![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](https://image.slidesharecdn.com/sarb-copy-121101014859-phpapp02/85/FUNCTION-APPROXIMATION-18-320.jpg)
FUNCTION APPROXIMATION
- 1. PRESENTED BY : Ankita Pandey ME ECE - 112616
- 2. CONTENT Learning Paradigm • Supervised Learning • Unsupervised Learning • Learning Rules Function Approximation System Identification Inverse Modeling Summary References
- 3. LEARNING PARADIGM Training data • A sample from the data source with the correct classification/regression solution already assigned. Two Types of Learning • SUPERVISED • UNSUPERVISED
- 4. 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.
- 5. LEARNING PARADIGM Unsupervised learning: Learning without training data. Data clustering : Dimension Divide input reduction data into groups techniques. of similar points
- 6. Learning Task Pattern Pattern Function Beam Approximation Controlling Filtering Association Recognition forming
- 7. 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.
- 8. Function Approximation 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.
- 9. Function Approximation 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
- 10. Function Approximation 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.
- 11. FUNCTION APPROXIMATION SYSTEM INVERSE IDENTIFICATION MODELING
- 12. SYSTEM BLOCK DIAGRAM IDENTIFICATION di UNKNOWN SYSTEM Input Vector ei xi Σ NEURAL NETWORK MODEL yi
- 13. System Identification 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.
- 14. System Identification • 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.
- 15. INVERSE MODELING BLOCK DIAGRAM Error ei System Output Model Input UNKNOW di Output xi Vector INVERS N xi SYSTEM E MODEL yi Σ f(.)
- 16. 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.
- 17. 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.
- 18. 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