**************************************************************************
* How Nonfuzzy Thresholds and Sets in One Universe of Discourse *
* become Fuzzy Thresholds and Sets in Another Universe of Discourse. *
* *
* A fuzzy threshold or set (for, e.g., `tall') in the universe UX of *
* exactly measured attribute (height) values is the result of a nonfuzzy *
* threshold or set in the universe US of estimated attribute values *
* combined with different sources of fuzziness assumed by a subject who *
* performs a MU experiment. *
**************************************************************************
In answer to my email letter of June 14th entitled
`The MEANING of fuzzy AND and OR'
the mysterious, unnamed ca314159@bestweb.net writes:
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The example you give of 'composite experiments' uses disjoint
(and apparently crisp) sets with a subjective determination of
set membership as a probability that is normalized on the
number of these disjoint sets. These don't sound like true
fuzzy sets ?
In this case (A AND B) = 0 because the two sets are completely
distinct in terms of whether a sample will wind up in one or
the other. The fact that you probabilistically assign membership
does not change this essential point that each sample winds up
in only one set. Hence these sets are disjoint.
This is not as general a case as was being considered in which
the sets were not necessarily disjoint. That is, one "calue"
(crisp value?) sample might simultaneously be a member of A and B to
various degrees. In this case (A AND B) >=0 and we say that these
sets are not necessarily orthogonal but that there is a degenerate
case where they can be. This degenerate case is when there is
certainty in the disjointness of the sets. The measure of disjointness
of these sets is a measure of their distinctness, orthogonality or
degree of overlap, which can be in the range:
max(A+B-1,0) <= (A AND B) <= min(A,B) [1]
if A and B are not disjoint and A + B <=2.
end citation from ca314159@bestweb.net's letter
-----------------------------------------------------------------------------
You are right that the question of disjointness is intimately related
to the connective problem. It is also connected with the question of
nonfuzzy versus fuzzy thresholds and sets, which are again connected
with the universe of discourse to which one refers.
As an example, consider the set `tall', (more specifically `tall man').
Suppose that a subject S1 is asked to label an object as either `tall', or
`medium' or `small'. We shall then say that S1 is asked to perform
a `labeling experiment'. We shall also assume that S1 is told the
exactly measured height value `ux' of the object. She then performs an
`exact labeling experiment.' (In [5] `ux' is denoted by `u' with a
superscript `ex' for `exact'. In the fuzzy set literature it is
usually denoted by `u'.)
In this situation, S1 has no choice but to set a nonfuzzy threshold for
`tall' in the universe UX of exactly measured height values; for example
180 cm. Similarly she must set a nonfuzzy threshold for the boundary
between `small' and `medium'.
When S1 is represented with a whole set of objects, the labeling
experiment will then result in the partitioning of this set into the
three nonfuzzy or crisp sets labeled `tall', `medium', `small'.
----------------------------------------
Alternatively, suppose now that S1 is asked to perform not an
`exact labeling experiment', but an `exact MU experiment'.
The adjective `exact' implies again that S1 is told (or can measure)
the height `ux' of the object. Performing a MU experiment means
that S1 is asked to assign to the object a grade of membership in the
set `tall'. The membership value must be in the real interval [0,1].
An intelligent S1 can then take into account that in everyday life
other subjects S2 are not acquainted with the exactly measured height
of objects which they encounter. They will therefore estimate this height.
We shall use the symbol `us' for the estimated height value. The
numerical value of `us' may differ from the value of `ux', the true,
measured height value. (`us' is denoted by `u' in [5]).
When the (by S1) imagined subject S2 is to perform a labeling experiment,
she has no choice but to set nonfuzzy thresholds in the universe US
of estimated height values `us'. S1 realizes that in everyday life,
in which objects may be observed under different conditions, the object
with the height ux (or two objects having the same true height ux)
may be estimated to have different height values `us'. Thus an object
with true height 179cm may sometimes be estimated to have a height of 182cm
and thus be labeled `tall'. At other times it may be estimated to have a
height of 179cm and be labeled `medium'.
Suppose that S1 thinks that
in 40% of all cases an object of true height ux=179cm
will be estimated to have a height us>=180cm, and
in 60% of all cases an object of height ux=179cm
will be estimated to have a height us<180cm.
When asked to perform a MU experiment,
S1 will then assign the membership value mu=0.4 to the object in `tall',
and the membership value mu=0.6 to the object in `NOT tall'.
The numerical value of mu according to S1 thus depends on S1's
estimate of the probability of error ux-us made by subjects under
everyday conditions of observation for the given ux.
This is how membership values and functions (of the true height value ux)
are generated by subjects S1 performing a MU experiment.
THE NONFUZZY THRESHOLD FOR `TALL' IN `US' IS CONVERTED TO A FUZZY ONE IN UX.
AND THE CRISP SET TALL(US) IS CONVERTED TO A FUZZY SET IN UX.
The mathematical treatment shows that
S1's membership function (of ux) in `tall'
should be the convolution
of the nonfuzzy, step-shaped, threshold curve in US
(whose value is 1 above 180cm and 0 below it)
with the error curve assumed by S1; resulting in an S-shaped threshold
(membership) curve.
This result is derived in references [5, fig. 4], [6]
and in more detail in [3], [4], and [1].
The above `source of fuzziness #1' (due to S1's taking errors of estimation
of the attribute value into account) is not the only one.
E.g., `fuzziness #3' is due to S1's taking into account that different
subjects S2 may have slightly different thresholds for `tall'.
The effect of this fuzziness will then be superimposed on the effect
of `fuzziness #1' (see [2]). Other details have not been mentioned here.
For example,
(1) in some special cases a labeling experiment must be
distinguished from a YES-NO experiment [3],
(2) a MU experiment must refer either to a labeling or a YES-NO
experiment [3],
(3) both a YES-NO, and a labeling,and a MU experiment
must refer to a given label set [3].
(4) the membership curve elicited in an inexact MU experiment
is a fuzzified version of
the membership curve elicited in an exact MU experiment
(see [2, section 3]).
The papers in the reference list below make use of quantized functions.
Parallel derivations can, however, be carried out for continuous functions,
making use of probability densities.
-------------------------------------
Coming back to the subject of ca314159@bestweb.net's letter
(see beginning of present letter):
It is not true that in this TEE model
the sets of exact ux values for labeling an object as `tall' and `medium'
respectively are disjoint. We have seen that subjects S2 may label objects
of the same height ux=179cm sometimes as `medium' and sometimes as `tall'.
Thus one of the results of the TEE model is that the membership of
`tall AND medium' is not necessarily 0. (See [5], eqs, (11), (14), (17)).
The membership curves of labels with connectives are derived with the
aid of the concept of a `composite experiment' consisting of two
successive labeling experiments performed on each object.
(For fuzziness #1, S1 can imagine that the object is observed under
different conditions in experiment1 and experiment2. For fuzziness #3,
S1 imagines that the object is observed in experiment1 and experiment2
by different subjects S2 with generally different thresholds.)
(See [1], section 10. An abbreviated version is found in [5], section 2.9)
The grade of membership of, e.g., `tall AND medium' for a given ux
is defined in [1] as S1's estimate of the probability that a subject S2
will label an object (of height ux) `tall' in one of these experiments
and `medium' in the other. See {5}, equations (10)-(19), for the
formulas for grades of member of labels with connectives. There exist
three sets of formulas for such labels, depending on the situation to
which S1 and the composite experiment refer.
Instead of ca314159@bestweb.net's eq.(1) above, the TEE model
therefore finds three possible numerical values for the membership
value of a label with a connective (for a given ux); one value for
each reference situation imagined by S1 for a composite experiment.
I have compared
(1) the numerical interval for the membership of a label
with an AND connective according to ca314159@bestweb.net's eq. (1)
with
(2) the numerical values obtained from [5, eqs, (11), (14), (17)],
for three specific cases of numerical values of A and B
(A = membership value of the first component of the AND label,
B = membership value of the second component of the AND label.):
A | 1 | 0.5 | 0.25 |
B | 0 | 0.5 | 0.25 |
---------------------------------|----------|----------|
ca314159@bestweb.net's eq.(1)| 0 | [0, 0.5] | [0, 0.25]|
---------------------------------|----------|----------|
Eq.(14) in [5] | 0 | 0.5 | 0.25 |
Eq.(17) in [5] | 0 | 0.25 | 0.125 |
Eqs.(11), (10) in [5] | 0 | 0 | 0 |
All membership values refer to the same, given ux.
The three membership columns on the right hand side of the table
show that ca314159@bestweb.net's interval includes the numerical values
of the three formulas of [5] for AND.
The difference between ca314159@bestweb.net's formula and those of the
TEE model is that the latter describes the reference situation to which
S1's imagined composite experiment refers, as well as the meaning of the
membership concept.
REFERENCES
[1]
@techreport{64,
author = {Hisdal, E.},
title = {A Theory of Logic Based on Probability},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1984},
type = {Research Report},
note = {ISBN~82-90230-60-5},
number = {64} }
[2]
@article{inf1p2,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.2. {D}ifferent Sources of Fuzziness
and Uncertainty},
journal = {Int. J. Man-Machine Studies},
year = {1986},
volume = {25},
pages = {113-138} }
[3]
@techreport{inf1p3,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.3. {R}eference Experiments and
Label Sets},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1988,1990},
type = {Research Report},
note = {ISBN~82-7368-053-3.
Can also be found on
http://www.ifi.uio.no/$\sim$ftp/publications/research-reports/Hisdal-3.ps},
number = {147} }
[4]
@techreport{inf1p4,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.4. {T}he {TEE} Model},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1988,1990},
type = {Research Report},
note = {ISBN~82-7368-054-1.
Can also be found on
http://www.ifi.uio.no/$\sim$ftp/publications/research-reports/Hisdal-4.ps},
number = {148} }
[5]
@article{are,
author = {Hisdal, E.},
title = {Are Grades of Membership Probabilities?},
journal = {Fuzzy Sets and Systems},
year = {1988},
volume = {25},
pages = {325-348} }
[6]
@inproceedings{explanatory,
author = {Hisdal, E.},
title = {Explanatory versus Postulate Fuzzy Set Theory},
booktitle={MEPP'92: Proceedings of the International Seminar on
Fuzzy Control},
year = {1992},
editor = {Eklund, Patrik},
pages = {42-52},
address = {\AA bo Akademi University, Department of Computer
Science, DataCity, Lemmink\"ainengatan 14-18, SF-20520,
\AA bo, Finland} }
Ellen Hisdal
---------------------------------------------------------------------
Address, etc.:
Ellen Hisdal | Email: ellen@ifi.uio.no
(Professor Emeritus) |
Mail: Department of Informatics | Fax: +47 22 85 24 01
University of Oslo | Tel: (office): 47 22 85 24 39
Box 1080 Blindern |
0316 Oslo, Norway | Tel: (secr.): 47 22 85 24 10
Location: Gaustadalleen 23, |
Oslo | Tel: (home): 47 22 49 56 53
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