A question about fuzzy number

[feilian]

Hi all
I have begin some study in Fuzzy regressions. I am not sure about the
meaning of fuzzy numbers.

I know that for a fuzzy set A, the member function A(x) means the
possibility x belongs to set A.
For a LR fuzzy triangle fuzzy number A=(a,s), what does this fuzzy
number mean? Does it means a interval[a-s,a+s]? And what does it's
member function u(x)=|x-a|/s mean? Does it means the possibility x
belonging to interval [a-s,a+s]?

Thanks!

[Dmitry A. Kazakov]

On Mon, 11 Aug 2008 21:32:08 -0700 (PDT), feilian wrote:

> I have begin some study in Fuzzy regressions. I am not sure about the
> meaning of fuzzy numbers.
>
> I know that for a fuzzy set A, the member function A(x) means the
> possibility x belongs to set A.

Fuzzy real number is a fuzzy subset of R.

> For a LR fuzzy triangle fuzzy number A=(a,s), what does this fuzzy
> number mean? Does it means a interval[a-s,a+s]? And what does it's
> member function u(x)=|x-a|/s mean?

It is the interpretation of a fuzzy set, If A is a fuzzy number A:R->[0,1],
then A(x) tells how much x belongs to A.

> Does it means the possibility x belonging to interval [a-s,a+s]?

These are two questions:

1. Yes, if the meaning of truth values is possibility, then, well, it is
possibility.

2. No. A triangular-shaped membership function is not rectangular
(interval).

The possibility of x being in an interval I is 1 if x in the interval, 0
otherwise. (You probably meant something else.)

A final note, the set of fuzzy numbers with rectangular membership
functions reaching 1 is fully equivalent to the classical interval
arithmetic. Fuzzy numbers of any shape can be considered a continuation of
that.

--
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de

[feilian]

Thank you .

I still have a another question:
in fuzzy linear regression model, if x is a crisp number, the
coefficient A is a fuzzy number A=(c,s);
then y would be y=(yc,ys)=(cx,s|x|). In this model, the spread of y(x)
and y(-x) is the same.It would
be not suitable for some case.

Then the question is : can I define yc and ys separately:
yc=fc(x);
ys=fs(x);
subject to :
fs(x)>=0 , for every x of interest

Thanks again.