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Greener approaches for chemical synthesis: ball mill and microwave assisted synthesis of fluoxetine and duloxetine and enantioselective catalysed addition of organometallic reagents to aldehydes

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O rder N um ber 8906233
N ew m ethodologies in the design of a general purpose fuzzy
expert system : A pplications w ith A I based precipitation
retrieval designed for satellite microwave m easurem ents
Oh, Kyung Whan, Ph.D.
The Florida State University, 1988
UMI
300N.ZeebRd.
Ann Aibor, MI 48106
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THE FLORIDA STATE UNIVERSITY
COLLEGE OF ARTS AND SCIENCES
NEW METHODOLOGIES IN THE DESIGN OF
A GENERAL PURPOSE FUZZY EXPERT SYSTEM:
APPLICATIONS WITH AI BASED PRECIPITATION RETRIEVAL
DESIGNED FOR SATELLITE MICROWAVE MEASUREMENTS
by
KYUNG WHAN OH
A Dissertation submitted to the
Department of Computer Science
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Approved :
Fall Semester, 1988
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
NEW METHODOLOGIES IN THE DESIGN OF
A GENERAL PURPOSE FUZZY EXPERT SYSTEM:
APPLICATIONS WITH AI BASED PRECIPITATION RETRIEVAL
DESIGNED FOR SATELLITE MICROWAVE MEASUREMENTS
(Publication No.
______ )
Kyung Whan O h , P h .D .
THe Florida State University,
1988
Major Professor: Abraham Kandel, Ph.D.
In the design of expert systems, management of uncertainty is
related to a computational analysis
premises to the conclusion.
In
of uncertainty
this work
from the
we propose
a new
reasoning method with the equivalence operator instead of the
implication operator in modus ponens.
It demonstrates a fuzzy-
logic-based computational framework. Based on this method, we
introduce
process.
the concept
Through
the
of
coimplication
coimplication
representation scheme and
concept,
an inference engine
Although several ad hoc models
with vagueness,
in
there has been
have been
the
inference
a
knowledge
are designed.
developed
to deal
a strong need for a globally
applicable method of dealing with vagueness in expert systems.
The General Purpose Fuzzy Expert System (G P F E S ) is an attempt
to model uncertainty in the general domain of expert systems.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Precipitation retrieval by means of remote sensing
long standing problem
in
radiometry from space
has
Satellite
great
Meteorology.
potential
for
is a
Microwave
estimating
rainfall because upwelling radiation over a cloud is directly
responsive to precipitation microphysics.
simulation
provide
studies
an
various
been carried out in an attempt
interpretation
measurements
of
remotely
sensed
to
microwave
in precipitating atmospheres and to explore the
feasibility
retrieval.
have
Therefore,
of passive multi-channel microwave precipitation
Numerical
retrieval is
studies have
shown that precipitation
an ill-conditioned problem
from a mathematical
perspective because of problems with non-unique relationships
between the radiation signals
From
be
and
a theoretical perspective,
overcome
by
There remains,
simply
however,
analytical framework.
adopting
to
intensity.
some of these problems, can
a multispectral strategy.
a number of
gaps
within
a purely
We propose to close these gaps with an
expert system approach which
coimplication
precipitation
handle
uses
our new technique
uncertainty
called
in general domains of
expert systems.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
To Miran, Minhu and my mother Kuemsoon
iv
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Acknowledgements
I am deeply indebted to my major professor,
for his encouragement,
Dr. A. Kandel,
guidance and constructive criticism.
His effort and thought to my professional development shall
not be forgotten. My gratitute is also due to the other
members of my supervisory committee: Drs. H. Levitz,
L. Hawkes, and E.A. Smith for their assistance.
I am
especially grateful to Dr. E.A. Smith for the construction
of the precipitation retrieval knowledge base.
My gratitute is also extended to my friendly colleagues
Dr. H. Cooper, M. Smith, and A. Mehta for their helpful
collaboration,
to Dr. Frank Wentz for providing the SSM/I
data set, and to The National Science Foundation for their
assistance in obtaining the INSAT data.
This dissertation is dedicated to my wife, Miran, my son,
Minhu, and my mother, Keumsoon.
It is no exaggeration to say
that without Miran's support, encouragement and love over the
past life in Tallahassee,
this work would not have been
accomplished.
This research has been supported by NASA grant NAGW-991
and DOE grant DE-FC05-85ER250000.
A portion of the
computational support has been provided by the Supercomputer
Computations Research Institute (S C R I ), at the Florida State
University, under the above DOE Contract.
v
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Contents
1.
2.
3.
Introduction
1
1.1 Problem Statement -----------------------------------
1
1.2 Literature Survey -----------------------------------
5
Background
8
2.1 Fuzzy S e t s -------------------------------------------
8
2.2 Possibility Theory ----------------------------------
17
2.3 Fuzzy logic and Approximate Reasoning --------------
20
Classical Expert Systems
33
3.1 Knowledge Representation ---------------------------
37
3.2 Inference Engine ------------------------------------
41
3.3 Reasoning with uncertainty in expert systems ------
43
3.3.1 Problem definition ---------------------------
43
3.3.2 Bayesian M e t h o d -------------------------------
46
3.3.3 Certainty Factor -----------------------------
49
3.3.4 Dempster-Shafer Theory of Evidence -----------
53
3.4 Development
of Expert Systems ----------------------
4. Building a General Purpose Fuzzy Expert SystemtGPFES)
57
67
4.1 Development
toward fuzzy expert systems ----------
67
4.2 Development
of the c o i m p l i c a t i o n -------------------
74
4.3 Knowledge representation in GPFES -----------------
82
4.4 Coimplication and Resolution procedure ------------
91
4.5 GPFES inference engine -----------------------------
93
vi
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5. Application of GPFES to microwave precipitation
retrieval technigue
107
5.1 I n t r o d u c t i o n ----------------------------------------- 107
5.2 The use of satellite remote s e n s i n g ----------------
111
5.3 Review of Passive Microwave methods to retrieve
r a i n f a l l ---------------------------------------------
117
5.4 Estimation of rainfall rate from multiple microwave
frequencies using GPFES ----------------------------
121
5.5 Data set for a case study with SSM/I measurements
125
5.6 Knowledge Base to retrieve Precipitation ---------
138
5 .7 R e s u l t s ----------------------------------------------
155
6. Conclusions
157
References
161
Appendices
A. System Configuration for Image Processing to retrieve
precipitation from Satellite Microwave Measurements
B. Interactive sample sessions
vii
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181
185
List of Tables
4.1
Fuzzy implication operators --------------------------
72
4.2
Knowledge schema ---------------------------------------
88
5.1
Temporal and spatial resolution of SSM/I channels —
127
5.2
Rules based on coimplication in the second tier ----
146
5.3
Asymmetrical difference for each r u l e ---------------
147
5.4
Membership functions used in the second tier -------
147
5.5
Rain rate i n d e x ---------------------------------------- 149
viii
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List of Figures
3.1
Major components of an expert system ----------------
34
4.1
Knowledge database design through schemas -----------
87
4.2
Diagram for G P F E S ----------------------------------
4.3
diagram of GPFES inference engine --------------------
4.4
A detailed flow diagram of fuzzy inference engine
in
G P F E S -----------------------------------------------
104
5.1
87
96
Relationship between brightness temperature and rainfall
rate over w a t e r ------------------------------------
123
5.2
Footprint geometry for the four SSM/I frequencies
—
129
5.3
SSM/I orbit and scan g e o m e t r y ---------------------- 131
5.4
An example of SSM/I orbit and scan geometry --------- 133
5.5
INSAT-IR i m a g e --------------------------------------
5.6
Composite images of
136
a. vertically polarized brightness temperature at 19 GHz
---------------------------------------------------------- 136
b. vertically polarized brightness temperature at 37 GHz
---------------------------------------------------------- 137
c. vertically polarized brightness temperature at 85 GHz
---------------------------------------------------------- 137
5.7
A schematic example of coimplication rule 1 --------- 154
5.8
The image, representing rainfall rate, obtained from
G P F E S -----------------------------------------------
156
ix
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1
Chapter 1
Introduction
1.1
Problem Statement
The analysis of expert systems consists mainly of
developing
a measure of uncertainty and establishing mechanisms of infe­
rence.
Because much of the information in the knowledge base
of a typical expert system is imprecise and vague, management
of uncertainty is an important issue.
expert systems use two-valued
However,
logic
Many of
the
existing
and probability theory.
it is widely recognized that such methods have some
shortcomings.
Zadeh[Z183a]
points out that conventional approaches to
the management of uncertainty in expert systems are intrinsic­
ally inadequate because
the uncertainty
they
fail to consider that
much of
in such systems is possibilistic rather than
pobabilistic in nature.
Expert systems are the fastest growing and the most vis­
ible
field of Artificial Intelligence [BF82,Gw82,Md79].
Al­
though several ad hoc models have been developed to deal with
vagueness,
there
is a strong need for a globally applicable
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2
method of dealing
with
models developed
vagueness
thus far have
in
expert
systems.
The
no theoretical base and have
been inadequately analyzed.
The objective of
of an expert in
expert systems is to get the knowledge
a specific problem
an appropriate structure,
domain,
and distribute
in the same problem domain.
represent it in
it to other
To accomplish
this,
users
management
of uncertainty is needed together with knowledge acquisition,
knowledge
representation, an inference engine,
some control
strategies and a user interface. Little development has
made
thus far towards the representation
been
and propagation of
uncertainty.
Management of uncertainty is related to a
analysis
of
computational
uncertainty from the premises to the conclusion
[Z183a]. Uncertainties in the premises will be transmitted to
the conclusion.
In other words,
the premise and the conclusion
We attempt
to establish
there is a relation between
in an inference.
such a computational
framework
based on fuzzy logic to deal with uncertainty in fuzzy expert
systems.
In a fuzzy conditional inference
in
fuzzy
expert
systems the problem is that if we are given values for A -> B
and
A',
we
modus ponens.
have to find
In
the real
a consistent value for B ’ through
fuzzy logic system we propose to
use the equivalence relation for
modus
ponens of the infer­
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3
ence on fuzzy expert systems instead of implication, survey a
property in equivalence
roximate reasoning
and introduce a new approach to app­
using
this new concept
Based on the equivalence method,
of dealing with
we have
of modus ponens.
developed a theory
vagueness in expert systems
which is called
coimplication.
Approximate reasoning is implemented using a "generalized
modus
ponens"
logics,
modus ponens
allows {Y is B} to be derived from {X is A} and
{(X is A) ->
(Y is B)}.
[Z177c].
In
classical
If A and B are described by fuzzy values, we have
to define the fuzzy
implication
match the meaning of any given
and
we need a mechanism to
situation with the premise of
the rule. The generalized modus ponens gives a definition for
such a mechanism. The method we propose will also give such a
mechanism.
The purpose of this research is
to provide
gui­
dance for modeling general domains.
The goals of this research are :
1. to
develop
the concept of
coimplication
based on the
equivalence operator.
2. to use coimplication
to develop
the structures,
facts
and rules which represent vague information and propaga­
tion of uncertainty in expert systems,
and to design an
inference engine for handling them.
3. to design a general purpose
fuzzy expert system (GPFES)
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4
based on the concept of the coimplication in a fuzzy en­
vironment
and
to
use
the concept
of the schema-like
knowledge representation for the management of the know­
ledge data structure in the implementation of GPFES.
4. to apply
GPFES,
based on
the construction
the coimplication method, to
of a knowledge base for passive micro­
wave based precipitation retrieval [BT85,Md79].
Precipitation retrieval by means
long standing problem in Meteorology.
of remote sensing is a
It is an ill conditioned
problem from a mathematical perspective because of non-unique
relationships between the radiation signals and
tation intensities.
Mugnai [SM88]
Mugnai
and Smith [MS88]
have developed
a theoretical
the precipi­
and
Smith and
foundation
for
avoiding some of the problems based on a multi-spectral pass­
ive microwave approach.
theoretical
However,
calculations that
channel framework,
it is
even in
evident
from
their
the case of a multi­
there remain ambiguities in the relation­
ships between microwave brightness temperatures.
We have developed a knowledge base for
testing
the co­
implication technique specifically for precipitation
retrie­
val from multi-channel passive microwave radiometer
ments obtained from the Air Force SSM/I satellite.
plication theorem
is
designed to remove
measure­
The coim­
the ambiguities in
the functional relationships in order to assign specific rain
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5
rate intensities
temperatures.
to
a 4 channel
The test is
combination
conducted
of brightness
over the
Indian ocean
within the Southwest-East Asian monsoon domain.
1.2
Literature Survey
There are four main reasons for the presence of vagueness
in an expert system[BT85].
1) The reliability of information : the factual knowledge
is a result of ill-defined concepts in the observation, or in­
accurate and poor instruments used,
and
the
result of weak implication when the expert or
ineer
is
unable
to
rule
set is a
knowledge eng­
establish a strong correlation between
premise and conclusion.
2) The inherent imprecision of
language : If rules
are
the rule
not expressed in a formal language,
their meaning cannot be interpreted exactly.
matching is no longer adequate to compare
with the premise.
representation
This requires a
Thus, a lexical
subsets
"semantic"
of
facts
matching com­
paring the approximate meaning of facts and premise.
3) Incomplete information : In this case we need to par­
tially match facts and premise.
Approaches to deal with this
kind of uncertainty range from default reasoning with consis­
tent assumptions[Rr80] to analogical reasoning[Wp80].
4) The
aggregation of
rules
from different
knowledge
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6
sources or different experts
: There are four possible errors
that can occur in knowledge represented
[SSS81]; conflicting!
redundant,
as production
rules
subsuming, and missing rules,
i) conflicting rules : They succeed under the same circum­
stances,
but make
contradictory conclusions and increase
the level of uncertainty by creating inconsistencies,
ii) redundant rules
: They succeed under
the same
circum­
stances, make the same conclusion and may create an inflated
assessment of the certainty of the conclusion,
iii) a subsumed rule : The premise
subset of the second
of
and creates an
the
first rule is a
over-estimate of the
certainty of the common conclusion,
iv) missing rules : They fail
sion
under the right
to provide a
needed conclu­
circumstances and create incomplete
information.
Expert systems have modelled uncertainty and imprecision
in various ways. Many systems,
including the PROSPECTOR mine­
ral exploration system[DHN78], have used the Bayesian approach
for the relative likelihood ratio to quantify the strength of
a given rule.
Shortliffe devised a scheme
based on
what he
called certainty factors (CFs) for
measuring
that could be placed
conclusion as a result of
in any given
the evidence so far[Se76,SB75].
A certainty
the confidence
factor
is
the
difference between two component measures; CF[h:e] = MB[h:e]-
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7
MD[h:e], where CF[h:e] is certainty of the hypothesis h given
evidence e,
MD[h:e]
MB[h:e]
is a measure
is a measure of
function [Sg76] has
of belief in h given e and
disbelief in h
been
given e.
developed within
The belief
the framework of
Dempster’s work on upper and lower probabilities induced by a
multivalued mapping[Da67].
These models have several problems when applied to general
domains[BT85].
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8
Chapter 2
Background
2.1
Fuzzy sets
In this section some basic concepts
discussed.
in fuzzy set theory
The theory of fuzzy sets,
1965[Z165],
introduced by Zadeh in
is a generalization of abstract set
attempted to generalize the classical notion of
proposition
sense.
are
to accomodate uncertainty
in the
In classical set theory an item either
theory.
a set
and a
nonstochastic
belongs to
set or it does not. It can not partially belong to a set.
the concept of
the set.
fuzzy sets,
Fuzzy set theory
founded basis for
is
used in
an item may
provides
the modeling
He
a
In
partially belong to
a systematic and well-
of imprecision.
Imprecision
the sense of vagueness rather than lack of know­
ledge about the value of a parameter as in tolerance analysis.
Fuzzy
in
set theory
which
provides
vague conceptual
rigorously studied.
a strict
phenomena
mathematical framework
can be
precisely and
It can also be considered as a
modeling
language well suited for situations in which fuzzy relations,
criteria, and phenomena exist.
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9
Let
us
consider
person is heavy.
the
problem
Someone who
sidered heavy by some people.
of
determining when a
is 180 pounds may not be con­
However
he
is certainly not
light.
In classical set theory one might say he
was
almost
heavy
and allow him a grade of membership of 0.8 in the set
of heavy people. This seems to be clearer.
Let
us
now
turn to the formal framework of fuzzy set
theory.
Definition 2.1:
Let U be a collection of objects denoted generically by x.
Then a fuzzy set F in U is a set of ordered pairs:
F =
X
{(x,X (x))
! x is a member of U } .
F
(x) is called the membership function or grade
of
memberF
ship of x in F which maps U to the closed interval [0,1].
The grade-of-membership value
X
(x) of an object x in F
F
can be interpreted as the degree of compatibility of the pre­
dicate associated with F and the object x.
ble to interpret
It is also possi­
X
(x) as the degree of possibility that x is
F
the value of a parameter fuzzily restricted by F.
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10
Example 2.1:
F = "Real numbers considerably larger than 20"
F = {(xfpf(x))
F
! x is a member of R) , where
0
if x < 20
-2 -1
if x > 20
1+(x - 2 0 )
The
assignment
of the membership function of a fuzzy set is
subjective in nature and,
in general, reflects the context in
which the problem is viewed.
Definition 2.2:
The support of
which
%
a
fuzzy set F is the set of all x 6 U at
(x) is positive.
F
Example 2.2:
Let the universe
as temperature.
be the interval
[0,120], with x interpreted
A fuzzy subset F of
U
labeled
WARM may be
defined by a grade of membership function such as
0
if
x < 60
if
60 < x
< 70
if
70 < x
<.80
if
80 < x i 110
/x - 60 \
V
X
20
/
2
<*> =
F
tx
-80^
'
20
1 - 2
/
Then, the support of WARM is the interval
(60,110].
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11
Definition 2.3
The height of a fuzzy set F is the supremum of
X
(x) over the
F
universe U.
hgt(F) = sup [X (x)}
x feU F
In the Example 2.2, the height of old is effectively 1.
Definition 2.4:
The
crossover
point
of
a fuzzy set F is
the point in the
universe U whose grade of membership in F is 0.5.
Definition 2.5:
A fuzzy set F is said to be normal if its height is unity,
that is, if
In
sup
xeU
Example 2.2,
X
=
F
the crossover point of WARM is 70
fuzzy set WARM is normal.
fuzzy sets,
To
and
simplify the representation of
it is convenient to use the following notation:
a finite fuzzy set F on U is expressed as
n
F =
the
%
(xl)/xl + ....
F
+
/C (xn)/xn
F
X
=
j=l
(xj)/xj.
F
when U is not finite, we can use the notation
f = £ r p <x>/x.
u
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12
Definition 2.6:
A fuzzy set F is convex iff ^xlfeU, ^ x 2 6 U ,
%
F
When
we
belongs
[Axl + (1 - X )x2] > mint# (xl)
F
want
to exhibit
to a fuzzy set F,
an element
X.6[0,l],
(x2) ] .
F
x6U
that
typically
we may demand that its membership
value be greater than some threshold.
Definition 2.7:
The o^-cut Fot, of F is the ordinary set of elements
their membership values are greater
than
or
such
equal
that
to some
threshold o(£(0,l].
F* = {x <£, U,
X
F
Similarly,
<x > > o O •
~
the strong o(-cut is defined by
F~ = {x <= U,
X
<x > ><*> •
F
The membership function of a fuzzy set F can be expressed
terms of the characteristic functions of its
to the formula
%
(x) =
F
where
X
-cuts according
sup min[o< ,
(x) ] ,
*€[0,1]
Hx
(X) =
F<*
1
if x 6 F ^
0
otherwise,
s
in
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13
The o^-cut sets are used for setting decision thresholds in an
expert system environment. At some point a yes/no decision is
needed in all expert systems.
We
will now define unions and
intersections of fuzzy sets.
Definition 2.8:
Let
A and B be
A and
fuzzy sets of U. Then the union of fuzzy sets
B is denoted by A U B and is defined
A U B = \
J
(X (x)
V
%
A
by
(x)) / x ,
B
U
where V is the symbol for maximum.
Definition 2.9:
Let A and B be fuzzy sets
of
U.
fuzzy sets A and B is denoted by A
A
AB
f\ B
the
intersection of
and is defined by
(x))/x, where A is the symbol for minimum.
= \
J
U
Then
A
B
Definition 2.10:
The complement of a fuzzy set A is denoted by A and is defined
by A =
1 -
PC
(x) )/x.
U
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14
Definition 2.11:
A linguistic
hedge
or
a
modifier
is
an
operation, which
modifies the meaning of a term, or more generally, of a fuzzy
set.
If
F is a fuzzy set
then the modifier m generates the
composite term G = m(F).
Mathematical models used for modifiers are :
:X
Concentration
(x) =
KX
Con(F)
Dilation
:
Z
<x) = (
Die(F)
2(X
Contrast
intensification
F
(x))
F
X
1/2
(x) )
F
if#
<*>>
1 - 2(1 N
(x)
e
[0,0.5]
F
X
(x))
otherwise
F
Generally the following linguistic hedges are associated with
the above
mentioned
mathematical operators.
If F is a fuzzy
set then
very F = Con(F)
more or less F = Dil(F).
Definition 2.12:
The product of A and B is denoted by AB and is defined by
AB = \ * ( x )
A
U
X
(x) /x.
B
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15
Definition 2.13:
If Fl,..,Fk are
Cartesian
defined
fuzzy subsets of Ul,..,Uk,
product
as a
fuzzy
respectively,
the
of Fl,..,Fk is denoted by Flx..xFk and is
subset
of
U l x . .xUk
whose
membership
function is expressed by
y
(xl,..,xk) =
Fix..xFk
X
(xl)A**A
FI
X
(xk).
Fk
Example 2.3:
Let A be the set of weights in pounds of five people.
A = {(Anna,140),(H a r r y ,175),(Na n c y ,165),(B i l l ,150),
(Eric,190)}.
Let
JC
(x),
the membership function of the fuzzy set
Heavy
Heavy, be defined on the following expression:
Heavy = 0.2/Anna + 0.8/Harry + 0.6/Nancy + 0.4/Bill
+ l./Eric.
Let B be the set of heights in inches of people.
B = {(Anna,57),(Harry,73),(Nancy,67),(Bill,62),(Eric,80)).
Let
Tall,
fC.
(x), the membership function of the fuzzy set
Tall
be defined on the following expression:
Tall s 0.3/Anna + 0.78/Harry + 0.7/Nancy + 0.45/Bill
+ l./Eric.
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16
Then
Heavy U Tall = 0.3/Anna + 0.8/Harry + 0.7/Nancy + 0.45/Bill
+ l./Eric
Heavy
Tall = 0.2/Anna + 0.78/Harry + 0.6/Nancy + 0.4/Bill
+ 1./Eric
Heavy = 0.8/Anna + 0.2/Harry + 0.4/Nancy + 0.6/Bill
+ O./Eric
Heavy Tall = 0.06/Anna + 0.62/Harry + 0.42/Nancy + 0.18/Bill
+ l./Eric
2
Heavy
= 0.04/Anna + 0.64/Harry + 0.36/Nancy + 0.16/Bill
+ l./Eric
0.5 Heavy = 0.1/Anna + 0.4/Harry + 0.3/Nancy + 0.2/Bill
+ 0.5/Eric
Con(Tall) = 0.09/Anna + 0.61/Harry + 0.49/Nancy + 0.2/Bill
+ l./Eric
Dil(Tall) = 0.56/Anna + 0.88/Harry + 0.84/Nancy + 0.67/Bill
+ l./Eric
Example 2.4:
Let XI = X2 = {2,4,6} and let A1 = 0.5/2 + 1/4 + 0.6/6 and
A2 = 1/2 + 0.6/4.
Then AlxA2 = 0.5/(2,2) + 1/(4,2) + 0.6/(6,2) + 0.5/(2,4)
+ 0.6/(4,4) + 0.6/(6,4).
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17
2.2 Possibility theory
The subject of possibility has been introduced by Zadeh[Z178].
Let X be a variable taking values in U.
Then
a
possibility
distribution, |T , associated with X may be viewed as a fuzzy
X
constraint on the values that may be assigned to X.
Such a
distribution
is characterized
by a possibility distribution
function TT :U -> [0,1] which associates with each xfi U the
X
degree of the possibility that X may take x as a value.
One
of the central concepts
of
possibility distribution.
distribution,
possibility theory is
In order to
define
it is convenient to introduce
the
that of a
a possibility
notion of a
fuzzy restriction.
Definition 2.14:
Let X be
a variable taking
values in
U characterized by
a
membership function ^
variable
(u).
F is a fuzzy restriction on the
F
X if it acts as an elastic constraint on the values
that may be assigned to X in the sense that the assignment of
value u to X has the form:
X = U :
X
<u > •
F
Whether a fuzzy set can be considered as a fuzzy restric­
tion
or not
obviously depends
is only the case if it acts
on its interpretation.
This
as a constraint on the values of
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18
a variable, which might take the form of linguistic term or a
classical variable.
Let R(X) be a fuzzy restriction associated with X as de­
fined in definition 2.14.
assignment equation
Then R(X)=F is called a relational
which
assigns
the fuzzy set
fuzzy restriction R ( X ) . For instance,
F
to the
A(X)=age of Jack and F
is the fuzzy set "young". The proposition "Jack is young" can
be expressed as R(A(X))
= F.
Definition 2.15:
Let
F
be a fuzzy set in an universe of discourse U which is
characterized
by
its
membership
function
X
(u), which in
F
turn is interpreted as the compatibility of u £ U with F. Let
X be a variable taking values in U and F act as a fuzzy rest­
riction, R(X), associated with X.
F",
which translates
distribution,JT
Then the proposition "X is
into R(X) = F associates a possibility
i which is postulated to be equal.to R(X).
X
The possibility distribution function, f|" (u), characterX
izing the possibility distribution || is defined to be nuX
merically equal to the membership function
(u) of F,
i.e.,
F
=
X
xF .
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19
Definition 2.16:
Let
F be a fuzzy set in the universe U and ]"[" a possibility
X
distribution associated with a variable X which takes values
in U. The
possibility measure,
X
IC (A) = sup
u 6A
natural
U is defined by
(u)
F
Zadeh[Z177b] proposes,
meaning in
TTL(A), of A C
in studying the representation of
language,
the
concept of a possibility
m easure.
Example 2.5:
Let us consider the possibility distribution which is induced
by the proposition "X is a small integer" .
jT
— {(1,1),(2,1),(3,.8),(4,.6),(5,.4),(6,.2)} and
X
Then the possibility measure IT(A) of A is
A—{3,4,5}•
1C (A ) = max (.8,.6,.4) = .8
Similar to probability theory there exist also conditio­
nal
possibilities. Such a conditional possibility distribut­
ion can be defined [Z181].
Definition 2.17:
Let
X
and
Y be variables in the universes U and V, respec­
tively. The conditional possibility distribution of Y,
X, is then induced by a
proposition
of
given
the form "If X is F
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20
then Y is G" and is denoted by
TT(Y/X) (v/u).
If we let A and B be fuzzy subsets of U, then the possi­
bility measure of the union is defined as
1C(B) and the possibility
and
B
is
related
means maximum and
by
measure
1C(A U B) = TC(A) V
of the
TC(A A B) <
tt:(A)
intersection of A
/\
TC(B),
where
V
means minimum.
2.3 Fuzzy logic and approximate reasoning
In Boolean logic truth values can be 0(false) or l(true)
and by
means of these truth values the operators are defined
via truth tables.
Zadeh
Fuzzy logic [Z175b]
has been
proposed by
and is an extension of set theoretic multivalued
in which the
discussion
truth values are linguistic
variables.
logic
In this
we limit consideration to possibilistic interpre­
tations of linguistic
variables
and will discuss mainly the
role of fuzzy logic and approximate reasoning for the manage­
ment of uncertainty in expert system.
Fuzzy
logic is essentially considered as an application
of possibility
theory to logic.
For
numerical truth values
v(A) and v(B), the logical operators are defined as
v(A and B) = min(v(A),v ( B ) )
v(A or
B) = max(v(A),v ( B ))
v(A ->
B) = min(1,l-v(A)+v(B))
v(not A)
s 1 - v(A).
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21
The employment
of
fuzzy ' logic
as a framework for the
management of uncertainty in expert systems makes it possible
to
consider a number of
issues
effectively or correctly
For example:
which
cannot be dealt with
by conventional techniques [Z183a].
1) If x is small then Y is large
with CF = 0.8,
in which the antecedent "X is small" and the consequent "Y is
large" are fuzzy
predicates
propositions because the denotations of the
small and large are fuzzy subsets SMALL and LARGE
of the real line. 2) Since
the
number of rules in an expert
system is usually relatively small (i.e. of
hundred),
there
such as "X is A ’"
the order of two
are likely to be many cases in which a fact
may
not
match exactly
the antecedent of
any rule of the form "If X is A then Y is B with CF = a". The
conventional
treat it
rule-based
systems usually avoid this issue or
in an ad hoc manner
because
partial matching does
not lend itself to analysis within the confines of two-valued
logic.
fuzzy
By contrast, the gradation of truth and membership in
logic
provides a natural way of
dealing with partial
matching through the use of the compositional rule
ence and interpolation.
of infer­
3) In many cases, the antecedent and
/or the consequent of a rule
contain
fuzzy quantifiers such as most, many,
implicit
or
explicit
few, many more, usually,
much of, etc.
Consider the disposition
"Student is young",
which may
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22
be interpreted
which in turn
as the proposition "Most students are you n g " ,
may be
as the conditional
expressed as a rule, or, equivalently,
proposition "If x is a student then it is
likely that x is young.". Here a disposition is a proposition
with implicit
fuzzy quantifiers
typical expert system are,
The
loping
in effect, disposition[Z183b].
approximate measures of uncertainty and
rules
of inference,
i.e.,
bilistic
expert
In statisti­
semantic evaluations are in the form of proba­
logic
relations
establishing
systems consisting of axioms,
and deductions based recursively on them.
systems,
ried
the rules in a
analysis of expert systems consists mainly of deve­
theories
cal
and many of
for
which
supplies
the calculus
the
of
basic
operators
probability.
In
and
existing
systems, the computation of certainty factors is car­
out through a combination of methods which are based on
two-valued
nized
logic and probability theory.
that such
methods have
It is widely recog­
serious shortcomings and, for
the most part, are hard to rationalize.
Zadeh [Z183a] shows
to
the
management
of
intrinsically inadequte
that
the
conventional
uncertainty
because
in
they
expert
approaches
systems are
fail to come to grips
with the fact that much of the uncertainty in such systems is
possibilistic rather than probabilistic
suggests
in
nature.
He also
a fuzzy-logic-based computational framework be
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em-
23
ployed to deal
with both possibilistic and probabilistic un­
certainty within a single conceptual system.
Fuzzy logic provides a natural framework for the manage­
ment of uncertainty in expert systems because
its main
pur­
pose is to provide a systematic basis for representing and in­
ferring from imprecise knowledge
ledge.
The greater
rather
than precise
expressive power of
from the fact that it contains
tional two-valued as well as
know­
fuzzy logic derives
as special
multi-valued
cases the
logics.
tradi­
The main
features of fuzzy logic which are of relevance to the manage­
ment of uncertainty in expert systems are the following[Z183a].
(a) In fuzzy logic, the truth values are allowed to range over
the fuzzy subsets of T. For example,
if T is the unit
inter­
val, then a truth-value in fuzzy logic, e.g., very true,
be interpreted as a fuzzy
defines the
possibility
subset of the unit interval
distribution
associated
may
which
with
the
truth-value in question. A fuzzy truth-value may be viewed as
an imprecise
charaterization of an intermediate truth-value.
(b) In fuzzy logic, the predicates may be crisp, e.g., mortal,
even,
father
of, etc. or,
more
generally,
tired, large, tall, much heavier, etc.
as multi-valued logics allow
some. However,
fuzzy,e.g.,
ill,
(c) Two-valued as well
only two quantifiers,
all
and
fuzzy logic allows the use of fuzzy quantifiers
exemplified by most, many, several,
few, much of, frequently,
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24
occasionally, about, etc. Such quantifiers may be interpreted
as fuzzy numbers which provide an imprecise
characterization
of the cardinality of one or more fuzzy or nonfuzzy sets,
(d)
Fuzzy logic provides a method for representing the meaning of
both nonfuzzy and fuzzy
predicate-modifiers
exemplified
by
not, very, more or less, extremely, slightly, much, a little,
etc. This leads to a
system for
variables[Z175a], i.e.,
computing
variables
with
linguistic
whose values are words or
sentences in a natural or synthetic language. For example, Age
is
a
linguistic variable when its values are assumed to be:
young, old, very
young,
not very old, etc., with each value
interpreted as a possibility distribution over the real line,
(e) In
two-valued logical
qualified,
principally
systems, a proposition, p, may be
by associating with p a truth-value,
true or false : a modal operator such as
possible or
neces­
sary: and an intensional operator such as know, believe, etc.
In fuzzy logic,
the
three
are : truth qualification,
principal modes of qualification
expressed as p is 7* , in which
is a fuzzy truth-value ; probability qualification,
as p is
X
» i-n which \
y
expressed
is a fuzzy probability ; and possi­
bility qualification, expressed
fuzzy possibility, e.g., quite
as p is 1C , in which
possible,
K.
is a
almost impossible,
etc.
Since
the
inference
processes
in
existing
expert
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25
systems are based
on
two-valued
logic
and/or
probability
theory, the principal tools for inference are the modus ponens
and/or its probability analog-Bayes’ rule and its variations,
the validity of these inference processes is open to question
since most of the information in the knowledge base of a typ­
ical expert system consists of a collection of
fuzzy
rather
than nonfuzzy propositions.
Among
viewed as
the traditional rules of inference - which may be
special
cases of the compositional rule of infer­
ence - is the modus ponens.
derive
To establish this fact, we shall
from the compositional rule of inference a more gene­
ral version of the modus ponens which in fuzzy logic
is ref­
erred to as the generalized modus ponens[MG81,Z177c].
Consider a pair of propositions {P1,P2} of the form
PI s
If X is F then Y is G,
P2 = X is F ’,
in which F, F ’ and G are fuzzy sets (or fuzzy predicates).
PI and P2 may be expressed as
PI ->
TT
= H
(Y|X)
P2 ->
where
TT (YlX)
tion
of Y given X,
X iu,v)
H
s1
A
|T = F ’ ,
X
denotes the conditional
(1
H
is
->£(u)
F
defined in
+X
possibility
terms of
distribuF and
G by
(v)), u and v are generic values
G
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26
of X and Y respectively and A is the min operator.
pC
:UxV -> [0,1] is the membership function of H.
H
By applying the compositional rule
and P2,
the possibility
distribution
of
of
inference
Y
to PI
is found to be
given by
TTY
= HoF*
where the right-hand side represents the composition of H and
F ’with respect to X.
More concretely,
X
Y
(v) = sup<;t (u ,v ) /\
u
H
= s u p < 2 (u) A
F’
x
F’
(1 -
X
<u) +
X
F
(v))).
G
This conclusion may be stated in the form of the syllogism
If X is F then Y is G
X is F*
Y is H o F ’
where HoF* is defined by
X
,
(v )* This syllogism expresses the
Y
generalized
modus
ponens.
The
generalized
modus
ponens
differs from its classical version in two respects. First, F ’
is not required to be
identical
classical case. And second,
the
with
F,
predicates
as IX
is in
the
F, G and F ’ are
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27
not required to be crisp.
and
the predicates
It can be verified that when F s F ’
are crisp,
HoF' deduces G and the above
syllogism becomes
If X is F then Y is G
X is F
Y is G
Standard formulations
both an
of
inference
processes
involve
implication and an inference rule, like modus ponens
or modus tollens.
In the modelization of approximate reasoning
by fuzzy sets, most researchers have basically focussed their
attention on the study of implication.
some difficulties identifying
Generally,
the statement
"If A
there are
then B " ,
even when dealing with classical logic. One of the reasons is
that the statement is understood
i.e.
the
whole
as
B will occur if A does,
modus ponens schema is
implicitely assumed
[TV85].
Trillas and Valverde [TV85] are devoted to the
modus ponens,
study of
especially in the context of multivalued logic
used to modelize
approximate reasoning
by
means
of
fuzzy
logic. After an overview of the concept and uses modus ponens
in some logics,
it
is
pointed out
that
procedure implied by modus ponens may be
in many cases the
described
by means
of a function.
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28
We now discuss the concept and meaning of logical rules.
A rule can be a directive which expresses
the assertion of
a formula [Rh80].
a permission as to
Consequently,
requirement for a rule of inference should be that
from true statements to true statements."
an inference is
not a statement
[Yr85].
the
only
"It leads
Therefore,
like an implication,
but a
procedure which uses implication to get true statements. Such
a procedure can be described only in a rule and
expressed in a schema. Thus,
symbolically
in the case of modus ponens, the
schema is
a -> b
a
b
that is,
if
a
Since a rule is
is true and a -> b
is
a procedure and not
true then
b is true.
a formula, it requires a
justification which follows from the truth table defining the
implication connective.
antecedent
"a"
is
Except
both
the truth value of the
false, by applying this procedure to the
whole table we can determine
quent "b" given
when
the
the truth value
of the
conse­
truth values of "a" and "a -> b " .
Modus ponens is
identified with the
the probability
of the sentence
"b", given both the probabil­
ities
sentences
and "a ->b", according to the
of
the
"a"
procedure
to determine
following schema
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29
I
P(a)
£
P(a->b) 1
7j
P(b) > m(£,lj)
where the function m depends on the
If P(a->b)
,
definition
is defined as P(b a) , then
be checked by applying the
=
of
€.Yj
P(a->b).
, as it can
law of total probabilities,
i.e.,
P(b) = P(b|a) P ( a ) + P(b|la) P(Ta) 2.P(b|a) P(a).
If P(a->b) = P(Ta V b ) , then we will have
P(a->b)
= P(1a V b)
= 1 - P (a ) + P(b) - P(ia
f\ b)
1 1 - P(a) + P ( b )
i.e.
£+7j -
= max(0,
From v(a->b)
m should be
variable.
also
A
±
1).
v(a->b*) if v(b) < v|b'),
non-decreasing
function
m
it follows that
with respect to the second
satisfying MP1 through MP4,
where
MP1 through MP4 are defined in the end of this section,
will
be called the modus ponens generating function. The nature of
the domain of a modus
ponens
generating
function
may vary
from the one system to another. Thus in Boolean, probabilistic
and multivalued
set of truth
logics, this domain will be VxV, V being the
values.
In fuzzy logic m is a function from
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30
U
UxV
V
[0,1] X [0,1]
into [0,1].
We digress
to
the analysis
of
the existence
of modus
ponens generating functions for some of the implication func­
tions of
the
multivalued
logic
system
introduced
in the
framework of fuzzy sets theory.
Fuzzy
implication functions are functions I from [0,1]X
[0,1] into [0,1],
defined
by I(x,y) = G(n(x),y),
continuous t-conorm and n a strong negation
G being a
function.
Among
strong implications, the most known and used are
IL(x,y) = min(l-x+y,l)
generated by G(x,y) = min(x+y,l) and N(x) = 1 - x,
IM(x,y) = max(l-x,y)
generated by G(x,y) = max(x,y) and N,
and
IP(x,y) = 1 - x + xy
generated
by G(x,y) = x +y - xy and
N.
Definition 2.18:
Triangular or T-Norms[SS63] are functions used
tional
conjunction
with domain
for
proposi-
in the unit interval of the
real line satisfying the axioms:
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31
(i)
T(0,0) = 0, T(a,1) = T(l,a)
= a
(ii)
T(a,b) <. T(c,d) whenever a
<_c , b < d
(iii) T (a ,b ) = T(b,a)
(iv)
T(T(a,b),c)
Similarly,
= T(a,T(b,c))
formulas for disjunction are triangular cono­
rms i.e. T-conorms.
T-Norms
with
T-conorms
also
the exception of the
satisfy
first
the axioms
axiom,
which
of
is
modified to be
(i) T ( 1,1) = 1, T(a,0) = T(0,a)
= a.
In many cases, the procedure modus ponens may be descri­
bed by a function m from [0,1]X[0,1]
among others, the two
basic
into [0,1] which satisfy,
conditions
m(x,x->y)
<_ y
and
m ( 1,1)=1.
According to the previous discussion
of
a modus ponens
generating function, the function m satisfies MP1 through MP4,
in such a way that
m(v(a,v(a->b))) is a lower bound for
the
validity of "b", i.e.
MP1:
m(v(a),v(a->b)) < v ( b ) ;
MP2:
■(1,1) = 1,
which guaranties the preservation of classical modus ponens;
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32
MP3:
m(0,y) = 0,
which may be
rephrased as "from a false
statement, anything
can be inferred", and;
MP4:
If x <L x ’ then m(x,y)
<. m(x' ,y),
which enables us to consider the chaining of inferences i.e.
m(m(v(a),v(a->b)),v(b->c)) < m ( v ( b ),v(b->c)) £ v ( c ) .
Modus ponens has been treated as a
be represented
procedure
which may
by means of functions to get valid statements
from valid conditional
statements.
Modus
ponens generating
functions for some fuzzy implication functions of multivalued
logic systems have been characterized.
In some way, the repr­
esentation of a conditional statement by means of an implica­
tion connective may be viewed as a sort of
formal
construc­
tion which is justified precisely because of their participa­
tion in a rule of inference which works.
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33
Chapter 3
A Review of Expert Systems
Computer scientists have tried
that perform remarkable
to develop computer
functions,
in
a way that would
considered intelligent if carried out by
research effort,
processing and
humans.
including robotic devices,
expert systems,
programs
be
The entire
natural language
is usually called artificial
intelligence(AI). A collection of AI techniques, which enables
computers to assist people in analyzing
decisions,
problems
and making
is called expert systems. An expert system is one
that handles real-world, complex problems requiring an expert’s
interpretation and solves problems using
a computer model of
expert human reasoning, reaching the same conclusion that the
human expert would reach.
The most
expert systems is that they
systems that can perform
significant
thing about
are highly successful. These are
skilled
medical
diagnosis,
mass-
spectrogram interpretation, predict crop disease, prospect for
minerals to give just a few examples.
based
on
an extensive
body of
An
knowledge
problem area. This knowledge is organized
rules which allow the system
to draw
expert
system
is
about a specific
as a collection of
conclusions from given
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34
data or premises.
There are in fact four essential
fledged expert system :
knowledge
components
of a
base, inference
knowledge-acquisition module and explanatory
ful-
engine,
interface.
The
structure of an expert system is shown in Figure 3.1.
Figure 3.1
Major components of an expert system
users^------------- explanatory
interface
^------ inference
engine
domain----------- ^knowledge <
experts
acquisition
module
The
knowledge base
1
) knowledge
base
is the program’s store of facts and
associations it "knows" about a subject
area
(such as medi­
cine). A critical design decision is how such knowledge is to
be represented within the program.
There are
many
choices.
For MYCIN, they chose to represent
knowledge
as conditional
statements, or rules, of the following form :
If A and B, then C.
The
inference
mechanism can
take many forms. We often
speak of the control structure to reflect the fact that there
are different controlling strategies for the system. A set of
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35
rules may be chained as follows:
If A then B
( rule 1 )
If B then C
( rule 2 )
A
C
This
is
input
( conclusion
called
forward
)
chaining,
or
data-directed
inference, because the data that are known drive
the
infer­
ences from left to right in rules, with rules chaining together
to deduce a conclusion C. In MYCIN, a backward
goal-directed
a
system
control strategy is used.
chaining or a
In backward chaining
starts with a statement of the goal to achieve and
works "backward" through
inference
rules i.e. from right
left, to find the data that establish that goal
Find
out about C
(goal
If B
Then C
( rule
1 )
IF A
then B
( rule
2 )
If A
then C
( implicit
to
:
)
rule )
Question : Is A true ? ( data )
Since there are many
rule chains and many pieces
ofdata
about Which the system needs to inquire, we say that MYCIN is
an evidence gathering program.
Knowledge acquisition is a bottleneck in
expert
system
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36
development.
Typically
with an expert,
the knowledge engineer must sit down
in many long
sessions, and extract the human
expert's domain knowledge for use in the expert
knowledge
expert’s
engineer
area
will often not
must
have
some
system.
understanding
the
so that he can talk to the expert. The expert
be
able
to
provide general problem solving
heuristics. Example problems and
the heuristics in
cess of the expert solving them must be presented.
ledge gathered
of
The
tends to
be
incomplete
the pro­
Some know­
and often not quite
correct.
In order to develop an expert system,
a domain expert for the knowledge
initial stages with the expert
we have
acquisition
to find
process.
The
will be an intense process of
finding the relevant information and how to get it. The know­
ledge engineer is responsible
for
weeding out the inconsis­
tencies in the knowledge base. Some tools have been developed
to aid the
knowledge acquisition process,
[DL82] which aids knowledge
the expert system
acquisition
such as TEIRESIAS
and
building EMYCIN [Vw79].
refinement for
SEEK[PZM81] gives
advice about rule refinement during the development of a diag­
nostic-type expert system.
A
computer
program
that functions like an expert in a
given domain is more likely to be accepted by experts in that
domain, and by nonexperts seeking
its
advice,
if the system
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37
can explain its actions.
need
not
be
a
A
consultative
psychological
model,
rule-based
imitating
system
a human’s
reasoning process. The important point is that the system and
a human
expert
arrive at the
use
same
the
same knowledge about the domain to
conclusions
knowledge base contains the
expert as well as
consideration.
facts
to
given
conditions.
domain-specific knowledge
about
a particular
problem
The
of an
under
When a rule is used, its actions make changes
to the internal data base, which contains the system’s decis­
ions. Explanations
require displays of how the rules use the
information provided by the user to make various intermediate
deductions
and
information
finally
to
arrive
at
the answer.
If the
contained in these rules adequately shows why an
action was taken, an explanation can simply entail displaying
each rule or a free-text translation.
The
purpose of the explanation
capacity is to give the
user access to as much of the system’s knowledge as possible.
It should be easy for a user
to
get a complete, understand­
able answer to any sort of question about the system’s
know­
ledge and operation.
3.1
Knowledge representation
There are several methods of
any
of
which
can
be
used
knowledge
alone
or in
representation,
conjunction with
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38
others
to build expert systems.
Each
method
provides
the
program with certain benefits, such as making it more effici­
ent, more easily understand, or more easily modified. A thor­
ough
description of them is given in "The Handbook of Arti­
ficial Inteligence"
[BF82].
The three main knowledge
repre­
sentation schemes are rules, semantic networks and frames.
Rule-based
premise THEN
knowledge
representation uses the form
"IF
conclusion". When the current problem situation
matches the condition part
it is performed.
This
of a rule, the conclusion part of
conclusion
part
may
direct program
control or may instruct the system to reach a real conclusion,
i.e. add a new fact to the database, and so on. Rules provide
a
natural
way
and changing
for describing processes driven by a complex
environment. A set of rules can specify how the
program should react to the
detailed advance knowledge
changing
about
the
data without requiring
flow of
control. The
term rule has a much narrower meaning than it does in ordinary
language.
It
refers
to the most
popular type of knowledge
representation technique[WH78].
Rules provide
tions, directives,
a
formal way of representing recommenda­
or strategies. The matching of a r u l e ’s con­
dition to the facts can produce
what
are
called
inference
chains. The inference chain indicates how the system used the
rules to infer the identity of a final conclusion. An
expert
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39
system's inference chains can be displayed to the user to help
explain how the system reaches its conclusions.
Semantic networks
were
originally
as psychological models of human
memory
developed
for
use
but are a
standard
representation method for AI and expert systems[Br79]. CASNET
[WKS78]
is an important example of using the semantic network
method. The semantic network consists of points called nodes,
and arcs. The latter connect those nodes which are related by
some semantic relation.
The
nodes
in
stand for objects, concepts, or events.
representing hierarchies include
a
semantic
network
Common arcs used for
isa and
has-part.
Given a
semantic network, because we know about the properties of the
relations linking the nodes,
we can infer
a conclusion from
the network as in the rule-based method. The isa relation and
the has-part relation establish a property inheritance hierar­
chy in the
network.
Items
lower in the network can inherit
properties from items higher up in the network. This can save
space
be
since information about similar nodes d o e s n ’t have
to
repeated at each node. Semantic networks are a useful way
to represent knowledge in domains that use taxonomies to simp­
lify problem solving. Also, the semantic network
representa­
tion is useful because it provides a standard way of analyzing
the meaning of a sentence and have been used in natural lang­
uage
research to represent complex English sentences [NR75].
However,
semantic
networks
can
suffer
from combinatorial
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40
explosion as well as over generalization!for X isa bird, birds
have-property flies, thus X flies; but X is an ostrich.)
Frame-based
knowledge
representation uses a network of
nodes connected by relations, and organized into a hierarchy.
Each node
represents a
concept
that
may be
attributes and values associated with the
the hierarchy inherit properties of
term
frame
refers
to
network.
Frames
A frame is
node. Nodes low in
higher-level nodes.
are
organized
The
the method employed by
much
like
a semantic
In a frame system each node is defined by a collec­
tion of attributes(e.g. size,
attributes(e .g . large, tall
ed
by
a special way of representing common
concepts and situations.
CENTAUR[Aj83].
described
height),
and values
of
those
), where the attributes are call­
slots. Each slot can have procedures attached to it which
are executed when the
Each slot can have
information
in
the slot
any number of procedures
Three useful types of
procedures
attached to
is changed.
attached to it.
slots
are as
follows:
i)
if-added : when new information is placed in the slot
ii)
if-removed : when information is deleted from the slot
iii) if-needed : when information
is
needed
from the slot,
but the slot is empty.
Frame
systems
expectations about
are
the
useful
for
problem
domains where
form and content of the data play an
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41
important role in problem solving such as interpreting visual
scenes or understanding speech.
3.2
Inference engine
The
inference
knowledge base to
engine
solve
is
an interpreter that uses the
the problem at hand. The main point
of interest in inference engine
is
that
structure information must be kept in the
all of the control
inference
engine.
The explicit division between a knowledge base and an infere­
nce
engine offers a degree of domain-independence. Knowledge
base
content is strongly
influenced by the control strategy
used in the inference engine. The inference engine and knowl­
edge base are not
cases.
necessarily
totally
independent
in many
It is responsible for determining what piece of know­
ledge to
inference
use next and for scheduling necessary actions.
The
engine also is responsible for determining when to
ask the user a question and when to search the knowledge base
for the information.
It must
deal with imprecise and uncer­
tain information.
Inference engines operate in
chaining,
backward
chaining
one of
or
both.
two
ways:
Forward
forward
chaining
begins with some data and continues along the inference chain
until it reaches a final conclusion. Backward chaining begins
with a final conclusion and continues backward until it finds
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42
a complete path of evidence for one of the satisfactory states.
EMYCIN[Vw79] is an example of a backward chaing method.
[Mj82a]
is
a forward chaining expert system.
systems which
use
a combination
of
chaining methods; e.g. SPERILL[OFY84].
use meta-knowledge,
which
is
forward
XCON
There are some
and
backward
The inference
knowledge about
engine
the system’s
knowledge i.e. how best to utilize various knowledges,
which
pieces of knowledge to use first, whether a piece of knowledge
should be inferred
and
when
knowledge may be in the same
to stop processing.
The meta­
knowledge base or in a separate
knowledge base.
EMYCIN[Vw79],
MYCIN[Se76],
basically a domain-independent version of
is an appropriate skeletal system for developing
a consultation program that can request data about a case and
provide
an interpretation
or
analysis.
EMYCIN
MYCIN inference engine to be applied to a new
whose problem-specific
MYCIN rule language,
knowledge
providing
can
all
allows the
problem domain
be represented in the
of M Y C I N ’s
features as
well as a human-engineered system-building environment, which
greatly facilitates entering and debugging the knowledge base.
In EMYCIN, the initial goal determines the value of the
level goal attribute.
top-
EMYCIN works on the goal of establish­
ing the value of the attribute of some object. To do this,
retrieves
a
it
precomputed list of rules whose consequents are
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43
known to
bear
on
that goal,
and
rules until it either
establishes
certainty or exhausts
the
deduced
it
attempting
truth
resorts
to
of its
to
rule
it attempts to apply the
the
value
list.
asking
the
If
with complete
no
value can be
user for the value.
In
apply a rule, EMYCIN must first establish the
antecedent,
certainty of each of its
which
requires
conditions.
that
the
To do this, the system
typically has to establish the values of
objects. This sets up subgoals
determining
other attributes of
are
addressed by using
the same mechanism recursively.
In KAS[Rr81],
At any given time
there is
no formal top-level hypotheses.
KAS is either trying
to identify the best
top-level hypothesis to pursue or trying to identify the best
question to ask the user to
first mode will be called
establish
that
hypothesis. The
the goal-selection
mode
and
the
second, the question-asking mode. Goal selection is guided by
information the user supplies during the session.
3.3
Reasoning with uncertainty in expert systems
3.3.1 Problem definition
Uncertainty just in data can be compounded when aggregat­
ing
uncertain
data
in
the
premise
certainty measures to the conclusion.
and
when propagating
Suppose
that "X is A"
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44
has a degree of
confidence
xl and there is a rule "(X is A)
-> (Y is B) with a degree of confidence rl".
Then
we should
be able to decide yl, the degree of confidence related to the
conclusion
"Y is B",
yl must be a function
of xl
and
rl.
Bonissone and Tong[BT85] have stated the questions related to
uncertainty management for reasoning as follows:
How can we elicit
from
the
expert
consistent
values
of rl ?
How can we compute or otherwise provide the value of xl ?
What are
the forms by which rl and xl should be repres­
ented : a scalar, an interval, a distribution, a lingui­
stic expression ?
Are
they
defined on an absolute scale or on a relative
one ?
In
a more
of multiple
general case,
clauses,
how
when the premise is
can
we
aggregate
composed
the degree of
certainty of the facts comprising the clauses of the premise !
i.e. what is the function T(xl,,,xn)
that determines xp, the
degree of certainty of that conclusion ?
Prade[Ph77] has modeled this kind of uncertainty using a
fuzzy extension of modal logic,
necessity and possibility.
the degree of
certainty
based on Zad e h ’s concepts of
It has also been
is
rather than a scalar[Bp83].
suggested
that
represented by a fuzzy interval
The degree of necessity[Ph77] is
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45
represented by the lower bound of the fuzzy interval
degree of possibility by its upper bound.
and the
Necessity and pos­
sibility respectively represent the amount of support for the
hypothesis, and the amount of failure to refute the hypothes­
is provided by the evidence. The value of the necessity of an
hypothesis is
always
smaller than or
equal to the value of
its possibility. Violation of this constraint during aggrega­
tion of
conclusions
from
inconsistencies among the
different experts
knowledge
suggests
sources.
rule
Our method to
handle uncertainty reasoning will be discussed in Chapter 4.
In the numerical
expert system,
sed in
representation
of
uncertainty in the
there are several methods and will be discus­
this section.
The
Bayesian
approach,
employed
in
PROSPECTOR(D 7 ), uses an effective likelihood ratio to quanti­
fy
the
strength
measures
the
of
a
given rule.
sufficiency
This
likelihood ratio
of a piece of evidence to prove a
given hypothesis.
The certainty factor (CF)
in MYCIN[Se76].
belief and a
The CF is the difference between a degree of
degree of disbelief in a given hypothesis after
supporting evidence is found.
preted as
approach [SB75] has been used
the relative
The CF
increase
was originally
or
inter­
decrease of a ratio of
probability.
The
belief
function,
proposed
by
Shafer [Sg76],
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was
46
developed within
and
lower
the framework
probabilities
of
Dempster’s work on upper
induced by
a multivalued
mapping
[Da67]. In this context, the lower probability was associated
with a degree
of
belief
degree of plausibility.
this approach.
and
the upper
probability with a
However, there are two problems with
A computational problem has been encountered,
since the evaluation of the degree of belief and plausibility
requires the exponential in the cardinality of the hypothesis
set.
The second problem results from a normalization process
present in both Dempster’s work and Shafer’s. Zadeh[Z179] has
argued that this normalization process
can lead to incorrect
and counter-intuitive results.
3.3.2 Bayesian Method
Many systems,
including the PPOSPECTOR mineral explora­
tion system[DHN78],
have used Bayes’s theorem as
the thread
for tying together information from disparate sources. Bay e s ’s
rule provides for computation of relative likelihoods between
competing
hypotheses
on
the
strength
of the evidence.
It
depends on the formula
L R (H :E ) = P(E:H) / P(E:H’),
where
the
of the event
likelihood ratio LR is defined as the probability
or
evidence
E given a particular hypothesis H
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47
divided by the probability of
the evidence given the falsity
of that hypothesis (H’). Thus if we know the
the
evidence
probability
of
given the hypothesis, and its negation, we can
work out the likelihood of the hypothesis in the light of the
evidence. For
given
instance,
chicken-pox
chicken-pox
if we know the probability of spots
and
the
patients,
probability of spots among non­
we can
compute
the
probability
of
chicken-pox given spots.
This
piece
of
likelihood
evidence
ratio
to
measures
prove
the
a given hypothesis. Pednault,
Zucker and Muresan [PZM81] concluded that
conditional
independence
hypotheses
and
inconsistent
under
of
the
the
negation
could
be
assumption of
both
under the
of the hypotheses were
with the other assumptions of an exhaustive and
theories rest on the belief that for
unlikely
the
evidence
mutually exclusive space of hypotheses.
how
sufficiency of a
it
true.
is,
there
no
matter
probability about some hypo­
thesis there must be some evidence
our views(beliefs) on the matter.
there
remaining forever unchanged.
everything,
is a prior probability that it
Given a prior
process would stop right
Essentially, Ba y e s ’s
we can call on
to adjust
If there were not, then the
with the
prior
probability
Given relevant evidence, we can
modify this prior probability to produce a posteriori
proba­
bility of the same hypothesis given some new evidence.
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48
Let P(H) be the prior probability of some hypothesis and
P(H:E) be the
posterior
probability
of the same hypothesis
given some item of relevant evidence E. By definition,
P(H and E)
P(H:E)
and
P(E and H)
=
P(E:H)
=
P(E)
Thus, we get
P(H)
P(E:H)P(H)
•P(H:E)
=
P(E)
One
it assumes
important criticism
independence of the variables being used. This is
quite a serious point.
symptoms.
on the Bayesian method is that
The
temperature".
first
For example, suppose that we have two
was
"fever"
There would be no
and the second was "high
point
in having both items
because they mean the same thing. They are exactly correlated.
But, in some cases,
twice and
therefore
probability of
we would have included the same evidence
either incremented, or decremented, the
influenza
more
than we should have done and
the final posterior probabilities would be wrong.
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49
3.3.3 Certainty Factor
Shortliffe devised a scheme
based on
what he
called a
certainty factor (CF) for measuring the confidence that could
be placed in any given conclusion as a result of the evidence
so far [Se76].
A certainty factor is
the difference between
two component measures:
CF[h:e]
where CF[H:e]
is the certainty of the hypothesis h given evi­
dence e, MB[h:e]
MD[h:e]
= M B [h :e ] - MD[h:e],
is
a measure
of
belief
in h
is a measure of disbelief in h given e.
from -1 (false)
to 1 (true)
meaning ignorance.
with
CF can range
values
and 0
MB and MD can range between 0 and 1.
CF reflects a simple balancing
The above
fractional
given e and
formula does
of evidence for
not permit
conflict of evidence as opposed
to
The
and against.
any distinction
between
lack of evidence i.e. MB
and MD both high, or MB and MD both low, respectively. MB and
MD obey some of the axioms of probability but are not derived
from a population sample of any kind and therefore cannnot be
given a statistical interpretation.
CF is an alternative to conditional probability that has
several
advantages in M Y C I N ’s domain. CF is in the statement
of decision rules
themselves.
In
corresponds to the conclusions in
this
the
case the evidence E
premise
of the rule.
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50
x
Thus A and B and C -> D is a representation of the
statement
CF[D: A and B and C] = x.
In accordance with subjective probability theory, it may
be argued that the expert’s personal probability P(h) reflects
his or her belief in h at
any
given time. Thus 1 - P(h) can
be viewed as an estimate of the expert’s
the truth of h.
tion
If P(h|e)
disbelief regarding
is greater than P(h),
the observa­
of e increase the expert’s belief in h while decreasing
disbelief regarding the truth of h. In fact, the proportionate
decrease in disbelief is given by the following ratio:
P ( h !e ) - P(h)
1 - P(h)
This ratio
is called the measure of increased belief in
h resulting from the observation of e, i.e. MB[h:e]. Suppose,
on the other hand,
observation
while
of
that P(h|e)
e would
were less than P(h). Then the
decrease
the expert’s belief in h
increasing his or her disbelief regarding the truth of
h. The proportionate decrease in belief in this case is given
by the following ratio:
(
P(h)-P(h!e))/P(h).
We call this ratio the measure of increased disbelief in
h resulting from the observation of e, i.e., MD[h:e]. To sum­
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51
marize these results,
we
consider
the measure of increased
belief, MB[h:e] to be the proportionate decrease in disbelief
regarding the hypothesis h that results from the
e.
observation
Similarly, the measure of increased disbelief, MD[h:e] is
the proportionate decrease in belief regarding the hypothesis
h that results from the observation e, where belief is estim­
ated by P(h) at any
time
and
disbelief is estimated by 1 -
P(h) .
The above
definition may be specified formally in terms
of conditional and a priori probabilities
:
if P(h) = 1
M B [h :e ]
m a x [P (h | e ),P(h)] - P(h)
o . w.
m a x [1,0] - P (h )
if P ( h )
M D [h :e ]
=
' min[P(h|e),P(h)] - P(h)
o.w
min[l,0] - P(h)
*
And we
find
a
measure termed a certainty factor (CF),
that combines the MB and MD :
CF[h:e]
= M B [h :e ] - MD[h:e].
The certainty factor is an artifact for combining degree
of belief and
disbelief
into a single number. Such a number
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52
is needed
in order
to facilitate comparisons of the eviden­
tial strength of competing hypotheses.
A property of CF is CF[h:e] + CF[ not H:e] = 1,
confirms h.
That is,
although
where e
evidence may support a hypo­
thesis with degree x, it does not support the negation to the
hypothesis with degree 1 -
x.
CF[not h:e] = MB[not h:e] - MD[not h:e]
P(not h|e) - P(not h)
=
0
-
- P(not h)
[1 - P ( h | e )] - [1 - P(h)]
1 - P(h)
P(h) - P ( h | e )
1 - P(h)
CF[h:e] = M B [h :e ] - MD[h:e]
P(h|e) - P(h)
=
-
0
1 - P(h)
Thus CF[h:e] + CF[not h:e] = 0.
This
result
occurs because
for any h and any e MB[h:e] = MD[not h:e]. This is intuitive­
ly
appealing
since
it states that evidence that supports a
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53
hypothesis
disfavors
the
negation
of the hypothesis to an
equal extent.
The
CF model is equivalent in part to the simple proba­
bility model.
The
empirical
success
stands in spite of theoretical
of CF
objections.
model in MYCIN
The fact that in
trying to create an alternative to probability theory or rea­
soning,
Shortliffe
and Buchanan duplicated the use of stan­
dard theory demonstrates the difficulty of
creating a useful
and internally consistent system that is not
isomorphic to a
portion of probability theory [Aj76].
3.3.4
Dempster-Shafer Theory of Evidence
The theory was first developed by
and extended by
Glenn
Shafer.
Dempster in the 1960s
He published "A mathematical
theory of Evidence"[Sg76]. Recently researchers have begun to
investigate applications of the theory to expert systems[Bj81,
F 1 8 1 ,G LF81].
The advantage
lity to model the
of the Dempster-Shafer theory is its abi­
narrowing
of
accumulation of evidence
and
diagnostic reasoning
medicine
in
a
the hypothesis set with the
process
and
that characterizes
expert
reasoning in
general.
An expert uses evidence that, instead of bearing on
a single
hypothesis
bears on
a
larger
in
the original hypothesis set,
subset
of this set.
The
often
functions and
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54
combining rule of the Dempster-Shafer
theory are well suited
to represent this type of evidence and its aggregation.
The generality
of
the Dempster-Shafer belief functions
is an avoidance of the Bayesian restriction that the
ment of belief to a hypothesis impies commitment of
commit­
the
re­
maining belief to its negation i.e., that P(h) = 1 - P(not H).
As in the CF model,
the
beliefs
original set need not sum
allotted to
subsets
of
to 1:
the
in each hypothesis
some
of the
original
in the
belief can be
hypothesis set.
The
Dempster-Shafer model includes many of the features of the CF
model but is based on a firm mathematical foundation. This is
a clear advantage over the ad hoc nature of CF m o d e l .
The Dempster-Shafer
[0,1] to indicate
evidence.
belief
This number
supports the hypothesis.
Shafer
model
theory
in
a
is the
Unlike
uses
a number in the range
hypothesis given a piece of
degree to
which the evidence
the CF model, the Dempster-
avoids the use of negative numbers. The impact
of each distinct piece of evidence on the subsets of 0 , which
is a
frame of discernment (hypothesis s e t ) , represented by a
function called a
is a generalization
basic probability assignment (bpa).
of
the traditional
A bpa
probability density
e
function. Using 2, the enlarged domain of all subsets of 0, a
bpa denoted m assigns a number in [0,1] to every subset
such that the numbers sum to 1. The
of @
quantity m(A) is exactly
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55
A,
where
A is an element
of 2
&
and the total
belief is 1.
This portion of belief cannot be further subdivided among the
subsets
of A and does not include portions of belief commit­
ted to subsets
of A.
Since
entails belief in subsets
belief
in
containing
a
subset certainly
that subsets,
be useful to
define a function that
computes a
of belief in
A. This quantity would include not only
it would
total
amount
belief
committed exactly to A but belief committed to all subsets of
A.
Such
a
function,
called a belief function denoted Bel,
corresponding to a specific bpa, m, assigns to every subset A
of 0 the sum
of A by m.
of the beliefs committed exactly to every
Thus,
Bel(A)
subset
is a measure of the total amount of
belief in A and not of the amount committed precisely to A by
the evidence giving rise to m.
The Dempster
combination
rule
differs
from the MYCIN
combining function in the pooling of evidence supporting mut­
ually exclusive hypotheses.
Let Bell and Bel2
and ml and m2 denote two belief func­
tions and their respective b p a ’s.
new
bpa,
denoted ml ® m2,
Dempster’s rule computes a
which
represents
the
combined
effect of ml and m2. The corresponding belief function, deno­
ted Bel ® Bel2, is easily computed from ml ® m2 by the defin­
ition of a
belief function.
If
we
sum all products of the
form ml(x)m2(Y), where X and Y run over all subsets of
0,
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the
56
result is 1
by
elementary algebra and
the
definition of a
bpa:
^Eml(X)m2(Y) = ^ m l (X )2lm2 (Y ) = 1 x 1
The
= 1.
bpa representing the combination of ml and m2 apportions
this number 1,
the total amount of belief, among the subsets
of 0 by assigning ml(X)m2(Y)
There are
typically
several
to the intersection of X and Y.
different
subsets of
0
whose
intersection equals that of X and Y. Thus, for every subset A
of 0, Dempster’s rule defines ml@m2(A)
to
be the sum of all
products of the form ml(X)m2(Y), where X and Y
run
over all
subsets whose intersection is A.
In the most general situation, a given piece of evidence
supports many of the subsets of 0,
each
to varying degrees.
The simplest situation is that in which the evidence supports
one
subset to a certain degree and
the
assigned to d. Because of the modular way
remaining belief is
in which knowledge
is captured and encoded in MYCIN, this latter situation appl­
ies
in
the case of MYCIN rules. If the premises confirm the
conclusion
of
a rule with degree s, where s is above thres­
hold value, then the r u l e ’s
of ©
effect
can be represented by a bpa.
singleton corresponding
on belief in the subsets
This bpa assigns s to the
to the hypothesis
in the conclusion
of the rule, call it A, and assigns 1 - s to 0.
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57
The Dempster-Shafer theory is appealing in its potential
for handling
evidence
bearing
on categories of diseases as
well as on specific disease entities.
It facilitates the
ag­
gregation of evidence gathered at varying levels of detail or
specificity. Thus collaborating experts
that refer
to
semantic
concepts
at
could
specify rules
whatever level in the
domain hierarchy is most natural and appropriate.
The
theory is probability
based and it has been specu­
lated to be useful in building expert systems. A problem with
this theory is that rules of
evidence combination may create
a too large certainty measure of the
combined evidence about
a fact if a normalization is used to eliminate or hide a con­
tradiction [Z184a,Z184b].
3.4 Development of Expert Systems
The DENDRAL [BF78] program is
the first
AI
program to
emphasize the power of specialized knowledge over generalized
problem-solving methods.
It was started
in the mid-1960s by
Lederberg and Feigenbaum as an investigation of the use of AI
techniques for hypothesis formation.
tions of empirical data
It constructed explana­
in organic chemistry,
specifically,
explanations of analytic data about the molecular structure of
an unknown organic chemical-compound. By the mid-1970s, there
were several
large
programs,
collectively
called DENDRAL,
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58
which interacted to help organic chemists elucidate molecular
structures.
The programs are knowledge-intensive,
require very specialized
knowledge of chemistry in
i.e., they
order to
produce plausible explanations of the data. Thus a major con­
cern in
research on DENDRAL was how to represent specialized
knowledge of a domain like chemistry so
that a computer pro­
gram could use it to solve complex problems.
MACSYMA[MF71]
auspices of the MAC
has been developed
project.
at M.I.T.
under
the
It is able to provide symbolic
solutions to calculus problems and various algebraic problems.
Some will argue that it is
makes use of algorithms
it has achieved high
not
an
expert system
since
it
to symbolically solve problems.
But
competence in the symbolic computations
associated with applied
analysis.
Many
mathematicians seek
its assistance in algebraic computation. The roots of MACSYMA
can be traced.back to early algebraic
manipulation
programs
for general mathematical simplications[Ht61,Kk65,W d 6 3 ].
MYCIN[SD75]
is
one of the first medical expert systems.
It gives consultative advice on diagnosis
and
theraphy
for
infections diseases that compares favorably with advice given
by experts in infections disease.
MYCIN's
is represented in terms of production
tainty
ing.
rules
medical knowledge
involving
cer­
factors, which help accommodate probabilistic reason­
The rules are invoked using a backward-chaining control
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59
strategy
since
that
it
effectively
works
backward
makes
from
MYCIN
hypothesis-driven,
its conclusions.
factors are loosely based on probability theory.
Certainty
They are an
ad hoc method
of uncertainty handling and have not proved to
be
uniformly
applicable.
in
the
sense
MYCIN is
an outgrowth of DENDRAL
that many of the lessons learned in the
struction of DENDRAL were used in the design
tion of MYCIN.
erberg
and
applied to
and implementa­
The senior members of the DENDRAL team,
Feigenbaum,
Buchanan that
con­
AI
ideas
had convinced
that
made
themselves and Bruce
DENDRAL
a problem of medical import.
Led-
work
could be
MYCIN has developed
the idea of making many uses of one data base. Using backwardchaining of rules,
MYCIN can give the reasons for its
sions in terms of its rules and can
carry on
asking the user for information needed
ward chaining.
deci­
a dialogue
by
to continue the back­
MYCIN has led to a more general framework for
expert systems such as EMYCIN [VSB84], where EMYCIN is called
Essential MYCIN or often
is called
Empty MYCIN.
used as an expert system building tool
if you fill its knowledge base
with the
EMYCIN
is
under the theory that
appropriate
knowledge it will then be an expert in that domain.
domain
It is ess­
entially an expert system shell. When you fill the shell with
the appropriate
knowledge
a full
fledged
expert system is
born. But EMYCIN has been successful for limited problem types.
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60
TEIRESIAS[Dr77] facilitates automatic acquisition of new
knowledge
in the MYCIN system.
of meta-level knowledge
domain.
It is concerned with the use
that is not specific to MYCIN or its
TEIRESIAS does assume that rule-like units
ledge are used and
of know­
that a chain of such rules suffices as an
explanation of system behavior.
Metaknowledge
is
knowledge
about how MYCIN knowledge is represented and used.
TEIRESIAS
has rule models and rules about the structure of
such meta-rule
dictates
that
rules
rules.
One
mentioning the culture
site of an organism should also mention the organism’s portal
of entry.
Such a rule
enables
TEIRESIAS
to detect
faulty
rules as they enter the system. Using the context of the dia­
logue
and its own knowledge of what a rule should look like.
TEIRESIAS can
fill
TEIRESIAS was
Randy D avis’dissertation research
original
in
much of a new
rule
for an
expert.
and its two
goals were the development of an intelligent assis­
tant, and to develop a set of tools for knowledge base
truction and
maintenance in order
to abstract
cons­
from them a
methodology applicable to a range of systems.
Rl [Mj82b]
is
one of the first
commercially sucessful
expert systems developed by John McDermott at Carnegie-Mellon
University.
It configures
tal Equipment Corporation.
much
uncertainty
and
is
VAX 11/780 computers for the Digi­
It does
able
not have to reason with
to effectively use a match,
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61
backtrack search method on the knowledge base. It has evolved
into the more versatile expert system called XCON[Mj82a].
It
gave rise to the expert system building language 0PS5 [Fc81].
Several speech-understanding systems have been developed
during the 1970s. HEARSAY-II[EHL80] is one of them and used a
global
working
memory ("blackboard"),
types and levels of
uniform structure.
information
in
which
different
were all integrated
into a
HEARSAY-II provided ways of focussing, so
the system could shift its attention
appropriately
area of the interpretation problem to another.
from one
This
idea of
independent knowledge modules has been used frequently in ex­
pert systems and contributes to the ability to modify and ac­
quire knowledge.
University.
[REF73].
It has been
HEARSAY-II
is
Unfortunately,
developed
a higher
at
Carnegie-Mellon
version
of
HEARSAY-I
it only obtained the expertise of a
10 year old in some very narrow domains. With several differ­
ent knowledge sources,
which works via a blackboard,
be thought of as several small expert systems,
to form a larger system.
cessors,
thereby
laying
it may
loosely coupled
It was implemented on separate pro­
the
ground work
for
distributed
expert systems. When a blackboard pattern matched the domain’s
schema, the system triggers a knowledge source.
HEARSAY
has
attacked a very large domain, unlike most expert systems.
Earlier
we mentioned
EMYCIN
which is an expert system
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62
building tool.
EMYCIN was
used
to build
PUFF [AKS83],
an
example of an expert system obtained from filling out a shell.
PUFF is a rule-based
any medical
expert
expert system
and is as successful
system to date.
It has been
used
as
as a
teaching system at a San Francisco Hospital.
CASNET(Casual
Associational Network program) was devel­
oped during
the early
and
primary use
has
in
been
midseventies [WKS77, WKS78].
the
glaucoma, and ophthalmologists
close to expert.
handling,
diagnosis
theraphy
of
have rated its performance as
It introduces its own method of uncertainty
which has a resemblance to both
fuzzy methods.
and
Its
probabilistic and
This expert system shows that LISP is not the
only possible language for development. They developed CASNET
in FORTRAN.
The
program
provides
modeling diseases instead of
program was one of
attempts
expert
development of EXPERT [WK79],
applied
to
glaucoma.
model
theumatology
diseases.
determines the
system
Within this
effects
of
systems,
a general
and
The CASNET program
framework for
merely modeling glaucoma.
the first
framework for building
a general
This
to provide a general
and
it led
tool that
endocrinology
as
to the
has been
well as to
introduced a casual network to
network the program reasons to
a theraphy or diease.
represents the disease as
a
dynamic
Thus
process
the
about
which it reasons and can explain to the user.
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63
SOPHIE[BB75] acts as an electronics laboratory instructor
that interacts
with a student
who attempts to debug a piece
of malfunctioning equipment. Responding quickly in restricted
English,
it is expert enough for
actual instruction.
employs a simulation model of electronics.
It
SOPHIE
also has
de­
clarative knowledge of a particular circuit, encoded in a se­
mantic
network,
and routines
that reason with this form of
knowledge.
PROSPECTOR[DGH79], structured similarly to MYCIN,
expert
advice
Networks
are
on finding ore deposits from geological data.
used
both judgmental
to
express the knowledge in PROSPECTOR,
knowledge
which
is expressed
static knowledge about domain objects.
The
a knowledge acquisition system(KAS) that
as rules and
program contains
facilitates the ac­
quisition of all types of knowledge in PROSPECTOR.
tinually
gives
prompts
the user until
structure are filled in.
KAS
con­
all missing parts of a new
This process is driven by an exter­
nal grammar that can be changed without difficulty, making it
easy to modify KAS as PROSPECTOR evolves.
a network
PROSPECTOR
editor
and
that
understands
The core of KAS is
various
mechanisms
in
gives the user a limited ability to edit new
knowledge in terms of content rather than form.
AGE[ABN81]
for attempt
is
an expert system building tool. It stands
to generalize.
It
is an
ambitious
attempt to
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64
develop a tool
that is
capable
of being
used
expert system for any desired application.
by the user for specific applications.
to build an
It may be modified
In addition its infer­
ence engine can be configured in various ways.
CENTAUR[Aj83], which has been developed by Aikens,
example of a second
generation expert system.
offspring of PUFF.
is an
CENTAUR is an
It also diagnoses pulmonary disfunctions.
Its questioning is more directed than P U F F ’s and it is a bet­
ter diagnostician.
The improvements are accomplished by
plicating the control
knowledge.
The
knowledge representation technique
use
ex­
of frames as its
allows
the control know­
ledge to be put in one of the frame slots.
Hence it combines
rules and frames,
showing them to be a powerful combination.
CADUCEUS[PW84],
consultation system
domain of
which is called INTERNIST,
is a medical
that attempts to make a diagnosis in the
internal medicine.
The
program
displays
expert
performance in about 85 percent of internal medicine,
so its
knowledge base is one of the largest
The diagnosis problem is
in
any
expert system.
complicated because a
have more than one disease,
which
makes
patient
can
the number of pos­
sible combinations enormous. Like CASNET, CADUCEUS represents
its medical knowledge in a structure, the disease tree, about
which
the
program
reasons
dynamically.
data-drivened and hypothesis-directed
CADUCEUS combines
reasoning
in the same
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65
framework. The patient data are used first
theses,
and these
tions that must
to predict
hypo­
are then used to predict other manifesta­
either
be confirmed
or used
to change the
hypotheses.
Kohout and Bandler[BK80b]
propose
methods to represent
and manipulate knowledge in fuzzy expert systems,
brief introduction to the concept
of
and give a
a fuzzy expert
system
[KB82].
Yager[Yr83]
fuzzy
sets,
introduces a robot planning methodology using
which may be seen as a first step in developing
an expert system for robot planning.
SPERILL-II[OFY84]
is an expert system designed to perform
damage assessment of existing structures. They use fuzzy sets
to represent imprecise data. Dempster and Shafer’s theory for
combining fuzzy sets with
certainty factors is used in doing
inexact inference.
FINDEX[WS85]
appropriate
products.
is an interactive expert system and suggests
techniques
It is a
for forecasting
linguistic
sales of commercial
fuzzy production rule system.
They have made some extensions to the wenstop language [Wf79]
of
linguistic
variables
and
incorporated
this
into
the
FESS[H186], which has been developed by HALL,
is a
re­
reasoning process of an expert system.
usable fuzzy
expert
system.
It provides
a methodology
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to
66
develop
a multi-knowledge
porates fuzzy reasoning
source expert system which incor­
techniques.
In FESS,
a
blackboard
concept is used for communication and the developed methodology
of it allows the construction of an expert system which has no
domain knowledge built into it.
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67
Chapter 4
Building a General Purpose Fuzzy Expert System (GPFES)
4.1
Development toward fuzzy expert systems
In a domain with little imprecision an expert system can
likely be developed. But, the development of an expert system
is very difficult in a domain with a large amount of impreci­
sion. There are several sources of imprecision and uncertain­
ty in expert system areas and when a solution is imprecise it
must be presented to the system user in a manner which
cates the uncertainty in it.
ally lead to
an answer
system must be able to
indi­
An imprecise question will usu­
which is also
phrase
imprecise.
questions in
captures their somewhat vague meanings
An expert
an manner which
when such arise.
The
process of acquring knowledge is quite imprecise. It is like­
ly that the knowledge
acquired
doesn't
expert's since the expert is often
exactly capture the
unaware of all
the tools
used in the reasoning process. Fuzzy reasoning techniques can
provide the basis
for representing
in an expert’s knowledge.
the imprecision inherent
Incomplete information gives another
uncertainty to the expert system.
Common sense reasoning in-
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68
eludes many inferences from
incomplete
information.
It has
been attempted with fuzzy reasoning according to Zadeh[Z184c].
Uncertainty also
have weak
occurs in
implications.
accept and use
the weak
the knowledge
The expert
base where we may
system must
implications
be able to
together
with
other
sources of information to come to a conclusion about the prob­
lem.
Uncertainty
also
arises in knowledge
sources i.e. experts and references, etc.
from
There may be
flicting, redundant, subsuming or missing knowledge.
be noted
of
different
con­
It should
that some of the applications with the least amount
imprecision
attempted.
in
expert systems
have
been
successfully
But those with large amounts of imprecision
not been as successfully solved.
have
Imprecision and uncertainty
must be handled theoretically.
There are several different ways that imprecision may be
handled in
etical
methods
an expert system.
basis
of
i.e.
handling
In a sense they have no theor­
have been handled
imprecision
in ad hoc ways.
Most
are based on probability.
Experts tend to think in terms of much, usually, always, some­
times.
They d o n ’t use probabilistic values.
introduced
several
methods
those methods are effective in
to
handle
specific
guaranteed to be effective in extreme
ral domains.
In Chapter 3, we
uncertainty.
cases,
While
they are not
situations or to gene­
Fuzzy set theory gives us a theory-based method
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69
of dealing with uncertainty.
By fuzzy expert system we mean an expert system which is
based on fuzzy sets and fuzzy logic
in handling
imprecision
and uncertainty for the inference process and knowledge represenation scheme.
The theory of fuzzy sets and fuzzy logic is
well-founded and strong.
The theoretical basis behind
fuzzy
techniques will allow us to deal with uncertainty in a manner
that is well-supported.
SPERILL-II[OFY84] is an expert system for damage assess­
ment of existing structures
imprecise data.
and uses fuzzy sets to represent
Knowledge may be represented in the proposi-
tional calculus but it is not good enough for general applica­
tions. Predicate calculus is a more useful one. But those are
very precise representation formalisms and have been used
in
in most expert systems. The fuzzy logic is good for imprecise
applications
pervasive
form.
That
but it is
still somewhat cumbersome.
The most
form of knowledge representation is the rule-based
is
because
it is a natural form
and
easy
to
understand.
In the design
of expert
systems
recognition of the need for graded
use of degree of implication,
there
is
increasing
production rules i.e. the
leading to degree of certainty
or possibility attached to the conclusions. This accords with
imprecise,
incomplete and faulty input information.
We
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have
70
surveyed
probabilistic
methods
in the previous chapter.
In
this section, we survey a method of fuzzy implication for re­
presenting a
relation
between
premise
and conclusion in a
rule.
Bonissonne [BB85]
represent
and
proposes
using
linguistic qualifiers or
conclusions
of
a rule-based
fuzzy
numbers
to
variables in the premise
knowledge
representation
scheme. A t-norm and t-conorm pair is chosen and they provide
guidence on the choice
Conclusions
of an appropriate implication scheme.
are given not with single-valued certainties but
as ranges, or, more likely, qualified as truth by some appro­
priate linguistic qualifier.
REVEAL [Jp84]
is
uses fuzzy reasoning.
a general decision support tool which
In REVEAL,
fuzzy sets take the role of
adjectives or qualifiers in natural languages.
production rules for performing inference.
tors
It uses fuzzy
Functional opera­
such as very and fairly are available both with default
values and those values defined by the user.
FLOPS[BST86]
built in
is viewed as a fuzzy
expert system.
the expert system building language OPS5. FLOPS has
been applied to be classification of regions
in
grams. It makes use of fuzzy numbers as well as
confidence values
linguistic
It is
for
descriptors
system conclusions.
echocardio­
multi-valued
It can translate
into fuzzy sets and back. The system
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71
uses
fuzzy
production
rules
for
knowledge
represen­
tation .
ARIES [AR85] provides
mechanisms for the representation
and manipulation of multiple
as defined by
several
degrees of propositional truth,
criteria of truth, belief, or likeli­
hood. It has been implemented as a general
purpopse approxi­
mate inference subsystem capable of being easily incorporated
into a wide variety cf information systems. Degrees of truth,
which may be interpreted in
ARIES as either
several ways, are represented in
classical or
interval of the real
line.
fuzzy
intervals
These intervals
of the [0,1]
represent
con­
straints on the possible truth values of facts or conditional
propositions (rules).
Implication relations are used for inference chaining in
the expert system.
The
implication
premise and a conclusion.
relation
consists of a
The premise is made up of a number
of clauses. Each clause has the form "X is Y " , where X
and Y
represent variable linguistic statements. Y may contain fuzzy
qualifiers such as "almost" or "very".
The premise
consists
of one to n claues. The clauses are connected by the conjunc­
tion and
disjunction
operators/\, V.
Any t-norm,
t-conorm
pair may be used ; see Definition 2.18.
The conclusion is a set of one or more clauses connected
by the linguistic connective and, which is used as an uniform
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72
1. S # Standard Sharp
a —♦> b —
1 iff
a
1 or 6 = 1
otherwise
0
2. S Standard Strict
a -*j
1
if.fa<b
0
otherwise
1
iff a <
6
otherwise
6 —1
3. S* Standard Star
a. — b —
6
4. G43 Gaines 43
4*. G43* Modified Gaines 43
a — b = m in(l,
-— -)
a l— o
5. L Lukasiewicz
a —»s 6 = m tn(l, 1 —a + b)
5.5 KDL Kleene-Dienes-Lukasiewicz
a -♦ 5 , 5
6
6
=
1
~ c + ob
. KD Kieene-Dienes
a —>8 6 = (1 —o) V 6
7. EZ Early Zadeh
a
—* 7
5 = (a A b) V (1 —a)
Table 4.1 Fuzzy im plication operators
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73
conjunctive operator. That means that each clause in the con­
clusion holds with the same strength as the overall conclusion.
Each relation is given some a priori strength which we call a
certainty.
This
with terms,
certainty may be represented linguistically
such as
usually,
sometimes, or occasionally.
It
may be represented with discrete numerical values in the real
-valued interval
[0,1] into
which
the
linguistic terms are
currently translated.
There are
which
may be
at least nine different implication operators
used in fuzzy implication relations which have
been classified by Bandler and Kohout [BK80b]. They are shown
in Table 4.1. Oh and Bandler[OB87] survey these fuzzy implica­
tion operators.
given
i.e.
In fuzzy expert systems,
the problem is that,
x->y and x, we must find a value y which is consistent
to find
the value of the conclusion
of an implication
relation by the use of fuzzy modus ponens[BK84].
What is done with the conclusion when it has been determ­
ined with some truth value ? It is used in the continuance of
the reasoning process.
system
has come
The reasoning process
halts when the
to a conclusion which provides the informa­
tion that the user of the expert system seeks or there are no
more relations available to determine an answer for the user.
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74
4.2
Development of coimplication
Fuzzy logic has been proposed by
Zadeh [Z175b]
and
is
based on the concepts of fuzzy sets[Cc88,Cs71,Gj67,Mp69,Z165].
Based on [Z175b] we have investigated an equivalence property
of fuzzy logic and applied it to the new approach to approxi­
mate reasoning
using
a fuzzy logic
which uses the infinite
multivalued logic of Lukasiewicz[Rn69]. Oh and Bandler [OB87]
have defined the difference between the truth degrees
Q and
Q -> P i.e.
d = v(P->Q)-v(Q->P),
of P ->
and have investigated
some properties regarding the difference in the fuzzy implica­
tion operators classified by Bandler and Kohout[BK80a].
Definition
4.1 shows the other cases of this difference
between two propositions
P and Q
with operator 5 in Bandler
and Kohout [BK80a]; (see Table 4.1).
Definition 4.1:
1) symmetrical difference: v(P | Q) = |v(P -> Q) - v(Q -> P)j
= Ip - q \
2) asymmetrical difference:
i)
v(P
ii)
v(P
|— Q ) = v(P -> Q) - v(Q -> P) = q - p
-\ Q)
= v(Q -> P) - v(P -> Q) = p - q
The concept of symmetrical difference
of
fuzzy sets is
discussed in Dubois and Prade[DP80], and Kandel[Ka86]. Theorem
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75
4.1
shows a relation between the equivalence and the symmetri­
cal difference of P and Q.
Theorem 4.1 :
v(P <-> Q) = 1
- | p - q| = 1 - v(P | Q) = v(not(P | Q ) )
pro o f :
v(P <->Q)=v(P -> Q and
Q -> P)
= min(min(l - p + q,l),
min(1 - q + p , 1))
i)
p
>
q : v(P
<->
Q)
= min(l
- p + q fl) = 1 - p + q
= i - Ip - q|
ii)
p
<
q : v(P
<->
Q)
= min(l|l - q + p )
= l - q + p
= 1 - | p - q|
Consider a method applying these operators to an infer­
ence through modus ponens. The classical and plausible modus
ponens is
the
formula
"IF P -> Q AND P THEN Q" which is a
tautology.
The
new
approach
is
which isa similarity relation
expert systems i.e.
the
to use the equivalence operator,
[Ka86], in
formula
modus ponens
"IF P <-> Q AND P THEN Q"
which is still a tautology.
Theorem 4.2:
1)
2)
in
(P <-> Q ) A P -> Q implies ( P -> Q ) /\ P ->
Q
P <-> Q implies P -> Q
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76
proof
trivial.
The following is a typical type of the fuzzy conditional inf­
erence :
Ant 1 : If x is A then y is B
Ant 2 :
x is A ’
Cons
y is B 1
Let us consider a specific example for
the fuzzy conditional
inference.
Ant
1 : If tomato is red (0.7) then tomato is ripe
Ant
2 :
tomato is red (0.8)
tomato is ripe
When we use
truth
(0.8)
Lukasiewicz’s
fuzzy
( ? )
implication operator,
the
degree of Ant 1 is 1; thus, given x -> y and x, then y
is determined.
In
this case, the truth degree of "tomato is
ripe" is greater than or equal to 0.8.
Consider
the
above
example by the equivalence operator :
Ant 1 : tomato is red (0.7) <-> tomato is ripe (0.8)
Ant 2 : tomato is red (0.8)
tomato is ripe ( ? )
If
we
use the equivalence operator then the truth degree of
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77
"tomato is
method
ripe" is 0.9.
is more specific
If the truth degree of
Therefore,
than 0.8 in the implication method.
"tomato is red"
truth degree of"tomato is ripe",
same
results.
0.9 in the equivalence
then
is
greater than the
both methods have the
Based on this equivalence method,
we develop
the coimplication concept. The definition of inference from a
set of premise[Yr85] can be improved in a fuzzy real environ­
ment system using the equivalence method.
Definition 4.2 :
In multivalued logic, we say that
(Pl,,,Pn) I—
atoms
in
Pl,,,Pn implies P, denoted
P i f , for every consistent interpretation of the
the propositions Pl,P2,,Pn and P, the relationship
| (PlA. «APn) 1 <_ | P I is true.
Note that this definition subsumes the inference rule in
binary logic which requires that ((PlA..APn) -> P) is a taut­
ology.
Definition 4.3 :
A fuzzy statement form which is always greater than or
to x, x £ (0,1], as a truth degree,
values
of its
equal
no matter what the truth
fuzzy statement letters may be,
is called
x-tautology.
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a
78
Definition 4.4 :
P
denoted (PI,..,Pn) H P,
Pl,P2,...Pn fuzzily coimplies P,
if
for every fuzzy consistent interpretation of the atoms in the
propositions Pl,,,Pn and P, p=sup(|v(Q) - v(P)|) and p <
where p is
Q and P, and
0.5,
PiQ
“
a maximum possible symmetrical difference between
Q = PlA. .APn.
In binary logic, the inference rule requires that
Pn)->P
tion
(PlA..A
is a tautology, but we use <-> instead of ->. Defini­
4.4 subsumes the coinference rule in binary logic which
requires that ((PlA. .APn)<->P)
is a tautology.
Theorem 4.3:
In binary logic,
0
(Pl,..,Pn) H P
if (P1A.A P n ) < - > P is a tautology.
Proof:
Assume (PIA. «APn)<->P is
a tautology.
Under
the assumption
that all the P i ’s have truth value one, i.e. v(PlA. .APn) = 1.
From the assumption, v((PlA. .APn)<->P) = l,
and then v(P)=l by
the equivalence property. Therefore p ^ 0 . 5 .
Under the assump­
tion that at least one of the P i ’s have truth value zero then
PlA. .APn have truth value 0.
From
the
assumption,
P has a
truth value 0. Therefore p <,0.5.
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79
Theorem 4.4:
In fuzzy logic,
1-x
(Pl,..,Pn) | | P
if (PlA. .APn)<->P is a x-tautology.
Pr o o f :
Assume (PlA. A P n )<->P is a x-tautology, where x>. 0.5. From the
property of <->, this requires that l-\v(PlA. .APn)-v( P)|
i.e. |v (P1A«-APn) - v(P)l
>_ x,
< 1 - x. Thus p = sup(\ v (P1A« *APn)~
Pi ,P
v(P) | ) = 1—x <_ 0. 5 .
Theorem 4.5:
0
Let P be a proposition. Then (P) 1
(P
proof
s u p (Iv(P) - v ( P ) I ) = 0
P
Theorem 4.6 (Syllogistic Reasoning):
Let P, Q and R be propositions.
pi
p2
If (P) |--- 1 Q, (Q) i
1 R and pi + p2 < 0.5 then (P)
pl+p2
1--\ R.
Pro o f :
Let p
-
v ( P ) , q = v (Q ) and
r = v(R). Then
p - q
< pi and
|q - r | 1 p2 .
-pi <_ p - q <_ pl and -p2 <
| P ” r I £.
pl + p 2 .
q - r < p2. Therefore
pl+p2
Thus (P ) I— -*— 1R
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80
Theorem 4.7:
pl
p2
)— 1 P 2 , (P2) 1---\P3,,
p
then (Pl)
— 1Pn
If (Pl)
pn-1
n-1
(Pn-1) I--- 1 Pn and p = 2 p i
i= 1
10.5
Theorem 4.8:
Assume (Pl,,,Pn)
is a set of premises and P is some conclusion.
Let (Pa,,,Pk) be any subset of the original premises (Pl,,,Pn).
P
P+d
If (Pa,..,Pk) V— H P then (Pl,..,Pn) )■--- 1 P and p + d £ 0.5,
where d =
min(Pa,,,Pk) - min(Pl,,,Pn)
.
proof
- ( p + d ) l ~ ( p +
min(Pa,..,Pk) - min(Pl,..,P n ))
1 m i n ( P l ,..,Pn) - v(P)
< p + min(Pl,..,Pn) - min(Pa,..,Pk)
<_ p + d.
Theorem 4.9:
Assume P and Q are any two propositions.
Ip-ql
i) (P,Q) I
1P
ip-ql
ii) (P) |
1 P v Q, where p = v(P) and q = v(Q).
proof
I min(P,Q) - p I = \min(0,q-p)|
0
i) q < P : (PiQ) M
P
p-q
ii) q <
p : (P,Q) |— | P
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81
The knowledge representation method of uncertainty, based
on the coimplication in fuzzy expert systems, will be discuss­
ed in the next
section.
fuzzy coimplication,
but
We
have
discussed the concept of
for an actual application in fuzzy
expert systems we are using the following definition.
Definition 4.5:
Pl,,Pn fuzzily coimplies P approximately with an asymmetrical
(d)
difference d, denoted
(Pl,..,Pn)
-[P, if for every fuzzy
consistent interpretaion of the atoms in the propositions Pl,,
Pn and P, pl = inf(v(P) - v(Q)) and p2 = sup(v(P) - v ( Q ) ), and
p2 - pl <^0.5
and d = (pl + p2) / 2,
where Q = PlA. *APn
and
d = v(P) - v(Q).
Theorem 4.10:
(dl)
If
(A)
V
1 C,
(d2)
(B) |--- 1 C
(min(dl,d2))
then (A V B) I------------ 1 C
Proof:
Let a = v(A), b = v(B) and c = v(C).
v(C) - v(A) a: dl and v(C) - v(B) fir d2.
Then v(C) - m a x (v (A ),v (B )) fi: min(dl,d2).
Using this concept, the four modes of inference in fuzzy
expert
systems
surveyed by Bandler and Kohout [BK84] can be
considered.
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82
Remarks:
1. Ponens
a -> b and a
:
b
(d)
a b— 1 b and a
b
cs a
+ d
2. Denial
a -> b and not a
:
not b
(d)
a \
-- 1 b and not a
not b s
1 - a + d
3. Confirmation
a -> b and b
:
a
(d)
a I— | b and b
a CS b - d
4. Tollens
a -> b and not b
:
not a
4.3
(d)
a H — \ b and not b
not a
is
d + 1 - b
Knowledge Representation in GPFES
One of the main purposes in this research is to
develop
the structure of facts and rules to represent vague informat­
ion and handle propagation of
uncertainty
in expert systems
using the coimplication.
Knowledge
representation
is one
parts in the expert system design.
of the most important
Syntactical structures of
knowledge representation were discussed in Chapter 3.
But we
emphasize the semantical structure in this section for GPFES.
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83
In syntactical types of knowledge representation,
there
are
semantic networks, production systems and frame (schema)-like
representation methods.
A knowledge base
knowledge about a problem domain.
facts about objects,
instructions) and
contains facts and
Examples
of knowledge are
events ( or actions ), performance ( or
meta-knowledge ( e.g. the
reliability
of
certain information ).
The
combination
powerful tool
of
of
rules
and schemas
can
provide a
the knowledge representation scheme. More­
over* the schema concept contains the concept of the semantic
network.
Since
network,
all
network
rules
three
can be easily described in a semantic
combinations of
the rule, the semantic
and the schema (frame)-like knowledge representation
would appear true.
When people are
faced
with a new
large amounts of highly structured
experience;
[Bf32] used
"schema"
word
they use
knowledge which they have
acquired from previous
the
situation,
Barlett [Bf32].
Barlett
to refer to this situation.
Marvin Minsky[Mm75] used the concept of a frame as a fundamen­
tal structure representing common concepts and
frame is organized much like
system (or schema system)
organized in a hierarchy,
general
a
semantic
situations. A
network.
A
frame
is a network of nodes and relations
where the topmost nodes
represent
concepts and the lower nodes more specific instances
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84
of those
concepts.
Davis and
Buchanan [DB77] introduce the
concept of a data structure schema,
a device that provides a
framework in which representations can be specified. The most
important
point
in
the
notion
specification of many kinds of
tions.
In the top level,
together,
of
schema is the detailed
information about representa­
a schema system links every schema
indicating which categories of data structure exist
in the system and the relationships among them.
level of organization,
there
are
individual
In the next
schemas,
the
basic units around which the information about representations
is organized.
Each schema indicates the structure and inter­
relationships of a single type of data structure. In the last
level, there are slots, which are the attributes (e.g., name,
c o l o r , size).
Use
of
the
schemas
in knowledge
acquisition process
relies on several ideas:
i)
Information
creating
in
a new
the
schema is viewed as a guide to
instance
of
the representation it
describes.
ii)
That guidance is supplied by
a) the structure description information,
in
is
the form of a prototype
and b) the relations information,
as
which
pointers
to
a number
which
is
interpreted
of structures that may
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85
require
updating
to
ensure that necessary data
structure interrelations are maintained,
iii)
The interrelation process drives the knowledge trans­
fer dialogue.
iv)
The advice present in the schema adds a level of
so­
phistication to the dialogue.
The main
knowledge
base,
purpose in using the schema is to maintain the
base systematically.
for example,
entation,
in the knowledge
adding a new instance of a known repres­
will not violate necessary
data structures.
base,
One change
Besides
the
relationships
between
maintenance of the
knowledge
Davis and Buchanan[DB77] offer a convenient
mechanism
for organizing and implementing
data structure
access
and
storage function. The main idea for the schemas is to utilize
them as points around which knowledge is organized.
facts
are represented with
represented
a
truth
In slots,
interval and rules are
with the asymmetrical difference
value for
co­
implication.
In individual schemas, the name of a coimplication
ation is contained.
of the coimplication
It contains the value for each
relation
rel­
attribute
and the asymmetrical
differ­
ence value for the coimplication.
The known
designing
set of facts we will call the data base.
a knowledge database using the schema-like
In
repre­
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86
sentation,
the
a communication scheme
knowledge
knowledge
base has been considered.
representation
the expert
between the database
and
The schemas in the
may be a conceptual description of
system for some application area. The term "know­
ledge schema"
is used
instead of the word "schema"
knowledge representation.
Figure 4.1 and Figure 4.2
in
show
the
a
knowledge database design scheme through schemas, and repres­
ents
a diagram for GPFES
interface
among
with
the communication
knowledge schema,
database
and
scheme or
knowledge
database, respectively.
Knowledge
the
conclusion
indicate
what
schemas
in any
are used to
store information about
reasoning process. Knowledge schemas
schemas give
evidence
about
the conclusion
and what schemas use the conclusion.
A coimplication
relation may have an a priori determin­
ed asymmetrical difference representing
interval
[-0.5,0.5],
which is
given
a certainty
by the
Coimplication relations are the primary
the
which
inference
can
be
process.
They
are
relations
An
the
domain expert.
very flexible
used for any application.
on
used
for
relations
example of the
coimplication relation is as follows:
(-0 .01)
Thunderstorm is strong)------- )There is a lightning.
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87
Figure 4.1
Knowledge database design through schemas
schema hierachy
I---------non-fuzzy schemas
fuzzy schemas
slots with attached
procedures
slots with fuzzy predicates
and attached procedures
Figure 4.2
Diagram for GPFES
Domain Expert
Conceptual
knowledge schema
4'
.1
Knowledge Engineer
GPFES
,
rt
I
i
l Knowledge database^I
»
-I
management system
I
<
I
I
I
Inference engine^.
^database schema
I
->•knowledge
database
— I
'I
users
FS: fuzzy schema
NF: non-fuzzy schema
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88
This is read as "If the thunderstorm is strong then there
a lightning"
as
in
(-0.01) indicates
the
that
normal
this
is
implication relation.
The
coimplication relation usually
holds. Table 4.2 shows an example of the knowledge schema.
Table 4.2
Knowledge schema
type
coimplication
name
lightning
clause-1 in premise
Thunderstorm
conclusion
there is lightning
a priori asymmetrical
difference
(-0.01) : This is assumed to be
a partitioned coimplication on
a subinterval which will be
discussed
premise value
[0.85,0.9]
For representation of facts the closed interval [0,1] of
truth degree is divided into n disjoint intervals;
I2=[b2,B2],,,In=[bn,Bn],
Il=[bl,Bl],
where Ii and Ij are disjoint (i=j),
and bl=0,Bl=b2,,,Bn-l=bn,Bn=l.
Let
{fi}
be
the
membership values in the
frequencies
interval
of
items
Ii (i=l,n)
fuzzy subset F. The basic form of an uncertain
cise fact is (predicate,F,{(Ii,fi,ui)},b,B),
are the
taking fuzzy
related
to a
and/or impre­
where
b and
lower bound and
B
the upper bound of the truth values
fi
of the fact, respectively, and ui=
(xk)/fi,
(x k )
; ui
k= 1 F
F
'Z.X
X
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89
is an usual value of the interval Xi; see Zadeh[Z186]. Suppose
that a universe of discourse U
has
100
elements,
i.e. xl,.
. ,xl00. A fuzzy subset F in U is characterized by a membership
function uF which takes the value in the interval [0,1], i.e.
X
:u -> [0,1]. Because U is discrete, F is represented as F =
F
^< xl)/xl
Let
+. . +
Ii
X
<xl00)/xl00.
and
Ij
be
the intervals containing b and B,
j
respectively, where i < j, and M = 2 ^ *
Then the certainty
=
k=l
degree p is approximately defined as p = 1 - P(UIk) +1/100 =
1 4 k
1 - M/100 + 1/100.
imprecise
truth
transform
them
fi upward
and
If b=B=l
value
and fn=l
i
then p = 1.
J
Given an
r and its certainty degree p, we can
into b and B. The basic idea is to count for
downward,
alternately,
from
a
partitioned
interval la containing r as 101 - lOOp.
In
the real world,
implies the conclusion
we are not sure that the antecedent
in
every point of truth degree, i.e.
in many cases, given an uncertain truth degree of the antece­
dent,
it is possible that the conclusion has several different
truth degrees.
plying
the
For solving
asymmetrical
this kind of problem, we are ap­
coimplication
to the rules on the
partitioned intervals.
A rule
conclusion,
is
and
composed of two parts, called antecedent and
usually
is denoted
clusion". A relation is suppoed
by "Antecedent -> Con­
to hold
between
them.
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The
90
approach to represent rules
is to use
the coimplication
(d)
(Pl,,Pn) |-- 1 P and, instead of finding a coimplication on the
closed
interval [0,1],
coimplication
of
the goal
is
to
find the pieces of
the predicate on partitioned intervals Ii,
i=l,n.
The
interval
into [Ii}i=l,n.
[0,1]
of
the
As in facts,
asymmetrical difference between
antecedent will be divided
we wish to find every possible
Ii
and
some
values in the
closed interval [0,1] of the conclusion.
Definition 4.18:
Let Di s [d:d=ci - a i , a i & I i
truth
degree
of
conclusion
approximate asymmetrical
and
from ai}. Then di is called an
difference on Ii
di = (inf(Di) + sup(Di))/2 if sup(Di)
Let
SDi s [sd:sd= ci - ai
possible truth degree
called
a
= sup(SDi)
of
ci is every possible
,
- inf(Di) ^ 0 . 5 .
aifeli
conclusion
and is defined by
and
ci
is
from ai}. Then
every
sdi is
symmetrical difference on Ii and is defined by sdi
if sdi <^0.5.
The structure of a rule in the knowledge
base
is
con­
structed by (A , C , [{(Ii,di,s d i )}i = s ,t ] ,b,B). In the next
sec­
tion, we will discuss how to infer the conclusion and to find
the truth interval of it on the
partitioned coimplication.
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91
4.4 Coimplication and resolution procedure
In Section 4.3,
we have discussed the knowledge
sentation scheme of vagueness through
repre­
the partitioned
coim­
plication. From those structures, we have to decide the truth
interval of the conclusion which is a resolvent of rules
and
facts (or user input) on the concept of coimplication.
Before
we
truth interval,
discuss how to resolve a conclusion with the
le t ’s
the disjunction of the
consider the conjunction operation and
propositions
on
the
antecedent and
splitting scheme of the truth interval into partitioned truth
intervals.
Corollary 4.1:
Let
truth
Pl,P2,..,Pn
be
propositions. Let [bl,B 1 ],..,[bn,Bn] be
intervals
of
Pl,..,Pn, respectively.
interval [b,B] of the
conjunction
b=inf(bi) and B=inf(Bi), and the
Then the truth
P1AP2A. .APn is defined by
truth interval
[b,B] of the
disjunction P1VP2V..VPn is defined by b=sup(bi) and B=sup(Bi).
In the resolution process,
the truth
first of all, we have to find
interval of the antecedent and find a subpartition
covering it.
Algorithm 4.1
shows the splitting procedure of
the truth interval of the antecedent la, calculated by Corol­
lary 4.1, into a subpartition.
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92
Algorithm 4.1: Splitting
1.
2.
Find
a minimum subpartition {Ii}i=s,t covering the truth
t
s-1
interval la of the antecedent such that la £ ( J Ii, U Ii f\
n
i=s
i=l
la =
and U Iif\Ia. = h.
i = t+l
Set the left bound of Is to the left bound of la and the
right bound of It to the right bound
of
la
i.e. a sub­
partition {Ii’}i=s»t
From
{Ii*} i=s,t,
difference between the
and
the
property
antecedent
of
asymmetrical
and the conclusion we can
infer that conclusion has the possible interval Ic=[bc,Bc] of
the truth degree of the conclusion as in Algorithm 4.2.
Algorithm 4.2: Resolution
1.
From {Ii’}i=s,t,
find {Ii"}i=s,t by Ii" = I i ’ + di
2.
Find the truth degree [be,Be] of the conclusion by be =
inf(U Ii") and Bc=sup(U Ii").
i=s,t
i=s,t
3.
Find the symmetrical difference SD by SD = sup(sdi).
i = s ,t
In
expert systems,
variable may be
several
conclusions
concerning
a
obtained from different rules. The structure
for facts and rules defined
in
the previous section will be
applied to compute a final truth degree of the conclusion and
combine several different pieces of information. Basically,
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93
(dl)
(d2)
(dn)
— \ C, A2 t- —( C, . . ,An |—— t C are combined
<d)
V. .V An |-- 1 C, where d = inf(d i )i = l,n .
AI
into AI V
A2
Algorithm 4.3 : Combination
(d)
A V B h 1C )
( Case of 2 rules i.e.
1.
Go
to Algorithm 4.1
with the disjunction interval
[b,B]
and find {Ii’}i=s,t.
2.
Find di[A]i=s,t and di[B]i=s,t, and if di does not exist
for some i, i=s,t then set di to 0.
3.
Calculate di[A V B] by min(di[A ] ,d i [ B ] ) for i=s,t.
4.
Go to Algorithm 4.2
for the
combined truth
interval of
C.
4.5
Fuzzy inference engine
The fuzzy
coimplication relation is
knowledge
representation
knowledge
formalism.
inference
chaining
consists of a premise
relation.
which
in GPFES.
The
are used for
coimplication
up of
variable linguistic
statements.
such as most,
or less.
clauses connected by
any
relation
and a conclusion like the implication
a type "0 is F " ,
more
used to model
Coimplication relations
The premise is made
Each clause has
can be
a new approach to
The
a number
where
0 and F
F can contain
conclusion
the connective A *
of clauses.
Each
represent
a qualifier
is a set
of
coimplication
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94
relation is given
some a priori
asymmetrical difference.
strength which we
call an
It is represented with a numerical
value in [-0.5, 0.5]. Classical modus ponens has the follow­
ing form:
IF x->y and x then y
We use a method based on the coimplication to determine
that a conclusion holds with some true value in the interval
[0,1].
The
d denote
problem may be stated in the following, letting
the
asymmetrical difference
relation, x denote
the
truth
value
of the coimplication
of the premise X and
_
y denote the truth value of the conclusion Y:
d
s? v(y)
<d)
or (X) |-- \ Y.
- v(x)
Given values for x and d, we must find a consistent value for
b.
That
Then we
may
find
conclusion.
be calculated easily by y = m i n ( 1 ,max<x+d,0)).
the
truth
degree of Y as a confidence in the
In Table 4.3,
modus
ponens
for the
fuzzy co­
implication method is shown.
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95
x
0 . 1 .2 .3 .4 .5 .6 .7 .8 .9 1.
-.5
-.4
-.3
-.2
-. 1
.0
.1
.2
.3
.4
.5
.0
.0
.0
.0
.0
.0
.1
.2
.3
.4
.5
.0
.0
.0
.0
.0
.1
.2
.3
.4
.5
.6
.0
.0
.0
.0
.1
.2
.3
.4
.5
.6
.7
.0
.0
.0
.1
.2
.3
.4
.5
.6
.7
.8
.0
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.
1.
.2
.3
.4
.5
.6
.7
.8
.9
1.
1.
1.
.3
.4
.5
.6
.7
.8
.9
1.
1.
1.
1.
.4
.5
.6
.7
.8
.9
1.
1.
1.
1.
1.
.5
.6
.7
.8
.9
1.
1.
1.
1.
1.
1.
y = min(1,max(x+d,0))
Table 4.3
Based on this simple method, given an a priori asymmetr­
ical
difference
as a certainty for a coimplication relation
of -0.1 and a certainty in the premise which lies anywhere in
the range [0.8,0.9], we find that the certainty value
conclusion is in the range [0.7,0.8].
The
of the
reasoning process
seems to be reasonable.
An inference
engine
Executing rules is also
is
needed
referred
to
to
execute the rules.
as
firing rules.
The
inference engine must determine which rules are relevant to a
given data memory and choose one to apply. This
selection or
control strategy is called conflict resolution. The inference
engine
can be
described
as
a
finite-state machine with a
cycle consisting of three action states : match-rules,
select-
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96
rules, and fire-rules.
In the state of match-rules,
the infe­
rence engine finds all of the rules that are satisfied by the
current contents of working memory according to some compari­
son
algorithms.
candidates
for
conflict set.
The rules
execution.
Algorithm 4.4
that are found are all potential
They
are
shows
referred
to
as
the
the match procedure.
the second state, select-rules, the machine determines
In
which
rules will be actually be executed and is showed in Algorithm
4.5.
the
It then transfers to the state of fire-rules, and fires
rules selected.
this execution
first state
Algorithm 4.6
or firing,
and
the
explains
machine
is ready to start
it.
cycles
over again.
Following
back to the
Figure
4.3
shows the architecture of the GPFES inference engine model.
Figure 4.3
The diagram of GPFES inference engine
$
data
knowledge
base
match-rules
■^working memory
select-rules
^ conclusion!s)
fire-rules
no
conclusions
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97
The inputs to the inference engine
are
the
user input
data and the knowledge base. The output is the conclusions of
rules that have been executed.
Working
memory
stores
during the problem-solving
the current
process
by
state of knowledge
holding symbols that
4k
represent
facts
assumption
about the
and goals.
domain
and the problem solver’s
The elements in working memory may be
created, modified, or removed for various reasons.
is too old to be of
intermediate
A variety
useful in
interest
computation
of
or
a
fact was
If a fact
the result of
then facts may be removed.
attributes of working memory elements are
determining
which
of
the
rule matches are most
relevant and should be selected for firing.
Algorithm 4.4: Match
Let CL be a list of matched rules i.e. the conflict set.
Let m be the number of rules in the knowledge base.
Let [blj,Blj] be a truth interval of the
element in working
memory.
Let [b2j,B2j]
be
a pre-defined
a priori truth interval of
elements in knowledge base.
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98
Let [b3jfB3j]
be a truth
interval
produced
by the
match
process.
( Pattern or Literal match )
1.
Set i = 1 and match = 0
2.
If rule i is already fired then
i = i + 1;
if i £ m then go to 2 endif;
endif;
if i > m then
go to Algorithm 4.5
endif;
3.
Get every clause of the premise in rule i
4.
Compare the compiled clauses
to the
elements in working
memory or data base of facts.
5.
If a pattern or literal match is made then
match = 1;
goto 6;
else
i
-
i + 1;
if i > m and match = 0 then
go to Algorithm 4.7
endif;
goto 2
endif
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99
( Truth interval match )
Let n be the number of clauses to be compared.
6.
for j = l ,n
If [blj,B1j] £ [ b 2 j , B 2 j ] then
set [b3j,B3j]=[blj,B1j];
MDj=l;
endif;
If [bl j ,B1 j ] Otl>2 j ,B2 j ] then
set [b3j,B 3 j ]=[b2jfB 2 j ];
MDj=(B2j-b2j)/(Blj-blj);
e nd i f ;
If [blj,Blj] f\[b2j,B2j] / 0 and not case 1 and not case 2
and blj <b2j then
set [b3j,B3j] = [b2j,Blj];
MDj=(Blj-b2j)/(Blj-blj);
endif;
If [bl j ,B1 j ] (\ [b2 j ,B2 j ] ^
<f) and
not case 1 and not case 2
and b2j < blj then
set [b3j,B3j] = [blj,B2j];
MDj = (B2j-blj)/(Blj-blj) ;
endif;
If [bl j ,B1 j ] A [ b 2 j ,B2 j ] =
f
then
MDj=0;
endif;
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100
7.
Find MD of the total match degree of the premise by
MDs V MDt = max(MDs,MDt) and MDs /\ MDt = min(MDs,MDt).
8.
Find the truth interval of the premise i.e.
[b3,B3] using
Corollary 4.1.
9.
Put
the
rule i
into
CL
with the match
degree MD and
[b3,B3].
10.
i=i+l; goto 2.
Algorithm 4.5: Selection
Let SL be the selected conflict set.
1.
If CL is
empty then ask the user to start again.
2.
Find the
rules with the match degree > 0.5
and
put the
rules into SL.
3.
If SL is
empty then find the rule with the
highest match
degree and put it into SL.
4. If MD in SL is equal to 0 then
ask the user to start again;
else
goto Algorithm 4.6;
endif
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101
Algorithm 4.6: Firing
Let FL be a list of the fired rules.
For each rule in SL,
1.
Get the conclusion of the rule.
2.
Find the truth interval of the conclusion and the
trical difference based on
symme­
the coimplication relation by
the method descibed in Algorithm 4.2.
3.
Place
the
fired
conclusion
with
the
rule
number on
working memory.
4.
Put the fired rule into FL.
5.
If
the rule is a conclusion then put the fired rule into
a conclusion list.
Algorithm 4.7: Combination
1.
Find the conclusions.
2.
If some conclusions are same then find a new truth inter­
val using Algorithm 4.3.
The
match
process
key words, which
means
the same semantically.
match degree.
working
memory
In order
matches
has two stages. Steps 1-5 match the
identical
keywords or that they are
Steps 6-10 do some evaluation for the
to
the
see
how
premise
well
the
elements in
with a priori defined
truth interval they evaluate the match degree.
The reasoning
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102
process continues until
the
system has come to a conclusion
which provides the information for which the user of the expert
system is searching, or there are no more coimplication relat­
ions to determine an answer for the user. GPFES will be imple­
mented in a forward chaining mode, but it will not be difficult
to
implement
in a backward
chaing
mode or
in
both modes
later. In the forward chaining or data-driven style, the sys­
tem begins
by choosing an initial coimplication relation for
the inference.
Given a certainty value of the premises,
the
conclusion will have a certainty degree through the new modus
ponens method.
If the conclusion is
satisfactorily,
determined to be certain
it is used to direct the choice
of the next
coimplication relation to be done. This conclusion will be in
the premise
evaluated as
of
the
the
continues until
next
best
the
coimplication
one
final
for
relation
processing.
conclusion
is
This
which is
process
reached with the
certainty value above a threshold or the intermediate conclu­
sion is determined to have a certainty value
below
a prede­
fined threshold. There can be several paths to reach the same
conclusion
Suppose that
and then we have to choose the best one of those.
there
are
n paths
{Pn} to reach a conclusion.
Then, for the coimplication relations of each path, they have
{sdij}
1<_ j<_ li, 11 i<_ n i.e. the symmetrical difference as a
certainty degree, where li is the path length of path i.
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For
103
determining the best one of many possible paths, we will take
ii
li
one with a smallest value of ( ^ tsdi.il } 1^ i<_ n w h e r e ^ L [sdijj
j= l
— ~
j= l
is assumed to be less than or equal to 0.5. If we have a path
with the match
degrees MDij, j=l,li for some i then we would
say that the conclusion have the match degree MDil X MDi2 X
.
.. X M D i l i .
Figure 4.4
is a detailed diagram of the fuzzy inference
engine.to show relationships among algorithms in this section.
The forward chaining scheduler in Figure 4.4 performs
a com­
munication interface between data base of facts and knowledge
base.
Although GPFES is not implemented in the backward chain­
ing mode,
let us
consider it.
first coimplication
tains a goal or
In
backward
chaining,
relation chosen will be one
final conclusion.
relation whose conclusion causes
That
which
the
con­
is
a coimplication
the system
to halt when it
has a satisfactory certainty value. If t*he system finds a co­
implication relation,
determine
then the premise
the uncertainty value of
the clauses
must be evaluated to
the conclusion.
in the premise will have some
certainty
Some of
values
supplied by the user, while other clauses will be the conclu­
sions of some
other coimplication relations.
several coimplication
relations
taining the clauses which
we
to
want.
There might be
reach a conclusion con­
This backward
chaining
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104
F ig u re 4 .4
A d e t a i l e d flo w diagram o f fu zzy i n f e r e n c e e n g in e i n GPFES
START
s c h e d u lin g queue; i
fo rw ard c h a in in g
{ f a c t u a l queries..
d a ta b ase
( fa c ts )
J d e c id e a n e x t f a c t u a l query
• t o be answ ered from u s e r
i
I
i a sk u s e r to answ er w ith
J a c e rta in ty (tru th in te rv a l)
,
J
i
i p u t a com piled answ er o f f a c t u a l
J query from a u s e r i n t o th e w orking
LnSSPSDL w ith_a J :r u tJ i_ in te r v a l
r knowledge base
( c o im p lic a tio n
based r u l e s )
p a t t e r n m atch
working
memory
match
(A lg o rith m 4 .4 )
MATCH
tr u th in te rv a l
m atch
s th e re
m atch
s th e re
i n a l con
[sto p or
■continue
■ "error" due to
^ Jp o o r knowledge
■base o r m issin g
[in fo rm a tio n s
■from u s e r
■
■
■
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Jput m atched r u l e s w ith m atch J
'd e g re e s i n t o a c o n f l i c t s e t
i
i
i
(c o n f l i c t
set
selection(Algorithm 4.5)
J.
J s e l e c t re a s o n a b le r u l e s w ith match d eg ree g r e a t e r th a n or
'e q u a l t o 0 .5 and p u t them i n t o th e s e le c te d c o n f l i c t s e t
[ selected conflict set
j
f ir in g ( A lg o r ith m 4 .6 )
J.
J p u t c o n c lu s io n s o f th e f i r e d r u le s
i i n to a c o n c lu s io n l i s t
T
c o n c lu s io n l i s t
I
combination(Algorithm 4.7)
I
ip u t c o n c lu s io n s o f th e combined r u l e s
Jw ith t r u t h i n t e r v a l and sy m m e tric a l
'd i f f e r e n c e i n t o w orking memory
in te r m e d ia te
o n c lu s io n
working
• memory
J
jgo to forw ard c h a in in g s c h e d u le r to |
a sk a n e x t f a c t u a l query
«
Igo to th e m atch s ta g e ” f o r " a "conJfirma"tiqn
'a s f i n a l c o n c lu s io n s
MATCH
START
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106
process will continue until a coimplication
relation,
which
has every clause supplied by the user or the facts, is found.
Then certainty
sions
and
they are propagated
paths to give us
which is
values are shown in
along the
a certainty value
our goal.
the intermediate conclu­
forward
for the final conclusion
As in the forward chaining mode,
choose a best path of many possible paths to
goal.
The essential building
structures
reach
of
reasoning scheme for GPFES have been described.
mentations are simple and flexible.
chaining
we can
the same
an approximate
Their imple­
Those concepts will give
a general guide to manage uncertainty in every domain in
expert system fields.
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the
107
Chapter 5
Application of GPFES to microwave precipitation
retrieval technique
5.1 Introduction
Precipitation
is
a vital exchange process
within the
hydrological cycle, and represents the net result of heating
from condensation in the atmosphere. The importance of prec­
ipitation monitoring is growing within the current trends in
environmental research. Rainfall studies are growing in sig­
nificance because of the steep upward curve in the demand of
human society
for water
supplies
as
standards and expectations have risen
populations,
in
living
many parts of the
world [BM81].
Precipitation varies
duration,
respect to
its
frequency,
intensity and spatial pattern, not to mention its
propensity to
sleet or snow.
zero,
with
fall in
various
forms
such as rain,
Rainrate intensities range from
to above 100 mm/h.
practically
Intensity, to some extent,
inverse function of duration.
Patterns
hail,
is an
of rainfall organi­
zation and distribution prompted by the overriding influence
of the general circulation
of
the atmosphere are
strongly
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108
influenced by global geography.
The sizes, surface configu­
ration, and dispositions of the land masses influence
their
own precipitation climatologies.
The
factors
the motion
properties,
governing
the occurrence of rainfall are
of cloud parcels and
their condensation
for these determine the concentration,
size distribution,
and nature
of cloud particles.
cloud has formed,
condensation and aggregation of
evolve a raindrop
spectrum.
is left uninterrupted,
initial
Once
a
droplets
precipitation
process
these microphysical processes steadily
increase the average radius
Whether particles grow
If the
nuclei
to
determined by air motion,
and
the
range
precipitation
of
drop size.
size
is
largely
through its control of macroscale
properties such as cloud-dimensions, water content, and life­
time. These latter conditions govern the roles of microscale
processes
and
and
therefore
the
length of time over which they operate,
the maximum
size
which
a cloud
drop
can
attain.
In view of the wide range of uses of rainfall data, mon­
itoring
rainfall with the desired
often difficult.
We first
detail and
summarize
accuracy
is
problems arising from
measurement of rainfall by conventional means.
"Conventional"
refers to those means which rely upon well-established types
of instruments:
not only
the traditional
in situ
devices
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109
(i.e. the raingauge),
but
also ground-based remote sensing
systems (principally weather radar).
Measurement
of rainfall by
gauges
interrelated factors of topography,
design.
site,
aspect,
cation in relation to hills and ridges.
raingauges
tropics are not
located
likely
on
to give
rainfall over the ocean in their
scale,
wind,
and gauge
The gauge catch may be representative of a small or
large area depending on slope,
example,
is affected by the
elevation,
and lo­
Taking
extreme
volcanic
an
islands
in the
readings representative of
vicinity.
nature
of the
local surface and the presence of nearby objects and
struc­
tures.
the gauge catch is influenced by the
On a very small
Wind is the single factor contributing most signifi­
cantly to errors in gauge measurements.
is well sited and exposed,
Even if a raingauge
owing to deflection of raindrops
in the disturbed airstream around the
orifice of the gauge,
monthly average winds as light as 5 m/s may result in errors
of
the
underestimation
monthly
as large as
rainfall
Raingauges designs are
comes
10 % in regimes where half
at rates
less than 100 mm/h.
intended to reduce wind effects.
those problems must be added a plethora
organizational difficulties,
To
of practical and/or
some of which are more
signi­
ficant in certain types of regions than others:
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110
(i) accessibility of desired raingauge locations,
(ii) The
availability
of suitable
personnel for reading
and servicing the gauges,
(iii) a suitable power supply
for some types of continuous
recording raingauges,
(iv) preserving
the
rainfall catch in accumulating rain-
gauges especially when read infrequently,
(v) security
of the rainfall
station from vandalism and
other da m a g e ,
(vi) access
to a suitable
data can be
sent
communication link so that the
quickly
to a central facility for
processing and archiving, and
(vii) transcription and transmission errors,
which tend to
introduce positive errors in reported rainfall.
In case of
weather radar for rainfall monitoring there
is a comparable range of problems which circumscribe the use
of such a system.
It is sufficient
for present purposes to
say that there are difficult problems as
yet not completely
solved relating to the proper relationship of
microwave energy
the
to drop size spectrum,
backscattered
partial filling of
radar beam, attenuation of the radar beam by intervening
drops,
absorption
and
reflection
propagation),
and
practice
calibration
the
by the ground (anomalous
signal calibration [B173,HAP79,Mb71].
In
of radars used for measurement of
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Ill
precipitation
includes an adjustment to match gauge measure­
ments of rainfall within the radar scan.
Radar
ding
in
has the
singular advantage over gauges of provi­
a spatialy continuous view.
support
nations
and
of raingauge
of North America,
the requisite
support
It is employed extensively
networks
in
some
more
advanced
Europe and Asia. However its cost
sophisticated
technical and
have limited the operational use
engineering
of radar for rain­
fall monitoring in many parts of the world. Despite the enor­
mous local value of radars,
on rainfall
5.2
most of the w o r l d ’s
information
continues to come from raingauges.
The use of satellite remote sensing
As
born
a scientific approach remote sensing was effectively
in
the
photography.
mid-nineteenth
As a tool in
century
with the invention of
rainfall monitoring it began with
the
development of meteorological radar after
The
term "remote sensing" was popularized after the birth of
space
exploration by
satellites which prompted the new dis­
cipline to take off in the late
rapidly expanding
World War II.
1960's
as
one of the
fields of scientific technological
most
endea­
vours .
There
namely
are two
types
of
Earth observation satellites,
"Earth resource" and "environmental and weather" sat­
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112
ellites.
In general
terms,
the differences between the two
types arise from design capabilities for the channel frequen­
cies,
and
the
ground spatial
resolutions
optics. For precipitation studies,
have relatively
an infrequent
repeat
cycle.
for rainfall
resource
of
equipped
environmental
is so highly
satellites
employ
On the other hand, although no
dedicated hydrological satellite has
number
the channel
Earth resource satellites
little to offer,
variable through time and Earth
of
and
yet to
be launched,
weather satellites have
a
been
with sensors which yield data of value for rainfall
monitoring.
Environmental
occupied two types
and
weather
satellites
have
of orbits and may be grouped
commonly
accordingly
into:
(a) polar-orbiting sun-synchronous satellites, and
(b) geostationary satellites.
The polar
orbiting satellites
occupy
level orbits (usually between 500-1500 km
of Earth),
orbit
Poles.
relatively
above
crossing the equator at high angles
takes such a satellite close
to the
Earth rotates on its polar axis within
the surface
so that each
North
Sun-synchronous orbits are organized so
and South
that
as the
the orbital ellipses,
each new orbit results in the presentation of a new
the global target to the satellite in
low-
such
a way
strip of
that
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the
113
relationship between Earth surface and sun angle is kept rela­
tively constant (i.e. satellite passage is invariant in local
time).
Characteristics orbital
periods
are 100 min,
orbits being required for each satellite to
entire globe under daylight conditions
course,
since such satellites
every
investigate
24
spend one-half of
over the nighttime side of the globe,
14-15
hours.
the
Of
each orbit
each unit area on
the
surface is thus viewed twice every 24 hours, once in daylight
and once at night.
Geostationary satellites are placed into orbit at approx­
imately 35,400 km above
the earth surface,
in the plane
of
the equator, and advance in the same direction as the rotation
of the Earth.
This type of orbit is geosynchronous,
satellite keeps
polar axis,
pace
with
i.e. the
the rotation of the earth on its
and geostationary in that it appears to be fixed
or stationary above a given
Such a satellite
point
on
the E a r t h ’s
surface.
is able to record the same geographic field
of view very frequently throughout the diurnal cycle, commonly
at intervals of 30 min.
The earlist geostationary satellites were the experimen­
tal Applications Technology Satellite(A T S ) of the early 1970s.
In 1979,
five
geostationary satellites provided
circumglobal coverage between latitudes
of
the equator
during
a complete
of 80 degree N and S
the Global Weather
Experiment(G W E ).
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114
These included three
American Geostationary
Operational E n ­
vironmental Satellites(GOES-E and W and GOES-1), one covering
the Americas and the western Atlantic,
Americas and
ocean.
eastern
Pacific,
and
a second
the western
a third over the Indian
The European Space Agency (ESA) satellite,
Meteosat,
was located over Europe-Africa; The Japanese Satellite(GMS-l)
was located over
the
western
Pacific.
Since
the GWE, the
Indian Space Agency has launched three INSAT satellites which
are three axis stabilized geostationary satellites.
Satellites have contributed
science of
enormously to
the related
meteorology and climatology through the last two
decades. The attributes of satellite systems which have made
application to atmospheric science problems possible are
as
follows:
(a) Satellite
coverage of data,
systems
can
provide
a complete global
thereby greatly extending our appreciation
of the atmosphere environment
particularly
in
data-remote
regions;
(b) Satellite
imaging systems yield spatially continu­
ous data,
contrasting strongly with those obtained from the
irregular
networks
(c) Satellites
of surface weather systems;
can
selected meteorological
investigate
parameters
the
distributions of
much more
than in situ observing networks which deploy
consistently
large
numbers
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115
of instrument packages;
(d) Geostationary
satellites
can
give
a much
higher
temporal frequency of information than commonly obtained from
surface and upper air weather stations;
(e) Satellites
provide a unique view of the atmosphere,
observing it from above,
rather than from within;
(f) Satellite data can be obtained for
broad
areas
in
near real time.
Satellite
data are
studies because they
and time.
problems
of
great
complement
utility in
atmospheric
conventional data
in space
On the other hand, satellite data have a number of
and associated difficulties.
(a) there
are
These include:
problems converting satellite parameters
into calibrated units;
(b) Elaborate
convert
parameters
transformation procedures are required to
satellite-observed
which
can be
radiances
integrated
into
with
meteorological
data provided by
conventional networks;
(c) The user of
from
the data
satellite
source;
processing operation
data
important
is
generally
procedures
of
removed
the
data
may remain obscure to him.
Further information on the role of satellites as remote
sensing systems for studies of the Earth’s atmosphere can be
found in [AV73,Be74,S S A70].
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116
Satellite
on data provided
sensors:
meters
retrieval
by
of rainfall has
a small
number
depended largely
of different types of
cross-track scanning radiometers, spin-scan radio­
and vertical profile radiometers. Cross-track
ning radiometers
are
flown
on polar-orbiting
satellites.
These monitor radiation
a revolving
mirror
satellite track.
which scans
revolution
adjacent
to the one before.
the
operational
target
via
the target across the sub­
As the satellite
each new
from
scan­
advances along its orbit
of the mirror yields
a new scan
line
The radiation collected by the
mirror is passed through a beam splitter and spectral filters
to give the desired wavelength separation.
The
wavelengths
usually used in rainfall studies are visible (VIS: radiation
with
the
wavelength
infrared radiation,
with
the
wavelength
0.5-0.7 /am),
infrared
(IR:
thermal
escaping through the atmospheric window
of 10.5-12. 5 ;um) , and
microwave
naturally emitted radiation at radar wavelengths from
1.55 cm).
Microwave
(MW:
0.33-
instruments employ antennas or antenna
reflectors to achieve their scan pattern. Examples of crosstrack scanning radiometers
are
the four
channel
Advanced
Very High Resolution Radiometer (AVHRR) on TIROS-N type NOAA
satellites and the Scanning Multispectral
meter (SMMR) on Nimbus-7,
Microwave
observing microwave
Radio­
radiation
five different wavelengths.
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at
117
Spin-scan
radiometers are flown on geostationary satel­
lites using spin stabilization of approximately 100 RPM.
coming
radiation is received
In­
by a fixed mirror angled at 45
s
degree to the optical axis of the satellite system,
which is
aligned parallel to the axis about which the satellite itself
spins. The spinning motion of each satellite provides a westto-east
scan motion across the target when the spin
parallel to
to West).
stepping
each
of
axis is
the polar axis of the Earth (METEOSAT spins East
Latitudinal coverage is
the scanning mirror
achieved
north
by sequentially
to south
at the end of
rotation(METEOSAT steps South to North).
Thus an image
the entire disc of the earth is built up over a period of
about 20 minutes. The INSAT satellite is three axis stabilized
and scans in a back and forth fashion.
5.3 Review of Passive Microwave methods to retrieve rainfall
There are several satellite rainfall monitoring methods;
cloud-indexing
methods [Be70],
GWB76,S077a, S 0 77b],
bi-spectral
life-history methods [SMS79,
and
cloud
[Lm67,DV73,Ga73,Wd79], rainfall monitoring
model
methods
from visible
and
infrared images[MS72,S077a,SMS79,WS71], and passive microwave
methods[Sd69].
In this section,
we discuss passive
microwave
to retrieve rainfall from remotely sensed data.
methods
COSPAR,
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the
118
Committee on Space Research,
assessed possibilities for mea­
suring rain from satellites,
and in 1967
endorsed
observa­
tions of passive microwave radiation as potentially
the most
viable approach.
first
In 1968 the
meteorological
ment [SH73].
USSR launched Cosmos 243,
the
satellite to carry a microwave instru­
The radiometer on Cosmos 243
Although the operating
lifetime
had four channels.
of each instrument was less
than 2 weeks, the observations they transmitted were suffici­
ent to construct latitudinal profiles of liquid water content
in the atmosphere[GD70,S m 7 2 ,SH73].
Nimbus-5
microwave
(NEMS)
instruments
and
(ESMR-5).
in 1972
an
carried
: a
Nimbus-E
Electrically
NEMS was
of
the first
American orbiting
Microwave
Scanning
Spectrometer
Microwave Radiometer
little use in measuring rainfall b e ­
cause of its 180km resolution and limited
coverage,
the ESMR acheived some success in oceanic rainfall
however
applicat­
ions. ESMR-5 measured passive microwave radiation at 1.55 cm.
The Nimbus-6, next satellite in this series, carried a conical
scanning version of the ESMR at a shorter wavelength i.e. 0.81
cm.
The Scanning Multichannel Microwave Radiometer (SMMR) on
Nimbus-7
and Seasat,
both launched in 1978,
employed a new
antenna design and provided brightness temperature information
at 5 microwave frequencies.
Resolution was improved to 20 km
at the smallest wavelength.
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119
The first
objective
attempt
to
microwave radiances to rain intensity
relate simple-channel
was
made
by
Wilheit
[WCR77], who developed a technique for quantitatively mapping
precipitation over the oceans by means
data at 1.55 cm.
This
study
of satellite radiance
has guided much subsequent re­
search on the methodology of associating microwave brightness
temperatures
to precipitation rates.
Kidder and Vonder Harr
[KV77] used data from the Nimbus 5 ESMR-5 in- conjunction with
Wilheit's freezing level approach to calculate seasonal preci­
pitation frequencies for the tropical oceans.
Rao et al.[RAT76] used all the available ESMR-5 data for
1973-1974 to
the oceans.
assemble
a global atlas of rainfall rates over
Their maps are
qualitatively
correct
in
many
areas, however, there are many instances in which the computed
precipitation amounts are quite different from actual
tological values.
While
discussing
clima-
the possible sources of
error for the rainfall atlas, Rao[Rm84] pointed out the over­
simplified treatment of
[WCR77] model
and
on freezing level.
to
the rain-cloud in the Wilheit et al.
the unrealistic dependence of the results
The latter problem
led Rao et al.[RAT76]
apply 'ad hoc* corrections to the climatological freezing
levels in order to produce realistic results for cold
pheres.
atmos­
Rodgers and Adler [RA81] later utilized measurements
from ESMR-5 to estimate latent
heat
release within tropical
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120
storm systems, however,
to avoid the uncertainties associated
with the Wilheit et al.[WCR77] freezing level approach,
utilized
an
empirical
relationship based on
they
radar derived
rainrates.
Wilheit et al.[WCK82] have compared simultaneous
obser­
vations of a precipitating storm from five microwave radiome­
ters aboard the NASA CV-990 aircraft.
Rodgers and Siddallin-
gaiah[RS83] have shown how the 10 and 18 GHz channels on SMMR
can be used to improve the rain.versus wet ground discrimina­
tion at 35 GHz in regards to the earlier Rodgers et al[RCW79]
study.
Spencer and Santek[SS85] have examined the use of two
frequencies (18 and 35 GHz) to
better
events over a land background.
The tendency
studies is
to use extra channels
select
precipitation
in these latter
to help select rain events
but to quantify the rain-rate with an individual channel.
Mugnai and Smith[MS88], and Smith and Mugnai[SM88]
recently extended the Wilheit et al.[WCR77]
include
a
time-dependent,
have
investigation
vertically-inhomogeneous
to
cloud
microphysical component in conjunction with the variable rain
layer.
To
dependent,
do so they have obtained the results from a timetwo-dimensional numerical cumulus model
developed
by Hall[Hw80], which provides detailed water drop spectra
the
horizontal
and
vertical directions,
hydrometeor-size domain.
in
as well as in the
The model has been integrated up to
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121
the time at which precipitation has just begun to fall through
the cloud base. The radiative calculations are based on a ver­
tically and angularly detailed plane-parallel radiative-transfer model for unpolarized microwave radiation.
A sequence of
seven microwave frequencies, which undergo both absorption and
scattering in a precipitation medium,
have
been considered.
They have thus explored the feasibility of using multichannel
passive
microwave techniques for estimating rainfall.
5.4 Estimation of
rainfall rate from multiple microwave fre­
quencies using GPFES
Precipitation retrieval by
conditioned
problem
remote
sensing
.is
an
ill
from a mathematical perspective because
of the non-unique relationships which exist between the radia­
tion
signals and precipitation intensity.
Mugnai
and Smith
[MS88], and Smith and Mugnai[SM88] have developed a theoreti­
cal foundation for avoiding
some of
application of a multispectral
a purely analytical
these gaps with a GPFES,
Figure
5.1
shows
problems
by the
approach using passive micro­
wave measurements. There remains,
within
these
however,
operation.
a number of gaps
We propose to
close
i.e. an expert system approach.
theoretical
relationships
between
brightness temperatures and rainfall rates, developed by Smith
and Mugnai[SM88] over ocean.
A selection of eight rain layer
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122
thicknesses[Flwc=0.0, 0.25, 0.50, 0.75, 0.85, 0.90, 0.95,
1.0]
are inserted into a model cloud environment at three different
evolution times during the cloud lifetime [Tc=1000,
1900 seconds].
The heights of the rain layers are defined in
conjunction
with
an integral property
water;
more
details
for
1500, and
of the
cloud liquid
see Smith and Mugnai[SM88].
rainrate ranges marked with
black
arrows
are
The
deemed to be
the most probable rainrate intervals associated with
a given
time step. The three time steps are generally related to light,
medium and heavy rainfall steps.
(BT) shown at each frequency
(i.e.
(V+H)/2,
temperature and
The brightness temperatures
correspond
to unpolarized B T ’s
where V is a vertical polarization brightness
H is
a horizontal
polarization
brightness
temperature).
When
considering
passive
remote
sensing
algorithms,
cloud development time and rain layer thickness are generally
not
known "a priori".
That is,
through brightness tempera­
tures we have to infer a rainrate, cloud development time and
rain layer thickness.
fall by
remote
Thus, the problem of estimating
sensing
is a far more complex process
suggested by studies which
are
based
on
rain­
than
single parameter,
invariant cloud microphysics.
At first glance,
approaches are
highly
Figure 5.1
suggests
ambiguous since
that one channel
the functional rela-
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123
tc»IQOOsec ,
tca 1900sec
A
.v = 19.35 GHz
i
i H>0
1
^
1
1
1
i
1
1
A
.-i.i*”
•
Lv = 22.235 GHz .
290'
!
260
w
3
I
1
1
kS— iJ
1
1
1
230
<____
«»
Q. 200
E
at 290
t
,V = 37.0 GHz ,
et
! ^
in
in
260
c
JC
g»
u
03
i
i
230
.
200'
290 '
r
i
!^
= 89.5 GHz ,
«
'260
t
_
_
1
230
i
1
1
« ------------> 1
1
1
200
1
,
„
10
<— >1
i
-
100
1
10
100
10
100
Rainfall rate (mm/hour)
Figure 5.1 Relationship between brightness temperature and ra in fa ll ra te over water
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
124
tionships between brightness
generally not
temperatures
single valued.
approach is incorporated,
However,
and rainrates are
when a multi-channel
the ambiguities can be reduced.
When considering the detection and measurement of preci­
pitation over water backgrounds,
it
microwave signatures induced by
contrasted to the background.
microwave frequencies,
is essential
that
the
the precipitating clouds are
In general, at standard passive
rain layers appear "hot" against
the
low emission "cold" ocean.
When considering the detection and measurement of preci­
pitation over
high
contrast arguments
detected,
emissivity
apply.
depending
on
land
backgrounds,
In general,
frequency,
the same
precipitation can be
if
the
background
is
obscured to the extent that rain clouds appear "cool" against
a "warm"
background.
This land effect was first pointed out
by Savage and Weinman [SW75]
at the 37 GHz frequency.
In order to unambiguously
microwave
measurements
backgrounds,
which
it is important
surface
variation,
of
emission
and
interpret frequency dependent
precipitating
to
clouds
understand
reflection,
contribute to the total
the
column
surface emissivity variation,
eliminate
i.e.
degree to
and its potential
brightness
peratures. Multi-channel differential techniques
for this problem because they
over land
the
are
tem­
useful
dependence
on
they isolate the relative
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125
effect of
the precipitating cloud regardless of the absolute
backgroud. Spencer and Santek[SS85] and Spencer et al.[SHS87]
have
discussed the use of 18-37
from
SMMR
from
effectively
measurements
discrimination
in
warmer
threshold
GHz temperature differences
screening intense precipitation
land backgrounds.
of 20 degrees
They utilized a
which was
obtained
empirically, and which they maintained was somewhat arbitrary
and could only be used to isolate intense precipitation.
5.5 Data set for a case study with SSM/I measurements
In order to test our knowledge base on passive microwave
measurements,
we have used a test case from a new SSM/I data
set aided
measurements
by
satellite (INSAT).
from
from
the Indian
geosynchronous
INSAT satellite measurements are obtained
the 22 km VHRR special data set prepared under the aus­
pices of the Indo-U.S. Science
Smith et al [SOS88].
and Technology
On June 19,
1987,
the
program,
see
Special Sensor
Microwave/Imager(S S M / I ) was launched aboard the Block 5D-2 F8
spacecraft,
a polar orbiting
satellite
from the Air
Force
Defense Meteorological Satellite Program!DMSP). This satellite
radiometer detects
19.35, 22.235,
ation on
microwave
radiation
37.0 and 85.5 GHz.
a variety of
at four frequencies
These data provide inform­
environmental
parameters,
atmospheric water, wind speed, and sea ice.
including
The data used in
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126
this
investigation have
been
produced
by
Remote
Sensing
Systems [Wf88]. The data set is a compacted, chronologicallyordered version of the Temperature Data Records (TDR’s) being
produced
by the Fleet
Numerical
Oceanographic Center.
The
Compacted SSM/I antenna temperature(TA) data are available on
a series of 6250
bpi
magnetic
tapes (8 tapes per
month of
d a t a ).
The DMSP orbit is circular,
polar,
sun-synchronous, and near-
with an altitude of 860 km and an inclination of 98.8
degrees. The orbital period is
102
minutes;
for the ascending equatorial crossing is
A.M..
SSM/I actually
radiometers,
each
incorporates
simultaneously
7
the local time
approximately
separate
measuring
6:00
total-power
the
microwave
emission coming from the Earth and the intervening atmosphere.
Table 5.1 gives the frequencies,
and spatial
polarization,
resplution of the 7 channels.
measurements are taken
vertical polarization is
at
19.35,
measured
37.0
at
and temporal
Dual-polarization
and
the
85.5 GHz; only
22.235 GHz water
vapor channel.
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127
Table 5.1
Temporal and spatial resolution of SSM/I channels
Frequency
(GHz)
Polarization
Integration
Period
3 dB Footprint Size
Along--track Cross--trs
19.35
vertical
7.95 ms
69 km
43 km
19.35
horizontal
7.95 ms
69 km
43 km
22.235
vertical
7.95 ms
50 km
40 km
37.0
vertical
7.95 ms
37 km
29 km
37.0
horizontal
7.95 ms
37 km
29 km
85.5
vertical
3.89 ms
15 km
13 km
85.5
horizontal
3.89 ms
15 km
13 km
To facilitate multi-spectral processing of the 4 individ­
ual frequencies,
the 22, 37, and 85 GHz channels are
voluted to match
the 19 GHz
diagram
given
footprint size.
The
decon-
schematic
in Figure 5.2 describes the raw antenna foot­
print geometry[GH78]
for the 4 indivisual SSM/I
frequencies.
The deconvolution scheme is based on the use of discrete, nonrecursive filters whose weights
correspond to the percentage
overlaps of the higher resolution/higher frequency footprints
with respect to the lowest resolution 19 GHz
footprint.
The
following four equations are used to produce spatially consis­
tent
B T ’s for
each (i,j)
grid point
in the 19 GHz domain,
where i is a down-track index and j is a cross-track index:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
BT(19)ltJ - BT(19)1(j
f
.
u
u
u
BT(22)ij a [6«BT(22)ltJ + BT(22)1_1 j
+ BT(22)1+1j ] / 8
F
_
U
U
BT<37)i.j » [4*BT(37)1(J + B T O T J j . i j
U
♦ BT(37)I+1>j
U
U
+ BT(37)ltJ_, + BT{37)1(J+1]/8
BT<85h , j - 1|BT{85)i j
+ BT(85)i_ltJ ♦ BT(85)i+1j
U
U
+ BTfBSJij.j + BT(85)IiJ+1
U
U
+ BT(85)|_2,j ♦ BT(85)I+2(j
0
U
+ BT(85)!-x,j-j ♦ BT(85)i+1,j-i
+ BT (8S >Y+1,J—1 ♦ BT(85)i+ 1 J + 1 }
U
U
+ 0.5 |BT(85)1_2(j _1 ♦ BT(85)I_2,j+1
U
U
+ BT(85)|+2> j_j + BT(85)i+2,j+i)
U
U
+ 0.25 (BT(85)|tj_2 + BT(85)i,j+2}]/ll.5
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129
FREQUENCY
>85
>19
>22
>37
GHz
GHz
GHz
GHz
= ->SCAN B
-->SCAN A
= ->SCAN B
> SCAN A
Figure 5.2 Footprint geometry for the four SEM/I frequencies
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130
The antenna
portion of the SSM/I consists of a parabolic
reflector of dimensions 61 by 66 cm, which focusses the micro­
wave radiation into a corrugated, broad-band,
feedhorn.
7-port
antenna
The radiometer and antenna spin as a unit about an
axis parallel to nadir.
The rotation period is 1.9 s. A cold
calibration reflector and a hot reference load
are
attached
to the spin axis and do not rotate. The rotating antenna feedhorn observes the fixed cold reflector and hot load once each
scan.
In this way
scan.
Earth observations are
segment
of
the
calibration
rotation
in the aft direction,
observations are taken every
taken
cycle
as is shown
during
when
a 102.4
degree
the SSM/I is looking
in Figure 5.3.
The 102.4
degree arc is centered on the spacecraft subtrack and corres­
ponds to a 1394 km wide swath on the Ear t h ’s surface.
During
each scan, the 85 GHz channels are sampled 128 times over the
102.4 degree arc.
The integration period for a single sample
is 3.89 ms. This sampling scheme results in 128 V - p o l (vertical­
ly polarized) footprints and 128 H-pol(horizontally polarized)
footprints having an effective 3-dB resolution of about 15 km.
Observations
at
the
lower
three frequencies are only
taken every other scan. Scans during which the lower channels
are
sampled
are
called
’A-scans';
called ’B-scans’, i.e. 85 GHz only.
arc of an A-scan,
64 samples
the
During
of each of
other
scans
are
the 102.4 degree
the lower channels
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131
6.58 KM /SEC
V E LO C IT Y
GROUND
TRACK
1 0 2 .° A CTIVE
SCAN ANGLE
19 A N D J
1394 KM
SWATH W ID TH
SL1^
85?”*'-
SCAN A
SCANS
1 2 5 hm >
- 1 .9 0 SECT
SCAN A
SC A N S
Figure 5.3 S3-1/I orbit and scan geometry ( f rom
[H LS87 ] )
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132
are taken, with the integration period being 7.95 ms.
5.2
also shows
Figure
the spatial resolutions of the samples taken
on the ’A-scans’ and 'B-scans'
[HLS87].
Figure 5.4
shows an
example of successive SSM/I orbits and their scan routes.
The predecessor to SSM/I was
the
Scanning Multichannel
Microwave Radiometer(S M M R ); different versions of this radio­
meter were flown aboard Seasat and Nimbus-7, both launched in
1978. The SSM/I sensor design has several advantages compared
to the SMMR design. First the parabolic reflector and feedhorn
spin as a unit. For SMMR, the feedhorn
was stationary, while
the parabolic reflector scanned back and forth. As the reflec­
tor scanned above the fixed feedhorn,
the orientation of the
reflected Ear t h ’s vertical and horizontal polarization vectors
were rotated relative to
Thus the mixture of V-pol
varied during the scan.
further
complicated
the
and
feedhorn polarization vectors.
H-pol radiation
continuously
The polarization rotation effect was
by spacecraft attitude variations.
DMSP spacecraft attitude
is also much more
The
stable than that
for SeaSat or Nimbus-7.
The
second
ibration of SSM/I.
ports
that go to
advantage is the simplified,
The SSM/I
7 separate
external cal­
has a single feedhorn
radiometers.
required
and all channels observe the same
sources.
In the SMMR design,
No
two
ferrite switches
with
7
switching is
calibration
were used to
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133
figure 5.4
An e x a m p l e
of SSM /I
orb it
and scan g e o m e t r y
wmmm
W:iM
iiiss
M
•• .y .:
m
k
\
xrnmm
i'!h .»«v\'v ' „■ . i » y “
-39
L
100
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134
switch back and
forth between
the cold horn and
the earth-
viewing horn introducing an extra transmission gate.
The
spatial
than that for the
and
temporal
Seasat
and
coverage of SSM/I is better
Nimbus-7 S M M R ’s.
The 1394 km
wide swath of SSM/I is about twice that of the SMMR. Further­
more,
in
contrast to
operations schedule,
ously.
the Nimbus-7
S M M R ’s
the SSM/I instrument
The periodic turning on and off
calibration problems; when the SMMR
every other day
operates continu­
of
the SMMR
warms up after
causes
turn-on,
calibration drift occurs.
The SSM/I antenna temperature data reside on a series of
6250 b p i , 2400-ft magnetic tapes.
Each data file corresponds
to a single SSM/I orbit. The beginning of an orbit is defined
as the ascending equator crossing
of
the
spacecraft (i.e.,
south-to-north crossing).
The data files (i.e., orbits)
chronologically
The
ordered.
orbit period is
are
102 minutes,
and thus there are 14.1 orbits per day. A tape contains three
or four days
of
antenna
temperatures.
For
computation of
calibrated brightness temperatures, we have used the algorithm
outlined
in [Wf88].
For
the case study used in this inves­
tigation of our research, we have choosen a 5 day period from
July 16 to July 20 during 1987. INSAT satellite data indicate
a major monsoon surge during this period. The area under study
is bounded by (40 S,40 N)
and (35 E,115 E).
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135
Figure 5.5
shows a composite
period under study;
INSAT-IR
image
for
the
the composite involves data from July 16,
18, and 20 and corresponds to SSM/I orbit swaths necessary to
generate full coverage.
One SSM/I
orbit pass is
missing on
July 20; therefore there is a missing data sector in the com­
posite imagery. Figure
5.6
shows
composite images of SSM/I
during the same period for 3 different channels of vertically
polarized brightness temperature.
brightness temperature
data
at
The
vertically
22 GHz have
because they show virtually the same pattern as
images
and
have been
Graphics
prepared
Enhancement
been
polarized
excluded
19 GHz.
The
on the Micro-based Image Display
Tool (MIDGET)
[SSA87,SOS88];
Appendix A.
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see
136
Figure
Figure
5.5
5.6.a
INSAT-IR
SSM/I-19 GHz
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137
Figure
5.6.b
SSM/1-37 GHz
Figure
5.6.C
SSM/l-85 GHz
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138
5.6
Knowledge base to retrieve precipitation
Numerical studies have shown that precipitation retrieval
is an ill-conditioned problem from a mathematical perspective
because of problems with non-unique relationships between the
radiation signals
and
precipitation intensity.
reduce some of these problems,
base
In order to
we have developed a knowledge
for precipitation retrieval incorporating multi-channel
passive
microwave radiometer measurements from the Air Force
SSM/I instrument for application
within
the
Southwest-East
Asian Monsoon domain. To aid the selection and classification
of appropriate targets, we also incorporate INSAT imagery.
The knowledge
system.
The
base
first
select cloud targets
lexes,
is developed
tier is
as
a three tier logic
"rule-based" and is designed to
(isolated clouds, meso-scale cloud comp­
or deep organized convective depressions) as possible
candidates for precipitation and
teristics.
The
VIS-IR channel
to describe general charac­
rule base structure
information
from
utilizes high frequency
INSAT along with the SSM/I
brightness temperatures to locate potential targets,
cribe their morphological structures,
ocean,
to
characterize
the residual
to
to des­
discriminate land-
polarization,
and to
identify large ice particles(graupel). This rule base is also
used to assign
most probable rainfall
categories
for light
and heavy rainfall(for both water and land background), which
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139
are assigned as the final solution if the tier 2
cannot provide a unique rainrate solution.
In
and 3 logic
the jargon of
fuzzy logic these would be referred to as possibility indices;
they are useful for situations in which paradoxical conditions
in a set of microwave brightness temperatures prevent an exact
solution.
All
facts
and
rules are
based on the behavior of the
SSM/I passive microwave measurements [19 (HV), 22 (V), 37 (HV)
and 85 (HV) GHz]
satellite.
along
with
VIS/IR
images
In the tier 1 rule definitions, H is
ly polarized
microwave channel;
microwave channel;
from the INSAT
a horizontal­
V is a vertically polarized
U = (H + V)/2 is an unpolarized estimate;
D = (V - H) is a degree of polarization;
F means a fuzziness
to represent the proximity of values which d o e s n ’t have fixed
boundaries or thresholds.
In the first tier of the knowledge base,
ions are made concerning the classification of
several decis­
precipitating
clouds. The following tests are carried out on the MIDGET(see
Appendix A for detailed procedures):
1. The first decision is to determine whether a cloud is pre­
sent based on a visible(VIS) channel image and/or an infrared
(IR) channel image.
2. The second decision concerns:
2.1 the cloud radiometric properties;
i.e., whether
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140
the cloud temperature is (cold, medium, warm) and
whether the cloud albedo is (high, medium,
2.2 the cloud
morphological
properties;
or appearence of cloud, specifically
low)
i.e., the shape
whether it is
cumuliform (convective) or stratiform(layered).
In
this discrimination test, we use the basic signatures
of cumuliform and stratiform clouds:
cumuliform:
unstable, large vertical versus horizontal
development,
short evolutionary
growth (minutes) and
cotton-like appearence,
stratiform:
stable(F), small vertical versus horizontal
development,
long evolutionary growth(hou r s ), homogene­
ous over large area.
3. The
land
third decision
or
water.
concerns
whether
the cloud is
over
There are four cases: cumuliform over land,
cumuliform over water, stratiform over
land
and
stratiform
over water.
4. The fourth decision concerns the residual polarization due
to the land background;
IF the 37 GHz signal is highly polar­
ized i.e., 37D brightness temperature exceeds about 18 degrees
(F) THEN the polarized signature of the
and rain is unlikely.
work for water;
The value of
surface
is dominant
"about 18 degrees"
will
it is different for land. However we have not
yet determined an optimum value.
This rule can be applied to
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141
both
cumuliform or stratiform clouds.
5. The
fifth decision
concerns the likely rainfall category
for over water situations; IF 19H
20 degrees (F) THEN
decide heavy,
- 37H does not exceed about
light or no rain:
5.1 IF the cloud is over water and 19U exceeds
about 240K
(F) THEN there is heavy rain;
5.2 IF the cloud is over water
about
and
19U
does
not exceed
240K (F) but is greater than or equal to about 195K
(F) THEN there is light rain;
5.3 IF the cloud
is over water and 19U
195K (F) THEN it
is not raining.
6. The sixth decision concerns
ice particles
the identification
(graupel) over land;
37H exceeds about 20 degrees
less than about 255K (F)",
is less than about
of
Let A be a clause
large
"19H -
(F)" and B be a clause "85PCT is
where 85PCT=1.82*85V - .82*85H:
6.1 IF the cloud is over land and A and B THEN
large
ice
particles are present;
6.2 IF the cloud is over land, and (A and not B) or (not A
and B) THEN go to a special care branch(under development);
6.3 IF the cloud is over land and
stratiform is the cloud
type and 6.1 is true THEN the situation is impossible, and
thus return to step
2.2 to
reevaluate
the appearence of
the cloud.
7. The seventh decision concerns the likely rainfall category
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142
for over land situations:
7.1 IF 6.1 is true THEN there is heavy rain;
7.2 IF 19H - 37H does not exceed about 20 degrees(F), 85PCT
is less than about 275K (F) and 37U is less than [.9(F)*ST
- 10(F)] THEN there is light rain,
where
ST is a surface
temperature (determined from INSAT IR temperature);
7.3 IF
The
neither 7.1 or 7.2 are true THEN it is not raining.
second tier
of the
inference network to remove
knowledge base incorporates an
the ill-conditioned
aspects
of
the problem based on the behaviour of the 4 microwave signals
at 19, 22, 37 and 85 GHz.
The
rules in
the second tier are
derived from Figure 5.1 over water. This portion of the know­
ledge base stresses the coimplication relation which is posed
as a form
of fuzzy logic.
We are currently
developing
co­
implication based
rules
over land to extend the application
of the technique.
An important feature of the second tier is
that brightness temperature boundaries or thresholds need not
be "fixed"
as
done
in
conventional
rainfall
algorithms.
Instead the methods of "fuzzy logic" are used which effective­
ly alter implications within the proximity of boundaries. The
elimination of ambiguities is the main advantage of
the coimplication method
in an ill conditioned
invoking
problem such
as rainfall retrieval.
The coimplication based rules over water
in tier 2
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are
143
classified in terms of
3 categories;
probable and low probable with
for the conclusion
high probable,
a reliability
medium
of rain
class
according to the development time-step of
the cloud. A coimplication based rule structure
the antecedent part,
consists
of
the conclusion part and the asymmetrical
differnces representing the coimplication
simplification of rule
representation,
relation.
For the
we have used several
notations:
1) General structure of Coimplication;
The "general structures of coimplication" notation
is
of
the form
(PI),,(Pn) I— | Q( j) ,
where Pi is a clause in the antecedent,
and j is a number
from
an asymmetrical
Q is a conclusion
difference array
representing the coimplication relation of the rule.
Table 5.2
provides
rules which describe
describe
the
the syntax
the behavior
behavior
of
for nine
coimplication
of Figure 5.1.
the non-linear
The rules
relationships in
Figure 5.1 interms of linear segments in which discrete ranges
of the B T ’s
are given high,
conjunction
with
heavy
their
rainfall intervals.
medium and low probabilities in
relationship
to light,
In the rule definitions,
medium and
the asym­
metrical difference transforms are given in Table 5.3.
differences represent partial
These
mapping between the antecedent
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144
and the conclusion in their truth values.
2) clause in the antecedent;
The "clause in antecedent" notation is of the form
(l,u,C,V(i)),
where 1 is a lower
bound of
brightness temperature, u is
an upper bound of brightness temperature,
frequency
and
V(i)
is
a fuzzy subset
C
is a channel
which is used to
evaluate the "truth degree" representing the characteristic
of the interval [l,u] which has membership function i.
The
degrees
5.4;
We
membership
in
functions used for evaluating the truth
the nine coimplication
have used
rules are given in Table
linear relationships
as the form of the
membership functions. The individual clauses used in the nine
rules are given explicitly in Table 5.2.
3) conclusion;
The "conclusion" notation is of the form
X(l,u,t) ,
where
X is
H (high probable) / M (medium probable)
L (low probable),
/
1 is a lower bound of rain index,
u is an upper bound of rain index and t is the development
time step.
Table 5.2 indicates the conclusions for the nine
rules;
The relationship between a rain index and a rainrate is given
in Table 5.5.
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145
To provide an example, a condition (267,281,85,high(H14))
means
that
a brightness temperature
within the interval [267,281]
H14.
The
high
denotes
that the BT interval is over a high
H14 ranges from 0.7 to 1.0.
temperatures
0.7, referred as
Therefore,
The
outside that interval would be zero.
e^-cut has been
discussed
where
The
in Section
2.1.
if a brightness temperature is 267 degrees,
then
most
obtained
truth degree for code
The truth degree for any bright­
the truth degree of this condition is
function,
the 85 GHz channel
has a membership function code
range of brightness temperatures.
ness
at
brightness
temperature
using a 0.7 o(-cut;
0.7 has
been fixed
0.7.
see
by
In
a membership
intervals have been
Definition
2.7 for o<-cut,
trial-and-error while finding
asymmetrical differences of the coimplication relation. Asym­
metrical
differences
are expressed by Definition 4.18.
value 0.7 could be changed later.
highly
index
probable
that there is light rain
interval [1,6]
seconds.".
An
H ( 1,6,1000)
example
at
the
of
development
means
with
step
The
"it is
a rain rate
time
1000
a complete coimplication rule is
(204,251,19,l o w ( H U ) ) , (246,268,22, medium low(H12)),
(240,273,37,medium low(H13)),
W
h
In words,
(267,281,85,h i g h (H 1 4 ))
(1,6,1000)(1).
it is interpreted as "IF BT(brightness temperature)
at 19 GHz is low and BT at 22 GHz is
medium low and BT at 37
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146
GHz is medium low and BT at 85 GHz is high THEN light rain is
highly probable in conjunction with
asymmetrical
difference
in rule 1.
Table 5.2
Rules based on coimplication in the second tier
RuleOl:
(204,251,19,low(Hll)), (246,268,22,medium low(H12)),
(240,273,37,medium low(H13)), (267,281,85,high(H14))
W
H ( 1,6,1000)(1)
Rule02:
(255,273,19,high medium(H21)), (265,274,22,high medium(H22)),
(253,273,37,high(H23)), (252,266,85,medium(H24))
M
H(4,8,1500)(2)
Ru le03:
(228,257,19,low(H31)), (227,255,22.medium low(H32)),
(225,251,37,low(H33)), (234,250,85,low(H34))
M
H(9,17,1900)(3 )
Ru le04:
(215,267,19,low medium(M41)), (252,273,22,high medium(M42)),
(265,276,37,high medium(M43)), (264,281,85,high(M44))
M
M ( 5,8,1000)(4 )
R u le05:
(254,258,19,high medium(M51)), (267,269,22,high medium(M52)),
(270,272,37,high(M53)), (261,266,85,high(M54))
W
M<1,4 , 1500)(5)
R u le06:
(242,277,19,not high medium(M61)), (243,277,22,not high
medium(M62)), (236,273,37,not medium high(M63)),
(248,266,85,medium low(M64))
M
(M(9,17,1500)(6)
R u le07:
(252,257,19.medium low(M71)), (249,255,22.medium low(M72)),
(239,251,37,medium low(M73)), (237,250,85,medium low(M74))
M
M(5,8 , 1900)(7)
R u le08:
(254,276,19,medium(L81)), (257,277,22,high(L82)),
(243,277,37,high(L83)), (259,281,85,high(L84))
M
L(9,17,1000)(8)
R u le09:
(257,257,19,medium(L91)), (255,255,22,m e dium(L 9 2 )),
(250,251,37,medium(L93)), (246,250,85.medium high(L94))
L(l,4 , 1 9 0 0 X 9 )
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147
Table 5.3
Asymmetrical Difference for each rule
itemxlOO
rule
No.
1
2
3
4.
5.
6.
7.
8.
9.
(70,75,3)(76,80,4)(81,85,0)(86,91,3)(92,95,0)(96,100,1)
(70,80,5)(81,85,6)(86,90,1)(91,95,0)(96,100,-3)
(70,75,14)(76,85,-5)(86,95,-2)(96,100,1)
(70,75,3)(76,85,8)(86,90,-2)(91,95,1)(96,100,-4)
(90,91,0)(92,92,1)(93,94,-1)(95,96,1)(97,100,-2)
(70,76,13)(77,91,-2)(92,94,-1)(95,100,1)
(70,75,14)(76,90,-1)(91,95,-3)(96,100,0)
(70,72,7)(73,75,10)(76,80,0)(81,84,2)(85,87,-1)(88,100,-5)
(90,99,2)(100,100,-5)
Table 5.4
Membership functions used in the second tier
In code type XNF, X means reliability class
(High-H; Medium-M; Low-L),
N is a rule number
in Tier-2,
and F means channel(1=19 GHz, 2=22 GHz, 3=37 GHz, 4=85 GHz)
or C=conclusion.
In the function definition x is a
brightness temperature or rain index, and y is a possibility
truth degree.
Code
HI 1
H12
H13
H14
H1C
H21
H22
H23
H24
H2C
Function
y=-3/470*(x-204)+0.7
y=-3/220*(x-246 )+0 .7
y=-3/330*(x -240)+0.7
y= 3/140*(x-267)+0.7
y=-3/50 *(x-l) + 1.
y=-3/180*(x-255)+l.
y= 3/40 *(x-265)+0.7
-3/50 * (x-269)+l.
y= 3/200*(x - 2 5 3 )+0.7
y= 3/140*(x-252)+0.7
y=-l/20 *(x - 4 ) +0.95
X
X
X
X
X
X
X
X
X
X
X
in
in
in
in
in
in
in
in
in
in
in
[204, 251]
[246, 268]
[240, 273]
[267, 281]
[1,6] and
[255, 273]
[265, 269]
[269, 274]
[253, 273]
[252, 266]
[4,8] and
and X is BT
and X is BT
and X is BT
and X is BT
x is rain index
and X is BT
and X is BT
and X is BT
and X is BT
x is rain index
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148
Table 5.4: Continued
H31
H32
H33
H34
H3C
M41
M42
M43
M44
M4C
M51
M52
M53
M54
M5C
M61
M62
M63
M64
M6C
M71
M72
M73
M74
M7C
L81
L82
L83
L84
L8C
L91
L92
L93
L94
L9C
y=-3/290*(x-228)+l.
y=-3/280*(x-227)+1.
y=-3/260*(x-225)+l.
y=-3/160*(x-234)+l.
y= 3/80 *(x - 9 ) +0.7
y=-3/520*(x-215)+l.
y=-3/210*(x-252)+l.
y=-3/110*(x-265)+l.
y= 3/170*(x-264)+0.7
y=-23/300*(x - 5 )+0.93
y=-l/40 *(x - 2 5 4 )+1.
y=-l/40 *(x - 2 6 7 )+1.
y=l/40 * (x-270)+0.95
y=l/50 *(x- 2 6 1 )+0.9
y=-l/30*(x - 1 ) + 1.
y=-3/280*(x-242)+l.
3/170*(x-270)+0.7
y=-3/260*(x-243)+l.
3/80 * (x-269)+0.7
y=-3/240*(x-236)+1.
3/130*(x-260)+0.7
y =-1/60 *(x- 2 4 8 )+ 1 .
y= 3/80 *(x - 9 ) + 0.7
y=-3/50 *(x - 2 5 2 )+1.
y=-3/60 *(x - 2 4 9 )+1.
y=-3/120*(x-239)+l.
y=-3/130*(x-237)+l.
y= 1/10 *(x - 5 ) + 0.7
y=-15/140*(x-254)+l.
= 0.85
y= 3/200*(x-257)+0.7
y= 3/340*(x-243)+0.7
y= 3/220*(x- 2 5 9 )+0.7
y=-3/80 * (x-9) + 1.
y= 1
y= l
y= l
y= 1/40 *(x - 2 4 6 )+0.9
y=-l/30 *(x-l) + 1.
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
in
[228,257] and X is BT
[227,255] and X is BT
[225,251] and X is BT
[234,250] and X is BT
[9,17] and x is rain index
[215,267] and X is BT
[252,273] and X is BT
[265,276] and X is BT
[264,281] and X is BT
[5,8] and x is rain index
[254,258] and X is BT
[267,269] and X is BT
[270,272] and X is BT
[261,266] and X is BT
[1,4] and x is rain index
[242,270]
[270,277] and X is BT
[243,269]
[269,277] and X is BT
[236,260]
[260,273] and X is BT
[248,266] and X is BT
[9,17] and x is rain index
[252,257] and X is BT
[249,255] and X is BT
[239,251] and X is BT
[237,250] and X is BT
[5,8] and x is rain index
[254,268]
[268,276] and X is BT
[257,277] and X is BT
[243,277] and X is BT
[259,281] and X is BT
[9,17] and x is rain index
[257,257] and X is BT
[255,255] and X is BT
[250,251] and X is BT
[246,250] and X is BT
[1,4] and x is rain index
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149
Table 5.5
Rain rate index
1
Index
2
3
4
5
6
7
8
R a t e (mm/hour)
0.2 0.4
0.8 1.0 2.0 5.0 8.0
Index
11
13 14
R a t e (mm/hour)
20. 30.
12
15
16
9
10
10. 12.
16.
17
40. 60. 80. 100 200
The final tier of the knowledge base uses Table 5.4
Table 5.5 as a "frame system" to uniquely
tation intensity to any combination of
and
assign
a precipi­
microwave
brightness
temperatures through the "Possibility Distribution Functions"
which are
the inverses of the membership functions.
ventional logic notation» these would be referred
In con­
to
as the
"Probability Distribution Functions" directly associated with
the nine
individual coimplication relationships.
In the ins­
tances that tier 2 cannot arrive at a unique solution, we say
that paradoxical conditions exist and the GPFES reverts
most
to a
probable rainfall category involving no, light or heavy
rain based on the tier 1 conclusions. Some sample interactive
runs to show relationships between tier 1 and tier 3 are given
in Appendix B.
We now formulate a retrieval algorithm based on how tier
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150
2 and tier 3 generate a unique
rainfall measurement.
Let
i
(i = l, 4) be
a channel number representing frequencies at
GHz, 22 GHz,
37 GHz and 85 GHz, respectively. Let j(j=l,9) be
a coimplication rule number in tier 2.
degree of a membership
function
19
Let P(i,j) be a truth
given
brightness temperatures BT(i) as input.
in
Let
Table
5.4
with
C(j) be a truth
degree evaluation from a combination of 4 individual frequency
membership function calculations based on a conjunction oper­
ation.
Let T(j) be the truth degree of a possible rain class
obtained from the asymmetrical difference adjusment.
Let
MT
be the maximum truth degree from the set of nine truth degrees
associated with the nine coimplication rules(only a subset of
these will have non-zero values):
In summary:
1. input BT(i)
(i=l,4)
2. find P(i,j)
(i=l,4 and j=l,9)
3. calculate by conjunction
C (j ) = min[P(l,j),P(2,j),P(3,j),P(4,j)]
(j=l,9)
4. find asymmetrical difference d(j) associated with
the subinterval containing C(j)
(j=l,9) in Table 5.3
5.
calculate T(j) = C(j) + d(j)
(j=l,9)
6.
find MT = max[T(j),j = l ,,9] and N s j of
7.
if (MT is greater than 0.0) then
MT
i) find a rain index from MT and rule number N using
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151
the membership function with conclusion given in
Table 5.4
ii) find a rain rate from a rain index using Table 5.5
8. paradoxical conditions:
if MT=0 then use tier 1 decision on none,
light or
heavy rain as the most probable rainfall category.
An example to retrieve a rain rate based on the algorithm
is as follows:
Let BT1 = 210, BT2 = 250, BT3 = 248 and BT4 = 277.
Then Pll = 0.96 ( H U ,
Table 5.2), P21 = 0.95
P31 = 0.93 (H13, Table 5.2), P41 = 0.91
(H 1 4 ,
P12 = o.
, P22 = 0.
P13 = o.
, P23 = 0.75, P33 = 0.73, P43 = 0.
P14 = 0.
, P24 = 0.
, P34 = 0.
, P44 = 0.93
P15 = o.
, P25 = 0.
, P35 = 0.
, P45 = 0.
P16 = 0.
, P26 = 0.91, P36 = 0.94, P46 = 0.
P17 = o.
, P27 = 0.95, P37 = 0.78, P47 = 0.
P18 = 0.
, P28 = 0 .
, P38 = 0.74, P48 = 0.95
P19
, P29 = 0 .
, P39 = 0 .
0.
, P32 = 0.
(H12,
, P42 = 0.
, P49 = 0.
Therefore, Cl = 0.91, C2 = C3 =...= C8 = C9 = 0. But Cl is in
[0.86,0.91] with a rule number 1 in Table 5.2
and dl = 0.03.
Thus T1 = Cl + dl, T2 = T3 =...= T8 = T9 = 0 .
Then MT = T1 =
max(Tl,,T9) and N = 1. Using H1C in Table 5.4, we find a rain
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152
index 2 (0.4 mm/h; see Table 5.5).
Thus if a set of B T ’s pass all the way through Tier 1
Tier 3,
rate.
then
to
the retrieval system can assign a unique rain-
Since there is no intersection point among
brightness
temperature intervals at 85 GHz associated with coimplication
rules 1-3 in Table 5.4,
=
i.e.
[267,281]A [252,266] ft[234,250]
we find at most only one coimplication rule with a truth
degree greater
than
different possible
or equal
to 0.7
as a combination of 4
brightness temperatures associated with a
high reliable conclusion of rain class. Similarly for 85 GHz,
the same argument
is applied to the case
conclusion of rain class because
brightness temperature
maximum
of
[259 ,251 ]
intervals
a low reliable
[246 ,250] =
for rules 8
and
9.
in
If a
truth value in the conclusion associated with a high
reliable class has the same value as the conclusion associated
with a low reliable class,
then we take the case with a high
reliable class to avoid a multivalued solution.
rules
4 and
7,
the same
[264, 281 ] C\[237 ,250] =
at 85 GHz.
argument
<f> for
can be applied
it is not
unique conclusion
examination,
however,
class involves
immediate obvious
with
a non-zero
note
because
brightness temperature intervals
But, the relationship among rules 4,
the medium reliable conclusion
Therefore
For cases of
that
5 and
6 of
intersection.
that there
truth degree.
exists a
On futhur
the intersection interval
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153
[264,281]fl[261,266]fl[248,266] of brightness temperatures with
rules 4, 5 and 6 at 85 GHz,
[264,266]
respectively,
we find a possible truth
rule number 4;
[0.96,1.0]
for rule number 6.
The
is [264,266].
From
interval [0.7,0.74]
for rule number 5;
and
for
[0.7,0.73]
value 0.72 is the only common inter­
section point of truth degrees(at brightness temperature 265)
for rules 4 and 6, and since the membership function at 85 GHz
of rule 4 is
an increasing
function
whereas the membership
function at 85 GHz of rule 6 is a decreasing function,
is only one intersection point at 0.72. Therefore,
there
there is a
guaranteed maximum certainty associated with the coimplication
rules of tier 2.
As a final illustration
of the tier 2 and tier 3 logic,
Figure 5.7 schematically presents the behaviour of the member­
ship functions of
step
by “««8tep
Hll, H12, H13, H14, and H1C along with the
calculations
brightness temperatures
of
the algorithm
which satisfy
The rainrate solution assumes
for a set of
coimplication rule 1.
the truth
degrees
associated
with coimplication rules 2-9 are all zero.
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154
Figure 5.7: A schematic example of coimplication rule 1
A rain-rate is assigned under the assumption that the truth degrees of
rules 2-9 are 0.0.
Rain
19 GHz
0.7Low
r
■5 204
3
4-1
251
277
[204,251]->[1.0,0.7]
y=-3/470*(x-204)+l
22 GHz
Light
1
6
17
[1,6]->[1.0,0.7]
y=-3/50*(x-l)+l
Asymmetrical Difference Transform
Medium
(0.7,0.75,0.03) (0.76,0.8,0.04)
(0.81,0.85,0.0) (0.86,0.91,0.03)
(0.92,0.95,0.0) (0.96,1.0,0.01)
Example
£
246
268 277
g
[246,268]->[1.0,0.7]
** y=-3/220*(x-246)+l
37 GHz
Medium
•f? 225
240
273 277
[240,273]->[l,0.7]
y=-3/330*(x-240)+l
BT1=210 BT2=250 BT3=248 BT4=277
Then we find fuzzy truth degrees.
Plla0.96 P21=0.95 P31=0.93 P41=0.91
Using conjunction operation,
Cl-0.91
Asymmetrical Difference Adjustment:
Cl=0.91 [0.86,0.91]->0.03
0.91+0.03=0.94
Optimal Rule Selection:
MAX[0.94,0.,0.,0.,0.,0.,0.,0.,0.]
Rainrate Conclusion:
v(Light Rain in [1,6])=0.94
Therefore,
x=50/3*(l-y)+l
=2->0.4 mm/hr
85 GHz
High
234
3
Ih
4J
r
267
281
[267,281]->[0.7,1]
y=3/140*(x-267)+0.7
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155
5.7 Results
Figure 5.8 demonstrates the results for the test case showing
precipitation areas
rain",
that have been
"light rain",
background.
Areas
classified
and "no rain" categories
over land
into
over
"heavy
a water
which qualify as possible rain
areas have also been identified.
The
figure indicates
that
there are infrequent instances when tier 2 identifies paradox­
ical conditions in which case the algorithm
tier 1 most probable rainfall category.
reverts
to
the
That is, if there is
no coimplication rule which a set of multi-channel brightness
temperatures satisfy,
no rain,
then
we have assigned to those points
light rain or heavy rain index based on tier 1 con­
clusions.
We are currently developing
a tier-2
system
for
land based precipitation.
Figure 5.8 shows good correlation to the temperature pat­
tern in the INSAT IR image previously shown in Figure 5.5. In
Figure 5.8,
we also
show a
small inset
which is the cloud
category map prepared on MIDGET during the classification
the cloud type (convective or stratiform);
see Appendix A.
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of
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 5.8 The image, representing rainfall rate, obtained from GPFES
blue:light rain, red:heavy rain, magen:most probable heavy rain
cyan:most probable light rain, green:possible rain areas over land, other:no rain
157
Chapter 6
Conclusions
The concept
of
a general purpose fuzzy expert system
which
makes use of fuzzy reasoning techniques based on coimplication
has
been
discussed
and incorporated into a General Purpose
Fuzzy Expert System (GPFES) together
with
an application to
passive microwave satellite precipitation retrieval. The GPFES
is
one in which no specific domain knowledge is incorporated
in
the inference engine.
vagueness in various ways
fuzzy techniques.
implication method.
Many
expert systems operate
although
they do
with
not make use of
We have modelled vagueness through the co­
The models developed
thus far
have
no
theoretical base and have been inadequately analyzed. We have
attempted to
provide guidance for modelling general domains.
Once GPFES was implemented,
its performance was necessary.
the GPFES performance,
some
method of
evaluating
In order to provide a test of
we have applied our technique
to the
problem of passive microwave precipitation retrieval.
A knowledge base to retrieve
precipitation
from satel­
lite microwave measurements has been developed and
in GPFES.
The
evaluated
investigation has been primarily attempted to
test the usefulness of coimplication on space-based
measure-
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158
merits
of passive
microwave
signals,
specifically
as they
pertain to the retrieval of precipitation over water and land
backgrounds
when
the BT-RR functions
exhibit problems with
non-uniqueness.
Any attempt to develop a quantitative formulation between
microwave brightness temperatures
sities
at one
considerations
and
or more frequencies
somewhat
beyond
precipitation
inten­
is adversely effected by
the control of contemporary
techniques. Mugnai and Smith[MS88], and Smith and Mugnai[SM88]
have developed
a theoretical foundation
for avoiding one of
the problems of "non-unique relationships between the radiat­
ion signals
and
precipitation
intensity".
They do this by
application of a multi-spectral approach using passive micro­
wave measurements.
There remains,
with a purely analytical
close these gaps
operation.
however, a number of gaps
We
have
attempted
to
with an expert system approach based on the
coimplication method.
A well-founded method of reasoning under uncertainty
been developed and implemented in GPFES.
mental system.
has
GPFES is an experi­
In GPFES, we h a v e n ’t considered natural lang­
uage facilities which are needed in real applications. Future
research will involve implementing natural language facilites
in GPFES. In the application to precipitation retrieval,
this
feature was not needed because GPFES
with
receives
answers
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159
respect to queries from
a numerical
data set and transforms
results into imprecise truth values.
The most important
problem
facing
the
expert
system
builders is to acquire knowledge from a domain expert.
a time consuming and
ledge acquisition
difficult process.
would enable
knowledge interactively.
It is
Automation of know­
the domain expert
to supply
The research and ideas described in
this dissertation are not
the final solution
to approximate
reasoning
in expert
system areas.
New methodologies are needed
if we are
to eventually develop a
general purpose fuzzy expert
system which involves principles for
in general domains. However,
modelling uncertainties
the results of this dissertation
clearly demonstrate that
1) a new reasoning method, based on the equivalence operator
instead of the implication oprator in modus ponens,
provides for a fuzzy logic-based computational framework
2) the concept of coimplication in the inference process is a
potential solution to model uncertainty in the general
domain
of expert
3) an expert
method,
system
systems
approach, based on the coimplication
is also a potential solution to handle an ill
conditioned problem in precipitation retrieval from
remotely sensed microwave measurements.
We note also some open problems in expert systems area related
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160
to this dissertation research as follow:
1) GPFES is a shell-like expert system tool. It has been
developed with a single application as a target. Future
work will involve having GPFES handle multiple knowledge
sources and support a communication interface among
different knowledge sources
2) The most important problem currently facing the builders
of expert systems is that of knowledge acquistion.
It is
a time consuming and difficult process. There is a great
need to automate this process. In GPFES,
the knowledge
representation scheme uses asymmetrical differences between
antecedent and conclusion,
in the partitioned truth
interval. To find asymmetrical differences makes the
knowledge acquisition process more difficult
3) Common sense knowledge is embodied in most human reasoning
and is very broad and hard to define.
It has not been
captured to any extent in expert system knowledge bases.
Zadeh[Z183b] has done some work on this topic. In GPFES,
the knowledge structure to handle common sense knowledge
must be described as poor
4) A great difficulty in expert systems development is that
of non-monotonic reasoning. Exceptions can be represented
in a knowledge base but when there are a large number,
space problems happen. We plan to solve this non-monotonic
reasoning problem using coimplication theorems.
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161
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181
Appendix A
System configuration for image processing to retrieve
precipitation from satellite microwave measurements
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182
1. MIDGET OCNTCGJRAHCN
8 MgHz PC/AT
WITH
32-BIT NUMBER NINE VIDEO BOARD
RGB MONITOR
HJ
ItM U '
STORAGE
_________
- m m
1.
2.
3.
4.
PUCK I TABLET
0.64 BASE M E M O R Y
I Mbyt* R A M M E M O R Y
20 Mbyt* HARO DISK
0.35 Mbyt* FLOPPY DISK
SOFTWARE CONTENTS
METEOROLOGY
DEPARTMENT
CY
930
GRAPHICS
WORKSTATION
SUN
3 /2 6 0
/xVAX
GPX 2
£tx
1.
2.
jtVAX
3.
GPX I
4.
HIGHER OROER 3.
MS-DOS
OP SYSTEM
8086 ASSEMBLER
FORTRAN, C COMPILERS
BSE EDITOR
NUMBER NINE GRAPHICS INTERFACE
LIBRARY
6. DECISION IMAGES GRAPHICS APPLICATIONS
•SYSTEM
7. MATHTEXT W O R O PROCESSOR
8. OECNET, TELNET COMMUNICATION PROTOCOLS
KEY
mm ETHERNET
EZZ3 LOOSELY COUPLEO
NETWORK (CDC)
C = 3 BROAD BAND CABLE
<^> BUFFERED REPEATER
EXTERNAL!
WORLD
CAMPUS
I
T
TERMINAL SERVER
SPAN
J
SUPERCOMPUTER
COMPUTATIONS
RESEARCH INSTITUTE
■
■
SUN
IRIS
3030
3 /2 6 0
COMPUTING CENTER
S U P E R T
COMPUTERS I
DEC
11/780
DEC
8700
CYBER
205
FRONTI
ENO I
ETA
10
I
CYBER CYBER
850
835
IBM
4381
BACK END
FILE STORAGE
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183
2. System Flow to retrieve precipitation
SSM/I ANTENNA TEMPERATURE TAPES
a t 4 frequencies
CYBER 205
EEA 10
'ssm/i bttemp :
IMAGES a t 4
^frequencies „
VAX 8700
INSAT IR IMAGES
VAX 11/780
SSM/I INSAT-H
OCMPOS. IMAGES
VBAOOBHL-J
see a menu in the next page
fo r cloud targ e ts selection
RXX: TABLET
RUN GEFES TO RETRIEVE FKEEEFTIA11CN
VAX 8700
FREapnmcN
CATEGORY IMAGE
MIDGET
DISPLAY
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184
3. MENU FCR OX3UD TARGETS SELECTION in MIDGET
*** DISPLAY OPTIONS ***
1
2
3
4
R
O
P
E
19GHz V/H/U
SSM/I
ULH SECTOR :
INSAT
SSM/I
URH SECTOR :
37GHz V/H/U
SSM/I
LLH SECTOR :
85GHz V/H/U
SSM/I
ULH SECTOR :
Resets 4 Frame Display
Draw Continental Boundary
Purge Overlay
Exit
CURSOR OPTIONS [MOVE CURSOR WITH PUCK]
STATS : M
VALUE : V
MARK CLOUD OPTIONS
STRATUS : S
CONVECTIVE : C
STDV
VAR
STATS AVG
11.57
133.86
136.42
LAT
VALUE SECTOR LIN ELE
-1.7
3
134
99
LDIM
MARK SECTOR LIN ELE
30
3
134
99
TVPOS RAS= 495
PIX= 286
MIN
MAX
164
118
COUNT
TEMP
LON
154
254
65.7
EDIM CLOUDTYPE
30
Con
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185
Appendix B
Interactive sample sessions
In this appendix I show some examples of how GPFES runs on the
Supercomputer Computations Research Institute
Florida State University. The
$
VAX 8700,
the
is the VAX system prompt. The
symbols are explained in Section 5.6 and dummy-x clauses
in
reasoning processes are used for the connection in the infer­
ence network.
$ run gpfes
Welcome to GPFES which is a first application to
retrieve precipitation from remote sensing
microwave measurements of Air Force satellite SSM/I
If you have to answer by a truth interval, then
use the free format in integer type of [0,100]:
e.g. 80 100,i.e. lower and upper bounds;
100 is yes, 0 is no and 50 is a medium certainty,
y: yes, n: no and q: question about symbols
continue ? (y/n)
y
there is a cloud over the point ? (y/n)
n
final conclusion;
no rain
truth;
100
100
match;
100 symm. dif=
0
Do you want to see the reasoning process ?(y,n)
n
If you have to answer by a truth interval, then
use the free format in integer type of [0,100]:
e.g. 80 100,i.e. lower and upper bounds;
100 is yes, 0 is no and 50 is a medium certainty,
y: yes, n: no and q: question about symbols
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
186
continue ? (y/n)
y
there is a cloud over the point ? (y/n)
y
area is water ? (y/n)
y
37V-37H exceeds about 18 ? (y/n/q or interval)
n
18H-37H exceeds about 20 ? (y/n/q or interval)
0 15
18U exceeds 195 ? (y/n/q or interval)
86 92
18U exceeds about 240 ? (y/n/q or interval)
q
U=(V+H)/2
18U exceeds about 240 ? (y/n/q or interval)
0 12
89U is high in the interval [267,281];1
90 93
19U is low in the interval [204,251];1
86 89
22U is medium in the interval [246,268];1
78 87
37U is medium in the interval [240,273] ;1
90 96
final conclusion;
rain is very light; H ( 1,6,1)
truth;
81
90
match;
100 symm. dif=
4
Do you want to see the reasoning process ?(y,n)
y
a set of clauses in the premise
continue
truth of premise^
100
match:
100 sym. dif:
100
0
intermediate conclusion;
dummy-0
truth :
100
100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a set of clauses in the premise
there is a cloud over the point
dummy-0
truth of premises
100
matchs
100 sym. difs
100
0
intermediate conclusion;
dummy-1
truth =
100
100
a set of clauses in the premise
dummy-1
area is water
truth of premises
100
matchs
100 sym. difs
100
0
intermediate conclusion;
dummy-8
truth s
100
100
a set of clauses in the premise
dummy-8
18H-37H is less than about 20
truth of premises
85
matchs
100 sym. difs
100
0
intermediate conclusion;
decide a rain class
truth s
85
100
a set of clauses in the premise
decide a rain class
18U exceeds 195
truth of premises
85
matchs
100 sym. difs
92
0
intermediate conclusion;
rain
truth s
85
92
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188
a set of clauses in the premise
rain
18U is less than about 240
truth of premise=
85
match:
100 sym. dif:
92
0
intermediate conclusion;
light rain over water; tier2
truth :
85
92
a set of clauses in the premise
light rain over water; tier2
truth of premise:
85
match:
100 sym. dif:
92
0
intermediate conclusion;
tier2
truth :
85
92
a set of clauses in the premise
tier2
89U is high in the interval [267,281];1
19U is low in the interval [204,251];1
22U is medium in the interval [246,268] ;1
37U is medium in the interval [240,273];1
truth of premise:
78
87
match:
100 sym. dif:
4
intermediate conclusion;
rain is very light; H ( 1,6,1)
truth :
81
90
If you have to answer by a truth interval, then
use the free format in integer type of [0,100]:
e.g. 80 100,i.e. lower and upper bounds;
100 is yes, 0 is no and 50 is a medium certainty,
y: yes, n: no and q: question about symbols
continue ? (y/n)
y
there is a cloud over the point ? (y/n)
y
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189
area is water ? (y/n)
n
do you know the cloud type (cumuli or strati)
n
? (y/n)
a tutoring session to help you about the cloud type !
homogeneous over large area ?(y/n)
n
stable ?(y/n)
n
v. vs. h. development is small ?(y/n)
n
long evolution ?(y/n)
n
cotton-like appearence ?(y/n)
y
cloud type is cumuliform
Now you are ready to answer
18H-37H exceeds about 20 ? (y/n/q or interval)
10 23
85PCT is less than 255 ? (y/n/q or interval)
8 19
ctype is cumuliform ? (y/n)
y
85PCT is less than about 275 ? (y/n/q or interval)
90 100
37U is less than .9(F)*ST-10 ? (y/n/q or interval)
87 93
final conclusion;
light rain over land
truth;
77
90
match;
100 symm. dif=
0
Do you want to see the reasoning process ?(y,n)
y
a set of clauses in the premise
continue
truth of premise=
100
match=
100 sym. dif=
100
0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
190
intermediate conclusion;
dummy-0
truth =
100
100
a set of clauses in the premise
there is a cloud over the point
dummy-0
truth of
premises
100
match=
100 sym. dif=
100
0
intermediate conclusion;
dummy-1
truth =
100
100
a set of clauses in the premise
area is land
dummy-1
truth of
premises
100
matchs
100 sym. difs
100
0
intermediate conclusion;
dummy-2
truth s
100
100
a set of clauses in the premise
"you d o n ’t know the cloud type" is true
dummy-2
truth of
premises
100
100
matchs
100 sym. difs
0
intermediate conclusion;
find a cloud type; go to tutoring section
truth s
100
100
a set of clauses in the premise
find a cloud type; go to tutoring section
truth of
premises
100
100
matchs
100 sym. difs
0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
191
intermediate conclusion;
dummy-3
truth =
100
100
a set of clauses in the premise
dummy-3
ctype is cumuliform
truth of
premises
match=
100 sym.
100
difs
100
0
intermediate conclusion;
dummy-5
truth s
100
100
a set of clauses in the premise
18H-37H is less than about 20
85PCT exceeds 255
truth of
premises
77
matchs
100 sym. difs
* ; not A
* ; not B
90
0
intermediate conclusion;
dummy-7
truth s
77
90
a set of clauses in the premise
dummy-7
truth of
matchs
premises
100 sym.
77
difs
go
0
intermediate conclusion;
(not A) and (not B)
truth s
77
90
a set of clauses in the premise
(not A) and (not B)
truth of
premises
matchs
100 sym.
77
difs
go
0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
192
intermediate conclusion;
dummy-6
truth =
a set of
77
90
clauses in
the premise
18H-37H is less than about 20
85PCT is less than about 275
37U is less than .9(F)*ST-10
truth of premises
77
matchs
100 sym. difs
*
*
*
not A
J
K
90
0
intermediate conclusion;
(not A) and J and K
truth s
77
a set of
clauses in
90
the premise
dummy - 6
(not A) and J and K
truth of premises
77
matchs
100 sym. dif=
90
0
intermediate conclusion;
light rain over land
truth s
77
90
If you have to answer by a truth interval, then
use the free format in integer'type of [0 ,1 0 0 ]:
e.g. 80 1 0 0 ,i.e. lower and upper bounds;
100 is yes, 0 is no and 50 is a medium certainty,
y: yes, n: no and q: question about symbols
continue ? (y/n)
n
Thank you !
FORTRAN STOP
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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