links-own [weight] ;; define the four node types breed [bias-nodes bias-node] bias-nodes-own [activation error] breed [input-nodes input-node] input-nodes-own [activation error] breed [output-nodes output-node] output-nodes-own [activation error] breed [hidden-nodes hidden-node] hidden-nodes-own [activation error] globals [ epoch-error ] ;;; ;;; SETUP PROCEDURES ;;; to setup clear-all ask patches [ set pcolor gray + 2 ] set-default-shape bias-nodes "bias-node" set-default-shape input-nodes "circle" set-default-shape output-nodes "output-node" set-default-shape hidden-nodes "output-node" setup-nodes setup-links propagate end to setup-nodes create-bias-nodes 1 [ setxy -5 5 ] ask bias-nodes [ set activation 1 ] create-input-nodes 1 [ setxy -5 -1 ] create-input-nodes 1 [ setxy -5 1 ] ask input-nodes [ set activation random 2 ] create-hidden-nodes 1 [ setxy 0 -1 ] create-hidden-nodes 1 [ setxy 0 1 ] ask hidden-nodes [ set activation random 2 set size 1.5 ] create-output-nodes 1 [ setxy 5 0 ] ask output-nodes [ set activation random 2 ] end to setup-links connect-all bias-nodes hidden-nodes connect-all bias-nodes output-nodes connect-all input-nodes hidden-nodes connect-all hidden-nodes output-nodes end to connect-all [nodes1 nodes2] ask nodes1 [ create-links-to nodes2 [ set weight random-float 0.2 - 0.1 ] ] end to recolor ask turtles with [breed != links] [ set color item (step activation) [black white] ] ask links [ set thickness 0.1 * abs weight ifelse weight > 0 [ set color red ] [ set color blue ] ] end ;;; ;;; TRAINING PROCEDURES ;;; to train set epoch-error 0 repeat examples-per-epoch [ ask input-nodes [ set activation random 2 ] propagate back-propagate ] tick set epoch-error epoch-error / examples-per-epoch plotxy ticks epoch-error end ;;; ;;; FUNCTIONS TO LEARN ;;; to-report target-answer ifelse target-function = "xor" [ report my-xor ] [ report my-or ] end to-report my-or ;; assumes exactly two input nodes ifelse [activation] of input-nodes = [0 0] [ report [0] ] [ report [1] ] end to-report my-xor ;; assumes exactly two input nodes let vals [activation] of input-nodes ifelse item 0 vals = item 1 vals [ report [0] ] [ report [1] ] end ;;; ;;; PROPAGATION PROCEDURES ;;; ;; carry out one calculation from beginning to end to propagate ask hidden-nodes [ set activation new-activation ] ask output-nodes [ set activation new-activation ] recolor end to-report new-activation ;; node procedure report sigmoid sum [[activation] of end1 * weight] of my-in-links end ;; changes weights to correct for errors to back-propagate ;; plot stats ;; computing error for output nodes ;; this assumes that the nodes-list will be in the same order that the list of the correct ;; answers is in let example-error 0 (foreach target-answer (sort output-nodes) [ ask ?2 [ set error activation * (1 - activation) * (?1 - activation) ] set example-error example-error + ( (?1 - [activation] of ?2) ^ 2 ) ] ) set epoch-error epoch-error + (example-error / count output-nodes) ask hidden-nodes [ set error activation * (1 - activation) * sum [weight * [error] of end2] of my-out-links ] ask links [ set weight weight + learning-rate * [error] of end2 * [activation] of end1 ] end ;;; ;;; MISC PROCEDURES ;;; ;; computes the sigmoid function given an input value and the weight on the link to-report sigmoid [input] report 1 / (1 + e ^ (- input)) end ;; computes the step function given an input value and the weight on the link to-report step [input] ifelse input > 0.5 [ report 1 ] [ report 0 ] end ;;; ;;; TESTING PROCEDURES ;;; ;; test runs one instance and computes the output to test ;; output the result ifelse test-success? input-1 input-2 [ user-message "Correct." ] [ user-message "Incorrect." ] end to-report test-success? [n1 n2] ask item 0 sort input-nodes [ set activation n1 ] ask item 1 sort input-nodes [ set activation n2 ] propagate report target-answer = map [step [activation] of ?] sort output-nodes end ; *** NetLogo 4.0 Model Copyright Notice *** ; ; Copyright 2006 by Uri Wilensky. All rights reserved. ; ; Permission to use, modify or redistribute this model is hereby granted, ; provided that both of the following requirements are followed: ; a) this copyright notice is included. ; b) this model will not be redistributed for profit without permission ; from Uri Wilensky. ; Contact Uri Wilensky for appropriate licenses for redistribution for ; profit. ; ; To refer to this model in academic publications, please use: ; Rand, W. and Wilensky, U. (2006). NetLogo Artificial Neural Net model. ; http://ccl.northwestern.edu/netlogo/models/ArtificialNeuralNet. ; Center for Connected Learning and Computer-Based Modeling, ; Northwestern University, Evanston, IL. ; ; In other publications, please use: ; Copyright 2006 Uri Wilensky. All rights reserved. ; See http://ccl.northwestern.edu/netlogo/models/ArtificialNeuralNet ; for terms of use. ; ; *** End of NetLogo 4.0 Model Copyright Notice *** @#$#@#$#@ GRAPHICS-WINDOW 222 10 538 275 8 -1 18.0 1 10 1 1 1 0 0 0 1 -8 8 -5 7 1 1 1 ticks CC-WINDOW 5 373 715 468 Command Center 0 BUTTON 133 31 214 64 setup setup NIL 1 T OBSERVER NIL NIL NIL NIL BUTTON 126 85 211 118 train train T 1 T OBSERVER NIL NIL NIL NIL BUTTON 557 64 620 97 test test NIL 1 T OBSERVER NIL NIL NIL NIL CHOOSER 556 105 648 150 input-1 input-1 0 1 0 CHOOSER 556 159 648 204 input-2 input-2 0 1 1 MONITOR 373 280 430 325 output [precision activation 2] of output-nodes 3 1 11 SLIDER 14 128 214 161 learning-rate learning-rate 0.0 1.0 0.2 1.0E-4 1 NIL HORIZONTAL PLOT 13 209 213 359 Error vs. Epochs Epochs Error 0.0 10.0 0.0 1.0 true false PENS "default" 1.0 0 -16777216 true BUTTON 13 85 107 118 train once train NIL 1 T OBSERVER NIL NIL NIL NIL SLIDER 14 168 213 201 examples-per-epoch examples-per-epoch 1.0 1000.0 500 1.0 1 NIL HORIZONTAL CHOOSER 223 281 361 326 target-function target-function "or" "xor" 1 TEXTBOX 16 38 133 56 1. Setup Neural Net 11 0.0 0 TEXTBOX 15 63 109 81 2. Train Net 11 0.0 0 TEXTBOX 556 32 706 50 3. Test Net 11 0.0 0 @#$#@#$#@ WHAT IS IT? ----------- This is a model of a very small neural network. It is based on the Perceptron model, but instead of one layer, this network has two layers of "perceptrons". That means it can learn operations a single layer cannot. The goal of a network is to take input from its input nodes on the far left and classify those inputs appropriately in the output nodes on the far right. It does this by being given a lot of examples and attempting to classify them, and having a supervisor tell it if the classification was right or wrong. Based on this information the neural network updates its weight until it correctly classifies all inputs correctly. HOW IT WORKS ------------ Initially the weights on the links of the networks are random. When inputs are fed into the network on the far left, those inputs times the random weights are added up to create the activation for the next node in the network. The next node then sends out an activation along its output link. These link weights and activations are summed up by the final output node which reports a value. This activation is passed through a sigmoid function, which means that values near 0 are assigned values close to 0, and vice versa for 1. The values increase nonlinearly between 0 and 1 with a sharp transition at 0.5. To train the network a lot of inputs are presented to the network along with how the network should correctly classify the inputs. The network uses a back-propagation algorithm to pass error back from the output node and uses this error to update the weights along each link. HOW TO USE IT ------------- To use it press SETUP to create the network and initialize the weights to small random numbers. Press TRAIN ONCE to run one epoch of training. The number of examples presented to the network during this epoch is controlled by EXAMPLES-PER-EPOCH slider. Press TRAIN to continually train the network. In the view, the larger the size of the link the greater the weight it has. If the link is red then its a positive weight. If the link is blue then its a negative weight. To test the network, set INPUT-1 and INPUT-2, then press the TEST button. A dialog box will appear telling you whether or not the network was able to correctly classify the input that you gave it. LEARNING-RATE controls how much the neural network will learn from any one example. TARGET-FUNCTION allows you to choose which function the network is trying to solve. THINGS TO NOTICE ---------------- Unlike the Perceptron model, this model is able to learn both OR and XOR. It is able to learn XOR because the hidden layer (the middle nodes) in a way allows the network to draw two lines classifying the input into positive and negative regions. As a result one of the nodes will learn essentially the OR function that if either of the inputs is on it should be on, and the other node will learn an exclusion function that if both of the inputs or on it should be on (but weighted negatively). However unlike the perceptron model, the neural network model takes longer to learn any of the functions, including the simple OR function. This is because it has a lot more that it needs to learn. The perceptron model had to learn three different weights (the input links, and the bias link). The neural network model has to learn ten weights (4 input to hidden layer weights, 2 hidden layer to output weight and the three bias weights). THINGS TO TRY ------------- Manipulate the LEARNING-RATE parameter. Can you speed up or slow down the training? Switch back and forth between OR and XOR several times during a run. Why does it take less time for the network to return to 0 error the longer the network runs? EXTENDING THE MODEL ------------------- Add additional functions for the network to learn beside OR and XOR. This may require you to add additional hidden nodes to the network. Back-propagation using gradient descent is considered somewhat unrealistic as a model of real neurons, because in the real neuronal system there is no way for the output node to pass its error back. Can you implement another weight-update rule that is more valid? NETLOGO FEATURES ---------------- This model uses the link primitives. It also makes heavy use of lists. RELATED MODELS -------------- This is the second in the series of models devoted to understanding artificial neural networks. The first model is Perceptron. CREDITS AND REFERENCES ---------------------- The code for this model is inspired by the pseudo-code which can be found in Tom M. Mitchell's "Machine Learning" (1997). Thanks to Craig Brozefsky for his work in improving this model. To refer to this model in academic publications, please use: Rand, W. and Wilensky, U. (2006). NetLogo Artificial Neural Net model. http://ccl.northwestern.edu/netlogo/models/ArtificialNeuralNet. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. In other publications, please use: Copyright 2006 Uri Wilensky. All rights reserved. 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