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Artificial neural netwoks2
- 1. Artificial Neural Netwoks Dr.Yosser ATASSI Lecture 2
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- 4. Perceptron x xo x x: class I(y = 1) o: class II (y = -1) x oo o x: class I (y = 1) o: class II (y = -1)
- 5. • Examples oflinearly inseparable classes - Logical XOR (exclusive OR) function patterns (bipolar) decision boundary x1 x2 y -1 -1 -1 -1 1 1 1 -1 1 1 1 -1 No line can separate these two classes, as can be seen from the fact that the following linear inequality system has no solution because we have b < 0 from (1) + (4), and b >= 0 from (2) + (3), which is a contradiction o xo x x: class I (y = 1) o: class II (y = -1) <++ ≥−+ ≥+− <−− (4) (3) (2) (1) 0 0 0 0 21 21 21 21 wwb wwb wwb wwb
- 6. – XOR canbe solved by a more complex network with hidden units Y z2 z1x1 x2 2 2 2 2 -2 -2 θ = 1 θ = 0 (-1, -1) (-1, -1) -1 (-1, 1) (-1, 1) 1 (1, -1) (1, -1) 1 (1, 1) (1, 1) -1
- 7.
- 8. • Perceptron learningalgorithm Step 0. Initialization: wk = 0, k = 1 to n Step 1. While stop condition is false do steps 2-5 Step 2. For each of the training sample ij: class(ij) do steps 3 -5 Step 3. compute net= w* ij Step 4. compute o=f(net) Step 5. If o != class(ij) wk := wk + µ ∗ ij * class(ij), k = 1 to n Notes: - Learning occurs only when a sample has o != class(ij) - Two loops, a completion of the inner loop (each sample is used once) is called an epoch Stop condition - When no weight is changed in the current epoch, or - When pre-determined number of epochs is reached
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- 11. [ ] [ ]5.29.0 1 0 2 1 5.01.04.06.0 5.0 1.0 4.0 6.0 1 5.0 1 1 *2.0 7.0 0 6.0 8.0 1)(1 1.2 1 5.0 1 1 7.006.08.0 * 3 3 22 2 222 <= − − −= − = − − + − = =≠−= −= − − −= = net onVerificati W iclasso net iWnet T
- 12. Notes Informal justification: Considero = 1 and class(ij) = -1 – To move o toward class(ij), w1should reduce net – If ij = 1, ij * class(ij) < 0, need to reduce w (ij *w is reduced ) – If ij = -1, ij * class(ij) >0 need to increase w (ij *w is reduced )
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- 19. [ ] [ ]6.0 1 0 2 1 1.015.01.04.0 1.0 15.0 1.0 4.0 1 5.0 1 1 *5.1*2.0 2.0 0 4.0 1.0 1)(1 5.0 1 5.0 1 1 2.004.01.0 * 3 3 22 2 222 −= − − −−−= − − − = − − + − − = =≠−= −= − − −−= = net onVerificati W iclasso net iWnet T
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- 21. (Relative concentration ofNO and NO2 in exhaust fumes as a function of the richness of the ethanol/air mixture burned in a car engine.)
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- 23. Error Backpropagation We wantto train a multi-layer feedforward network by gradient descent to approximate an unknown function, based on some training data consisting of pairs (x,t). The vector x represents a pattern of input to the network, and the vector t the corresponding target (desired output). Weight from unit j to unit i by wij.
- 24.