Download Introduction of Fuzzy Inference Systems

Introduction of
Fuzzy Inference Systems
By Kuentai Chen
Fuzzy Inference Systems
• Base on
– Fuzzy set theory
– Fuzzy If-Then rules
– Fuzzy Reasoning
Fuzzy Inference Systems
• Also named
– Fuzzy-rule-based system
– Fuzzy Expert system
– Fuzzy model
– Fuzzy associative memory
– Fuzzy logic controller
– Fuzzy system
Fuzzy inference
• The process of formulating the mapping from a
•
•
given input to an output using fuzzy logic.
The mapping then provides a basis from which
decisions can be made, or patterns discerned.
Fuzzy Logic Toolbox uses Mamdani-type and
Sugeno-type: Vary in the way outputs are
determined.
Applications
• Automatic control
• Data classification
• Decision analysis
• Expert systems
• Computer vision
Mamdani's fuzzy inference method
• Proposed in 1975 by Ebrahim Mamdani
– control a steam engine and boiler combination
– synthesizing a set of linguistic control rules
– obtained from experienced human operators.
• Based on Lotfi Zadeh's 1973 paper
• Fuzzy Logic Toolbox uses a modified
version
Fuzzy IF-THEN rules
• Mamdani style
If pressure is high then volume is small
high
small
• Sugeno style
If speed is medium then resistance = 5*speed
medium
resistance = 5*speed
Fuzzy inference
system (FIS)
If speed is low then resistance = 2
If speed is medium then resistance = 4*speed
If speed is high then resistance = 8*speed
MFs
low
medium
high
.8
.3
.1
2
Rule 1: w1 = .3; r1 = 2
Rule 2: w2 = .8; r2 = 4*2
Rule 3: w3 = .1; r3 = 8*2
Speed
Resistance = S(wi*ri) /
= 7.12
Swi
First-order Sugeno FIS
• Rule base
If X is A1 and Y is B1 then Z = p1*x + q1*y + r1
If X is A2 and Y is B2 then Z = p2*x + q2*y + r2
• Fuzzy reasoning
A1
B1
w1
X
A2
x=3
Y
B2
X
z1 =
p1*x+q1*y+r1
y=2
w2
Y
P
z2 =
p2*x+q2*y+r2
z=
w1*z1+w2*z2
w1+w2
...
Fuzzy modeling
x1
Unknown target system
y
xn
Fuzzy Inference System
y*
• Given desired i/o pairs (training data set) of the form
(x1, ..., xn; y), construct a FIS to match the i/o pairs
• Two steps in fuzzy modeling
structure identification --- input selection, MF numbers
parameter identification --- optimal parameters
Data Clustering
Cluster analysis is a technique for grouping data and finding structures in
data. The most common application of clustering methods is to partition a
data set into clusters or classes, where similar data are assigned to the same
cluster whereas dissimilar data should belong to different clusters.
In real applications there is very often no sharp boundary between
clusters so that fuzzy clustering is often better suited for the data.
Membership degrees between zero and one are used in fuzzy clustering
instead of crisp assignments of the data to clusters.
Fuzzy clustering can be applied as an unsupervised learning strategy in order
to group data Another area of application of fuzzy cluster analysis is image
analysis and recognition. Segmentation and the detection of special
geometrical shapes like circles and ellipses can be achieved by so-called
shell clustering algorithms.
Types of Fuzzy Cluster Algorithms
Classical Fuzzy Algorithms (cummulus like clusters)
The fuzzy c-means algorithm
The Gustafson-Kessel algorithm
The Gath-Geva algorithm
Linear and Ellipsodial (lines)
The fuzzy c-varieties algorithm
The adaptive clustering algorithm
Shell (circles,ellipses, parabolas)
Fuzzy c-shells algorithm
Fuzzy c-spherical algorithm
Adaptive fuzzy c-shells algorithm
Fuzzy c-mean cluster analysis
The Fuzzy c-mean algorithm (FCM) recognizes spherical clouds of points in p-dimensional space.
Having a finite set of objects
and the number of cluster centers c to be
calculated, the assignment of the n objects to the c clusters is represented by the proximity
matrix
. With
and
,
expressing the fuzzy
proximity or affiliation of object to cluster center .
The fuzzy c-mean algorithm consists of the following steps:
1. Fix the number c of cluster centers to be calculated and a threshold for the stop condition in
step 4. Initialize the proximity matrix
.
2. Update the c cluster centers according to the actual proximity matrix
.
3. Update
to
according to the actual cluster centers .
4. Stop the algorithm if
is fulfilled, else go on with step 2.
ANFIS
• Fuzzy reasoning
B1
A1
A2
B2
w1
w2
z1 =
p1*x+q1*y+r1
z=
z2 =
p2*x+q2*y+r2
w1*z1+w2*z2
w1+w2
y
x
• ANFIS (Adaptive Neuro-Fuzzy Inference System)
A1
x
A2
B1
y
B2
P
w1
P
w1*z1
S Swi*zi
w2*z2
w2
S
Sw i
/
z
Four-rule ANFIS
• Input space partitioning
y
A2
A1
B2
x
B2
B1
B1
y
A1
A2
• ANFIS (Adaptive Neuro-Fuzzy Inference System)
x
y
A1
P
A2
P
B1
P
B2
P
w1
w1*z1
S Swi*zi
w4
Sw i
w4*z4
S
/
z
x