Transcript ANFIS (Adaptive Network Fuzzy Inference system)

ANFIS
(Adaptive Network Fuzzy
Inference system)
G.Anuradha
Introduction
• Conventional mathematical tools are quantitative
in nature
• They are not well suited for uncertain problems
• FIS on the other hand can model qualitative
aspects without employing precise quantitative
analyses.
• Though FIS has more practical applications it
lack behind
– Standard methods for transformation into rule base
– Effective methods for tuning MFs for better
performance index
So……
ANFIS serve as a basis for constructing a
set of fuzzy if-then rules with appropriate
membership functions to generate the
stipulated input-output pairs
Fuzzy if-then rules and Fuzzy
Inference systems
• Fuzzy if-then rules are of the form IF A
THEN B where A and B are labels of fuzzy
sets.
• Example
– “if pressure is high then volume is small”
Linguistic
variables
Linguistic values
Sugeno model
Assume that the fuzzy inference system has two
inputs x and y and one output z.
A first-order Sugeno fuzzy model has rules as the
following:
Rule1:
If x is A1 and y is B1, then f1 = p1x + q1y + r1
Rule2:
If x is A2 and y is B2, then f2 = p2x + q2y + r2
Fuzzy Inference system
Blocks of FIS
Steps of fuzzy reasoning
Types of fuzzy reasoning
• Type 1: The overall output is the weighted
average of each rule’s firing strength and output
membership functions.
• Type 2: The overall output is derived by applying
the “max” operation to the qualified fuzzy
outputs. The final crisp output can be obtained
using some defuzzification methods
• Type 3: Takegi and Sugeno fuzzy if-then rules
are used. The output of each rule is a linear
combination of input variables plus a constant
term and the final output is the weighted
average of each rule’s output
Adaptive Networks – Architecture
and Learning
Has
parameters
Has no parameters
Adaptive Networks – Architecture
and Learning
• Superset of all feedforward NN with supervised
learning capability
• Has nodes and directional links connecting
different nodes
• Part or all the nodes are adaptive(each output of
these nodes depends on parameters pertaining
to this node) and learning rule specifies how
these parameters should be changed to
minimize a error measure
Learning rule
• The basic learning rule is gradient descent
and chain rule
• Because of the problem of slowness and
being trapped in local minima a hybrid
learning rule is proposed
• This learning rule comes in two modes
– Batch learning
– Pattern learning
Architecture and basic learning
• An adaptive network is a multi-layer
feedforward network in which each node
performs a particular function on the
incoming signals
• The nature and the choice of the node
function depends on the overall inputoutput function
• No weights are associated with links and
the links just indicate the flow
Architecture and basic learning
Contd…
• To achieve desired i/p-o/p mapping the
parameters are updated according to
training data and gradient-based learning
procedure
Gradient based learning procedure
• Given adaptive network has L layers
• k-th layer has #k nodes
• (k,i)- ith node in the kth layer
Node function- ith node in the k-layer
Node output depends on its incoming signals and its parameter set
and a,b,c etc. are parameters pertaining to this node
Learning paradigms for Adaptive
networks
• Batch learning:-Update action takes place
only after the whole training data set has
been presented(After an epoch)
• On-line learning:-parameters are updated
immediately after each input-output pair
has been presented.
Hybrid Learning Rule-Batch-Off line
learning rule
• Combines gradient method and least
square estimator to identify parameters
Where I is a set of input variables and S is the set of parameters
If there exists a function H such that the composite function HoF is linear in some
of the elements of S, then these elements can be identified by the least square
Method.
• Using least square estimator we have
For systems with changing characteristics, X can be
iteratively calculated with the formulae given below. Usually
used for online version
Si is the covariance matrix. The initial conditions to the equation
are X0=0 and
where
is a positive large number
and I is the identity matrix
ANFIS
(Adaptive Network based fuzzy
inference system)
• It is functionally equivalent to FIS
• It has minimum constraints so very
popular
• It should be feedforward and piecewise
differentiable