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Combining fuzzy and statistical uncertainty:
probabilistic fuzzy systems
and their applications
prof.dr.ir. Jan van den Berg
- TUDelft: Faculties of TPM and EWI
- Cyber Security Academy The Hague
[email protected]
http://tbm.tudelft.nl/index.php?id=30084&L=1
1
Summary of the talk
• Two complementary conceptualizations of uncertainty will be
discussed: statistical and fuzzy uncertainty.
• These uncertainties can be combined into one theory on
probabilistic fuzzy events. Using this theory,classical fuzzy systems
can be generalized to probabilistic fuzzy systems (PFS).
• PFS can be induced using both expert knowledge and data enabling
both interpretability and accuracy (despite the fact there remains an
accuracy-interpretability dilemma we have to deal with in practice).
• To finalize, one or two examples of PFSs we developed will be
shown: next time!
2
Agenda
•
•
•
•
•
•
•
Probabilistic/statistical uncertainty
Fuzzy uncertainty
Fuzzy systems
Probabilistic fuzzy theory
Probabilistic fuzzy systems
Applications (next time)
Conclusions
3
Probabilitic/statistical uncertainty
• Probabilitic/statistical uncertainty:
well-known notion
• Crisp events occur with a certain probability
• These probabilities can be assessed statistically
• E.g., frequentist approach: by repeating experiments
• A lot of theory on unbiased estimation, ML estimation, etc.
• In continuous outcome spaces, probability distributions are used
• Mathematical statistics: descriptive and inferential statistics
(the latter on drawing conclusions from data using some model for the data)
4
Agenda
•
•
•
•
•
•
•
Probabilistic/statistical uncertainty
Fuzzy uncertainty
Fuzzy systems
Probabilistic fuzzy theory
Probabilistic fuzzy systems
Applications
Conclusions
5
Crisp sets
Collection of definite, well-definable objects (elements) to form a
whole, having crisp boundaries
Representations of sets:
• list of all elements
A = {x1, ,xn}, xj  X
• elements with property P
A={x|x satisfies P }, x  X
• Venn diagram
• characteristic function
fA: X  {0,1},
fA(x) = 1,  x  X
fA(x) = 0,  x  X
Real numbers larger than 3:
1
A
0
X
3
6
Fuzzy sets
• Sets with fuzzy, gradual boundaries (Zadeh 1965)
• A fuzzy set A in X is characterized by its
membership function mA: X  [0,1]
A fuzzy set A is completely
determined by the set of
ordered pairs
Real numbers about 3:
1
mA(x)
A={(x,mA(x))| x  X}
X is called the domain or
universe of discourse
0
X
3
7
Fuzzy sets on discrete universes
• Fuzzy set C = “desirable city to live in”
X = {SF, Boston, LA} (discrete and non-ordered)
C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}
• Fuzzy set A = “sensible number of children”
Membership Grades
X = {0, 1, 2, 3, 4, 5, 6} (discrete universe)
A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}
Number of Children
8
Fuzzy sets on continuous universes
• Fuzzy set B = “about 50 years old”
X = Set of positive real numbers (continuous)
mB(x) 
1
 x  50 
1 

 10 
2
Membership Grades
B = {(x, mB(x)) | x in X}
Age
9
Fuzzy partition
Membership Grades
• Fuzzy partition formed by the linguistic values “young”,
“middle aged”, and “old”
• For any age: sum of membership values = 1
Young
Middle Aged
Old
1
0.5
0
0
10
20
30
40
50
60
70
80
90
Age
10
Operations with fuzzy sets
A  X  A  mA ( x )  1  m A ( x )
C  A  B  mc ( x)  max( m A ( x), m B ( x))  m A ( x)  m B ( x)
C  A  B  m c ( x)  m A ( x)  m B ( x )  m A ( x ) m B ( x)
C  A  B  mc ( x)  min( m A ( x), m B ( x))  m A ( x)  m B ( x)
C  A  B  m c ( x)  m A ( x)  m B ( x)
Note the multiple definitions!
11
Set theoretic operations, examples
(b) Fuzzy Set "not A"
(a) Fuzzy Sets A and B
1
A
B
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(d) Fuzzy Set "A AND B"
(c) Fuzzy Set "A OR B"
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
maximum
minimum
0.2
0
12
Agenda
•
•
•
•
•
•
•
Probabilistic/statistical uncertainty
Fuzzy uncertainty
Fuzzy systems
Probabilistic fuzzy theory
Probabilistic fuzzy systems
Applications
Conclusions
13
Fuzzy Modeling (expert-driven)
•FM can be defined based on expert knowledge:
• Many human concepts (big, long, high, very much,
adequate, satisfactory, …) are defined in a (context
dependent) quantitative way, describing state of
nature
• Examples of facts about the world:
• The president is middle-aged
• The water supply is insufficient
• Birth weight of Romanian children is quite low
 Need for formalization: linguistic variable
14
Linguistic variable
• A numerical variable takes numerical values:
Age = 65 (defines a crisp event)
• A linguistic variables takes linguistic values:
Age is old (defines a fuzzy event)
• A linguistic value is defined by a fuzzy set (enabling a
characterization with gradual transitions):
Membership Grades
Ex: A fuzzy partition of linguistic cariable ‘Age’ formed with
linguistic values “young”, “middle aged”, and “old”:
Young
Middle Aged
Old
1
0.5
0
0
10
20
30
40
Age
50
60
70
80
90
15
Expert-driven fuzzy modeling, cont.
•Experts can express their knowledge in
• Facts about the world (see above), and
• Fuzzy IF-THEN rules
•Example fuzzy rules:
• IF Nutrician state is poor AND Birth weight is medium AND
Respiration disease is absent THEN Child mortality rate is
medium
• IF Nutrician state is medium AND Birth weight is not too low
AND Respiration disease is absent THEN Child mortality rate is
rather low
•Need for Reasoning/Inference mechanism
16
Fuzzy (Inference) System (FS)
Inference engine
output
Rule base
(knowledge base)
Defuzzifier
Fuzzifier
Fuzzifier (interface: from crisp to fuzzy)
Rule base (enabling interpretability/transparency)
Inference engine (implements fuzzy reasoning)
Defuzzifier (interface: from fuzzy to crisp)
Note: Fuzzifier or defuzzifier may be absent
input
•
•
•
•
•
17
Example Fuzzy System: Mamdani model
•Five major steps:
1.
2.
3.
4.
5.
Fuzzification
Degree of fulfillment
Inference
Aggregation
Defuzzification
• Computations according to Mamdani reasoning apply a
generalized form of classical modus ponens:
Given: x is A' and If x is A, then y is B,
Conclude: y is B'
18
Mamdani reasoning - example
19
Resulting FS: a smooth non-linear mapping
20
Inducing a fuzzy model from data


If income is Low then tax is Low
If income is High then tax is High
21
Bias-variance dilemma (!)
Interpretability-accuracy dilemma (!)
Algorithms exist to
gradually induce more
and more rules from
data 
• Bias-variance dilemma to
find models of right
complexity
• Interpretability-accuracy
dilemma is another key
issue (of Data Mining)
22
Agenda
•
•
•
•
•
•
•
Probabilistic/statistical uncertainty
Fuzzy uncertainty
Fuzzy systems
Probabilistic fuzzy theory
Probabilistic fuzzy systems
Applications
Conclusions
23
Probability of a crisp and fuzzy events
• Crisp
Pr( A)  
xA
p( x) dx    A ( x) p( x) dx
X
• Fuzzy (Zadeh, 1968)
Probability of a fuzzy event
is the expectation of its
membership function!
Pr( A)   m A ( x) p( x) dx
X
•
Conditional probability:
Pr( A  B)
Pr( A | B) 

Pr( B)

X
m A ( x) m B ( x) p( x) dx

X
m B ( x) p( x) dx
Satisfies P(A|A) = 1
24
An example, continuous domain
calculating the probability that a
randomly selected Indian woman is tall
Answering the question:
3.5
“What is the probability
that a randomly selected
Indian woman is tall?”
3
2.5
tall
2
1.5
Pr( A) 



m A ( x) f ( x) dx
 E ( m A ( x))
pdf of length
1
fuzzy pdf
0.5
0
-0.5
1.50
1.68
2.0
25
Probabilistic fuzzy events, discrete case
• 2 discrete probabilistic fuzzy events:
A1 = (m (x ), m (x )) = (m, 1 - n) and A2 = (m ’ (x 1), m ’ (x )) = (1 – m,n
1
• If
x
2
2
occurs, then
A1 occurs with degree m
1
and
A2 occurs with degree 1- m
• Similarly, if
x
2
occurs…
• E.g., A1 means “tall” and
A2 = “small”
where x 1 and x 2 are two
values of the x -variable length
• Pr(A1) = mp + (1 – n)(1 – p), and
Pr(A 2) = (1 –m )p + n (1 – p)
26
Simple estimation of probabilities
• Let x1, …, xn be a random sample on a domain
X
• The probability of a crisp event A can be estimated by
1
 A (x k )

n k
• The probability of a fuzzy event Ai can be estimated by
1
m Ai (x k )

n k
assuming that X is well-formed (~ fuzzy partition), i.e.
x k
m
Ai
(x k )  1
i
27
Deterministic and probabilistic rules
Rule Rq : If x is Aq then y is Cq
Model parameters:
Aq , Cq
Ex: If current returns are large, then future returns will be large
If current returns are large,
then future returns will be large with probability r1
future returns will be small with probability 1-r1
Linguistic
vagueness
Probabilistic
uncertainty
28
Probabilistic fuzzy rules
Rule Rq : If x is Aq then
y is C1q with Pr( C1q|Aq ) and ...and
y is C jq with Pr( C jq|Aq ) and ...and
y is C Jq with Pr( C Jq|Aq )
Model parameters:
Aq , C jq , Pr(C jq|Aq )
29
A ‘crazy’ theoretical sidestep
•Probabilistic Fuzzy Entropy
•To be used to induce fuzzy decision trees
30
Definition of PF Entropy, discrete source
• Given a (well-formed) sample space with a fuzzy partition of fuzzy
events A1, …, AC defined by membership functions m1 ( x),, m C ( x)
occurring with probabilities Pr(A1), …, Pr(AC ), the PFE is defined
as:
H sf (m , p)    Pr( Ac ) log 2 (Pr( Ac ))  E ( log 2 (Pr( Ac )))
C
c 1
C
  E ( mc ( x)) log 2 ( E ( mc ( x)))  E ( log 2 ( E ( mc ( x)))
c 1
•  PFE is a probabilistic type of entropy defined in a
fuzzily partitioned (sample) space!
31
A very special information source
Consider 2 discrete strictly complementary statistical fuzzy events:
A1 = (m (x1), m (x2)) = (m, 1 - m) , A2 = (m ’ (x1), m ’ (x2)) = (1 – m,m)
32
A very special information source, cont.
Since A1 = (m, 1 - m), A2 = (1 – m, m), Pr (x1 ) = p , and Pr (x2 ) = 1 – p,
it follows that:
Pr(A1) = mp + (1 – m)(1 – p) = 2mp + 1 – m – p = 1 – Q
(1)
Pr(A2) = p (1 – m) + (1 – p) m = m + p – 2mp = Q
(2)
33
Entropy of information source generating 2
strictly complementary stat fuzzy events
• Using definition of PFE (sh. 31 ) and equations (1) and (2) from previous
sheet, it follows that
Hsf (m,p ) = - Q log2 Q - (1 - Q ) log2 (1 - Q ) where
Q (m,p ) = m + p - 2 m p
• Q (like 1 – Q) relates to the
combined uncertainty of the
probabilistic fuzzy events based on
their fuzziness m and the
probability of occurrence p:
34
Further interpreting Q
• Q (m,p ) = m + p - 2 m p
•
Q = 0 or 1  no uncertainty;
Q = 0.5  highest uncertainty
•
Illumination and interpretation:
- m = p = 0 or 1,  Q = 0
two crisp events, one of which
occurs with probability 1
- m = 0, p = 1 or m = 1, p = 0 
Q = 1, same explanation!
- p = 0.5 or m = 0.5  Q = 0.5
two fuzzy events having equal
prob, or
two non-distinguishable fuzzy
events!!
35
Interpretation of Hpf (m,p)
Hpf (m,p ) = - Q log2 Q - (1 - Q ) log2 (1 - Q )
• PFE quantifies the combined
uncertainty
• Illumination:
- Q = 0 or 1 ~ no uncertainty ~
H (m,p ) = 0, in 4 corners
- p = 0.5 or m = 0.5 ~ Q = 0.5 ~
two fuzzy events having equal prob,
or two non-distinguishable fuzzy
events ~ highest uncertainty ~
H (m,p ) = 1
- if m = 0 or 1: classical ‘crisp’ entropy
- if p = 0 or 1: ‘fuzzy’ entropy only
36
Agenda
•
•
•
•
•
•
•
Probabilistic/statistical uncertainty
Fuzzy uncertainty
Fuzzy systems
Probabilistic fuzzy theory
Probabilistic fuzzy systems
Applications
Conclusions
37
Probabilistic fuzzy systems, with
discrete probability distribution

Consists of a set of rules whose antecedents are fuzzy
conditions and whose consequents are probability
distributions
consequent
antecedent
fuzzy set
where,
and
(3)
38
Deterministic vs. probabilistic FS
Y
B3
If x is A4 then y is B2
y is B1 with probability p(B1 | A4),
and y is B2 with probability p(B2 | A4),
and y is B3 with probability p(B3 | A4).
B2
A1
B1
A3
A4
A5
A2
X
39
Probabilistic fuzzy system, with
continuous probability distribution
Rule Rq : If x is Aq then f ( y)  f ( y | Aq )
Additive reasoning:
Q
mA
f ( y | x) 
q 1
(x) f ( y | Aq )
q
Q
 m Aq (x)
Q
   q (x) f ( y | Aq )
q 1
q 1
Q
y  E( y | x)    q (x) E( y | Aq )
q 1
40
Probability distribution characterization
• In general, different characterizations can be used for
the conditional probability density in the rule
consequents
• This characterization could be an approximation with
a histogram or an explicit model for density, e.g., a
normal or other distribution
• In PFS, we can select a fuzzy histogram
characterization
41
Histograms, classical crisp case
• Let x1, …, xn be a random sample from a univariate distribution
with pdf f(x)
• Let the characteristic functions i (x) (defining crisp bins/intervals Ai)
constitute a crisp partitioning:
x   :   i ( x)  1  i  j :  i ( x). j ( x)  0
i
• A histogram estimates f (x) (from data xk ) as follows:
1
 i ( x)(   i ( xk ))
^
 i ( x) pi
n k
f ( x)  

| Ai |
i
i
 i ( x)dx
42
Fuzzy histograms
• Let x1, …, xn be a random sample of size n from a (univariate)
distribution with pdf f(x)
• Let the membership functions mi (x) (defining fuzzy bins Ai) constitute
a fuzzy partitioning:
x   :  mi ( x)  1
i
• A fuzzy histogram estimates f (x) (from data xk ) as follows:
^
f ( x)  
i
mi ( x) pi
| Ai |

i
1
mi ( xk ))

n k
 mi ( x)dx
mi ( x)(
43
Crisp vs. fuzzy histogram
pdf normal distribution
Membership values
Crisp histogram
Fuzzy histogram
1.2
1
A1
A2
0.8
A3
0.6
A4
A5
0.4
A6
0.2
A7
3.5
3.15
2.8
2.45
2.1
1.75
1.4
1.05
0.7
0.35
-0
-0.4
-0.7
-1.1
-1.4
-1.8
-2.1
-2.5
-2.8
-3.2
-3.5
0
44
PFS: fuzzy histogram model
45
Fuzz IEEE 2013, Hyderabad India
45
Fuzzy histogram model
46
Fuzz IEEE 2013, Hyderabad India
46
Probabilistic Mamdani systems
Rule Rq : If x is Aq then
y  C1 with Pr( C1|Aq ) and ...and
y  C j with Pr( C j|Aq ) and ...and
y  C J with Pr( C J |Aq )
Q
Reasoning:
J
y  E( y | x)    q (x) Pr(C j|Aq ) z j
q 1 j 1
Centroid of fuzzy consequent set Cj
47
Probabilistic fuzzy output model
48
Probabilistic TS systems
Zero-order probabilistic Takagi-Sugeno
Rule Rq : If x is Aq then
y  y1 with Pr( y1|Aq ) and
y  y 2 with Pr( y 2|Aq ) and ...and
y  y J with Pr( y J |Aq )
In essence, consequents are crisp sets centered around
Q
Reasoning:
yj
J
y  E( y | x)    q (x) y j Pr( y j|Aq )
q 1 j 1
49
Relation to deterministic FSs
• Zero-order Takagi-Sugeno system
Q
y*    q (x)cq
Takagi-Sugeno reasoning
q 1
Q
c.f.
J
y  E( y | x)    q (x) Pr( y j|Aq ) z j
q 1 j 1
J
Select cq   Pr( y j|Aq ) z j
j 1
50
Probabilistic fuzzy systems
summary
• Essentially a fuzzy system that estimates a probability density
function, i.e. the fuzzy system approximates a p.d.f.
• Usually p.d.f. is conditional on the input
• Linguistic information is coded in fuzzy rules
• Combine linguistic uncertainty with probabilistic uncertainty
• Different types of fuzzy systems can be extended to the PFS
equivalent (e.g. Mamdani fuzzy systems, Takagi-Sugeno fuzzy
systems)
51
PFS design
• Identifying mental world vs. observed world
(van den Eijkel 1999)
• Mental world: linguistic descriptions, fuzzy conceptualization,
experts’ knowledge
• Observed world: data measurements, probability density
functions, optimal consequent parameters
• Optimal design given a mental world: application of conditional
probability measures for fuzzy events
• Optimal design given an observed world: nonlinear
optimization techniques
52
Parameter determination
Sequential Method
Part 1
MF
parameters
Maximum Likelihood
Method
Part 1 – Initialization
Sequential Method
Part 2 - Optimization
Part 2
Probability
parameters
MF
parameters
Probability
parameters
MF – Membership function
53
Sequential method



Part 1: Finding the membership function (MF) parameters
is a fuzzy set defined by a membership function
E.g. – Gaussian MF parameters:
o
o
v – center of MF
σ – width of MF
FCM Clustering
54
MF determination
• For the first part of the sequential method, well-known
techniques from fuzzy modeling can be applied
•
•
•
•
•
•
•
Fuzzy clustering in input-output product space
Fuzzy clustering in input and output space
Expert-driven design
Similarity-based rule-base simplification
Feature selection
Heuristic approaches
Etc.
55
Sequential method

Part 2: Finding the probability parameters - Pr(ωc|Aj)

Set the parameters Pr(ωc|Aj) equal to estimates of the
conditional probabilities - conditional probability estimation
56
56
Estimation of probability parameters
• Conditional probabilities Pr(Cj | Aq) can be assessed directly by using the
definition of the probability of joint events:
Pr(C j | Aq ) 
Pr( Aq  C j )
Pr( Aq )


( x k , yk )
m A (x k ) mC ( yk )

q
xk
j
m A (x k )
q
• This method does not provide maximum likelihood estimates of the
probability parameters.
57
57
Maximum likelihood method

Part 2: Optimization of vj, σj and Pr(ωc|Aj)
Likelihood of a
data set
Minimization
of the negative loglikelihood
Optimize parameters vj, σj and Pr(ωc|Aj) that minimize the error function
Constrained optimization problem
(probability parameters Pr(ωc|Aj) must satisfy summation conditions)
58
58
Maximum likelihood method
Constrained
optimization
problem
using ujc
Unconstrained
optimization
problem
 Unconstrained minimization of vj, σj and ujc
 Gradient descent optimization algorithm is used to minimize the objective
function – i.e. the available classification examples are processed one by
one and updates are performed after each sample
59
59
Experimental comparison (1)
• Use Gaussian membership functions
d

( xl  cql ) 2
m Aq ( x)  exp 

2
 

l

1
ql





• The centers cql are determined using fuzzy c-means clustering
• The widths σql are set equal to σql = minj′ ≠ j ||cq – cq′||
60
Fuzz IEEE 2013, Hyderabad India
60
Experimental comparison (2)
• Misclassification rates
Sequential method
Wisconsin breast
cancer
Wine
0.261
(0.036)
0.034
(0.048)
• Calculated using ten-fold cross-validation
0.029
Maximum
likelihood
• Standard deviations reported within parentheses
(0.021)
0.023
(0.041)
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Fuzz IEEE 2013, Hyderabad India
61
Future research directions
• New estimation methods for the model parameters
• Joint estimation
• Information-theory based techniques
• Better optimization methods
•
•
•
•
•
•
Interaction linguistic knowledge and data-driven estimation
Optimizing model complexity, model simplification
Interpretability of probabilistic fuzzy models
Linguistic descriptions of probability density functions
Equivalence to other systems: e.g. fuzzy Markov models
Density estimation using more complex models as rule
consequents: e.g. fuzzy GARCH models
• New applications
62
Fuzz IEEE 2013, Hyderabad India
62
Agenda
•
•
•
•
•
•
•
Probabilistic/statistical uncertainty
Fuzzy uncertainty
Fuzzy systems
Probabilistic fuzzy theory
Probabilistic fuzzy systems
Applications
Conclusions
63
Applications
• Next time!
64
Agenda
•
•
•
•
•
•
•
Probabilistic/statistical uncertainty
Fuzzy uncertainty
Fuzzy systems
Probabilistic fuzzy theory
Probabilistic fuzzy systems
Applications
Conclusions
65
Concluding remarks
• Probabilistic fuzzy systems combine linguistic uncertainty and
probabilistic uncertainty
• Very useful in applications where a probabilistic model (pdf
estimation) has to be conditioned (or constrained) by linguistic
information
• Good parameter estimation methods exist and the added value
of these models has been demonstrated in various applications
66
Conclusions
• Fuzzy models usually show smooth non-linear behavior
• If certain measures are taken, fuzzy models are interpretable
• Fuzzy models can be induced from
• expert knowledge (grid selection and definition of fuzzy rules): this expert-
driven approach was very successful in control theory)
• a data set: the data-driven approach
• The accuracy-interpretability dilemma is prominent!
• The bias-variance dilemma is prominent!
67
Selected bibliography
• J. van den Berg, W. M. van den Bergh, and U. Kaymak. Probabilistic and statistical fuzzy
set foundations of competitive exception learning. In Proceedings of the Tenth IEEE
International Conference on Fuzzy Systems, volume 2, pages 1035–1038, Melbourne,
Australia, Dec. 2001.
• J. van den Berg, U. Kaymak, and W.-M. van den Bergh. Probabilistic reasoning in fuzzy
rule-based systems. In P. Grzegorzewski, O. Hryniewicz, and M. A. Gil, editors, Soft
Methods in Probability, Statistics and Data Analysis, Advances in Soft Computing, pages
189–196. Physica Verlag, Heidelberg, 2002.
• J. van den Berg, U. Kaymak, and W.-M. van den Bergh. Fuzzy classification using
probability-based rule weighting. In Proceedings of 2002 IEEE International Conference
on Fuzzy Systems, pages 991–996, Honolulu, Hawaii, May 2002.
• U. Kaymak, W.-M. van den Bergh, and J. van den Berg. A fuzzy additive reasoning
scheme for probabilistic Mamdani fuzzy systems. In Proceedings of the 2003 IEEE
International Conference on Fuzzy Systems, volume 1, pages 331–336, St. Louis, USA,
May 2003.
• U. Kaymak and J. van den Berg. On probabilistic connections of fuzzy systems. In
Proceedings of the 15th Belgium-Netherlands Artificial Intelligence Conference, pages
187–194, Nijmegen, Netherlands, Oct. 2003.
• J. van den Berg, U. Kaymak, and W.-M. van den Bergh. Financial markets analysis by
using a probabilistic fuzzy modelling approach. International Journal of Approximate
Reasoning, 35: 291–305, 2004.
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Selected bibliography
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L. Waltman, U. Kaymak, and J. van den Berg. Maximum likelihood parameter
estimation in probabilistic fuzzy classifiers. In Proceedings of the 14th Annual
IEEE International Conference on Fuzzy Systems, pages 1098–1103, Reno,
Nevada, USA, May 2005.
D. Xu and U. Kaymak. Value-at-risk estimation by using probabilistic fuzzy
systems. In Proceedings of the 2008 IEEE World Congress on Computational
Intelligence (WCCI 2008), pages 2109–2116, Hong Kong, June 2008.
R. J. Almeida and U. Kaymak. Probabilistic fuzzy systems in value-at-risk
estimation. International Journal of Intelligent Systems in Accounting, Finance
and Management, 16(1/2):49–70, 2009.
J. Hinojosa, S. Nefti, and U. Kaymak. Systems control with generalized
probabilistic fuzzy-reinforcement learning. IEEE Transactions on Fuzzy
Systems, 19(1):51–64, February 2011.
R. J. Almeida, N. Basturk, U. Kaymak, and V. Milea. A multi-covariate semiparametric conditional volatility model using probabilistic fuzzy systems. In
Proceedings of the 2012 IEEE International Conference on Computational
Intelligence in Financial Engineering and Economics (CIFEr 2012), pages 489–
496, New York City, USA, 2012.
J. van den Berg, U. Kaymak, and R.J. Almeida. Function approximation using
probabilistic fuzzy systems. IEEE Transactions on Fuzzy Systems, 2013.
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