The Cardioid


Problem 1
Create a New Project called CardioidAuto to build an appropriate training set for learning of the cardioid r = 1 + cos(θ) using a ANN for auto-association. Use 2000 training cases and 64 inputs. The input training set must include noisy cardioids and the target must include clean cardioids as shown in the figure. The training set must include the same phase in eight different training cases.

AutoAssociation

Solution 1
After editing the fileRunRun click the button to execute the code.If you do not have any errors, the training set will be generated and displayed on the variable list and the file list. Click on the file to see its contents. The training set input is a noisy sine signal; the training set target is a clean sine signal of the same phase as the input.

CardioidAuto\BuildTrainSet.lab
int numCases = 2000;
int numInputs = 64;
double deltaInput = 6.28/numInputs;
double delta = 6.28/(numCases-1);
int i;
int j;
double phase = 0.0;
//____________________ trainSetTarget
Matrix trainSetTarget ;
trainSetTarget .Create(numCases, numInputs);
for(i=0; i<numCases; i++)
{
     phase = i*delta;
     for (j=0; j<numInputs; j++)
     {
          trainSetTarget [i][j] = (1.0+cos(j*deltaInput + phase));
     }
}
trainSetTarget .Save();
//____________________ trainSetInput
Matrix noise;
noise.CreateRandom(numCases, numInputs, 0.0, 2.0);
Matrix trainSetInput= 0.9*trainSetTarget + 0.1*noise;
trainSetInput.Save();


TrainSetInput

TrainSetTarget

Problem 2
Edit the BuilValidSet.lab file to build the validation set using the training set. Contaminate the target training set with 10% of noise to create the input validation set. The target of the validation set is the same as the target of the training set.

Solution 2
After editing the file, RunRun click the button to execute the code. If you do not have any errors, the validation set will be generated and displayed on the variable list and the file list. Use the scroll bar at the right to visualize each validation case. Each case should be a noisy sine wave at a different phase. Observe that in this specific case the validation set input and target are the same.

CardioidAuto\BuildValidSet.lab
//_______________________ Load the training set
Matrix trainSetTarget;
trainSetTarget.Load();

int numRows = trainSetTarget.GetRowCount();
int numCols= trainSetTarget.GetColCount();

Matrix noise;
noise.CreateRandom(numRows, numCols, 0.0, 2.0);

Matrix validSetInput = 0.9*trainSetTarget + 0.1*noise;
validSetInput.Save();

Problem 3
Edit the Train.lab file to design and train an ANN for the cardioid. Use one hidden layer. Train the ANN using simulated annealing and the conjugate gradient method. Because of the number of outputs, it is not possible to use the method of Levenberg Marquardt.

Solution 3
After editing the file, RunRun click the button to execute the code. You will be asked to save the file, set the file name to Train.lab. If you do not have any errors, the ANN will be trained when the code execution stops. Double click the network in the variable list to see the network. Then, select the trainSetInput.csv file for input, and the trainSetTarget.csv file for target. You will be able to review the network behavior in real time by moving the scrollbar at the right.

CardioidAuto\Train.lab
int numInputs = 64;
int i;
//_________________________ Network Setup
LayerNet net;
net.Create(numInputs, 13, 0, numInputs);
for(i = 0; i<numInputs; i++)
{
     net.SetInScaler(i, 0, 2); // Input values are from 0 to 2
     net.SetOutScaler(i, 0, 2); // Output values are from 0 to 2
}

//________________________ Load and set the training set
Matrix trainSetInput;
trainSetInput.Load();
Matrix trainSetTarget;
trainSetTarget.Load();
net.SetTrainSet(trainSetInput, trainSetTarget, false);

//________________________ Train
net.TrainSimAnneal(100, 100, 15, 0.01, false, 4, 1.0e-12);
net.TrainConjGrad(2000,1.0e-12);

//_____________________________ Save the trained network
net.Save();

NetSimulation

Problem 4
Edit the CheckTraining.lab file to check the training: (a) Compute the mean squared error for the ANN using the training set. (b) Plot the error for each training case. (c) Save the plot as a vector image (checkTraining.pdf and checkTraining.emf)

Solution 4 (a)
After editing the file, RunRun click the button to execute the code. If you do not have any errors, the mse will be displayed in the variable list. Click the variable called output to review the network behavior. If you network was trained properly, you should be able to see that the output of the network is a perfect sine wave for all of training cases (use the scroll bar at the right to verify the output for other training cases).

CardioidAuto\CheckTraining.lab
//_________________________________________ Load the training set
Matrix trainSetInput;
trainSetInput.Load();
Matrix trainSetTarget;
trainSetTarget.Load();
//_________________________________________ Load the ANN
LayerNet net;
net.Load();
//_________________________________________ Run
Matrix output = net.Run(trainSetInput);
double mse = ComputeMse(output, trainSetTarget);
//_________________________________________ Relative Error
Vector error = ComputeRelError(output, trainSetTarget);
XyChart checkTraining;
checkTraining.SetCaption("case", false, "Relative Error", false);
checkTraining.AddGraphY(error, "Relative Error", 2, 1, 0, 255, 0);
checkTraining.SetLogScale(false, true);
checkTraining.SetLimits(0, trainSetTarget.GetRowCount(), 1.0e-5, 1.0);
checkTraining.SetColorMode(2);
checkTraining.SaveAndShow();
checkTraining.SavePDF(0.0);
checkTraining.SaveEMF();

CheckTraining1

CheckTraining2

Problem 5
Edit the Validation.lab file to perform the validation of the ANN. (a) Compute the mean squared error for the ANN using the validation set. (b) Plot the error for each validation case. (c) Save the plot as a vector image (validation.pdf and validation.emf)

Solution 5 (a)
After editing the file, RunRun click the button to execute the code. If you do not have any errors, the mse will be displayed in the variable list. As the value of the mse for both the training set and the validation is somehow similar, the network was properly designed and trained. Click the variable called output to review the network behavior, observe how the network is able to considerably eliminate the noise of the input signal for each training case.

CardioidAuto\Validation.lab
//_________________________________________ Load the validation set
Matrix validSetInput;
validSetInput.Load();
Matrix trainSetTarget;
trainSetTarget.Load();
//_________________________________________ Load the ANN
LayerNet net;
net.Load();
//_________________________________________ Run
Matrix output = net.Run(validSetInput);
//_________________________________________ Compute the mean squared error
double mse = ComputeMse(output, trainSetTarget);
//_________________________________________ Relative Error
Vector error = ComputeRelError(output, trainSetTarget);
XyChart validation;
validation.SetCaption("case", false, "Relative Error", false);
validation.AddGraphY(error, "Relative Error", 2, 1, 0, 255, 0);
validation.SetLogScale(false, true);
validation.SetLimits(0, trainSetTarget.GetRowCount(), 1.0e-5, 1.0);
validation.SetColorMode(2);
validation.SaveAndShow();
validation.SavePDF(0.0);
validation.SaveEMF();

Validation1

Validation2

Problem 6
Generate a report in Microsoft Word. Write some conclusions in the report focusing on the problems that were faced during the simulation and how these problems were or could be solved.

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