PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint...
Transcript of PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint...
![Page 1: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/1.jpg)
Z-NUMBERS—A NEW DIRECTION IN THE ANALYSIS
OF UNCERTAIN AND COMPLEX SYSTEMS
Lotfi A. Zadeh
Computer Science Division
Department of EECS
UC Berkeley
7th IEEE International Conference on Digital Ecosystems and
Technologies
July 24, 2013
Menlo Park
Research supported in part by ONR Grant N00014-02-1-0294, Omron Grant, Tekes
Grant, Azerbaijan Ministry of Communications and Information Technology Grant,
Azerbaijan University of Azerbaijan Republic and the BISC Program of UC
Berkeley.
Email: [email protected]
URL: http://www.cs.berkeley.edu/~zadeh/ LAZ 7/23/2013 1/60
![Page 2: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/2.jpg)
LAZ 7/23/2013 2/60
![Page 3: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/3.jpg)
BASIC IDEAS
Z-number = number + its
reliability
Z-number-based models of
complex systems—especially
economic systems—are more
realistic than traditional
number-based models.
LAZ 7/23/2013 3/60
![Page 4: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/4.jpg)
PREAMBLE
In large measure, science and
engineering dwell in the world of
measurements and numbers. In this
world, a basic question which arises
is: How reliable are the numbers
which we deal with? This question
plays a particularly important role in
decision analysis, planning,
economics, risk assessment, design
and process analysis. LAZ 7/23/2013 4/60
![Page 5: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/5.jpg)
THE CONCEPT OF A Z-NUMBER (ZADEH
2011)
The concept of a Z-number is
intended to provide a basis for
computation with numbers which
are not totally reliable. More
concretely, a Z-number, Z=(A,B), is
an ordered pair of two fuzzy
numbers. The first number, A, is a
restriction on the values which a
real-valued variable, X, can take. LAZ 7/23/2013 5/60
![Page 6: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/6.jpg)
CONTINUED
The second number, B, is a
restriction on the reliability that X
is A. Typically, A and B are
described in a natural language.
Z= (fuzzy value, fuzzy reliability)
NL NL
LAZ 7/23/2013 6/60
![Page 7: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/7.jpg)
EXAMPLES
A=approximately 2 million dollars
B=very likely
A=approximately 1 hour
B=usually
A=high
B=surely LAZ 7/23/2013 7/60
![Page 8: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/8.jpg)
PRINCIPAL APPLICATIONS OF Z-
NUMBERS
Of particular importance are
applications in economics,
decision analysis, risk
assessment, prediction,
anticipation, planning, biomedicine
and rule-based manipulation of
imprecise functions and relations.
LAZ 7/23/2013 8/60
![Page 9: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/9.jpg)
THE CONCEPT OF Z-VALUATION
A Z-valuation is an ordered triple of
the form (X, (A,B)), where X is a
real-valued variable and (A,B) is a
Z-number.
Example.
(unemployment next year, sharp
decline, unlikely)
LAZ 7/23/2013 9/60
![Page 10: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/10.jpg)
CONDITIONAL Z-VALUATION (ZADEH 2011)
If (X, A, B) then (Y, C, D)
Simpler version
If (X, A) then (Y, B, C)
Equivalently,
If (X is A) then (Y iz (B, C))
Z-rule
LAZ 7/23/2013 10/60
![Page 11: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/11.jpg)
Z-INFORMATION
In real-world settings, much of
the information in an
environment of uncertainty and
imprecision may be
represented as a collection of
Z-valuations and conditional Z-
valuations—a collection which
is referred to as Z-information. LAZ 7/23/2013 11/60
![Page 12: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/12.jpg)
EXAMPLES OF Z-INFORMATION
Usually it is difficult to find parking
near the campus in the early
morning (finding parking
near the campus in the early
morning, difficult, usually)
Usually it takes Robert about an
hour to get home from work
(travel time from work to home,
about one hour, usually)
LAZ 7/23/2013 12/60
![Page 13: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/13.jpg)
NOTE
In the analysis of economic
systems, much of the information
is Z-information.
Existing economic theories do not
reflect this fact.
Existing models of economic
systems are unreliable, E. Derman.
“Models Behaving Badly” 2011.
LAZ 7/23/2013 13/60
![Page 14: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/14.jpg)
SEMANTICS OF A FUZZY IF-THEN RULE
A concept which plays a pivotal
role in uncertain systems analysis
is that of a fuzzy if-then rule (Zadeh
1973).
If X is A then Y is B (X, Y) is A×B
fuzzy set fuzzy set
LAZ 7/23/2013 14/60
![Page 15: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/15.jpg)
CONTINUED
X/u
Y/v
0
1
µB
A×B
µA
µA×B(u,v)=µA(u)ΛµB(v)
LAZ 7/23/2013 15/60
![Page 16: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/16.jpg)
FUZZY RULE SETS
A fuzzy rule set is a collection of
fuzzy if-then rules.
Principally, fuzzy rule sets are
used for two purposes: (a)
representation of imprecisely
defined functions and relations;
and (b) approximation to precisely
known functions and relations.
LAZ 7/23/2013 16/60
![Page 17: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/17.jpg)
BASIC FUZZY RULE SET
There are many kinds of fuzzy rule
sets. A simple fuzzy rule set which
is employed in many applications
of fuzzy control is expressed as:
(If X is A1 then Y is B1, … If X is An
then Y is Bn)
LAZ 7/23/2013 17/60
![Page 18: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/18.jpg)
FUZZY LOGIC (FL) GAMBIT
0
Y f
0
S M L (large)
L
M
S
granule
X
granulation
Fuzzy logic gambit underlies many important
applications of fuzzy logic. In essence,
FL gambit = deliberate value imprecisiation (v-
imprecisiation) through granulation, followed
by meaning precisiation (m-precisiation)
v-imprecisiation
precisely specified function
LAZ 7/23/2013 18/60
![Page 19: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/19.jpg)
CONTINUED
if X is small then Y is small
if X is medium then Y is large
if X is large then Y is small 0 X
f *f : granulation
Y
*f (fuzzy graph)
medium × large (M×L)
summarization
*f =
if X is small then Y is small
if X is medium then Y is large
if X is large then Y is small
*f = m-precisiation
small×small+medium×large+large×small
mathematical summary of f LAZ 7/23/2013 19/60
![Page 20: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/20.jpg)
A BASIC PROBLEM (INTERPOLATION)
(ZADEH 1976)
If X is A1 then Y is B1
…
If X is An then Y is Bn
X is A
Y is ?B
Y is m1 Λ b1+…+mn Λ bn,
where mi is the degree to which
A matches Ai, i=1, …, n. LAZ 7/23/2013 20/60
![Page 21: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/21.jpg)
SEMANTICS OF A Z-RULE
If (X is A) then Y iz (B, C)
(X×Y) iz (A×B, C)
LAZ 7/23/2013 21/60
![Page 22: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/22.jpg)
SEMANTICS OF Z-RULE SET /
INTERPOLATION
If X is A1 then Y iz (B1C1)
…
If X is An then Y iz (BnCn) ?
(X, Y) iz ((A1×B1,C1) +···+ (X, Y) iz
(An×Bn,Cn))
Interpolation?
LAZ 7/23/2013 22/60
![Page 23: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/23.jpg)
SEMANTICS OF Z-RULE SET /
INTERPOLATION
X
Y
Bi
Ai A
LAZ 7/23/2013 23/60
![Page 24: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/24.jpg)
COMPUTATION WITH Z-NUMBERS
(approximately 1 hr, usually) +
(approximately 45 min, usually)
What is the square root of
(approximately 100, very likely?)
(A, B)(C, D)=?(E, F)
f(A, B)=?(C, D)
LAZ 7/23/2013 24/60
![Page 25: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/25.jpg)
COMPUTATION WITH Z-NUMBERS
(CONTINUED)
There are two concepts which play
an essential role in computation
with Z-numbers—the concept of a
restriction and the concept of
extension principle. These
concepts are briefly discussed in
the following.
LAZ 7/23/2013 25/60
![Page 26: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/26.jpg)
LAZ 7/23/2013 26/60
![Page 27: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/27.jpg)
THE CONCEPT OF A RESTRICTION
The centerpiece of the calculus of Z-
numbers is a deceptively simple
concept—the concept of a restriction.
The concept of a restriction has greater
generality than the concept of interval,
set, fuzzy set, rough set and probability
distribution. An early discussion of the
concept of a restriction appeared in
Zadeh 1975 “Calculus of fuzzy
restrictions.”
LAZ 7/23/2013 27/60
![Page 28: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/28.jpg)
GENERALITY OF RESTRICTIONS
number
interval
set
fuzzy set
Z-restriction
probability distribution
related concepts: generalized
constraint, generalized
assignment statement,
granule.
set-valued theory fuzzy set-
valued theory restriction-
valued theory
rough set
restriction
LAZ 7/23/2013 28/60
![Page 29: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/29.jpg)
WHAT IS A RESTRICTION?
Informally, a restriction, R(X), on a
variable, X, is an answer to a question of
the form: What is the value of X?
Examples
X=5
X is between 3 and 7
X is small
It is likely that X is small
Usually X is much larger than
approximately 5. LAZ 7/23/2013 29/60
![Page 30: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/30.jpg)
CANONICAL FORM OF A RESTRICTION
The canonical form of a restriction is
expressed as
R(X): X isr R,
where X is the restricted variable, R is the
restricting relation and r is an indexical
variable which defines the way R restricts
X.
Note. In the sequel, the term restriction
is sometimes applied to R.
LAZ 7/23/2013 30/60
![Page 31: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/31.jpg)
SINGULAR RESTRICTIONS
There are many types of
restrictions. A restriction is
singular if R is a singleton.
Example. X=5. A restriction is
nonsingular if R is not a
singleton. Nonsingularity
implies uncertainty.
LAZ 7/23/2013 31/60
![Page 32: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/32.jpg)
INDIRECT RESTRICTIONS
A restriction is direct if the
restricted variable is X. A
restriction is indirect if the
restricted variable is of the form
f(X). Example.
is an indirect restriction on p.
R p : μ u p u du is likely.
b
a
LAZ 7/23/2013 32/60
![Page 33: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/33.jpg)
PRINCIPAL TYPES OF RESTRICTIONS
The principal types of
restrictions are: possibilistic
restrictions, probabilistic
restrictions and Z-
restrictions.
LAZ 7/23/2013 33/60
![Page 34: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/34.jpg)
POSSIBILISTIC RESTRICTION (r=blank)
R(X): X is A,
where A is a fuzzy set in a space, U,
with the membership function, µA. A
plays the role of the possibility
distribution of X,
Poss(X=u)=µA(u).
LAZ 7/23/2013 34/60
![Page 35: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/35.jpg)
EXAMPLE—POSSIBILISTIC RESTRICTION
X is small
restricted variable
The fuzzy set small plays the role
of the possibility distribution on X.
restricting relation (fuzzy set)
LAZ 7/23/2013 35/60
![Page 36: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/36.jpg)
PROBABILISTIC RESTRICTION (r=p)
R(X): X isp p,
where p is the probability density
function of X,
Prob(uXu+du) = p(u)du.
LAZ 7/23/2013 36/60
![Page 37: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/37.jpg)
EXAMPLE—PROBABILISTIC RESTRICTION
X isp 1
2πexp(−(X−m)
2/2σ2).
restricted variable restricting relation
(probability density
function)
LAZ 7/23/2013 37/60
![Page 38: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/38.jpg)
Z-RESTRICTION (r=z, s is suppressed)
X is a real-valued random variable.
A Z-restriction is expressed as
R(X): X iz Z,
where Z is a combination of
possibilistic and probabilistic
restrictions defined as
Z: Prob(X is A) is B,
LAZ 7/23/2013 38/60
![Page 39: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/39.jpg)
NATURAL LANGUAGE ≈ SYSTEM OF
POSSIBILISTIC RESTRICTIONS
A natural language may be
viewed as a system of
restrictions. In the realm of
natural languages, restrictions
are predominantly possibilistic.
For simplicity, restrictions may
be assumed to be trapezoidal.
LAZ 7/23/2013 39/60
![Page 40: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/40.jpg)
Note. Parameters are context-dependent.
40 60 45 55
μ
1
0
definitely
middle-age
definitely not
middle-age
definitely not
middle-age
43
0.8
core of middle-age
membership function
of middle age
Age
TRAPEZOIDAL POSSIBILISTIC RESTRICTION
ON AGE
LAZ 7/23/2013 40/60
![Page 41: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/41.jpg)
BACK TO Z-NUMBERS
X iz (A, B)
The Z-number (A, B) is a restriction
on X.
X is a random variable with
unknown probability density
function, p. p is referred to as the
underlying probability density
function of X.
LAZ 7/23/2013 41/60
![Page 42: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/42.jpg)
THE UNDERLYING PROBABILITY DENSITY
FUNCTION
Probability measure of A may be
expressed as (Zadeh 1968)
μA u p u du
or, more simply as,
µA·p,
where · is the scalar product.
LAZ 7/23/2013 42/60
![Page 43: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/43.jpg)
IMPORTANT OBSERVATION
In a Z-number, (A, B), B is an
indirect possibilistic restriction on
the probability measure of A. More
concretely,
µA·p is B.
This expression relates p to B.
If X=(A, B), then (A, B) is a direct
Z-restriction on X. LAZ 7/23/2013 43/60
![Page 44: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/44.jpg)
NOTE
when there is a need for placing in
evidence the underlying probability
density function, p, a Z-number
may be expressed as
(A, p, B),
with the understanding that
µA·p is B.
LAZ 7/23/2013 44/60
![Page 45: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/45.jpg)
LAZ 7/23/2013 45/60
![Page 46: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/46.jpg)
COMPUTATION WITH Z-NUMBERS
Computation with Z-numbers plays
an essential role in computation
with Z-information. Stated in
general terms, computation with Z-
numbers involves evaluation of an
n-ary function, f, whose arguments
are Z-numbers. For simplicity, in
the following, n is assumed to be 1
or 2. LAZ 7/23/2013 46/60
![Page 47: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/47.jpg)
SIMPLE EXAMPLES OF COMPUTATION
(about 1 hour, usually) + (about 2
hours, very likely)
(small, usually) × (about 4, very
likely)
the square root of (about 30, likely)
LAZ 7/23/2013 47/60
![Page 48: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/48.jpg)
COMBINATION OF TWO Z-
NUMBERS
Let X=(AX,BX) and Y=(AY,BY) be
Z-numbers in the space of real
numbers. Let Z be a
combination of X and Y, Z=XY.
Example. =+ More concretely,
Z= (AX,BX)(AY,BY)
LAZ 7/23/2013 48/60
![Page 49: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/49.jpg)
COMBINATION OF FUZZY NUMBERS
Consider a generic combination,
Z=(AZBZ), of two fuzzy numbers X=(AX,
BX) and Y=(AY, BY).
Z=XY.
More explicitly,
(AZ, BZ)=(AX, BX)(AY, BY)
The goal of computation is
determination of AZ and BZ.
LAZ 7/23/2013 49/60
![Page 50: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/50.jpg)
DEFINITION OF AZ AND BZ
AZ=AXBX
BZ=μAZ· pZ,
where pZ is the underlying
probability density function of Z.
More concretely,
pZ=pX°pY,
LAZ 7/23/2013 50/60
![Page 51: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/51.jpg)
CONTINUED
where ° is the counterpart of for
probability density functions. Simple
example. When =+, ° =convolution.
Computation of pZ would be a
simple matter if we knew pX and pY.
What we know are possibilistic
restrictions on pX and pY.
Specifically,
LAZ 7/23/2013 51/60
![Page 52: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/52.jpg)
CONTINUED
μAX· pX is BX
μAY· pY is BY
At this point, computation of BZ
reduces to computation of
pZ=pX °pY,
with the above restrictions on pX
and pY. LAZ 7/23/2013 52/60
![Page 53: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/53.jpg)
EXTENSION PRINCIPLE
Computation of pZ requires the use
of a basic version of the extension
principle (Zadeh 1965, 1975b).
Assume that Z is a function of X
and Y
Z=f(X,Y)
in which X and Y are subject to
possibilistic restrictions
LAZ 7/23/2013 53/60
![Page 54: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/54.jpg)
CONTINUED
g(X) is A
h(Y) is B,
where A and B are fuzzy sets.
In this version of the extension
principle, Z may be expressed as
the solution of the variational
problem
μZ(w)=supu,v(μA(g(u))˄μB(h(v)))
LAZ 7/23/2013 54/60
![Page 55: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/55.jpg)
CONTINUED
subject to
w=f(u,v)
Applying this version of the extension
principle to the computation of pZ, we
arrive at the following definition of the
combination of two Z-numbers
(AXBX)(AY,BY)=((AXAY),μAXAY ·pZ
where μpZ is given by
LAZ 7/23/2013 55/60
![Page 56: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/56.jpg)
CONTINUED
μpZ(w)=supu,v(μBX
(μAX ·u)˄
(μBY
(μAY ·v)
subject to
w=uv,
where u, v, and w take values in the
space of probability density
functions. LAZ 7/23/2013 56/60
![Page 57: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/57.jpg)
OBSERVATION
What we see is that
computation of a combination
of two Z-numbers is not a
simple matter. Complexity of
computation reflects a basic
fact—construction of realistic
models of complex systems
carries a price.
LAZ 7/23/2013 57/60
![Page 58: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/58.jpg)
CONCLUDING REMARK
Computation with Z-numbers is a move
into uncharted territory. A variety of
issues remain to be explored. One
such issue is that of informativeness of
results of computation. What is of
relevance is that a restriction is a
carrier of information. Informativeness
of a restriction relates to a measure of
the information which it carries.
Informativeness is hard to define
because it is application-dependent.
LAZ 7/23/2013 58/60
![Page 59: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/59.jpg)
CONTINUED
In general, informativeness of a
combination of Z-numbers is lessened
by combination. To minimize the loss
of informativeness and reduce the
complexity of computations, it may be
expedient to make simplifying
assumptions about the underlying
probability density functions. A
discussion of this issue may be found
in Zadeh 2011, A Note on Z-numbers.
LAZ 7/23/2013 59/60
![Page 60: PowerPoint Presentationdest2015.digital-ecology.org/images/pdf/DEST 2013-Z... · Title: PowerPoint Presentation Author: flaca Created Date: 7/30/2013 3:35:47 PM](https://reader033.fdocuments.net/reader033/viewer/2022060517/6049f192066d3534dc5e4cf8/html5/thumbnails/60.jpg)
CONTINUED
This presentation focuses on the basics of
the calculus of Z-numbers. No applications
are discussed. Applications of Z-numbers
are just beginning to get developed. A start-
up, Z-Advanced Systems, is focusing on the
development of applications of Z-numbers.
A team of researchers headed by Professor
Rafik Aliev is exploring applications of Z-
numbers at the Azerbaijan State Oil
Academy. Papers by other researchers are
in preparation.
LAZ 7/23/2013 60/60