Practical Considerations

When t, the supp , then there will be a value of t when supp folds, it becomes multi-w-to-one-x mapping

~x

~x

Practical Considerations

Then the maximum of all candidate membership value of w is the membership of x.

Practical Considerations

If supp occupies [-1,1], x [-1,1] in the state of complete fuzziness.

~x 1

~

xx

Fuzzy Numbers

We define a normal, convex fuzzy set on a real line to be a fuzzy number

Let and be fuzzy numbers

is a real line in universe Y

* Is a set of arithmetic operations

Z = *

~I

~I

~J

~I

,,,

~I

~J

ZYX

JIJI yxZ

**

~~~~

Fuzzy Numbers

Example:

4/2.03/2.02/11/2.00/2.0

4

2.0,2.0min

3

2.0,1min,1,2.0minmax2

2.0,2.0min,1,1min,2.0,2.0minmax1

2.0,1min,1,2.0minmax

0

2.0,2.0min

22/2.01/10/2.02/2.01/10/2.011

2/2.01/10/2.01

~~~

~

Fuzzy Numbers

supp 0|~~

xxI I

supp (z) = supp * supp~I

~J

~~* JI

= I * J

(crisp intervals)!

JIJI **~~

They are intervals!

Interval analysis in arithmetic

Fuzzy Numbers

I1 = [a,b] a < bI2 = [c,d] c < d

I1 * I2 = [a,b] * [c,d]

[a b] + [c d] = [a+c b+d]

[a b] – [c d] = [a-d b-c]

[a b] [c d] = [min(ac,ad,bc,bd) max(ac,ad,bc,bd)]

[a b] ÷ [c d] = [a b] [1/d 1/c] 0 [c,d]

[a b] > 0[a b] =

[b a] > 0

I(J+K) I J + I K

Note: 0~~ AA

Fuzzy Numbers

-3[1,2] = [-6,-3]

[0,1] – [0,1] = [-1,1]

[1,3] [2,4] = [min(1.2,1.4,3.2,3.4)max(1.2,1.4,3.2,3.4)]

=[2,1.2]

[1 2] ÷ [1 2] = [1 2] [1/2 1] = [1/2 2]

If I = [1,2] J = [2,3] K = [1,4]

I (J-K) = [1,2] [-2,2] = [-4,4]

IJ – IK = [1,2][2,3] – [1,2][1,4] = [2,6] – [1,8] = [-6,5]

[-4,4] [-6,5]

Approximate Methods of Extension

When discretization of continuous-valued function, it may

have irregular and error membership values, which will be

propagated from input to output by extension principle.

To overcome the above problem, several methods are studied.

Approximate Methods of Extension

Vertex Method

Combining the -cut and standard interval analysis.

For

We can decompose A into a series of -cut and standard interval I. If f(x) is continuous and monotonic on I = [a,b] the interval representing at a particular .

B = f(I) = [min(f(a),f(b)) max(f(a),f(b))]

~~AfB

~B

Approximate Methods of Extension

If y = f(x1,x2,…,xn)

Each input variable can be described by an interval Ii

Ii = [ai bi] i = 1,2,…,n

Approximate Methods of Extension

As seen in the fig. above, the endpoint pairs of each interval intersect in the 3D space and form the vertices (corners) of the Cartesian space. The coordinates of these vertices are the values used in the vertex method when determining the output interval for each -cut. The number of vertices, N, is a quantity equal to N = 2n, where n is the number of fuzzy input variables. When the mapping y = f(x1,x2,…,xn) is continuous in the n-dimensional Cartesian region

Approximate Methods of Extension

Nj

cfcf

IIIfB

jjj

n

,...,2,1

max,min

,...,, 21

If there are extreme points

where j = 1,2,…,N and k = 1,2,…,m for m extreme points in the region.

kj

kjkj

kjEfcfEfcfB ,max,,min

,,

Approximate Methods of Extension

Example:

We wish to determine the fuzziness in the output of a simple nonlinear mapping given by the expression, y = f(x) = x(2 – x), seen in the fig 6.7a, where the fuzzy input variable, x, has the membership function shown in fig 6.7b.

Approximate Methods of Extension

We shall solve this problem using the fuzzy vertex method at three -cut levels, for = 0+,0.5,1. As seen in fig 6.7b, the intervals corresponding to these -cuts are I0 = [0.5,2], I0.5 = [0.75,1.5], I1 = [1,1] (a single point).

1,01,0,75.0max,1.0,75.0min

11211

02222

75.05.025.01

11,22,5.01

2,5.0

0

0

B

Ef

cf

cf

Ecc

I

Approximate Methods of Extension

1)12(121

1211

1,11

1,75.0

1,75.0,9375.0max,1,75.0,9375.0min

75.05.125.12

9375.075.0275.01

11,5.12,75.01

5.1,75.0

5.0

5.0

Efcfcf

Ecc

I

B

cf

cf

Ecc

I

DSW Algorithm

1. Select a , 0 < < 1

2. Find the interval(s) in the input membership function(s) corresponding to .

3. Using standard binary interval operations, compute the interval for the output membership function for the selected -cut.

4. Repeat 1 – 3 for different values of

Example:

DSW Algorithm

33,31,12,21,11,12

1,1

25.5,0625.2

25.2,5625.03,5.15.1,75.05.1,75.02

5.1,75.0

8,25.12,5.02,5.02

2,5.0

2

221

1

225.0

5.0

22

0

0

2

B

I

B

I

B

I

xxy

DSW Algorithm

Example 2:

Suppose the domain of the input variable x is changed to include negative numbers… the computations for each -cut will be as follows:

3,11,02,11,01,5.02

1,5.02

0

0

B

I

Note: The zero marked with the arrow is taken as the minimum, since (-0.5)2 > 0; because zero is contained in the interval [-0.5,1] the minimum of squares of any number in the interval will be zero.

DSW Algorithm

0,00,00,02

0,0

25.1,5.025.0,01,5.05.0,05.0,25.02

5.0,25.0

1

1

25.0

5.0

B

I

B

I

Restricted DSW Algorithm

I = [a,b]

J = [c,d] a,b,c,d > 0

No Subtraction

Then, I J = [a,b] [c,d] = [a c,b d]

I/J = [a,b] ÷ [c,d] = [a/d,b/c]

Comparisons

Comparisons

Comparisons

Comparisons

Comparisons

Comparisons

Comparisons

Comparisons

Fuzzy Vectors

Formally, a vector, , is called a fuzzy vector

if for any element we have 0 < aI < 1 for I = 1,2,…,n.

Similarly, the transpose of a fuzzy vector ,denoted by, is

a column vector if is a row vector, i.e.,

naaaa ,...,, 21~

~a Ta

~

n

T

a

a

a

a2

1

~

Fuzzy Vectors

Let us define and as fuzzy vectors of length n, and define

as the fuzzy inner product of the two fuzzy vectors and

as the fuzzy outer product of the two vectors

~a

~b

ii

n

i

T baba 1~~

ii

n

i

T baba 1~~

Fuzzy Vectors

Example:

Find the inner and outer product of the given fuzzy vectors of length 4

4.04.19.5.1.4.3.19.7.5.3.

7.01.3.7.3.1.4.3.19.7.5.3.

1.0

3.0

9.0

5.0

4.0,1,7.0,3.0

1.0,3.0,9.0,5.0

4.0,1,7.0,3.0

~~

~~

~

~

ba

ba

b

a

Note: outer product is different from the one in inner algebra.