Crisp description of fuzzy sets

Crisp description of fuzzy sets

Post by Andrzej Pownu » Sun, 10 Aug 2003 05:20:18



Dear fuzzy specialists,

On the UAI mailing list (http://www.auai.org)
there was discussion about fuzzy sets theory.
One of the participants in this discussion was prof. Zadeh.

This is a copy of one of his e-mails (2003-07-31).

*******************************************************
Dear Andrzej

Your examples illustrate the point I made in my message(7-l6-03),
namely, that standard probability theory, PT, does not address problems
in which, as in your examples, we encounter partiality of truth and/or
partiality of possibility. Thus, in the proposition, "Robert is half-
German, quarter- French and quarter- Italian," the numbers 0.5, 0.25 and
0.25 are not probabilities but grades of membership or, equivalently,
truth values.

Cordially yours,

Lotfi
*******************************************************

In prof. Zadeh's example the grade of membership can be calculated quite
precisely.
(for example the numbers 0.5, 0.25 has very clear interpretation).
Additionally he accepts examples which were presented by me
as more or less relevant to the problem.

I created extended summary of the conclusion from this discussion.

http://zeus.polsl.gliwice.pl/~pownuk/uncertainty/Crisp_description_of...

or

http://zeus.polsl.gliwice.pl/~pownuk/uncertainty/Crisp_description_of...

I will be very grateful for any comments.

Regards,

Andrzej Pownuk

-----------------------------------
 Ph.D., research associate at:
 Chair of Theoretical Mechanics
 Faculty of Civil Engineering
 Silesian University of Technology
 URL: http://zeus.polsl.gliwice.pl/~pownuk

-----------------------------------

 
 
 

Crisp description of fuzzy sets

Post by Dmitry A. Kazako » Wed, 13 Aug 2003 17:50:53


On Fri, 8 Aug 2003 22:20:18 +0200, "Andrzej Pownuk"


>Dear fuzzy specialists,

>On the UAI mailing list (http://www.auai.org)
>there was discussion about fuzzy sets theory.
>One of the participants in this discussion was prof. Zadeh.

>This is a copy of one of his e-mails (2003-07-31).

>*******************************************************
>Dear Andrzej

>Your examples illustrate the point I made in my message(7-l6-03),
>namely, that standard probability theory, PT, does not address problems
>in which, as in your examples, we encounter partiality of truth and/or
>partiality of possibility. Thus, in the proposition, "Robert is half-
>German, quarter- French and quarter- Italian," the numbers 0.5, 0.25 and
>0.25 are not probabilities but grades of membership or, equivalently,
>truth values.

>Cordially yours,

>Lotfi
>*******************************************************

>In prof. Zadeh's example the grade of membership can be calculated quite
>precisely.
>(for example the numbers 0.5, 0.25 has very clear interpretation).
>Additionally he accepts examples which were presented by me
>as more or less relevant to the problem.

It is difficult to comment. So let me give another example:

The house I see from my window is 0.1 tall as Empire State Building,
50.0 wide as my car, its color is 200 red and 80 percent covered with
graffiti by some bastards. Definitely these numbers are not
probabilities etc.

Note that Prof. Zadeh's carefully avoids to call these arbitrary
numbers possibilities. They are neither possibilities nor
probabilities, but they could be under some conditions. So Prof. Zadeh
is right, but you overstretch his arguments to get more from that.

Quote:>I created extended summary of the conclusion from this discussion.

>http://zeus.polsl.gliwice.pl/~pownuk/uncertainty/Crisp_description_of...

>or

>http://zeus.polsl.gliwice.pl/~pownuk/uncertainty/Crisp_description_of...

>I will be very grateful for any comments.

You are trying to define some additive measure which will be treated
as the values of a membership function. So far so good.

The first objection is. How it differs from what the probability
theory does. It also defines an additive measure, and this is where
the problems of the probability theory start.

Note that the probability theory is a well established one. Its
additive measure is additive because there is a set of elementary
outcomes which are independent and incompatible. Which is not a fact,
but a premise on which the whole building stands. Remove that premise
and it would collapse. On the contrary, you just claim that something
of unknown origin can be added and *is* a membership grade. The
question is why? Is anything that can be added a membership grade? How
to mix grades of different origin not knowing what they are? Isn't it
apples and oranges hidden under a cover of nice words?

Then observe that the probability theory does not claim that the
probability of, for instance, hitting a mark *is* a relation of the
areas of the mark and the rest. It can be *numerically* equal to the
relation under *definite* curcumstances. Probability is neither an
area nor a relation of areas. Why do you think that fuzzy membership
grade should be one?

Even if you define some additive measure correctly, then the most
interesting question remains. What is the gain? An additive measure is
inherently incompatible with set operations defined in terms of
membership functions. In the example from your paper F U F will not be
equal to F. It contain "Robert" with 1, while F does it with 0.5!

---
Regards,
Dmitry Kazakov
www.dmitry-kazakov.de

 
 
 

Crisp description of fuzzy sets

Post by Andrzej Pownu » Wed, 13 Aug 2003 22:42:36


Dear Dmitry,

> >*******************************************************
> >Dear Andrzej

> >Your examples illustrate the point I made in my message(7-l6-03),
> >namely, that standard probability theory, PT, does not address problems
> >in which, as in your examples, we encounter partiality of truth and/or
> >partiality of possibility. Thus, in the proposition, "Robert is half-
> >German, quarter- French and quarter- Italian," the numbers 0.5, 0.25 and
> >0.25 are not probabilities but grades of membership or, equivalently,
> >truth values.

> >Cordially yours,

> >Lotfi
> >*******************************************************

> It is difficult to comment. So let me give another example:

> The house I see from my window is 0.1 tall as Empire State Building,
> 50.0 wide as my car, its color is 200 red and 80 percent covered with
> graffiti by some bastards. Definitely these numbers are not
> probabilities etc.

I agree with that.
This is not probability.

Probability theory is based on "yes" or "no" questions
because of that this "fuzzy" description is beyond
of the definition of the probability theory.
That was my final conclusion in the discussion with Bayesian specialists
and they do not give me any argument that my way of thinking is wrong.
Of course they don't agree with that :)

> Note that Prof. Zadeh's carefully avoids to call these arbitrary
> numbers possibilities. They are neither possibilities nor
> probabilities, but they could be under some conditions. So Prof. Zadeh
> is right, but you overstretch his arguments to get more from that.

Well, maybe .

> You are trying to define some additive measure which will be treated
> as the values of a membership function. So far so good.

I am not sure that these measures are always so additive.
In a real life we always use some measures, levels etc.
I am not sure that these descriptions are always additive.
***********************************
There is only one requirement.
The decryption should be compatible
with the phenomena which is just described.
***********************************
How to do that?
It depends on the application.

That is why I am so skeptical about
t-norms and existing fuzzy logic.
In existing fuzzy logic we can do the same operations
no matter what is going on.
Well if the reality is like that, then no problems.
But I am not sure that it is so simple.

> The first objection is. How it differs from what the probability
> theory does. It also defines an additive measure, and this is where
> the problems of the probability theory start.

If we consider the example of Robert (who know 50% of answers) the question
"Does Robert know passwords?" is ambiguous.
It is not convenient to answer to that question "yes" or "no".
We need another answer.
However in traditional sense there is no another answer.
Maybe I am wrong
but I think that because of that it is not possible
to find the answer to that question by using probability theory.
The answer "Robert know 50% of question" is precise
but this is not the answer in "classical" sense
(because the answer in classical sense doesn't exists).

This is the main difference between the "fuzzy" theory
and the probability theory.

Probability theory cannot find the answers to the ambiguous questions.
Probability theory need "yes" or "no" answers.

> Note that the probability theory is a well established one. Its
> additive measure is additive because there is a set of elementary
> outcomes which are independent and incompatible. Which is not a fact,
> but a premise on which the whole building stands. Remove that premise
> and it would collapse.

OK.
My description is not related to crisp problems.
In probability theory there is no such things the "half events"
of "quarter events".
My way of thinking is in completely other direction.
There are no events in my examples. No randomness etc.

> On the contrary, you just claim that something
> of unknown origin can be added and *is* a membership grade.

Well, I don't know what this thing really is.

> The question is why?

As far as I know the definition of membership function is completely
subjective.
We don't know how works our brain then the definition
of the membership function actually doesn't exists .

> Is anything that can be added a membership grade?

The meaning of grade of something depends on the problem.
Additivity is completely not necessary to that problem.
What is necessary?
Difficult to say in general.
Everything depends on the problem.
The decryption should be compatible

with the phenomena which is just described.
There is only one condition that have to be satisfied.

Grade of temperature grade at school, grade of height.

I have example of grade which is probably not additive measure.
This example is a little drastic
(I am not going to hurt anybody).
During the second word war in Germany there were grade of  "Germans".
Each person in Germany has some "grade".
Some persons are full Germans other has only different degree of
"Germnannes".
I cannot you give much details how these degree were calculated but
as far as I know they have precise meaning
(and they are not related to probability).

> How to mix grades of different origin not knowing what they are? Isn't it
> apples and oranges hidden under a cover of nice words?

Excellent questions.
Tell me that.

The only thing that I know at this moment
is that we use different grade in everyday life very often.
There is nothing extraordinary in that phenomenon.
How to crate general theory of grades?
I don't know unfortunately.

Fuzzy set theory is trying to do that.
However I see some weakness in this theory and this is
the reason why I wrote this e-mail.

> Then observe that the probability theory does not claim that the
> probability of, for instance, hitting a mark *is* a relation of the
> areas of the mark and the rest. It can be *numerically* equal to the
> relation under *definite* circumstances. Probability is neither an
> area nor a relation of areas. Why do you think that fuzzy membership
> grade should be one?

What fuzzy membership should be?
According to prof. Zadeh:
""approximately a" given u, where u is a real number. In the fuzzy set
interpretation, the grade of membership of u in the fuzzy set would be an
answer to the question:  On the scale  from 0 to l, what is the degree to
which u fits your perception of "approximately a."

The problem is that this is a "magic" definition.
In my opinion in some cases we can train our brain
in order to accept some ideas.
We don't have to use some undefined statement like
"subjective answer to the question on the scale from 0 to 1".

In some cases this scale is very crisp.
There is no need to use subjective answers.
The number of passwords is very precise.
However the question "Does Robert know passwords?" is ambiguous.

The amount of water in a glass is also very precise.
However the question "Is the glass full?" is also ambiguous
(if the glass contain 80% of water).

The source of that kind of ambiguity cannot be removed.
One word cannot describe precisely the infinite number
of different amounts of water in the glass.

In the case of word "full"
(which is related to the amount of water in the glass)
we can describe this problem by using percent.

percent of water = (volume of water)*100/(volume of glass)

The problem is whiter we accept this scale
as a appropriate scale in the answer to the question:
"Is this glass full?".
As far as I know this is a problem of agreement.
In other words, people define the meaning of the words.
As far as I know this is quite acceptable solution.

We know "what we are talking about"
only in that cases when we know the definition.

The basic question is:

*************************************

How well particular object "fit"

to the word which describes that object?

*************************************

Because the words are discrete

and the world is continuous (sometimes) and very complicated

then the word description will be always imprecise.

However the process of that fitting

can be described sometimes quite precisely.

Additionally we can make some assumptions about that.

The process of giving names is based on some assumption.

We can also make some assumptions about degree

which describe the process of "fitting the words to some real objects".

This is also one more phenomena that need a name and description.

The name that problem is "fuzzy sets theory" - quite good name .

If we are talking about how to describe the process of

"fitting the words to some real phenomena" .

This is more complicated problem.

My mail is only one more attempt to clear the current situation .

Let us consider another example.
Let us consider a car without three wheels, engine
and with broken windows.

Is this a car?
Both answers "yes" and "no" are not appropriate.
Well we can say this is a car with degree 0.7.
However what that number mean?
We don't know.
This is something that generate our brain but this number
has no clear interpretation.
However we can say:
this is a car without three wheels, engine
and with broken windows.
In this description I applied some vector of degrees.

By the way, is the word "broken" additive measure of the engine?

Additionally in classical fuzzy logic there are no vector-valued membership
functions.
In a real life there are many problems which need more precise decryption
than just one number.

As far as I know people this kind of description very often
in the case when we have to find an answers to ambiguous questions.

Of course there are other problems.
Is that picture beautiful?
I have no idea how to create some objective measure of "beauty".
In such cases we have no choice.
We have to use some subjective degree.

However in many other cases the degree has very clear interpretations.

> Even if you define some additive measure correctly, then the most
> interesting question remains. What is the gain?

This is already done.
There are ...

read more »

 
 
 

Crisp description of fuzzy sets

Post by Dmitry A. Kazako » Thu, 14 Aug 2003 04:34:13


Dear Andrzej,

You catched me (:-)), so I have to explain my view on the subject. Maybe you
will find it conroversial, but anyway ...

Andrzej Pownuk wrote:
> Probability theory is based on "yes" or "no" questions
> because of that this "fuzzy" description is beyond
> of the definition of the probability theory.

Hmm, the probability of a set is in [0,1]. So it is not just about "yes" and
"no".

>> You are trying to define some additive measure which will be treated
>> as the values of a membership function. So far so good.

> I am not sure that these measures are always so additive.

But all your examples are. When you are saying that "the bottle is
half-full" <=> "50% of its volume is filled" <=> "it is full to 0.5
degree". This implicitly assumes that the degree is a volume rate which is
additive. It is a fallacy. Of course you can define a degree as a volume
rate, but in this case it will not be applicable to cars.

There is a direct analogy with the probability theory. A random variable is
by no means a probability. This is also true for the possibility theory.
Compare:

             Probability   Possibility
             theory        theory / classical fuzzy sets

Measure of   Pr(S)         Pos(S)
some set     (probability) (possibility)

Measurable   (random       (some free
function     variable)     function)

Now in the case of a bottle, 50% is not a measure, it is a measurable
function. If F is the set of full bottles, and H is the given bottle, which
is 50% full, you cannot just claim that Pr(F|H) = 0.5. It is a clear
fallacy. Similarly you cannot claim that Pos(F|H) = 0.5. Or WHATEVER
measure m() you take.

What you should do, is to define a set of all bottles for which the function
V gives the answer 50%. Then you can probably measure that set using m()
m({x|V(x)=50%}). You can define and measure the set of full bottles. Then
you can measure an intersection of both sets. These three values being
mixed, might constitute a conditional measure m(F|H), which then is a sort
of degree to which H "is in" F.

> In a real life we always use some measures, levels etc.
> I am not sure that these descriptions are always additive.

The measure above has other meaning than physical measures you refer to.
Those are measurable functions, usually random variables. You might have an
unlimited number of functions, but only one measure. Otherwise, it will
fall apart.

> ***********************************
> There is only one requirement.
> The decryption should be compatible
> with the phenomena which is just described.
> ***********************************
> How to do that?
> It depends on the application.

There is only a very limited number of "interesting" measures. Then they
(probablity, possibility) are universally applicable. It is same as integer
numbers. Their algebra does not depend on the application. It would be
silly to question whether 2+2 be 4, at 1 Apr 2005, 10:00 afternoon.

> That is why I am so skeptical about
> t-norms and existing fuzzy logic.

Me too. The problem is that people are trying to substitute measurable
functions for a measure. This trick won't work.

>> The first objection is. How it differs from what the probability
>> theory does. It also defines an additive measure, and this is where
>> the problems of the probability theory start.

> If we consider the example of Robert (who know 50% of answers) the
> question "Does Robert know passwords?" is ambiguous.
> It is not convenient to answer to that question "yes" or "no".
> We need another answer.
> However in traditional sense there is no another answer.
> Maybe I am wrong
> but I think that because of that it is not possible
> to find the answer to that question by using probability theory.
> The answer "Robert know 50% of question" is precise
> but this is not the answer in "classical" sense
> (because the answer in classical sense doesn't exists).

It is pretty easy to answer if you formulate it properly. For instance, what
is the probability that a respondent would know all passwords provided that
it is Robert. The answer is 0. But should you ask, what is the probability
that Robert would know one randomly chosen password, you will become your
magical number 0.5.

> This is the main difference between the "fuzzy" theory
> and the probability theory.

There is no difference because the possiblity of that is 0. And again,
playing with questions I can supply you with any desired number.

> Probability theory cannot find the answers to the ambiguous questions.

No theory can. Ambiguios questions are ambiguios, if you mean illegal
questions.

>> Note that the probability theory is a well established one. Its
>> additive measure is additive because there is a set of elementary
>> outcomes which are independent and incompatible. Which is not a fact,
>> but a premise on which the whole building stands. Remove that premise
>> and it would collapse.

> OK.
> My description is not related to crisp problems.
> In probability theory there is no such things the "half events"
> of "quarter events".

It is not a problem. You can split an elementary event into two. Why not?

> My way of thinking is in completely other direction.
> There are no events in my examples. No randomness etc.

There must be some starting point to apply a measure to the things of the
real world.

>> The question is why?

> As far as I know the definition of membership function is completely
> subjective.

No more than arithmetics of natural numbers.

> We don't know how works our brain then the definition
> of the membership function actually doesn't exists .

As well as the natural numbers.

>> How to mix grades of different origin not knowing what they are? Isn't it
>> apples and oranges hidden under a cover of nice words?

> Excellent questions.
> Tell me that.

You define a set of events, a measure and all sorts of measurable functions
representing the grades. See above.

>> Then observe that the probability theory does not claim that the
>> probability of, for instance, hitting a mark *is* a relation of the
>> areas of the mark and the rest. It can be *numerically* equal to the
>> relation under *definite* circumstances. Probability is neither an
>> area nor a relation of areas. Why do you think that fuzzy membership
>> grade should be one?

> What fuzzy membership should be?
> According to prof. Zadeh:
> ""approximately a" given u, where u is a real number. In the fuzzy set
> interpretation, the grade of membership of u in the fuzzy set would be an
> answer to the question:  On the scale  from 0 to l, what is the degree to
> which u fits your perception of "approximately a."

> The problem is that this is a "magic" definition.

There HAS to be a magic in. It is all fuzziness and randomnes are about.
Both theories describe UNCERTAINTY in a process of obtaining data. You
CANNOT look into the box, whether it be a human brain or an electron. Yet,
you can describe, predict, communicate etc with the box. Open it and you
will need no theory of uncertainty anymore.

> The basic question is:
> *************************************
> How well particular object "fit"
> to the word which describes that object?
> *************************************
> Because the words are discrete
> and the world is continuous (sometimes) and very complicated
> then the word description will be always imprecise.

> However the process of that fitting
> can be described sometimes quite precisely.
> Additionally we can make some assumptions about that.
> The process of giving names is based on some assumption.
> We can also make some assumptions about degree
> which describe the process of "fitting the words to some real objects".

> This is also one more phenomena that need a name and description.
> The name that problem is "fuzzy sets theory" - quite good name .
> If we are talking about how to describe the process of
> "fitting the words to some real phenomena" .
> This is more complicated problem.
> My mail is only one more attempt to clear the current situation .

Well, I do not think comp.ai.fuzzy is a right place for philosophical
excercises. I feel myself a bit unprepared to challenge that. However, the
word Pi very precisely describes a number, which cannot be written down.
But, better ask philosophers.

> Let us consider another example.
> Let us consider a car without three wheels, engine
> and with broken windows.

> Is this a car?
> Both answers "yes" and "no" are not appropriate.
> Well we can say this is a car with degree 0.7.
> However what that number mean?

It means nothing as long as there is no model of an experiment which would
produce this number. You have to build a set of lingustics variables like
"car", "truck", "donkey" etc.

> We don't know.
> This is something that generate our brain but this number
> has no clear interpretation.
> However we can say:
> this is a car without three wheels, engine
> and with broken windows.
> In this description I applied some vector of degrees.

This is another model. You have [fuzzy] features (=measurable functions)
such as "number of weels", "state of engine" etc. You have a set of
classes: "car", "not-car". And you want to classify some given object after
measuring its features. That is to build conditional measures
m(car|object). AKA Bayesian classifier if m()=Pr(), fuzzy subset if
m()=Pos() etc.

> By the way, is the word "broken" additive measure of the engine?

No. It is a measurable function.

> Additionally in classical fuzzy logic there are no vector-valued
> membership functions.

Because there is no need in that.

> Of course there are other problems.
> Is that picture beautiful?
> I have no idea how to create some objective measure of "beauty".

Any physical measure is as objective as the instrument you hold in hand. If
you measure something using opinions of human beings, then here you are.

- Show quoted text -

>> An additive measure is
>> inherently incompatible with set operations defined in terms of
>> membership

...

read more »

 
 
 

Crisp description of fuzzy sets

Post by Phil Diamo » Thu, 14 Aug 2003 10:10:07


You can also have the best of both worlds by having fuzzy-set-valued
random variables. It is not easy to apply this theory to practical
applications, but there has been several to estimation of mineral
deposits. This theory closely follows G. Matheron's work on random sets
and kriging.

Cordially,  Phil

#######################################################################

Department of Mathematics, University of Queensland, Brisbane,AUSTRALIA
4072.


>Dear Dmitry,
>> >*******************************************************
>> >Dear Andrzej

>> >Your examples illustrate the point I made in my message(7-l6-03),
>> >namely, that standard probability theory, PT, does not address problems
>> >in which, as in your examples, we encounter partiality of truth and/or
>> >partiality of possibility. Thus, in the proposition, "Robert is half-
>> >German, quarter- French and quarter- Italian," the numbers 0.5, 0.25 and
>> >0.25 are not probabilities but grades of membership or, equivalently,
>> >truth values.

>> >Cordially yours,

>> >Lotfi
>> >*******************************************************

>> It is difficult to comment. So let me give another example:

>> The house I see from my window is 0.1 tall as Empire State Building,
>> 50.0 wide as my car, its color is 200 red and 80 percent covered with
>> graffiti by some bastards. Definitely these numbers are not
>> probabilities etc.

--
###############################################################################

Department of Mathematics, University of Queensland, Brisbane,AUSTRALIA 4072.
Tel +61 7 3365 3253 Fax +61 7 3365 1477
 
 
 

Crisp description of fuzzy sets

Post by Andrzej Pownu » Thu, 14 Aug 2003 20:04:46


Dear Dmitry,

> When you are saying that "the bottle is
> half-full" <=> "50% of its volume is filled" <=> "it is full to 0.5
> degree". This implicitly assumes that the degree is a volume rate which is
> additive. It is a fallacy. Of course you can define a degree as a volume

rate

Then what is wrong?

> but in this case it will not be applicable to cars.

Because in order to describe cars we have to apply another functions.
We have to measure temperature in degree Celsius,
and volume in m^3.
I don't think that we can measure everything by using one universal way.

> There is a direct analogy with the probability theory. A random variable
is
> by no means a probability. This is also true for the possibility theory.
> Compare:

>              Probability   Possibility
>              theory        theory / classical fuzzy sets

> Measure of   Pr(S)         Pos(S)
> some set     (probability) (possibility)

> Measurable   (random       (some free
> function     variable)     function)

There is a good Paper about that:

Gert de Cooman, 1995,
The formal analogy between possibility and probability theory.
http://www.mat.univie.ac.at/~andrzej/papers/measpos.pdf

but I am not sure that this is important here.

> Now in the case of a bottle, 50% is not a measure, it is a measurable
> function.

OK. You are right.
However I am not very interested in the axioms at this moment.
I just would like to describe the phenomena (i.e. fuzziness) correctly.

> If F is the set of full bottles, and H is the given bottle, which
> is 50% full, you cannot just claim that Pr(F|H) = 0.5. It is a clear
> fallacy. Similarly you cannot claim that Pos(F|H) = 0.5. Or WHATEVER
> measure m() you take.

> What you should do, is to define a set of all bottles for which the
function
> V gives the answer 50%. Then you can probably measure that set using m()
> m({x|V(x)=50%}). You can define and measure the set of full bottles. Then
> you can measure an intersection of both sets. These three values being
> mixed, might constitute a conditional measure m(F|H), which then is a sort
> of degree to which H "is in" F.

If in your opinion this is a good description of the "full bottle" then
grate.
However in real life I suspect that
you are just using a percent in order to describe
what is going on with the water in bottle.

Your method is a little too complicated but you can use it.
We live in a free world (I hope .).

> > In a real life we always use some measures, levels etc.
> > I am not sure that these descriptions are always additive.

> The measure above has other meaning than physical measures you refer to.
> Those are measurable functions, usually random variables. You might have
an
> unlimited number of functions, but only one measure. Otherwise, it will
> fall apart.

What is so wrong in using different way of description in different
situations?
From the other hand I cannot understand
why I have to use one methodology to all possible phenomena in the word.

> > ***********************************
> > There is only one requirement.
> > The decryption should be compatible
> > with the phenomena which is just described.
> > ***********************************
> > How to do that?
> > It depends on the application.

> There is only a very limited number of "interesting" measures. Then they
> (probablity, possibility) are universally applicable. It is same as
integer
> numbers. Their algebra does not depend on the application. It would be
> silly to question whether 2+2 be 4, at 1 Apr 2005, 10:00 afternoon.

Well temperature, volume, height, weight are also quite interesting.
and they have very precise interpretation.
By using these measures we can describe what is going on
in many different situations.

From the other hand possibility measure has very unclear interpretation.
I know that we can just say that Poss(x|F)=0.5
and treat that as fact without any explanation.
We can also assume some algebraic operation on such things.

But in that case I can also invent algebra on UFO.
We don't need any explanation we just assume something
and from mathematical point of view everything is OK.

We can for example say

m(UFO1 and UFO2)=sqrt(m(UFO1)+m(UFO2))

We can also develop general theory of integrals on UFO.
Topology on UFO, differential equations on UFO,
UFO-random variables etc.

I can accept that.
Everything is formally OK.
However I have no Idea how to apply these things in practice.
I need clear interpretation
of all operations from UFO theory.

> > That is why I am so skeptical about
> > t-norms and existing fuzzy logic.

> Me too. The problem is that people are trying to substitute measurable
> functions for a measure. This trick won't work.

I am not so sure that measure theory is so powerful
that is able to reject all new ideas.
As far as I now the world is much more complicated than just measure theory.

For example from classical mechanics point of view
relativistic mechanics and quantum mechanics are wrong.

I care only about experiments and relation to reality.
I don't care about the fact that some specialist
are angry after looking at my e-mails.

> >> The first objection is. How it differs from what the probability
> >> theory does. It also defines an additive measure, and this is where
> >> the problems of the probability theory start.

> > If we consider the example of Robert (who know 50% of answers) the
> > question "Does Robert know passwords?" is ambiguous.
> > It is not convenient to answer to that question "yes" or "no".
> > We need another answer.
> > However in traditional sense there is no another answer.
> > Maybe I am wrong
> > but I think that because of that it is not possible
> > to find the answer to that question by using probability theory.
> > The answer "Robert know 50% of question" is precise
> > but this is not the answer in "classical" sense
> > (because the answer in classical sense doesn't exists).

> It is pretty easy to answer if you formulate it properly. For instance,
what
> is the probability that a respondent would know all passwords provided
that
> it is Robert. The answer is 0. But should you ask, what is the probability
> that Robert would know one randomly chosen password, you will become your
> magical number 0.5.

Well, nice trick.
In the same way we can calculate the percentage volume of water.
We can just choose randomly some area in the bottle
and calculate the probability that we find a water.
This is Monte-Carlo method.

We can apply Monte Carlo method to calculation of the values of integrals.
However that does not mean that there is something random in the integrals.

I know that modern probability theory is just a part of measure theory.
Well from mathematical point of view everything is perfect.

Probability theory was invented in order to describe randomness.
Due to some paradoxes people introduce axiomatic probability theory.
Axiomatic probability theory is just a part of measure theory.
This theory doesn't need randomness at all.
Now we call all things which satisfy some axioms probability.
I now that mathematics is a "queen of science"
but maybe some relation to reality is also important ...

However I am only interested describing "fuzziness" correctly.
I can call that problem probability if I will have to
(due to some naming conventions).
However there is no randomness in this problem at all.

> Ambiguous questions are ambiguous, if you mean illegal questions.

In a nice theory of course you are right.

Let us consider a glass.
If I apply high temperature and appropriate tools
I can change the shape of that glass and create an ashtray,
after that I can create a plate.
This is a continuous process.
I can apply some very strict definition of "glass" but after very small
change
I will get something which is almost a glass.
Of course "almost glass" is not a glass but everybody know
that "almost glass" is closer to the "glass" than "plate".
The problem is how to describe that problem.
In my opinion this is a main problem in fuzzy sets theory in this case.

In a real life we deal with "almost glasses" very often.
In such cases the question "Is this a glass?" is ambiguous.
Of courses in classical sense the answer is simply "no".
If something is not a glass then we have to use such answer.
However in a real life we say rather.
"This is a glass with holder" or we use similar description
which tells us how far this object is from "ideal glass".

> > My way of thinking is in completely other direction.
> > There are no events in my examples. No randomness etc.

> There must be some starting point to apply a measure to the things of the
> real world.

I don't have to use events in order to describe
all phenomena in a real world.
There are no such restrictions in science (I hope .).
I can also use other methods (I hope .).

- Show quoted text -

> >> Then observe that the probability theory does not claim that the
> >> probability of, for instance, hitting a mark *is* a relation of the
> >> areas of the mark and the rest. It can be *numerically* equal to the
> >> relation under *definite* circumstances. Probability is neither an
> >> area nor a relation of areas. Why do you think that fuzzy membership
> >> grade should be one?

> > What fuzzy membership should be?
> > According to prof. Zadeh:
> > ""approximately a" given u, where u is a real number. In the fuzzy set
> > interpretation, the grade of membership of u in the fuzzy set would be
an
> > answer to the question:  On the scale  from 0 to l, what is the degree
to
> > which u fits your perception of "approximately a."

> > The problem is that this is a "magic" definition.

> There HAS to be a magic in. It is all fuzziness and randomnes are about.
> Both theories describe UNCERTAINTY in a process of obtaining data. You
> CANNOT look into the box, whether it be a human brain or an electron. Yet,
> you can describe, predict, communicate etc with the box. Open it and you
> will need no theory of uncertainty anymore.

I don't agree ...

read more »

 
 
 

Crisp description of fuzzy sets

Post by Dmitry A. Kazako » Thu, 14 Aug 2003 22:34:14


On Wed, 13 Aug 2003 13:04:46 +0200, "Andrzej Pownuk"

<pon...@poczta.onet.pl> wrote:
>Dear Dmitry,

>> When you are saying that "the bottle is
>> half-full" <=> "50% of its volume is filled" <=> "it is full to 0.5
>> degree". This implicitly assumes that the degree is a volume rate which is
>> additive. It is a fallacy. Of course you can define a degree as a volume
>> rate

>Then what is wrong?

Because this degree is not unversally applicable.

>> but in this case it will not be applicable to cars.

>Because in order to describe cars we have to apply another functions.
>We have to measure temperature in degree Celsius,
>and volume in m^3.
>I don't think that we can measure everything by using one universal way.

So why are you trying to do it? You are doing a wrong thing, admit it,
and keep asking why it does not work! (:-))

>> There is a direct analogy with the probability theory. A random variable
>> is by no means a probability. This is also true for the possibility theory.
>> Compare:

>>              Probability   Possibility
>>              theory        theory / classical fuzzy sets

>> Measure of   Pr(S)         Pos(S)
>> some set     (probability) (possibility)

>> Measurable   (random       (some free
>> function     variable)     function)

>There is a good Paper about that:

>Gert de Cooman, 1995,
>The formal analogy between possibility and probability theory.
>http://www.mat.univie.ac.at/~andrzej/papers/measpos.pdf

Yes it is good. But the analogy it shows is not formal. It is an
*essential* analogy.

>but I am not sure that this is important here.

It is decisive here! Without a formal and precise definition of what
we are doing we will round in circles.

You can disagree with my approach, which is obviously equivalent to
one described in the paper. Well you can, but then you have to provide
another.

>> Now in the case of a bottle, 50% is not a measure, it is a measurable
>> function.

>OK. You are right.
>However I am not very interested in the axioms at this moment.
>I just would like to describe the phenomena (i.e. fuzziness) correctly.

Egh, how? It is same as to correctly describe the phenomenon of
counting without mentioning numbers.

Fuzzy reasoning isn't fuzzy. It is just about fuzzy things! (:-))

- Show quoted text -

>> If F is the set of full bottles, and H is the given bottle, which
>> is 50% full, you cannot just claim that Pr(F|H) = 0.5. It is a clear
>> fallacy. Similarly you cannot claim that Pos(F|H) = 0.5. Or WHATEVER
>> measure m() you take.

>> What you should do, is to define a set of all bottles for which the
>function
>> V gives the answer 50%. Then you can probably measure that set using m()
>> m({x|V(x)=50%}). You can define and measure the set of full bottles. Then
>> you can measure an intersection of both sets. These three values being
>> mixed, might constitute a conditional measure m(F|H), which then is a sort
>> of degree to which H "is in" F.

>If in your opinion this is a good description of the "full bottle" then
>grate.
>However in real life I suspect that
>you are just using a percent in order to describe
>what is going on with the water in bottle.

It is how you defined it.

>Your method is a little too complicated but you can use it.
>We live in a free world (I hope .).

>> > In a real life we always use some measures, levels etc.
>> > I am not sure that these descriptions are always additive.

>> The measure above has other meaning than physical measures you refer to.
>> Those are measurable functions, usually random variables. You might have
>> an unlimited number of functions, but only one measure. Otherwise, it will
>> fall apart.

>What is so wrong in using different way of description in different
>situations?

Nothing. Different ways are well represented by different measurable
functions. If you do not trust me, ask statisticians whether they
develop a new probability theory (= another measure) each time they
need to solve a problem. You might not believe, but they do not. Why
should we?

>From the other hand I cannot understand
>why I have to use one methodology to all possible phenomena in the word.

Because mathematics is a silly thing. Try to develop an alternative
theory of natural numbers, and you will see. Again there is a very
small number of interesting measures. This issue is well studied and,
I would say, closed.

- Show quoted text -

>> > ***********************************
>> > There is only one requirement.
>> > The decryption should be compatible
>> > with the phenomena which is just described.
>> > ***********************************
>> > How to do that?
>> > It depends on the application.

>> There is only a very limited number of "interesting" measures. Then they
>> (probablity, possibility) are universally applicable. It is same as
>integer
>> numbers. Their algebra does not depend on the application. It would be
>> silly to question whether 2+2 be 4, at 1 Apr 2005, 10:00 afternoon.

>Well temperature, volume, height, weight are also quite interesting.
>and they have very precise interpretation.
>By using these measures we can describe what is going on
>in many different situations.

They all are not measures, but measurable functions!

>From the other hand possibility measure has very unclear interpretation.
>I know that we can just say that Poss(x|F)=0.5
>and treat that as fact without any explanation.
>We can also assume some algebraic operation on such things.

>But in that case I can also invent algebra on UFO.
>We don't need any explanation we just assume something
>and from mathematical point of view everything is OK.

Exactly. It is an existential question. Mathematics does not answer
such. Religion and philosophy do.

- Show quoted text -

>> >> The first objection is. How it differs from what the probability
>> >> theory does. It also defines an additive measure, and this is where
>> >> the problems of the probability theory start.

>> > If we consider the example of Robert (who know 50% of answers) the
>> > question "Does Robert know passwords?" is ambiguous.
>> > It is not convenient to answer to that question "yes" or "no".
>> > We need another answer.
>> > However in traditional sense there is no another answer.
>> > Maybe I am wrong
>> > but I think that because of that it is not possible
>> > to find the answer to that question by using probability theory.
>> > The answer "Robert know 50% of question" is precise
>> > but this is not the answer in "classical" sense
>> > (because the answer in classical sense doesn't exists).

>> It is pretty easy to answer if you formulate it properly. For instance,
>what
>> is the probability that a respondent would know all passwords provided
>that
>> it is Robert. The answer is 0. But should you ask, what is the probability
>> that Robert would know one randomly chosen password, you will become your
>> magical number 0.5.

>Well, nice trick.
>In the same way we can calculate the percentage volume of water.
>We can just choose randomly some area in the bottle
>and calculate the probability that we find a water.
>This is Monte-Carlo method.

>We can apply Monte Carlo method to calculation of the values of integrals.
>However that does not mean that there is something random in the integrals.

>I know that modern probability theory is just a part of measure theory.
>Well from mathematical point of view everything is perfect.

And what is not perfect?

Well, I only tried to highlight a very intuitive fact that to get
meaningful answers one should ask meaningful questions. In the case:
"what meaning could have the number 0.5 combined with a bottle?" The
only answer is ANY.

>Probability theory was invented in order to describe randomness.
>Due to some paradoxes people introduce axiomatic probability theory.

Due to continuous questions "what does probability mean?". The answer
is: it probably does nothing. (:-))

>Let us consider a glass.
>If I apply high temperature and appropriate tools
>I can change the shape of that glass and create an ashtray,
>after that I can create a plate.
>This is a continuous process.
>I can apply some very strict definition of "glass" but after very small
>change
>I will get something which is almost a glass.
>Of course "almost glass" is not a glass but everybody know
>that "almost glass" is closer to the "glass" than "plate".
>The problem is how to describe that problem.
>In my opinion this is a main problem in fuzzy sets theory in this case.

I do not think so. Rational numbers is the thing which corresponds to
the phenomenon. Between any two, there is another. You do not need
fuzzy to describe it.

Fuzzy appears when a human being gets involved.

>In a real life we deal with "almost glasses" very often.
>In such cases the question "Is this a glass?" is ambiguous.

Because it is wrong. A correct one could be, "what a human would say
about this glass" and how we could model this.

>I don't have to use events in order to describe
>all phenomena in a real world.
>There are no such restrictions in science (I hope .).
>I can also use other methods (I hope .).

Good luck.

- Show quoted text -

>> >> Then observe that the probability theory does not claim that the
>> >> probability of, for instance, hitting a mark *is* a relation of the
>> >> areas of the mark and the rest. It can be *numerically* equal to the
>> >> relation under *definite* circumstances. Probability is neither an
>> >> area nor a relation of areas. Why do you think that fuzzy membership
>> >> grade should be one?

>> > What fuzzy membership should be?
>> > According to prof. Zadeh:
>> > ""approximately a" given u, where u is a real number. In the fuzzy set
>> > interpretation, the grade of membership of u in the fuzzy set would be
>an
>> > answer to the question:  On the scale  from 0 to l, what is the degree
>to
>> > which u fits your perception of "approximately a."

>> > The problem is that this is a "magic" definition.

>> There HAS to be a magic in. It is all fuzziness and randomnes are about.
>> Both theories describe

...

read more »

 
 
 

Crisp description of fuzzy sets

Post by Andrzej Pownu » Fri, 15 Aug 2003 02:14:12


Dear Dmitry,

Instead answering your question
I have very simply example (see below).

This program creates the answers to the following questions:

1) Is the bottle full?
2) Is the bottle empty?

I think that it works quite reasonably.
The program has the same problems in answering the questions like human.
Fuzziness exists also in "artificial intelligence":)

Fuzziness i.e. problems with choosing appropriate words when
we deal with continuous phenomena
are very similar in the case of humans and computers.

Conclusion: fuzziness does not need human brain.

Regards,

Andrzej Pownuk

//--------------------------------------------------------------------------
-
#include <iostream.h>
#include <string.h>
#include <conio.h>
#pragma hdrstop
//--------------------------------------------------------------------------
-
#pragma argsused
int main(int argc, char* argv[])
{
  cout<<"\n Drinker's adviser v.1.0.0 ";
  cout<<"\n";
  cout<<"\n Choose question: ";
  cout<<"\n 1) Is the bottle full? ";
  cout<<"\n 2) Is the bottle empty?";
  cout<<"\n : ";
  int numberOfQuestion;
  cin>>numberOfQuestion;
  if((numberOfQuestion>=1)&&(numberOfQuestion<=2))
  {
    cout<<"\n What is amount of water in percent?";
    cout<<"\n : ";
    int percentOfWater;
    cin>>percentOfWater;
    switch(numberOfQuestion)
    {
      case 1:
      {
        if(percentOfWater==100)
          cout << " The bottle is full ";
        else
          cout << " The bottle is full in "<< percentOfWater <<" % ";
      };break;
      case 2:
      {
        if(percentOfWater==0)
          cout << " The bottle is empty ";
        else
          cout << " The bottle is empty in "<< 100 - percentOfWater <<" % ";
      };break;
      default:
      {
      };
    };
  }
  else
  {
    cout<<"\n ERROR : question doesn't exist ";
  };
  getch();

  return 0;

Quote:}

//--------------------------------------------------------------------------
-
 
 
 

Crisp description of fuzzy sets

Post by Andrzej Pownu » Fri, 15 Aug 2003 05:48:45


Dear Dmitry,

Here you have another example "fuzzy" description of "car".
I hope that I didn't make to much mistakes here ...

The main idea of that program is to give more or less precise description
what is going on around the "ideal car".
Of course this example is too simple and a little strange
but it is good enough in order to show the idea of fuzziness.
(i.e. in the world we have much more stages
and levels than words in natural language
and that is the source of uncertainty).

Regards,

Andrzej Pownuk

//--------------------------------------------------------------------------
-

#include <iostream.h>
#include <conio.h>
#pragma hdrstop

//--------------------------------------------------------------------------
-

#pragma argsused
int main(int argc, char* argv[])
{
  cout<<"\n I can give you adivices which are related to cars ";
  cout<<"\n but first I would like to know some details.";
  cout<<"\n ";
  bool stop;
  int counter = 0;

  int numberOfWheels;
  do
  {
    stop = false;
    cout<<"\n How many wheels has this object? \n : ";
    cin>>numberOfWheels;
    counter++;
    if(counter>2)
    {
      cout<<"\n If you don't know how looks the wheels, then good by. ";
      getch();
      exit(0);
    };
    if(numberOfWheels<0)
    {
      cout<<"\n That is impossible. Let's try again.";
      stop=true;
    };
  }
  while(stop);

  counter = 0;
  int engine;
  do
  {
    stop = false;
    cout<<"\n Has this object an engine (1 - Yes,0 - No)? \n : ";
    cin>>engine;
    counter++;
    if(counter>2)
    {
      cout<<"\n You don't know what the engine is, then good by ";
      getch();
      exit(0);
    };
    if(!((engine==0)||(engine==1)))
    {
      cout<<"\n That is impossible. Let's try again.";
      stop=true;
    };
  }
  while(stop);

  if((numberOfWheels==4)&&(engine==1))
  {
    cout<<"\n That is the car!";
    getch();
  }else if(((numberOfWheels>=2)&&(numberOfWheels<=3)&&(engine==1))||
           ((numberOfWheels==4)&&(engine==0)))
  {
    cout<<"\n That may be a broken car.";
    getch();
  }else if((numberOfWheels>=0)&&(numberOfWheels<=2)&&(engine==1))
  {
    cout<<"\n That may be sirously broken car.";
  }else if((numberOfWheels>=0)&&(numberOfWheels<=3)&&(engine==0))
  {
    cout<<"\n That may car after accident or this is not a car.";
  };
  getch();

  return 0;

Quote:}

//--------------------------------------------------------------------------
-
 
 
 

1. Fuzzy Sets or defuzzified crisp values

Hi, i need some advise on this issue...
I am using linguistic terms for my expert system in wich each linguistic
term is associated with a fuzzy set, but, operations of reasoning becomes
quite more complicated if performed over fuzzy sets than if i use
defuzzyfied those sets before (with centroid of area for ex.) to get crisp
values and then operate on those.
My question is, in general, what do i loose if i defuzzyfy sets before and
use crisp values instead of fuzzy sets, what are the pros and cons...?
thanks.

Otto

=========================================
Otto Cordero
Centro de Tecnologas de Informacin - ESPOL
Campus  Prosperina
Telf: 269 - 775

http://www.cti.espol.edu.ec
=========================================

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