PART 7 Constructing Fuzzy Sets 1. Direct/one-expert 2. Direct/multi-expert 3. Indirect/one-expert 4. Indirect/multi-expert 5. Construction from samples

PART 7Constructing Fuzzy Sets

1. Direct/one-expert2. Direct/multi-expert3. Indirect/one-expert4. Indirect/multi-expert5. Construction from samples

FUZZY SETS AND

FUZZY LOGICTheory and Applications

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Direct/one-expert

• An expert is expected to assign to each given element x a membership grade A(x) that, according to his or her opinion, best captures the meaning of the linguistic term represented by the fuzzy set A.

It can be done by either 1. defining the membership function completely in

terms of a justifiable mathematical formula,

2. exemplifying it for some selected elements of X.

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Direct/multi-expert

• When a direct method is extended from one expert to multiple experts, the opinions of individual experts must be appropriately aggregated.

One of the most common methods is based on a probabilistic interpretation of membership functions.

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Direct/multi-expert

"x belongs to A" is either true or false, where A is a fuzzy set on X that represent a linguistic term associated with a given linguistic variable.

let ai (x) denote the answer of expert i. Assume that ai (x) = 1 when the proposition is valued by expert i as true, and ai (x) = 0 when it is valued as false.

n: number of expects )( ni N

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Direct/multi-expert

Generalize the interpretation A(x) by allowing one to distinguish degrees of competence, ci, of the individual experts.

where

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Direct/multi-expert

Let A and B are two fuzzy sets, defined on the same universal set X.

We can calculate A(x) and B(x) for each x X, and then choose appropriate fuzzy operators to calculate , , A U B, A ∩ B, and so forth.

Let

A B

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Direct/multi-expert

and

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Indirect/one-expert

• Given a linguistic term in a particular context, let A denote a fuzzy set that is supposed to capture the meaning of this term.

Let x1, …,n be elements of the universal set X for which we want to estimate the grades of membership in A.

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Indirect/one-expert

Our problem is to determine the values ai = A(xi). Instead of asking the expert to estimate values ai directly.

We ask him or her to compare elements x1, …,n in pairs according to their relative weights of belonging to A.

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Indirect/one-expert

• pairwise comparisons A square matrix P = [pij ], i,j Nn, which has positive

entries everywhere.

Assume first that it is possible to obtain perfect values pij . In this case, pij = ai /aj ; and matrix P is consistent in the sense that

for all i, j, k Nn, which implies that pii = 1 and pij = 1/ pji.

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Indirect/one-expert

Furthermore,

for all i Nn or, in matrix form,

where

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Indirect/one-expert

Pa = na means that n is an eigenvalue of P and a is the corresponding eigenvector. It also can be rewritten in the form

where I is the identity matrix.

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Indirect/one-expert

If we assume that

then aj for any j Nn can be determined by the

following simple procedure:

hence,

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Indirect/one-expert

The problem of estimating vector a from matrix P now becomes the problem of finding the largest eigenvalue λmax and the associated eigenvector. That is, the estimated vector a must satisfy the equation

Pa = λmax a,

where λmax is usually close to n.

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Indirect/multi-expert

• Let us illustrate methods in this category by describing an interesting method, which enables us to determine degrees of competence of participating experts.

It is based on the assumption that, in general, the concept in question is n-dimensional (based on n distinct features), each defined on R. Hence, the universal set on which the concept is defined is Rn.

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The full opinion of expert i regarding the relevance of elements (n-tuples) of Rn to the concept is expressed by the hyperparallelepiped

Where, denote the interval of values of feature ; that, in the opinion of expert i, relate to the concept in question (i Nm, j Nn).

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We obtain m hyperparallelepipeds of this form for m experts.

Membership function of the fuzzy set by which the concept is to be represented is then constructed by the following algorithmic procedure:

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Construction from samples

• Lagrange Interpretation

A curve-fitting method in which the constructed function is assumed to be expressed by a suitable polynomial form.

The function f employed for the interpolation of given sample data <xi, ai> for all x R has the form

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for all i Nn. Since values f (x) need not be in [0,1] for some x R, function f cannot be directly considered as the sought membership function A. We may convert f to A for each x by the formula

1 1 1

1 1 1

( )...( )( )...( )( )

( )...( )( )...( )i i n

ii i i i i i n

x x x x x x x aL x

x x x x x x x a

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An advantage of this method is that the membership function matches the sample data exactly.

Its disadvantage is that the complexity of the resulting function (expressed by the degree of the polynomial involved) increases with the number of data samples.

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• Least-square curve fitting

The method of least-square curve fitting selects that function f (x: α0, β0, • • •) from the class for which

reaches its minimum. Then,

for all x R.28

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An example of the class of bell-shaped functions is frequently used for this purpose.

where α controls the position of the center of the bell, (β / 2 )-2 defines the inflection points, and γ control the height of the bell (Fig. 10.4a).

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Given sample data <xi, ai>, we determine (by any effective optimization method) values α0, β0, γ0 of parameters α, β, γ, respectively, for which

reaches its minimum. Then, according to A(x), the bell-shape membership function A that best conforms to the sample data is given by the formula

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Another class of functions that is frequently used for representing linguistic terms is the class of trapezoidal-shaped functions,

The meaning of the five parameters is illustrated in Fig. 10.4b.

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Construction from samples• Neural networks

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Following the backpropagation learning algorithm, we first initialize the weights in the network. This means that we assign a small random number to each weight. Then, we apply pairs 〈 xp, tp 〉 of the training set

to the learning algorithm in some order.

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For each xp, we calculate the actual output yp and calculate the square error

Using Ep, we update the weights in the network according to the backpropagation algorithm described in Appendix A. We also calculate a cumulative cycle error,

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At the end, we compare the cumulative error with the largest acceptable error, Emax, specified by the user.

– E ≦ Emax : the neural network represents the desired membership function.

– E ＞ Emax : we initiate a new cycle.

The algorithm is terminated when either we obtain a solution or the number of cycles exceeds a number specified by the user.

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Exercise 7

• 7.1

• 7.2

• 7.3

• 7.4

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PART 7 Constructing Fuzzy Sets 1. Direct/one-expert 2. Direct/multi-expert 3. Indirect/one-expert 4. Indirect/multi-expert 5. Construction from samples