Theory of Fuzzy Sets-2
There's more to fuzzy theory than meets the eye, or doesn't meet the eye.
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Part 1 of the Theory of Fuzzy Sets deals with fuzzy extensions of ordinary set operations and includes fuzzy union, intersection, and complement of sets. When working with fuzzy sets, however, additional operations can be defined. These operations are uniquely fuzzy operations and have no counterpart in ordinary set theory.
The operations, as discussed by Kurt J Schmucker, are:
Concentration
is an operation that reduces even further the degree of membership of already peripheral members of a set and thus takes them well out of the picture, in effect concentrating on the central characters.Dilation
has the opposite effect of concentration in that it heightens the degree of membership of the peripherals and brings them more into the picture.Normalization
changes the degree of set membership of elements across the board in order to raise the membership status of at least one of them to the maximum of 1, rendering the element(s) totally in the set.Intensification
increases the degree of membership of all the elements of a set that are already half members or better and decreases the degree of those members less than half in the set. That is, it makes a fuzzy set less fuzzy.Fuzzification
operates in reverse of intensification and makes the set more fuzzy.-----------------------------------------------
The purpose of the concentration operator is to compress the members of a fuzzy set that are already practically full members and to separate further away the members that are already pretty much out of it -- strengthening the strong members and weakening the weak members. . This means increasing the degree of membership for the top bananas and decreasing the degree for the untouchables. You can do this by squaring the degree of membership of each member in the set.
Note that degree ranges from 0 on the low end to 1 on the high end. So when you square a degree it stays in the same range. But more importantly, if the degree is either 1 or 0, it remains the same, and if it lies in between 1 and 0 it gets smaller. That is:
x2 < x for 0 < x < 1.
For example, (1/2)2 = 1/4 < 1/2.
Now, higher degrees decrease less than lower degree. For instance, .7 goes to .49, when squared, so it drops less than 30%. But .2 goes to .04, which is 80% less than .2. Squaring pushes the lower members practically out of the set.
The concentrated form of fuzzy set, A, is thus given as:
CON(A) = {a2(x)/x | x is an element of U}
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The procedure for dilation is the same as for concentration, except it inverts the order. Instead of squaring the degree of membership for each member, you take the positive square root. This goes in the opposite direction from concentration. If you start with 1/2 and square it, you get 1/4. Start with 1/4, take the positive square root, you get 1/2.
For all values except 1 and 0, which remain the same in the operation, the result of taking the square root is to increase the degree of membership for all members. The positive square root of .01 is .1. The positive square root of 1/9 is 1/3. The positive square root of .25 is .5. And so on. Each positive square root is bigger than the original number.
The dilation set of a set, A, is:
DIL(A) = {SQRT(a(x))/x | x is an element of U}
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Suppose you have two sets of numbers that you'd like to compare but they are expressed in different scales so a direct comparison is difficult. One thing you can do is normalize the sets. This process reduces the sets to the same standard. It standardizes them. Now you can compare them more directly. But what is the process?
Say you have a number, a, that's less than 1. If you divide that number by itself you get a/a for your effort, and this is simply the value, 1. OK? Similarly, if you divide each member of a set by the largest number in the set, you get the value 1 as the new largest number in the set -- the maximum. So, for example, if you have the set:
{3, .6, and 1.2}
you can divide through by 3 and get the set:
{1, .2, .4).
In just this way, if you divide the degree of membership of the elements of a fuzzy set by the maximum -- the largest -- of all the degrees of membership in the set, you normalize the set. If the maximum degree of membership already happens to be 1, you of course change nothing by the division. Doing this to both of your sets thus puts them in the same scale for comparison. The formula for normalization is:
NORM(A) = {a(x)/m(x))/x | x is an element of U}
where m(x) = max{a(x)} for all x in U.
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Intensification combines concentration and dilation and raises the degree of membership of some elements of a fuzzy set and lowers the degree of membership of other elements of the set. In particular it raises the degree of membership of elements already having a degree of membership greater than .5, and it lowers the degree of membership of elements already having a degree of membership less than .5. In other words it heightens the contrast between the "haves" and the "have nots."
The intensification formula is:
INT(A) = {m(x)/x | x is an element of U}
where
m(x) = 2a2(x) for a(x) for x greater than or equal to 0 and less or equal to .5
m(x) = 1 - 2(1 - a(x))2 for x greater than .5 and less than or equal to 1.0
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Fuzzification of a fuzzy set, A, produces the fuzzy union of certain subsets of A. The result is a new set made up of possibly different elements of U, each of which may have a new degree of membership in the fuzzified A.
Each one of the subsets of A has selected elements from U, and each element in each subset has its degree of membership in the subset. The membership and the corresponding degree of membership in each subset are determined by the product of the degree of membership of the element in A and a function, K(x), which thereby acquires special importance. The formula representing this operation is:
Fuzzification of A = È {a(x)*K(x)}
Note that È is the fuzzy union, a(x) is a real number, and K(x) is a set. The rule for the multiplication, *, is for a(x) to form the product of the degree of membership of each x in A with the degree of membership of each occurrence of x in the K set. The idea is to fuzzify even further each fuzzy element of each particular subset.
Using an example from Zadeh, if U is the set:
U = {1, 2, 3, 4}
and the original fuzzy set is:
A = {.8/1, .6/2}
we can get another specific subset by applying a particular function, K(x). For example, K(1) maps the element 1 into A as follows:
K(1) = {1/1, .4/2}
This says that K maps 1 into 1 with degree of membership 1, and it also maps 1 into 2 with degree of membership .4.
Using the real number/set multiplication, *, we can multiply the degree of membership of the elements in A by the degree of membership of the elements of the subset K(1). That is,
a(1)*K(1) = .8{1/1, .4/2} = {.8/1, .32/2}
We can do the same thing with:
K(2) = {.4/1, 1/2, .4/3}
which maps the element, 2, into 1, 2, and 3. The result is:
a(2)*K(2) = .6{.4/1, 1/2, .4/3} = {.24/1, .6/2, .24/3}
Combining the two sets using fuzzy union, we get:
Fuzzified A = {.8/1, .6/2, .24/3}
Note that the degree of membership of 2 is determined by taking the larger of the two degrees of membership of 2 -- i.e., the maximum.
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