FUNCTION APPROXIMATION

PRESENTED BY
:
Ankita Pandey
ME ECE - 112616

CONTENT
Learning Paradigm
• Supervised Learning
• Unsupervised Learning
• Learning Rules

Function Approximation

System Identification

Inverse Modeling

Summary

References

LEARNING
PARADIGM
Training data

• A sample from the data source with the
correct classiﬁcation/regression solution
already assigned.

Two Types of Learning

• SUPERVISED
• UNSUPERVISED

LEARNING
PARADIGM
Supervised learning : Learning
based on training data.

Example:- Perceptron, LDA, SVMs,
1. Training step: Learn classiﬁer/regressor 2. Prediction step: Assign class
linear/ridge/kernel ridge regression are all
from training data. labels/functional values to test data.
supervised methods.

LEARNING
PARADIGM
Unsupervised learning: Learning
without training data.

Data clustering :
Dimension
Divide input
reduction
data into groups
techniques.
of similar points

Learning
Task

Pattern Pattern Function Beam
Approximation Controlling Filtering
Association Recognition forming

Function
Approximation

To design a neural network that
approximates the unknown function
f(.) such that the function F(.)
describing the input-output mapping
actually realized by the network, is
close enough to f(.) in a Euclidean sense
over all inputs.

Consider a non linear input – output
mapping described by the functional
relationship
d f x
where
Vector x is input.
Vector d is output.
The vector valued function f(.) is assumed to
be unknown.

To get the knowledge about the function
f(.), some set of examples are taken,
N
xi , di i 1
A neural network is designed to
approximate the unknown function in
Euclidean sense over all inputs, given
by the equation

F x f x

Where
• Ε is a small positive number.
• Size N of training sample is large
enough and network is equipped with an
adequate number of free parameters,
• Thus approximation error ε can be
reduced.

• The approximation problem discussed
here would be example of supervised
learning.

FUNCTION
APPROXIMATION

SYSTEM INVERSE
IDENTIFICATION MODELING

SYSTEM
BLOCK DIAGRAM
IDENTIFICATION
di
UNKNOWN
SYSTEM
Input
Vector ei
xi
Σ
NEURAL
NETWORK
MODEL yi

Let input-output relation of unknown memoryless MIMO
system i.e. time invariant system is
d f x
Set of examples are used to train a neural network as a model
of the system.
N
xi , di i 1
Where
Vector y i denote the actual output of the neural network.

• x i denotes the input vector.
• d i denotes the desired response.
• ei denotes the error signal i.e. the difference between
d i and y i .

This error is used to adjust the free parameters of the
network to minimize the squared difference between the
outputsof the unknown system and neural network in a
statistical sense and computed over entire training samples.

INVERSE MODELING
BLOCK DIAGRAM

Error
ei
System
Output Model
Input UNKNOW
di Output xi
Vector INVERS
N
xi
SYSTEM
E
MODEL yi
Σ
f(.)

Inverse Modeling

In this we construct an inverse model that
produces the vector x in response to the vector d.
This can be given by the eqution :
x f 1 d

Where
f 1 denote inverse of f .
Again with the use of stated examples neural
network approximation of f 1 is constructed.

Inverse Modeling
Here d i is used as input and x i as desired response.
is the error signal between and produced
e
ini response to . xi yi
di
This error is used to adjust the free parameters of
the network to minimize the squared difference
between the outputsof the unknown system and
neural network in a statistical sense and computed
over entire training samples.

References

[1] Neural Network And Learning Machines, 3rd Edition, By : Simon
Haykins.
[2] Satish Kumar – Neural Network : A classroom approach.
[3] Jacek M.Zurada- Artificial Neural Networks.
[4] Rajasekaran & Pai – Neural networks, Fuzzy logic and genetic
algorithms.
[5] www.slideshare.net
[6] www.wikipedia.org

FUNCTION APPROXIMATION

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