1 Pattern Recognition: Statistical and Neural Lonnie C. Ludeman Lecture 21 Oct 28, 2005 Nanjing University of Science & Technology

2 Lecture 21 Topics 1.Example – Analysis of simple Neural Network 2.Example - Synthesis of special forms of Artificial Neural Networks 3. General concepts of Training an Artificial Neural Network- Supervised and unsupervised,training sets 4. Neural Networks Nomenclature and Notation 5. Derivation and Description of the Backpropagation Algorithm for Feedforward Neural Networks

3 Example: Analyze the following Neural Network

4 Solution: Outputs of layer 1 ANEs

5 Output of layer 2 ANE is Thus from layer 1 we have - 2 ≥ 0 < 0

6

7 Final Solution: Output Function for Given Neural Network

8 Example: Synthesize a Neural Network Given the following decision regions build a neural network to perform the classification process Solution: Use Hyperplane-AND-OR structure

9 Each g k (x) specifies a hyperplane boundary

10 Hyperplane LayerAND LayerOR Layer all f(·) = μ (·) Solution:

11 Training a Neural Network “With a teacher” “Without a teacher”

12

13 Training Set x j are the training samples d j is the class assigned to training sample x j

14 Example of a training set: ( x 1 = [ 0, 1,2 ] T, d 1 = C 1 ), ( x 2 = [ 0, 1,0 ] T, d 2 = C 1 ), ( x 3 = [ 0, 1,1 ] T, d 3 = C 1 ), ( x 4 = [ 1, 0,2 ] T, d 4 = C 2 ), ( x 5 = [ 1, 0,3 ] T, d 5 = C 2 ), ( x 6 = [ 0, 0,1 ] T, d 6 = C 3 ), ( x 7 = [ 0, 0,2 ] T, d 7 = C 3 ) ( x 8 = [ 0, 0,3 ] T d 8 = C 3 ) ( x 9 = [ 0, 0,3 ] T d 9 = C 3 ) ( x 10 = [ 1, 1,0 ] T d 10 = C 4 ) ( x 11 = [ 2, 2,0 ] T d 11 = C 4 ) ( x 12 = [ 2, 2,2 ] T d 12 = C 5 ) ( x 13 = [ 3, 2, 2 ] T d 13 = C 6 ) { }

15 General Weight Update Algorithm x(k) is the training sample for the k th iteration d(k) is the class assigned to training sample x(k) y(k) is the output vector for the k th training sample

16 Training with a Teacher( Supervised) 1. Given a set of N ordered samples with their known class assignments. 2. Randomly select all weights in the neural network. 3. For each successive sample in the total set of samples, evaluate the output. 4. Use these outputs and the input sample to update the weights 5. Stop at some predetermined number of iterations or if given performance measure is satisfied. If not stopped go to step 3

17 Training without a Teacher( Unsupervised) 1. Given a set of N ordered samples with unknown class assignments. 2. Randomly select all weights in the neural network. 3. For each successive sample in the total set of samples, evaluate the outputs. 4. Using these outputs and the inputs update the weights 5. If weights do not change significantly stop with that result. If weights change return to step 3

18 Supervised Training of a Feedforward Neural Network Nomenclature

19 Output vector of layer m Output vector of layer L Node Number Layer m Node Number Layer L 1

20 Weight Matrix for layer m Node 1 Node 2 Node N m N NmNm

21 fix Layers, Nets, Outputs, Nonlinearities

22 Define the performance E p for sample x(p) as We wish to select weights so that E p is Minimized – Use Gradient Algorithm

23 Gradient Algorithm for Updating the weights p w(p)w(p) p x(p)x(p)

24 Derivation of weight update equation for Last Layer (Rule #1) Backpropagation Algorihm The partial of y m (L) with respect to w kj (L) is

25 General Rule #1 for Weight Update Therefore

26 Derivation of weight update equation for Next to Last Layer (L-1) Backpropagation Algorithm

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28 General Rule #2 for Weight Update - Layer L-1 Backpropagation Algorithm Therefore and the weight correction is as follows

29 where weight correction (general Rule #2) is w (L-1)

30 Backpropagation Training Algorithm for Feedforward Neural networks

31 Input pattern sample x k

32 Calculate Outputs First Layer

33 Calculate Outputs Second Layer

34 Calculate Outputs Last Layer

35 Check Performance E TOTAL (p)  ½  (d[x(p-i)] – f( w T (p-i)  x(p-i) ) 2 i = 0 N s - 1 E TOTAL (p+1) = E TOTAL (p) + E p+1 (p+1) – E p-Ns (p-N s ) Single Sample Error Over all Samples Error Can be computed recursively

36 Change Weights Last Layer using Rule #1

37 Change Weights previous Layer using Rule #2

38 Change Weights previous Layer using Modified Rule #2

39 Input pattern sample x k+1 Continue Iterations Until

40 Repeat process until performance is satisfied or maximum number of iterations are reached. If performance not satisfied at maximum number of iterations the algorithm stops and NO design is obtained. If performance is satisfied then the current weights and structure provide the required design.

41 Freeze Weights to get Acceptable Neural Net Design

42 Backpropagation Algorithm for Training Feedforward Artificial Neural Networks

43 Summary Lecture 21 1.Example – Analysis of simple Neural Network 2.Example - Synthesis of special forms of Artificial Neural Networks 3. General concepts of Training an Artificial Neural Network- Supervised and unsupervised,and description of training sets 4. Neural Networks Nomenclature and Notation 5. Derivation and Description of the Backpropagation Algorithm for Feedforward Neural Networks

44 End of Lecture 21