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  • 1. F UNCT IONAP ROXIM ION PSarbjeet Singh ATNITTTR chandigarh
  • 2. CONT NT E
  • 3. Learning Paradigms• Training data: A sample from the data source with the correct classification /regression solution already assigned.• Two Types of Learning- – SUPERVISED – UNSUPERVISED• Supervised learning = Learning based on training data.• Two steps:• 1. Training step: Learn classifier /regressor from training data.• 2. Prediction step: Assign class labels/functional values to test data.• Example:- Perceptron, LDA, SVMs, linear/ridge/kernel ridge regression are all supervised methods.
  • 4. Learning Paradigms Contd..• Unsupervised learning: Learning without training data.• Examples: • Data clustering. (Some authors do not distinguish between clustering and unsupervised learning.) • Dimension reduction techniques.• Data clustering: Divide input data into groups of similar points.• → Roughly the unsupervised counterpart to classification.• Note the difference: • Supervised case: Fit model to each class of training points, then use models to classify test points. • Clustering: Simultaneous inference of group structure and model.
  • 5. Learning Tasks• There are Six learning – Pattern Association – Pattern Recognition – Function Approximation – Controlling – Filtering – Beam forming
  • 6. Function ApproximationConsider a non linear input – output mapping described by the functional relationshipwhere d = f (x)Vector x is input.Vector d is output.The vector valued function f(.) is assumed to be unknown.
  • 7. Function ApproximationTo get the knowledge about the function f(.), some set of examples are taken, ℑ = {( xi , d i )} N i =1A neural network is designed to approximate the unknown function in Euclidean sense over all inputs, given by the equation F ( x) − f ( x) < ε
  • 8. Function ApproximationWhereΕ is a small positive number.Size N of training sample ℑ 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.
  • 9. SYST M IDE IF E NT ICATION B OCK DIAGRAM L di UNKNOW N SYST M EInput +Vector ei xi Σ − NEURAL NE W T ORK M ODEL yi
  • 10. 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. ℑ={( xi , d i )}i = N 1Where yiVector denote the actual output of the neural network.
  • 11. System Identificationxi denotes the input vector.d i denotes the desired response.ei denotes the error signal i.e. the differencebetween d i and yi .This error is used to adjust the free parameters of thenetwork to minimize the squared difference betweenthe outputsof the unknown system and neuralnetwork in a statistical sense and computed overentire training samples.
  • 12. INVERSE MODE ING L B OCK DIAGRAM L Error ei System Output ModelInput di Output xiVector UNK NOW INVERSE xi N SYST M f(.) E MODEL y Σ i
  • 13. 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 :Where x=f −1 (d ) f −1 ( • ) denote inverse of f ( • ) .Again with the use of stated−1examples neuralnetwork approximation of f ( • ) is constructed.
  • 14. Inverse Modeling• Here d i is used as input and xi as desired response.• ei is the error signal between xi and yi produced in response to d i .• This error is used to adjust the free parameters of the network to minimize the squared difference between the outputs of the unknown system and neural network in a statistical sense and computed over entire training samples.