few nodes from Let the expert E 1 choose to work with the nodes X 1 = { 1,, 3,, 8 } Let the expert E work with the nodes X = { 1,,, 4, } and the expert E 3 chooses to work with the nodes X 3 = {,, 10, 11, 4 } Let G 1, G and G 3 be the directed graphs associated with the experts E 1, E and E 3 respectiely gien in Figures 41, 4 and 43 G 1 = 1 3 Figure 41 8

10 K Thulukkanam, et al is the directed graph gien in Figure 41 of the FMs gien by the first expert G = 1 4 M 3 = Figure 4 Let G be the directed graph of the FMs gien by the second expert as gien in Figure 4 Let G 3 be the directed graph gien by the third expert which is gien in the following Figure 43: be the connection matrices associated with the directed graphs G 1, G and G 3 respectiely gien in Figures 41, 4 and 43 Now we see we cannot merge G 1 and G 3 as they do not hae a common node or an edge Now we can merge only in one way G 1 with G and G with G 3 which is the specially linked merged graph G gien in the following Figure 44 G 3 = G = 8 Let M 1 = , M = and Figure Figure 44 Let M 1 denote the specially linked merged matrix of the matrices M 1, M and M 3 of the directed graphs G 1, G and G 3 Using G we can find the integrated merged linked special connection matrix M of the

11 Two new Fuzzy Models---and Kosko Hamming Distance 3 specially merged linked matrices of the matrices M 1, M and M 3 which is also the connection matrix of the specially linked graph G of the dynamical system That matrix will sere as the dynamical system of the specially merged linked FMs model gien in the following The matrix M of G is got by merging M 1, M and M 3 is as follows: M 1 = Next we proceed on to define Kosko Hamming distance function in the next section Kosko Hamming Distance Function and Kosko Hamming Weight : We know Hamming distance measures the number of coordinates a row ector 1 n differs from another 1 n row ector So in the ector space V n = {(a 1,, a n ) a i F; 1 i n} defined oer the field F we can define the Hamming distance for any x, y V n as d(x, y) = number of coordinates in which x and y differ Howeer in case of Kosko Hamming distance function we can define the distance function d k only if these conditions are satisfied 1 We need to hae basically a fuzzy model say a FMs model We need atleast two experts working with the same number of concepts/nodes using only the same type of fuzzy model with same set of attributes on the same problem say for instance a FM model 3 We can define the Kosko Hamming distance function d k on the resultant state ectors gien by two experts that is on the hidden pattern of a same initial state ector X i used by both the experts using FMs model So the concept of Kosko Hamming distance cannot be defined for any resultant state ectors but only those resultant state ectors whose initial input state ector is the same Thus Kosko Hamming distance is distinctly different from the Hamming distance This d k also measures only the number of places in which the resultant state ectors gien by two experts for the same initial state ector differs Now we proceed onto define the notion of Kosko Hamming distance Definition 1: Let some t experts E 1, E,, E t work with the same problem using the same set of attributes or nodes = { 1,,, n } using FMs model Let M 1, M,, M t be the t (n n) connection matrices of the FMs associated

12 4 K Thulukkanam, et al with the experts E 1, E,, E t respectiely Suppose X be the initial state ector for which the hidden patterns using the dynamical system M 1, M,, M t is gien by Y 1, Y,, Y t respectiely Now the Kosko Hamming distance function d k is defined as d k (Y i, Y j ) = number of coordinates in which Y i is different from Y j ; i j; i i, j t; 0 d k (Y i, Y j ) n 1 The use of this function is that it measures the closeness or the non closeness of the resultant ectors of two experts for the same initial state ector X of the problem So if the deiation, that is the alue of d k (Y i, Y j ) is ery high we can analyse the experts E i and E j separately for that initial point X as well as for the other initial points also If the functional alue d k (Y i, Y j ) is small we accept that both the experts hold the same iew for the particular initial state ector X Definition : Let the experts, nodes and connection matrices be as in definition 1 Let the initial state ector be X, Y 1,, Y t be the hidden patterns gien by the dynamical systems M 1, M,, M t respectiely The Kosko Hamming weight function is defined and denoted by w k (Y i ) = d k (Y i, X); number of coordinates in which Y i differs from X; i = 1,, t This new notion of Kosko Hamming weight function helps one in finding the impact of the particular node/nodes in X which are taken in the on state oer the dynamical system that is oer the other nodes If w k (Y i ) is ery large it implies that the node has a ery high impact on the system that is on other nodes which it has made to on state by its on state If w k (Y i ) is small it implies the node/ nodes are taken in the on state has no impact on the other nodes or equialently no impact oer the system onclusions In this chapter we hae introduced two new mathematical tools to study and analyse the FMs model, the relation between the experts opinion and the impact on one node oer other nodes in that dynamical system Both the models can gie the integrated iew of all the experts This model thus saes time This model satisfies all experts because all are gien equal status about the problem 1-13 The first tool helps in replacing the combined FMs and makes the working easy by considering less number of attributes by getting the integrated merged FMs model or special integrated linked merged FMs model The former can gie more than one FMs model so that comparison can be made, howeer the latter technique can gie only one integrated merged FMs model Finally the tool of Kosko Hamming distance function and Kosko Hamming weight function can study the influence of a node oer other nodes and the comparison of experts iew on the problem References 1 Kosko, B Fuzzy ognitie Maps, Int J

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