public MinimizationResult Train()
{
Vector <double> theta = Vector <double> .Build.Dense(X.ColumnCount);
LinearRegression lr = new LinearRegression(this.X, this.y, this.Lambda);
var obj = ObjectiveFunction.Gradient(lr.Cost, lr.Gradient);
var solver = new BfgsMinimizer(1e-5, 1e-5, 1e-5, 200);
MinimizationResult result = solver.FindMinimum(obj, theta);
return(result);
}
private static (Vector <double> error_train, Vector <double> error_val) LearningCurve(
Matrix <double> X, Vector <double> y, Matrix <double> xVal, Vector <double> yVal, double lambda)
{
int m = X.RowCount;
Vector <double> theta = Vector <double> .Build.Dense(X.ColumnCount, 0);
Vector <double> error_train = Vector <double> .Build.Dense(m, 0);
Vector <double> error_val = Vector <double> .Build.Dense(m, 0);
LinearRegression lr = new LinearRegression();
MinimizationResult result;
for (int i = 0; i < m; i++)
{
var xset = X.SubMatrix(0, i + 1, 0, X.ColumnCount);
var yset = y.SubVector(0, i + 1);
lr.X = xset;
lr.y = yset;
lr.Lambda = lambda;
result = lr.Train();
System.Console.WriteLine("Iteration {0,5} | Cost: {1:e}", result.Iterations, result.FunctionInfoAtMinimum.Value);
lr.Lambda = 0;
error_train[i] = lr.Cost(result.MinimizingPoint);
lr.X = xVal;
lr.y = yVal;
lr.Lambda = 0;
error_val[i] = lr.Cost(result.MinimizingPoint);
}
System.Console.WriteLine();
return(error_train, error_val);
}
private static (Vector <double> Lamda_vec, Vector <double> error_train, Vector <double> error_val) ValidationCurve(
Matrix <double> X, Vector <double> y, Matrix <double> xVal, Vector <double> yVal)
{
Vector <double> lamda_vec = Vector <double> .Build.DenseOfArray(new [] {
0.0, 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1, 3, 10
});
int m = lamda_vec.Count;
Vector <double> theta = Vector <double> .Build.Dense(X.ColumnCount, 0);
Vector <double> error_train = Vector <double> .Build.Dense(m, 0);
Vector <double> error_val = Vector <double> .Build.Dense(m, 0);
LinearRegression lr = new LinearRegression(X, y, 0);
MinimizationResult result;
for (int i = 0; i < m; i++)
{
lr.Lambda = lamda_vec[i];
lr.X = X;
lr.y = y;
result = lr.Train();
System.Console.WriteLine("Iteration {0,5} | Cost: {1:e}", result.Iterations, result.FunctionInfoAtMinimum.Value);
lr.Lambda = 0;
error_train[i] = lr.Cost(result.MinimizingPoint);
lr.X = xVal;
lr.y = yVal;
lr.Lambda = 0;
error_val[i] = lr.Cost(result.MinimizingPoint);
}
return(lamda_vec, error_train, error_val);
}
static void Main(string[] args)
{
if (!System.Console.IsOutputRedirected)
{
System.Console.Clear();
}
CultureInfo.CurrentCulture = CultureInfo.CreateSpecificCulture("en-US");
System.Console.WriteLine("Regularized Linear Regression and Bias v.s. Variance ex.5");
System.Console.WriteLine("=========================================================\n");
var M = Matrix <double> .Build;
var V = Vector <double> .Build;
// =========== Part 1: Loading and Visualizing Data =============
// We start the exercise by first loading and visualizing the dataset.
// The following code will load the dataset into your environment and plot
// the data.
// Load Training Data
System.Console.WriteLine("Loading and Visualizing Data ...\n");
// Load from ex5data1:
// You will have X, y, Xval, yval, Xtest, ytest in your environment
Dictionary <string, Matrix <double> > ms = MatlabReader.ReadAll <double>("data\\ex5data1.mat");
Matrix <double> X = ms["X"];
Vector <double> y = ms["y"].Column(0);
Matrix <double> Xval = ms["Xval"];
Vector <double> yval = ms["yval"].Column(0);
Matrix <double> Xtest = ms["Xtest"];
Vector <double> ytest = ms["ytest"].Column(0);
// m = Number of examples
int m = X.RowCount;
GnuPlot.HoldOn();
PlotData(X.Column(0).ToArray(), y.ToArray());
Pause();
// =========== Part 2: Regularized Linear Regression Cost =============
// You should now implement the cost function for regularized linear
// regression.
Vector <double> theta = V.Dense(2, 1.0);
LinearRegression lr = new LinearRegression(X.InsertColumn(0, V.Dense(m, 1)), y, 1);
double J = lr.Cost(theta);
Vector <double> grad = lr.Gradient(theta);
System.Console.WriteLine("Cost at theta = [1 ; 1]: {0:f6} \n(this value should be about 303.993192)\n", J);
Pause();
// =========== Part 3: Regularized Linear Regression Gradient =============
// You should now implement the gradient for regularized linear
// regression.
System.Console.WriteLine("Gradient at theta = [1 ; 1]: [{0:f6}; {1:f6}] \n(this value should be about [-15.303016; 598.250744])\n", grad[0], grad[1]);
Pause();
// =========== Part 4: Train Linear Regression =============
// Once you have implemented the cost and gradient correctly, the
// trainLinearReg function will use your cost function to train
// regularized linear regression.
//
// Write Up Note: The data is non-linear, so this will not give a great
// fit.
//
// Train linear regression with lambda = 0
lr = new LinearRegression(X.InsertColumn(0, V.Dense(m, 1)), y, 0);
var result = lr.Train();
Vector <double> h = X.InsertColumn(0, V.Dense(m, 1)) * result.MinimizingPoint; // hypothesys
PlotLinearFit(X.Column(0).ToArray(), h.ToArray());
GnuPlot.HoldOff();
Pause();
// =========== Part 5: Learning Curve for Linear Regression =============
// Next, you should implement the learningCurve function.
//
// Write Up Note: Since the model is underfitting the data, we expect to
// see a graph with "high bias" -- Figure 3 in ex5.pdf
//
(Vector <double> error_train, Vector <double> error_val)res;
res = LearningCurve(X.InsertColumn(0, V.Dense(m, 1)), y, Xval.InsertColumn(0, V.Dense(Xval.RowCount, 1)), yval, 0);
PlotLinearLearningCurve(
Generate.LinearRange(1, 1, m),
res.error_train.ToArray(),
res.error_val.ToArray()
);
System.Console.WriteLine("# Training Examples\tTrain Error\tCross Validation Error\n");
for (int i = 0; i < m; i++)
{
System.Console.WriteLine("\t{0,2}\t\t{1:f6}\t{2:f6}", i, res.error_train[i], res.error_val[i]);
}
System.Console.WriteLine();
Pause();
// =========== Part 6: Feature Mapping for Polynomial Regression =============
// One solution to this is to use polynomial regression. You should now
// complete polyFeatures to map each example into its powers
//
int p = 8;
// Map X onto Polynomial Features and Normalize
Matrix <double> X_poly = MapPolyFeatures(X, p);
// normalize
var norm = FeatureNormalize(X_poly);
X_poly = norm.X_norm;
// add one's
X_poly = X_poly.InsertColumn(0, V.Dense(X_poly.RowCount, 1));
// Map X_poly_test and normalize (using mu and sigma)
Matrix <double> X_poly_test = MapPolyFeatures(Xtest, p);
for (int i = 0; i < X_poly_test.ColumnCount; i++)
{
Vector <double> v = X_poly_test.Column(i);
v = v - norm.mu[0, i];
v = v / norm.sigma[0, i];
X_poly_test.SetColumn(i, v);
}
// add one's
X_poly_test = X_poly_test.InsertColumn(0, V.Dense(X_poly_test.RowCount, 1));
// Map X_poly_val and normalize (using mu and sigma)
Matrix <double> X_poly_val = MapPolyFeatures(Xval, p);
for (int i = 0; i < X_poly_val.ColumnCount; i++)
{
Vector <double> v = X_poly_val.Column(i);
v = v - norm.mu[0, i];
v = v / norm.sigma[0, i];
X_poly_val.SetColumn(i, v);
}
// add one's
X_poly_val = X_poly_val.InsertColumn(0, V.Dense(X_poly_val.RowCount, 1));
System.Console.WriteLine("Normalized Training Example 1:\n");
System.Console.WriteLine(X_poly.Row(0));
Pause();
// =========== Part 7: Learning Curve for Polynomial Regression =============
// Now, you will get to experiment with polynomial regression with multiple
// values of lambda. The code below runs polynomial regression with
// lambda = 0. You should try running the code with different values of
// lambda to see how the fit and learning curve change.
//
double lambda = 0;
lr = new LinearRegression(X_poly, y, lambda);
var minRes = lr.Train();
GnuPlot.HoldOn();
GnuPlot.Set("terminal wxt 1");
PlotData(X.Column(0).ToArray(), y.ToArray());
PlotFit(X.Column(0).Minimum(), X.Column(0).Maximum(), norm.mu.Row(0), norm.sigma.Row(0), minRes.MinimizingPoint, p, lambda);
GnuPlot.HoldOff();
// learning curve
GnuPlot.Set("terminal wxt 2");
res = LearningCurve(X_poly, y, X_poly_val, yval, lambda);
PlotLinearLearningCurve(
Generate.LinearRange(1, 1, m),
res.error_train.ToArray(),
res.error_val.ToArray()
);
System.Console.WriteLine("# Training Examples\tTrain Error\tCross Validation Error\n");
for (int i = 0; i < m; i++)
{
System.Console.WriteLine("\t{0,2}\t\t{1:f6}\t{2:f6}", i, res.error_train[i], res.error_val[i]);
}
System.Console.WriteLine();
Pause();
// =========== Part 8: Validation for Selecting Lambda =============
// You will now implement validationCurve to test various values of
// lambda on a validation set. You will then use this to select the
// "best" lambda value.
//
var resVal = ValidationCurve(X_poly, y, X_poly_val, yval);
PlotValidationCurve(resVal.Lamda_vec.ToArray(), resVal.error_train.ToArray(), resVal.error_val.ToArray());
System.Console.WriteLine("# Lambda\tTrain Error\tCross Validation Error\n");
for (int i = 0; i < resVal.error_train.Count; i++)
{
System.Console.WriteLine("\t{0:f4}\t\t{1:f6}\t{2:f6}", resVal.Lamda_vec[i], resVal.error_train[i], resVal.error_val[i]);
}
System.Console.WriteLine();
Pause();
// =========== Part 9: Compute test set error =============
// Compute the test error using the best value of λ you
// found
lambda = 3.0;
lr = new LinearRegression(X_poly, y, lambda);
minRes = lr.Train();
theta = minRes.MinimizingPoint;
h = X_poly_test * theta;
m = X_poly.RowCount;
System.Console.WriteLine("Evaluating test-set:\n");
System.Console.WriteLine("# \tHypothesis\tExpected\tError\n");
for (int i = 0; i < m; i++)
{
System.Console.WriteLine("{0,3}\t{1:f6}\t{2:f6}\t{3:f6}", i + 1, h[i], ytest[i], h[i] - ytest[i]);
}
double mae = (h - ytest).L1Norm(); // Mean Absolute Error
double mse = (h - ytest).L2Norm(); // Mean Squared Error
double rmse = Math.Sqrt(mse); // Root Mean Squared Error
System.Console.WriteLine("\nMAE on test set: {0:F6}", mae);
System.Console.WriteLine("MSE on test set: {0:F6}", mse);
System.Console.WriteLine("RMSE on test set: {0:F6}\n", rmse);
Pause();
GnuPlot.HoldOn();
GnuPlot.Set("terminal wxt 3");
PlotData(Xtest.Column(0).ToArray(), ytest.ToArray());
PlotFit(Xtest.Column(0).Minimum(), Xtest.Column(0).Maximum(), norm.mu.Row(0), norm.sigma.Row(0), theta, p, lambda);
GnuPlot.HoldOff();
Pause();
}