malikmal: Back propagation

Neural Network (Part 7) : Cut and Paste Code

Oleh Unknown

Ok - so you don't like tutorials, and would rather just cut and paste some code.
This is for you.
http://www.openprocessing.org/visuals/?visualID=33991

Make sure to select "Source code" when you get there, otherwise it will be quite boring.
Here is an animated gif which shows the program in action.

See BELOW for the WHOLE screenshot so that you don't have to speed read.

If you want to know how it works, then you will have to go back and read part 1 to 6.

But as you can see from the example above:
Before the neural network is trained, the outputs are not even close to the expected outputs. After training, the neural network produces the desired results (or very close to it).

Please note, this neural network also allows more than one output neuron, so you are not limited to single yes no decisions. You can use this neural network to make classifications. You will soon see this with my LED colour sensor.

Feel free to use this Neural Network in your own projects, but please let me know if you do, just for curiosity sake.

Update: See below for the Processing Sketch (much easier than going to the open processing site). It is a bit long - but saves you from having to navigate to another site.

Processing sketch

/*Neural Network created by ScottC on 15th Aug 2011
 
Please visit my blog for a detailed explanation of my Neural Network
http://arduinobasics.blogspot.com/p/arduinoprojects.html
 
*/
   
   
 
void setup(){
  ArrayList myTrainingInputs = new ArrayList();
  ArrayList myTrainingOutputs = new ArrayList();
   
  float[] myInputsA={0,0};
  float[] myInputsB={0,1};
  float[] myInputsC={1,0};
  float[] myInputsD={1,1};
  float[] myOutputsA={1};
  float[] myOutputsB={0};
   
   
  println("TRAINING DATA");
  println("--------------------------------------------");
  myTrainingInputs.add(myInputsA);
  myTrainingOutputs.add(myOutputsA);
  println("INPUTS= " + myInputsA[0] + ", " + myInputsA[1] + "; Expected output = " + myOutputsA[0]);
  myTrainingInputs.add(myInputsB);
  myTrainingOutputs.add(myOutputsB);
  println("INPUTS= " + myInputsB[0] + ", " + myInputsB[1] + "; Expected output = " + myOutputsB[0]);
  myTrainingInputs.add(myInputsC);
  myTrainingOutputs.add(myOutputsB);
  println("INPUTS= " + myInputsC[0] + ", " + myInputsC[1] + "; Expected output = " + myOutputsB[0]);
  myTrainingInputs.add(myInputsD);
  myTrainingOutputs.add(myOutputsA);
  println("INPUTS= " + myInputsD[0] + ", " + myInputsD[1] + "; Expected output = " + myOutputsA[0]);
  println("--------------------------------------------");
  
  NeuralNetwork NN = new NeuralNetwork();
  NN.addLayer(2,2);
  NN.addLayer(2,1);
  
  println("Before Training");
  float[] myInputDataA1={0,0};
  NN.processInputsToOutputs(myInputDataA1);
  float[] myOutputDataA1={};
  myOutputDataA1=NN.getOutputs();
  println("Feed Forward:  INPUT = 0,0; OUTPUT=" + myOutputDataA1[0]);
   
  float[] myInputDataB1={0,1};
  NN.processInputsToOutputs(myInputDataB1);
  float[] myOutputDataB1={};
  myOutputDataB1=NN.getOutputs();
  println("Feed Forward:  INPUT = 0,1; OUTPUT=" + myOutputDataB1[0]);
   
  float[] myInputDataC1={1,0};
  NN.processInputsToOutputs(myInputDataC1);
  float[] myOutputDataC1={};
  myOutputDataC1=NN.getOutputs();
  println("Feed Forward:  INPUT = 1,0; OUTPUT=" + myOutputDataC1[0]);
   
  float[] myInputDataD1={1,1};
  NN.processInputsToOutputs(myInputDataD1);
  float[] myOutputDataD1={};
  myOutputDataD1=NN.getOutputs();
  println("Feed Forward:  INPUT = 1,1; OUTPUT=" + myOutputDataD1[0]);
 
  println("");
  println("--------------------------------------------");
   
  println("Begin Training");
  NN.autoTrainNetwork(myTrainingInputs,myTrainingOutputs,0.0001,500000);
  println("");
  println("End Training");
  println("");
  println("--------------------------------------------");
  println("Test the neural network");
  float[] myInputDataA2={0,0};
  NN.processInputsToOutputs(myInputDataA2);
  float[] myOutputDataA2={};
  myOutputDataA2=NN.getOutputs();
  println("Feed Forward:  INPUT = 0,0; OUTPUT=" + myOutputDataA2[0]);
   
  float[] myInputDataB2={0,1};
  NN.processInputsToOutputs(myInputDataB2);
  float[] myOutputDataB2={};
  myOutputDataB2=NN.getOutputs();
  println("Feed Forward:  INPUT = 0,1; OUTPUT=" + myOutputDataB2[0]);
   
  float[] myInputDataC2={1,0};
  NN.processInputsToOutputs(myInputDataC2);
  float[] myOutputDataC2={};
  myOutputDataC2=NN.getOutputs();
  println("Feed Forward:  INPUT = 1,0; OUTPUT=" + myOutputDataC2[0]);
   
  float[] myInputDataD2={1,1};
  NN.processInputsToOutputs(myInputDataD2);
  float[] myOutputDataD2={};
  myOutputDataD2=NN.getOutputs();
  println("Feed Forward:  INPUT = 1,1; OUTPUT=" + myOutputDataD2[0]);
  
   
}
 
 
/* ---------------------------------------------------------------------
A connection determines how much of a signal is passed through to the neuron.
--------------------------------------------------------------------  */
 
class Connection{
  float connEntry;
  float weight;
  float connExit;
   
  //This is the default constructor for an Connection
  Connection(){
    randomiseWeight();
  }
   
  //A custom weight for this Connection constructor
  Connection(float tempWeight){
    setWeight(tempWeight);
  }
   
  //Function to set the weight of this connection
  void setWeight(float tempWeight){
    weight=tempWeight;
  }
   
  //Function to randomise the weight of this connection
  void randomiseWeight(){
    setWeight(random(2)-1);
  }
   
  //Function to calculate and store the output of this Connection
  float calcConnExit(float tempInput){
    connEntry = tempInput;
    connExit = connEntry * weight;
    return connExit;
  }
}
 
 
/* -----------------------------------------------------------------
   A neuron does all the processing and calculation to convert an input into an output
--------------------------------------------------------------------- */
 
class Neuron{
  Connection[] connections={};
  float bias;
  float neuronInputValue;
  float neuronOutputValue;
  float deltaError;
   
  //The default constructor for a Neuron
  Neuron(){
  }
   
  /*The typical constructor of a Neuron - with random Bias and Connection weights */
  Neuron(int numOfConnections){
    randomiseBias();
    for(int i=0; i<numOfConnections; i++){
      Connection conn = new Connection();
      addConnection(conn);
    }
  }
   
  //Function to add a Connection to this neuron
  void addConnection(Connection conn){
      connections = (Connection[]) append(connections, conn);
  }
   
  /* Function to return the number of connections associated with this neuron.*/
  int getConnectionCount(){
      return connections.length;
  }
   
  //Function to set the bias of this Neron
  void setBias(float tempBias){
    bias = tempBias;
  }
   
  //Function to randomise the bias of this Neuron
  void randomiseBias(){
    setBias(random(1));
  }
   
  /*Function to convert the inputValue to an outputValue
  Make sure that the number of connEntryValues matches the number of connections */
   
  float getNeuronOutput(float[] connEntryValues){
    if(connEntryValues.length!=getConnectionCount()){
      println("Neuron Error: getNeuronOutput() : Wrong number of connEntryValues");
      exit();
    }
     
    neuronInputValue=0;
     
    /* First SUM all of the weighted connection values (connExit) attached to this neuron. This becomes the neuronInputValue. */
    for(int i=0; i<getConnectionCount(); i++){
      neuronInputValue+=connections[i].calcConnExit(connEntryValues[i]);
    }
     
    //Add the bias to the Neuron's inputValue
    neuronInputValue+=bias;
     
    /* Send the inputValue through the activation function to produce the Neuron's outputValue */
    neuronOutputValue=Activation(neuronInputValue);
     
    //Return the outputValue
    return neuronOutputValue;
  }
   
  //Activation function
  float Activation(float x){
    float activatedValue = 1 / (1 + exp(-1 * x));
    return activatedValue;
  }
   
}
 
class Layer{
  Neuron[] neurons = {};
  float[] layerINPUTs={};
  float[] actualOUTPUTs={};
  float[] expectedOUTPUTs={};
  float layerError;
  float learningRate;
   
   
  /* This is the default constructor for the Layer */
  Layer(int numberConnections, int numberNeurons){
    /* Add all the neurons and actualOUTPUTs to the layer */
    for(int i=0; i<numberNeurons; i++){
      Neuron tempNeuron = new Neuron(numberConnections);
      addNeuron(tempNeuron);
      addActualOUTPUT();
    }
  }
     
   
  /* Function to add an input or output Neuron to this Layer */
  void addNeuron(Neuron xNeuron){
        neurons = (Neuron[]) append(neurons, xNeuron);
  }
   
   
  /* Function to get the number of neurons in this layer */
  int getNeuronCount(){
    return neurons.length;
  }
   
   
  /* Function to increment the size of the actualOUTPUTs array by one. */
  void addActualOUTPUT(){
      actualOUTPUTs = (float[]) expand(actualOUTPUTs,(actualOUTPUTs.length+1));
  }
   
   
  /* Function to set the ENTIRE expectedOUTPUTs array in one go. */
  void setExpectedOUTPUTs(float[] tempExpectedOUTPUTs){
    expectedOUTPUTs=tempExpectedOUTPUTs;
  }
   
   
  /* Function to clear ALL values from the expectedOUTPUTs array */
  void clearExpectedOUTPUT(){
       expectedOUTPUTs = (float[]) expand(expectedOUTPUTs, 0);
  }
   
   
  /* Function to set the learning rate of the layer */
  void setLearningRate(float tempLearningRate){
    learningRate=tempLearningRate;
  }
   
   
  /* Function to set the inputs of this layer */
  void setInputs(float[] tempInputs){
    layerINPUTs=tempInputs;
  }
   
   
   
  /* Function to convert ALL the Neuron input values into Neuron output values in this layer, through a special activation function. */
  void processInputsToOutputs(){
     
    /* neuronCount is used a couple of times in this function. */
    int neuronCount = getNeuronCount();
     
    /* Check to make sure that there are neurons in this layer to process the inputs */
    if(neuronCount>0) {
      /* Check to make sure that the number of inputs matches the number of Neuron Connections. */
      if(layerINPUTs.length!=neurons[0].getConnectionCount()){
        println("Error in Layer: processInputsToOutputs: The number of inputs do NOT match the number of Neuron connections in this layer");
        exit();
      } else {
        /* The number of inputs are fine : continue
           Calculate the actualOUTPUT of each neuron in this layer,
           based on their layerINPUTs (which were previously calculated).
           Add the value to the layer's actualOUTPUTs array. */
        for(int i=0; i<neuronCount;i++){
          actualOUTPUTs[i]=neurons[i].getNeuronOutput(layerINPUTs);
        }
      }
    }else{
      println("Error in Layer: processInputsToOutputs: There are no Neurons in this layer");
      exit();
    }
  }
   
   
  /* Function to get the error of this layer */
  float getLayerError(){
    return layerError;
  }
   
   
  /* Function to set the error of this layer */
  void setLayerError(float tempLayerError){
    layerError=tempLayerError;
  }
   
   
  /* Function to increase the layerError by a certain amount */
  void increaseLayerErrorBy(float tempLayerError){
    layerError+=tempLayerError;
  }
   
   
  /* Function to calculate and set the deltaError of each neuron in the layer */
  void setDeltaError(float[] expectedOutputData){
    setExpectedOUTPUTs(expectedOutputData);
    int neuronCount = getNeuronCount();
    /* Reset the layer error to 0 before cycling through each neuron */
    setLayerError(0);
     for(int i=0; i<neuronCount;i++){
       neurons[i].deltaError = actualOUTPUTs[i]*(1-actualOUTPUTs[i])*(expectedOUTPUTs[i]-actualOUTPUTs[i]);
        
       /* Increase the layer Error by the absolute difference between the calculated value (actualOUTPUT) and the expected value (expectedOUTPUT). */
       increaseLayerErrorBy(abs(expectedOUTPUTs[i]-actualOUTPUTs[i]));
     }
  }
   
   
  /* Function to train the layer : which uses a training set to adjust the connection weights and biases of the neurons in this layer */
  void trainLayer(float tempLearningRate){
    setLearningRate(tempLearningRate);
     
    int neuronCount = getNeuronCount();
     
    for(int i=0; i<neuronCount;i++){
      /* update the bias for neuron[i] */
      neurons[i].bias += (learningRate * 1 * neurons[i].deltaError);
       
      /* update the weight of each connection for this neuron[i] */
      for(int j=0; j<neurons[i].getConnectionCount(); j++){
          neurons[i].connections[j].weight += (learningRate * neurons[i].connections[j].connEntry * neurons[i].deltaError);
      }  
    }
  }
}
 
/* -------------------------------------------------------------
   The Neural Network class is a container to hold and manage all the layers
   ---------------------------------------------------------------- */
 
class NeuralNetwork{
  Layer[] layers = {};
  float[] arrayOfInputs={};
  float[] arrayOfOutputs={};
  float learningRate;
  float networkError;
  float trainingError;
  int retrainChances=0;
   
  NeuralNetwork(){
    /* the default learning rate of a neural network is set to 0.1, which can changed by the setLearningRate(lR) function. */
    learningRate=0.1;
  }
   
   
   
  /* Function to add a Layer to the Neural Network */
  void addLayer(int numConnections, int numNeurons){
    layers = (Layer[]) append(layers, new Layer(numConnections,numNeurons));
  }
 
 
 
  /* Function to return the number of layers in the neural network */
  int getLayerCount(){
      return layers.length;
  }
   
   
   
  /* Function to set the learningRate of the Neural Network */
  void setLearningRate(float tempLearningRate){
    learningRate=tempLearningRate;
  }
   
   
   
  /* Function to set the inputs of the neural network */
  void setInputs(float[] tempInputs){
    arrayOfInputs=tempInputs;
  }
   
   
   
  /* Function to set the inputs of a specified layer */
  void setLayerInputs(float[] tempInputs, int layerIndex){
    if(layerIndex>getLayerCount()-1){
      println("NN Error: setLayerInputs: layerIndex=" + layerIndex + " exceeded limits= " + (getLayerCount()-1));
    } else {
      layers[layerIndex].setInputs(tempInputs);
    }
  }
   
   
   
  /* Function to set the outputs of the neural network */
  void setOutputs(float[] tempOutputs){
    arrayOfOutputs=tempOutputs;
  }
   
   
   
  /* Function to return the outputs of the Neural Network */
  float[] getOutputs(){
    return arrayOfOutputs;
  }
   
   
   
  /* Function to process the Neural Network's input values and convert them to an output pattern using ALL layers in the network */
  void processInputsToOutputs(float[] tempInputs){
    setInputs(tempInputs);
     
    /* Check to make sure that the number of NeuralNetwork inputs matches the Neuron Connection Count in the first layer. */
    if(getLayerCount()>0){
      if(arrayOfInputs.length!=layers[0].neurons[0].getConnectionCount()){
        println("NN Error: processInputsToOutputs: The number of inputs do NOT match the NN");
        exit();
      } else {
        /* The number of inputs are fine : continue */
        for(int i=0; i<getLayerCount(); i++){
           
          /*Set the INPUTs for each layer: The first layer gets it's input data from the NN, whereas the 2nd and subsequent layers get their input data from the previous layer's actual output. */
          if(i==0){
            setLayerInputs(arrayOfInputs,i);
          } else {
            setLayerInputs(layers[i-1].actualOUTPUTs, i);
          }
           
          /* Now that the layer has had it's input values set, it can now process this data, and convert them into an output using the layer's neurons. The outputs will be used as inputs in the next layer (if available). */
          layers[i].processInputsToOutputs();
        }
        /* Once all the data has filtered through to the end of network, we can grab the actualOUTPUTs of the LAST layer
           These values become or will be set to the NN output values (arrayOfOutputs), through the setOutputs function call. */
        setOutputs(layers[getLayerCount()-1].actualOUTPUTs);
      }
    }else{
      println("Error: There are no layers in this Neural Network");
      exit();
    }
  }
   
   
   
   
  /* Function to train the entire network using an array. */
  void trainNetwork(float[] inputData, float[] expectedOutputData){
    /* Populate the ENTIRE network by processing the inputData. */
    processInputsToOutputs(inputData);
     
    /* train each layer - from back to front (back propagation) */
    for(int i=getLayerCount()-1; i>-1; i--){
      if(i==getLayerCount()-1){
        layers[i].setDeltaError(expectedOutputData);
        layers[i].trainLayer(learningRate);
        networkError=layers[i].getLayerError();
      } else {
        /* Calculate the expected value for each neuron in this layer (eg. HIDDEN LAYER) */
        for(int j=0; j<layers[i].getNeuronCount(); j++){
          /* Reset the delta error of this neuron to zero. */
          layers[i].neurons[j].deltaError=0;
          /* The delta error of a hidden layer neuron is equal to the SUM of [the PRODUCT of the connection.weight and error of the neurons in the next layer(eg OUTPUT Layer)]. */
          /* Connection#1 of each neuron in the output layer connect with Neuron#1 in the hidden layer */
          for(int k=0; k<layers[i+1].getNeuronCount(); k++){
            layers[i].neurons[j].deltaError += (layers[i+1].neurons[k].connections[j].weight * layers[i+1].neurons[k].deltaError);
          }
          /* Now that we have the sum of Errors x weights attached to this neuron. We must multiply it by the derivative of the activation function. */
          layers[i].neurons[j].deltaError *= (layers[i].neurons[j].neuronOutputValue * (1-layers[i].neurons[j].neuronOutputValue));
        }
        /* Now that you have all the necessary fields populated, you can now Train this hidden layer and then clear the Expected outputs, ready for the next round. */
        layers[i].trainLayer(learningRate);
        layers[i].clearExpectedOUTPUT();
      }
    }
  }
   
   
   
   
   
  /* Function to train the entire network, using an array of input and expected data within an ArrayList */
  void trainingCycle(ArrayList trainingInputData, ArrayList trainingExpectedData, Boolean trainRandomly){
      int dataIndex;
       
      /* re-initialise the training Error with every cycle */
      trainingError=0;
       
      /* Cycle through the training data either randomly or sequentially */
      for(int i=0; i<trainingInputData.size(); i++){
        if(trainRandomly){
          dataIndex=(int) (random(trainingInputData.size()));
        } else {
          dataIndex=i;
        }
  
        trainNetwork((float[]) trainingInputData.get(dataIndex),(float[]) trainingExpectedData.get(dataIndex));
         
        /* Use the networkError variable which is calculated at the end of each individual training session to calculate the entire trainingError. */
        trainingError+=abs(networkError);
      }
  }
   
   
   
   
   
  /* Function to train the network until the Error is below a specific threshold */
  void autoTrainNetwork(ArrayList trainingInputData, ArrayList trainingExpectedData, float trainingErrorTarget, int cycleLimit){
    trainingError=9999;
    int trainingCounter=0;
     
     
    /* cycle through the training data until the trainingError gets below trainingErrorTarget (eg. 0.0005) or the training cycles have exceeded the cycleLimit
 
variable (eg. 10000). */
    while(trainingError>trainingErrorTarget && trainingCounter<cycleLimit){
       
      /* re-initialise the training Error with every cycle */
      trainingError=0;
       
      /* Cycle through the training data randomly */
      trainingCycle(trainingInputData, trainingExpectedData, true);
       
      /* increment the training counter to prevent endless loop */
      trainingCounter++;
    }
     
    /* Due to the random nature in which this neural network is trained. There may be occasions when the training error may drop below the threshold
       To check if this is the case, we will go through one more cycle (but sequentially this time), and check the trainingError for that cycle
       If the training error is still below the trainingErrorTarget, then we will end the training session.
       If the training error is above the trainingErrorTarget, we will continue to train. It will do this check a  Maximum of 9 times. */
    if(trainingCounter<cycleLimit){
       trainingCycle(trainingInputData, trainingExpectedData, false);
       trainingCounter++;
       
       if(trainingError>trainingErrorTarget){
         if (retrainChances<10){
           retrainChances++;
           autoTrainNetwork(trainingInputData, trainingExpectedData,trainingErrorTarget, cycleLimit);
         }
       }
        
    } else {
      println("CycleLimit has been reached. Has been retrained " + retrainChances + " times.  Error is = " + trainingError);
    }  
  }
}

The above code was formatted using hilite.me

Description: Neural Network (Part 7) : Cut and Paste Code Rating: 3.5 Reviewer: Unknown ItemReviewed: Neural Network (Part 7) : Cut and Paste Code

Neural Network (Part 6) : Back Propagation, a worked example

Oleh Unknown

A worked example of a Back-propagation training cycle.

In this example we will create a 2 layer network (as seen above), to accept 2 readings, and produce 2 outputs. The readings are (0,1) and the expectedOutputs in this example are (1,0).

Step 1: Create the network

NeuralNetwork NN = new NeuralNetwork();

NN.addLayer(2,2);

float[] readings = {0,1};

float[] expectedOutputs = {1,0};

NN.trainNetwork(readings,expectedOutputs);

This neural network will have randomised weights and biases when created.

Let us assume that the network generates the following random variables:

LAYER1.Neuron1

Layer1.Neuron1.Connection1.weight = cW111 = 0.3

Layer1.Neuron1.Connection2.weight = cW112 = 0.8

Layer1.Neuron1.Bias = bW11 = 0.5

LAYER1.Neuron2

Layer1.Neuron2.Connection1.weight = cW121 = 0.1

Layer1.Neuron2.Connection2.weight = cW122 = 0.1

Layer1.Neuron2.Bias = bW12 = 0.2

LAYER2.Neuron1

Layer2.Neuron1.Connection1.weight = cW211 = 0.6

Layer2.Neuron1.Connection2.weight = cW212 = 0.4

Layer2.Neuron1.Bias = bW21 = 0.4

LAYER2.Neuron2

Layer2.Neuron2.Connection1.weight = cW221 = 0.9

Layer2.Neuron2.Connection2.weight = cW222 = 0.9

Layer2.Neuron2.Bias = bW22 = 0.5

Step 2: Process the Readings through the Neural Network

a) Provide the Readings to the first layer, and calculate the neuron outputs

The readings provided to the neural network is (0,1), which go straight through to the first layer (layer1).

Starting with Layer 1:

Layer1.INPUT1 = 0

Layer1.INPUT2 =1

Calculate Layer1.Neuron1.NeuronOutput

ConnExit (cEx111) = ConnEntry (cEn111) x Weight (cW111) = 0 x 0.3 = 0;

ConnExit (cEx112) = ConnEntry (cEn112) x Weight (cW112) = 1 x 0.8 = 0.8;

Bias (bEx11) = ConnEntry (1) x Weight (bW11) = 1 x 0.4 = 0.4

NeuronInputValue11 = 0 + 0.8 + 0.4 = 1.2

NeuronOutputValue11 = 1/(1+EXP(-1 x 1.2)) = 0.768525

Calculate Layer1.Neuron2.NeuronOutput

ConnExit (cEx121) = ConnEntry (cEn121) x Weight (cW121) = 0 x 0.1 = 0;

ConnExit (cEx122) = ConnEntry (cEn122) x Weight (cW122) = 1 x 0.1 = 0.1;

Bias (bEx12) = ConnEntry (1) x Weight (bW12) = 1 x 0.2 = 0.2

NeuronInputValue12 = 0 + 0.1 + 0.2 = 0.3

NeuronOutputValue12 = 1/(1+EXP(-1 x 0.3)) = 0.574443

b) Provide LAYER2 with Layer 1 Outputs.

Now lets move to Layer 2:

Layer2.INPUT1 = NeuronOutputValue11 = 0.768525

Layer2.INPUT2 = NeuronOutputValue12 = 0.574443

Calculate Layer2.Neuron1.NeuronOutput

ConnExit (cEx211) = (cEn211) x Weight (cW211) = 0.768525 x 0.6 = 0.461115;

ConnExit (cEx212) = (cEn212) x Weight (cW212) = 0.574443 x 0.4 = 0.229777;

Bias (bEx21) = ConnEntry (1) x Weight (bW21) = 1 x 0.4 = 0.4

NeuronInputValue21 = 0.461115 + 0.229777 + 0.4 = 1.090892

NeuronOutputValue21 = 1/(1+EXP(-1 x 1.090892)) = 0.74855

Calculate Layer2.Neuron2.NeuronOutput

ConnExit (cEx221) = (cEn221) x Weight (cW221) = 0.768525 x 0.1 = 0.076853;

ConnExit (cEx222) = (cEn222) x Weight (cW222) = 0.574443 x 0.1 = 0.057444;

Bias(bEx22) = ConnEntry (1) x Weight (bW22) = 1 x 0.5 = 0.5

NeuronInputValue22 = 0.076853 + 0.057444 + 0.5 = 0.634297

NeuronOutputValue22 = 1/(1+EXP(-1 x 0.634297)) = 0.653463

Step 3) Calculate the delta error for neurons in layer 2

-Because layer 2 is the last layer in this neural network -

we will use the expected output data (1,0) to calculate the delta error.

LAYER2.Neuron1:

Let Layer2.ExpectedOutput1 = eO21 = 1

Layer2.ActualOutput1= aO21 = NeuronOutputValue21= 0.74855

Layer2.Neuron1.deltaError1 = dE21

dE21 = aO21 x (1 - aO21) x (eO21 - aO21)

= (0.74855) x (1 - 0.74855) x (1 - 0.74855)

= (0.74855) x (0.25145) x (0.25145)

= 0.047329

LAYER2.Neuron2:

Let Layer2.ExpectedOutput2 = eO22 = 0

Layer2.ActualOutput2 = aO22 = NeuronOutputValue22 = 0.653463

Layer2.Neuron2.deltaError = dE22

dE22 = aO22 x (1 - aO22) x (eO22 - aO22)

= (0.653463) x (1 - 0.653463) x (0 - 0.653463)

= (0.653463) x (0.346537) x (-0.653463)

= -0.14797

Step 4) Calculate the delta error for neurons in layer 1

LAYER1.Neuron1 delta Error calculation

Let Layer1.Neuron1.deltaError = dE11

Layer1.actualOutput1 = aO11 = NeuronOutputValue11 = 0.768525

Layer2.Neuron1.Connection1.weight = cW211 = 0.6

Layer2.Neuron1.deltaError = dE21 = 0.047329

Layer2.Neuron2.Connection1.weight = cW221 = 0.9

Layer2.Neuron2.deltaError = dE22 = -0.14797

dE11 = (aO11) x (1 - aO11) x ( [cW211 x dE21] + [cW221 x dE22] )

= (0.768525) x (1 - 0.768525) x ([0.6 x 0.047329] + [ 0.9 x -0.14797] )

= -0.01864

LAYER1.Neuron2 delta Error calculation

Let Layer1.Neuron2.deltaError = dE12

Layer1.actualOutput2 = aO12 = NeuronOutputValue12 = 0.574443

Layer2.Neuron1.Connection2.weight = cW212 = 0.4

Layer2.Neuron1.deltaError = dE21 = 0.047329

Layer2.Neuron2.Connection2.weight = cW222 = 0.9

Layer2.Neuron2.deltaError = dE22 = -0.14797

dE12 = (aO12) x (1 - aO12) x ( [cW212 x dE21] + [cW222 x dE22] )

= (0.574443) x (1 - 0.574443) x ([0.4 x 0.047329] + [ 0.9 x -0.14797] )

= -0.02793

Step 5) Update Layer_2 neuron connection weights and bias (with a learning rate (LR) = 0.1)

Layer 2, Neuron 1 calculations:

Let

Layer2.Neuron1.Connection1.New_weight = New_cW211

Layer2.Neuron1.Connection1.Old_weight = Old_cW211 = 0.6

Layer2.Neuron1.Connection1.connEntry = cEn211 = 0.768525

Layer2.Neuron1.deltaError = dE21 = 0.047329

New_cW211 = Old_cW211 + (LR x cEn211 x dE21)

= 0.6 + (0.1 x 0.768525 x 0.047329)

= 0.6 + ( 0.003627)

= 0.603627

Layer2.Neuron1.Connection2.New_weight = New_cW212

Layer2.Neuron1.Connection2.Old_weight = Old_cW212 = 0.4

Layer2.Neuron1.Connection2.connEntry = cEn212 = 0.574443

Layer2.Neuron1.deltaError = dE21 = 0.047329

New_cW212 = Old_cW212 + (LR x cEn212 x dE21)

= 0.4 + (0.1 x 0.574443 x 0.047329)

= 0.4 + (0.002719)

= 0.402719

Layer2.Neuron1.New_Bias = New_Bias21

Layer2.Neuron1.Old_Bias = Old_Bias21 = 0.4

Layer2.Neuron1.deltaError = dE21 = 0.047329

New_Bias21 = Old_Bias21 + (LR x 1 x de21)

= 0.4 + (0.1 x 1 x 0.047329)

= 0.4 + (0.0047329)

= 0.4047329

--------------------------------------------------------------------

Layer 2, Neuron 2 calculations:

Layer2.Neuron2.Connection1.New_weight = New_cW221

Layer2.Neuron2.Connection1.Old_weight = Old_cW221 = 0.9

Layer2.Neuron2.Connection1.connEntry = cEn221 = 0.768525

Layer2.Neuron2.deltaError = dE22 = -0.14797

New_cW221 = Old_cW221 + (LR x cEn221 x dE22)

= 0.9 + (0.1 x 0.768525 x -0.14797)

= 0.9 + ( -0.01137)

= 0.88863

Layer2.Neuron2.Connection2.New_weight = New_cW222

Layer2.Neuron2.Connection2.Old_weight = Old_cW222 = 0.9

Layer2.Neuron2.Connection2.connEntry = cEn222 = 0.574443

Layer2.Neuron2.deltaError = dE22 = -0.14797

New_cW222 = Old_cW222 + (LR x cEn222 x dE22)

= 0.9 + (0.1 x 0.574443 x -0.14797)

= 0.9 + (-0.0085)

= 0.8915

Layer2.Neuron2.New_Bias = New_Bias22

Layer2.Neuron2.Old_Bias = Old_Bias22 = 0.5

Layer2.Neuron2.deltaError = dE22 = -0.14797

New_Bias22 = Old_Bias22 + (LR x 1 x de22)

= 0.5 + (0.1 x 1 x -0.14797)

= 0.5 + (-0.014797)

= 0.485203

--------------------------------------------------------------------------

Step 6) Update Layer_1 neuron connection weights and bias.

Layer 1, Neuron 1 calculations:

Let

Layer1.Neuron1.Connection1.New_weight = New_cW111

Layer1.Neuron1.Connection1.Old_weight = Old_cW111 = 0.3

Layer1.Neuron1.Connection1.connEntry = cEn111 = 0

Layer1.Neuron1.deltaError = dE11 = -0.01864

New_cW111 = Old_cW111 + (LR x cEn111 x dE11)

= 0.3 + (0.1 x 0 x -0.01864)

= 0.3 + ( 0 )

= 0.3

Layer1.Neuron1.Connection2.New_weight = New_cW112

Layer1.Neuron1.Connection2.Old_weight = Old_cW112 = 0.8

Layer1.Neuron1.Connection2.connEntry = cEn112 = 1

Layer1.Neuron1.deltaError = dE11 = -0.01864

New_cW112 = Old_cW112 + (LR x cEn112 x dE11)

= 0.8 + (0.1 x 1 x -0.01864)

= 0.8 + (-0.001864)

= 0.798136

Layer1.Neuron1.New_Bias = New_Bias11

Layer1.Neuron1.Old_Bias = Old_Bias11 = 0.5

Layer1.Neuron1.deltaError = dE11 = -0.01864

New_Bias11 = Old_Bias11 + (LR x 1 x dE11)

= 0.5 + (0.1 x 1 x -0.01864 )

= 0.5 + (-0.001864)

= 0.498136

--------------------------------------------------------------------

Layer 1, Neuron 2 calculations:

Layer1.Neuron2.Connection1.New_weight = New_cW121

Layer1.Neuron2.Connection1.Old_weight = Old_cW121 = 0.1

Layer1.Neuron2.Connection1.connEntry = cEn121 = 0

Layer1.Neuron2.deltaError = dE12 = -0.02793

New_cW121 = Old_cW121 + (LR x cEn121 x dE12)

= 0.1 + (0.1 x 0 x -0.02793 )

= 0.1 + (0)

= 0.1

Layer1.Neuron2.Connection2.New_weight = New_cW122

Layer1.Neuron2.Connection2.Old_weight = Old_cW122 = 0.1

Layer1.Neuron2.Connection2.connEntry = cEn122 = 1

Layer1.Neuron2.deltaError = dE12 = -0.02793

New_cW122 = Old_cW122 + (LR x cEn122 x dE12)

= 0.1 + (0.1 x 1 x -0.02793)

= 0.1 + (-0.002793)

= 0.097207

Layer1.Neuron2.New_Bias = New_Bias12

Layer1.Neuron2.Old_Bias = Old_Bias12 = 0.2

Layer1.Neuron2.deltaError = dE12 = -0.02793

New_Bias12 = Old_Bias12 + (LR x 1 x de12)

= 0.2 + (0.1 x 1 x -0.02793)

= 0.2 + (-0.002793)

= 0.197207

----------------------------------------------------------------------

All done. That was just one training cycle. Thank goodness we have computers !

A computer can process these calculations really quickly, and depending on how complicated your neural network is (ie. number of layers, and number of neurons per layer), you may find that the training procedure may take some time. But believe me, if you have designed it right, it is well worth the wait.

Because once you have the desired weights and bias values set up, you are good to go, and as you receive data, the computer can do a single forward pass in a fraction of a second, and you will get your desired output, hopefully :)

Here is a complete Processing.org script that demonstrates the use of my neural network.

Neural Network (Part 7): Cut and Paste Code (click here).

If you liked my tutorial - please let me know in the comments. It is sometimes hard to know if anyone is actually reading this stuff. If you use my code in your own project, I am also happy for you to leave a link to a YouTube video etc in the comments also.

To go back to the table of contents click here

Description: Neural Network (Part 6) : Back Propagation, a worked example Rating: 3.5 Reviewer: Unknown ItemReviewed: Neural Network (Part 6) : Back Propagation, a worked example

Neural Network (Part 5): The Back Propagation process

Oleh Unknown

Back-propagation

Back propagation is the process by which you move backwards through the neural network to adjust the weights and biases so as to reduce the total error of the network. The total error of the network is essentially the difference between the end results (actual Outputs) and the expected results. If you expected to get a result of 1, but instead got a 0: you would go back through the network and tweak each of the weights (and bias) values so that your end result was a little bit closer to 1 than before.

The process of back-propagation is such that larger errors and larger weights and biases that create those errors are penalised more than their smaller counterparts. Bigger weights have a bigger influence on the final outcome than smaller weights, and are therefore penalised more for incorrect answers.

After many training cycles, the neural network reaches a stage of equilibrium (not quite, but close enough), whereby the tweaking is insignificant to the final outcome.

If you under-train, then you will get the wrong result more often than desired.

If you over-train, then the neural network will not be able to "think outside the sqaure", so to speak.

So how do you propagate backwards ??

Step 1: Feed-forward pass through:

Send some data through the network to populate all the variables. This feed-forward pass allows you calculate your actualOUTPUTs, which you will use to compare against your expectedOUTPUTs.

Step 2: Calculate delta-error for the neurons in the last layer (output layer).

The delta-error calculation for the neuron(s) in the last layer of the neural network is a liitle bit different than the other layers. You can work this out once you calculate the actualOUTPUTs from the feedforward pass.

Let Last Layer Neuron1.deltaError = LLN1.dE

Last Layer.actualOutput1 = aO1 <--- This is the same as the Neuron1 Output Value

Last Layer.expectedOutput1 = exO1

LLN1.dE = (aO1) x (1-aO1) x (exO1 - aO1);

Once you have calculated the deltaError for every neuron in the last layer (output layer), you can move onto the next step.

Step 3: Calculate the delta-error for the hidden layer neurons

The hidden layers for this neural network, is any layer in the neural network, that is not the last layer. However, each layer should sit like ducks in a row. And we are now going to calculate the delta error for the second last layer in the neural network. This could in theory be the first layer in the network (if this network only had 2 layers).

HLN = Hidden Layer Neuron,

LLN = Last Layer Neuron,

aO=actualOUTPUT,

dE=deltaError

HLN.dE = (HLN.aO) x (1-HLN.aO) x (Sum of [LLN.dE x LLN to HLN connection weight])

Keep moving back through the network layers until you reach the 1st layer (ie, you run out of layers).

Step 4: Update the weights of the connections and Bias of neuron.

a) Multiply the neuron's deltaError which was calculated in either step 2 or 3, by the learning rate (0.1), and by the connection's connEntry value.

b) Then add this calculated value (in Step (4a)) to the current weight of the connection.

neuron.connections[i].weight += (learningRate * neuron.connections[i].connEntry * neuron.deltaError);

The bias is like a connection with a constant connEntry of 1, therefore the calculation is

neuron.bias += (learningRate * 1 * neuron.deltaError);

Up Next: Neural Network (Part 6):

Back Propagation - a fully worked example.

To go back to the table of contents click here

Description: Neural Network (Part 5): The Back Propagation process Rating: 3.5 Reviewer: Unknown ItemReviewed: Neural Network (Part 5): The Back Propagation process