/// <summary> /// Linear regression simulation for homework Q5-Q6 of the /// 2nd week of the CS1156x "Learning From Data" at eDX /// </summary> static void RunQ5Q6Simulation() { const int EXPERIMENT_COUNT = 1000, N = 100; Random rnd = new Random(); double avgEin = 0, avgEout = 0; for (int i = 1; i <= EXPERIMENT_COUNT; i++) { //pick a random line y = a1 * x + b1 double x1 = rnd.NextDouble(), y1 = rnd.NextDouble(), x2 = rnd.NextDouble(), y2 = rnd.NextDouble(); double a = (y1 - y2) / (x1 - x2), b = y1 - a * x1; Func<double, double, int> f = (x, y) => a * x + b >= y ? 1 : -1; //generate training set of N random points var X = new DenseMatrix(N, 3); var Y = new DenseVector(N); for (int j = 0; j < N; j++) { X[j, 0] = 1; X[j, 1] = rnd.NextDouble() * 2 - 1; X[j, 2] = rnd.NextDouble() * 2 - 1; Y[j] = f(X[j, 1], X[j, 2]); } var W = X.QR().Solve(DenseMatrix.Identity(X.RowCount)).Multiply(Y); Func<double, double, int> h = (x, y) => W[0] + W[1] * x + W[2] * y >= 0 ? 1 : -1; //find Ein int count = 0; for (int j = 0; j < N; j++) if (h(X[j, 1], X[j, 2]) != Y[j]) count++; avgEin += (count + 0.0) / N; //find p: f != g const int P_SAMPLE_COUNT = 1000; count = 0; for (int j = 1; j <= P_SAMPLE_COUNT; j++) { double xx = rnd.NextDouble() * 2 - 1; double yy = rnd. NextDouble() * 2 - 1; if (f(xx, yy) != h(xx, yy)) count++; } avgEout += (count + 0.0) / P_SAMPLE_COUNT; } Console.Out.WriteLine("HW2 Q5:"); Console.Out.WriteLine("\tEin = {0}", avgEin / EXPERIMENT_COUNT); Console.Out.WriteLine("HW2 Q6:"); Console.Out.WriteLine("\tEout = {0}", avgEout / EXPERIMENT_COUNT); }
/// <summary> /// Run example /// </summary> public void Run() { // Format matrix output to console var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone(); formatProvider.TextInfo.ListSeparator = " "; // Solve next system of linear equations (Ax=b): // 5*x + 2*y - 4*z = -7 // 3*x - 7*y + 6*z = 38 // 4*x + 1*y + 5*z = 43 // Create matrix "A" with coefficients var matrixA = new DenseMatrix(new[,] { { 5.00, 2.00, -4.00 }, { 3.00, -7.00, 6.00 }, { 4.00, 1.00, 5.00 } }); Console.WriteLine(@"Matrix 'A' with coefficients"); Console.WriteLine(matrixA.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // Create vector "b" with the constant terms. var vectorB = new DenseVector(new[] { -7.0, 38.0, 43.0 }); Console.WriteLine(@"Vector 'b' with the constant terms"); Console.WriteLine(vectorB.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 1. Solve linear equations using LU decomposition var resultX = matrixA.LU().Solve(vectorB); Console.WriteLine(@"1. Solution using LU decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 2. Solve linear equations using QR decomposition resultX = matrixA.QR().Solve(vectorB); Console.WriteLine(@"2. Solution using QR decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 3. Solve linear equations using SVD decomposition matrixA.Svd(true).Solve(vectorB, resultX); Console.WriteLine(@"3. Solution using SVD decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 4. Solve linear equations using Gram-Shmidt decomposition matrixA.GramSchmidt().Solve(vectorB, resultX); Console.WriteLine(@"4. Solution using Gram-Shmidt decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 5. Verify result. Multiply coefficient matrix "A" by result vector "x" var reconstructVecorB = matrixA * resultX; Console.WriteLine(@"5. Multiply coefficient matrix 'A' by result vector 'x'"); Console.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // To use Cholesky or Eigenvalue decomposition coefficient matrix must be // symmetric (for Evd and Cholesky) and positive definite (for Cholesky) // Multipy matrix "A" by its transpose - the result will be symmetric and positive definite matrix var newMatrixA = matrixA.TransposeAndMultiply(matrixA); Console.WriteLine(@"Symmetric positive definite matrix"); Console.WriteLine(newMatrixA.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 6. Solve linear equations using Cholesky decomposition newMatrixA.Cholesky().Solve(vectorB, resultX); Console.WriteLine(@"6. Solution using Cholesky decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 7. Solve linear equations using eigen value decomposition newMatrixA.Evd().Solve(vectorB, resultX); Console.WriteLine(@"7. Solution using eigen value decomposition"); Console.WriteLine(resultX.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 8. Verify result. Multiply new coefficient matrix "A" by result vector "x" reconstructVecorB = newMatrixA * resultX; Console.WriteLine(@"8. Multiply new coefficient matrix 'A' by result vector 'x'"); Console.WriteLine(reconstructVecorB.ToString("#0.00\t", formatProvider)); Console.WriteLine(); }
/// <summary> /// Linear regressionsimulation with non-separable target function /// for homework Q8 of the 2nd week of the CS1156x "Learning From Data" at eDX /// </summary> static void RunQ8Simulation() { const int EXPERIMENT_COUNT = 1000, N = 100; Random rnd = new Random(); double avgEin = 0; for (int i = 1; i <= EXPERIMENT_COUNT; i++) { Func<double, double, int> f = (x1, x2) => x1 * x1 + x2 * x2 - 0.6 >= 0 ? 1 : -1; //generate training set of N random points var X = new DenseMatrix(N, 3); var Y = new DenseVector(N); for (int j = 0; j < N; j++) { X[j, 0] = 1; X[j, 1] = rnd.NextDouble() * 2 - 1; X[j, 2] = rnd.NextDouble() * 2 - 1; Y[j] = f(X[j, 1], X[j, 2]); //not exactly how it was defined in the problem statement, but shall be good enough if (rnd.NextDouble() < 0.1) Y[j] = -Y[j]; } var W = X.QR().Solve(DenseMatrix.Identity(X.RowCount)).Multiply(Y); Func<double, double, int> h = (x, y) => W[0] + W[1] * x + W[2] * y >= 0 ? 1 : -1; //find Ein int count = 0; for (int j = 0; j < N; j++) if (h(X[j, 1], X[j, 2]) != Y[j]) count++; avgEin += (count + 0.0) / N; } Console.Out.WriteLine("HW2 Q8:"); Console.Out.WriteLine("\tEin = {0}", avgEin / EXPERIMENT_COUNT); }
/// <summary> /// Non-linear-transformed linear regression simulation for homework Q9, Q10 of the /// 2nd week of the CS1156x "Learning From Data" at eDX /// </summary> static void RunQ9Q10Simulation() { const int EXPERIMENT_COUNT = 1000, N = 100; Random rnd = new Random(); double avgEout = 0; for (int i = 1; i <= EXPERIMENT_COUNT; i++) { Func<double, double, int> f = (x1, x2) => x1 * x1 + x2 * x2 - 0.6 >= 0 ? 1 : -1; //generate training set of N random points var X = new DenseMatrix(N, 3); var Y = new DenseVector(N); for (int j = 0; j < N; j++) { X[j, 0] = 1; X[j, 1] = rnd.NextDouble() * 2 - 1; X[j, 2] = rnd.NextDouble() * 2 - 1; Y[j] = f(X[j, 1], X[j, 2]); // Just flipping each Y with a 10% chance - // not exactly how it was defined in the problem statement, but shall be good enough if (rnd.NextDouble() < 0.1) Y[j] = -Y[j]; } var XX = new DenseMatrix(N, 6); for (int j = 0; j < N; j++) { XX[j, 0] = 1; XX[j, 1] = X[j, 1]; XX[j, 2] = X[j, 2]; XX[j, 3] = X[j, 1] * X[j, 2]; XX[j, 4] = X[j, 1] * X[j, 1]; XX[j, 5] = X[j, 2] * X[j, 2]; } var W = XX.QR().Solve(DenseMatrix.Identity(XX.RowCount)).Multiply(Y); Func<double, double, int> h = (x, y) => W[0] + W[1] * x + W[2] * y + W[3] * x * y + W[4] * x * x + W[5] * y * y >= 0 ? 1 : -1; //find p: f != g const int P_SAMPLE_COUNT = 1000; int count = 0; for (int j = 1; j <= P_SAMPLE_COUNT; j++) { double xx = rnd.NextDouble() * 2 - 1; double yy = rnd.NextDouble() * 2 - 1; int ff = f(xx, yy); if (rnd.NextDouble() < 0.1) ff = -ff; if (ff != h(xx, yy)) count++; } avgEout += (count + 0.0) / P_SAMPLE_COUNT; } Console.Out.WriteLine("HW2 Q10:"); Console.Out.WriteLine("\tEout = {0}", avgEout / EXPERIMENT_COUNT); }
/// <summary> /// Run example /// </summary> /// <seealso cref="http://en.wikipedia.org/wiki/QR_decomposition">QR decomposition</seealso> public void Run() { // Format matrix output to console var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone(); formatProvider.TextInfo.ListSeparator = " "; // Create 3 x 2 matrix var matrix = new DenseMatrix(new[,] { { 1.0, 2.0 }, { 3.0, 4.0 }, { 5.0, 6.0 } }); Console.WriteLine(@"Initial 3x2 matrix"); Console.WriteLine(matrix.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // Perform QR decomposition (Householder transformations) var qr = matrix.QR(); Console.WriteLine(@"QR decomposition (Householder transformations)"); // 1. Orthogonal Q matrix Console.WriteLine(@"1. Orthogonal Q matrix"); Console.WriteLine(qr.Q.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 2. Multiply Q matrix by its transpose gives identity matrix Console.WriteLine(@"2. Multiply Q matrix by its transpose gives identity matrix"); Console.WriteLine(qr.Q.TransposeAndMultiply(qr.Q).ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 3. Upper triangular factor R Console.WriteLine(@"3. Upper triangular factor R"); Console.WriteLine(qr.R.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 4. Reconstruct initial matrix: A = Q * R var reconstruct = qr.Q * qr.R; Console.WriteLine(@"4. Reconstruct initial matrix: A = Q*R"); Console.WriteLine(reconstruct.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // Perform QR decomposition (Gram–Schmidt process) var gramSchmidt = matrix.GramSchmidt(); Console.WriteLine(@"QR decomposition (Gram–Schmidt process)"); // 5. Orthogonal Q matrix Console.WriteLine(@"5. Orthogonal Q matrix"); Console.WriteLine(gramSchmidt.Q.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 6. Multiply Q matrix by its transpose gives identity matrix Console.WriteLine(@"6. Multiply Q matrix by its transpose gives identity matrix"); Console.WriteLine((gramSchmidt.Q.Transpose() * gramSchmidt.Q).ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 7. Upper triangular factor R Console.WriteLine(@"7. Upper triangular factor R"); Console.WriteLine(gramSchmidt.R.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 8. Reconstruct initial matrix: A = Q * R reconstruct = gramSchmidt.Q * gramSchmidt.R; Console.WriteLine(@"8. Reconstruct initial matrix: A = Q*R"); Console.WriteLine(reconstruct.ToString("#0.00\t", formatProvider)); Console.WriteLine(); }
/// <summary> /// Linear regression/Perceptron simulation for homework Q7 of the /// 2nd week of the CS1156x "Learning From Data" at eDX /// </summary> static void RunQ7Simulation() { const int EXPERIMENT_COUNT = 1000, N = 10; Random rnd = new Random(); double avgK = 0; for (int i = 1; i <= EXPERIMENT_COUNT; i++) { //pick a random line y = a1 * x + b1 double x1 = rnd.NextDouble(), y1 = rnd.NextDouble(), x2 = rnd.NextDouble(), y2 = rnd.NextDouble(); double a = (y1 - y2) / (x1 - x2), b = y1 - a * x1; Func<double, double, int> f = (x, y) => a * x + b >= y ? 1 : -1; //generate training set of N random points var X = new DenseMatrix(N, 3); var Y = new DenseVector(N); for (int j = 0; j < N; j++) { X[j, 0] = 1; X[j, 1] = rnd.NextDouble() * 2 - 1; X[j, 2] = rnd.NextDouble() * 2 - 1; Y[j] = f(X[j, 1], X[j, 2]); } var W = X.QR().Solve(DenseMatrix.Identity(X.RowCount)).Multiply(Y); double w0 = W[0], w1 = W[1], w2 = W[2]; Func<double, double, int> h = (x, y) => w0 + w1 * x + w2 * y >= 0 ? 1 : -1; //run Perceptron int k = 1; while (Enumerable.Range(0, N).Any(j => f(X[j, 1], X[j, 2]) != h(X[j, 1], X[j, 2]))) { //find all misclasified points int[] M = Enumerable.Range(0, N).Where(j => f(X[j, 1], X[j, 2]) != h(X[j, 1], X[j, 2])).ToArray(); int m = M[rnd.Next(0, M.Length)]; int sign = f(X[m, 1], X[m, 2]); w0 += sign; w1 += sign * X[m, 1]; w2 += sign * X[m, 2]; k++; } avgK += k; } Console.Out.WriteLine("HW2 Q7:"); Console.Out.WriteLine("\tK = {0}", avgK / EXPERIMENT_COUNT); }
/// <summary> /// Run example /// </summary> /// <seealso cref="http://en.wikipedia.org/wiki/Transpose">Transpose</seealso> /// <seealso cref="http://en.wikipedia.org/wiki/Invertible_matrix">Invertible matrix</seealso> public void Run() { // Format matrix output to console var formatProvider = (CultureInfo)CultureInfo.InvariantCulture.Clone(); formatProvider.TextInfo.ListSeparator = " "; // Create random square matrix var matrix = new DenseMatrix(5); var rnd = new Random(1); for (var i = 0; i < matrix.RowCount; i++) { for (var j = 0; j < matrix.ColumnCount; j++) { matrix[i, j] = rnd.NextDouble(); } } Console.WriteLine(@"Initial matrix"); Console.WriteLine(matrix.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 1. Get matrix inverse var inverse = matrix.Inverse(); Console.WriteLine(@"1. Matrix inverse"); Console.WriteLine(inverse.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 2. Matrix multiplied by its inverse gives identity matrix var identity = matrix * inverse; Console.WriteLine(@"2. Matrix multiplied by its inverse"); Console.WriteLine(identity.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 3. Get matrix transpose var transpose = matrix.Transpose(); Console.WriteLine(@"3. Matrix transpose"); Console.WriteLine(transpose.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 4. Get orthogonal matrix, i.e. do QR decomposition and get matrix Q var orthogonal = matrix.QR().Q; Console.WriteLine(@"4. Orthogonal matrix"); Console.WriteLine(orthogonal.ToString("#0.00\t", formatProvider)); Console.WriteLine(); // 5. Transpose and multiply orthogonal matrix by iteslf gives identity matrix identity = orthogonal.TransposeAndMultiply(orthogonal); Console.WriteLine(@"Transpose and multiply orthogonal matrix by iteslf"); Console.WriteLine(identity.ToString("#0.00\t", formatProvider)); Console.WriteLine(); }
public static int solve(double[,] A, double[] fitz, CenterArrayNode CenterNode, IRBFPolynomial Poly, bool MathNet) { var matrixA = new DenseMatrix(A); var vectorB = new DenseVector(fitz); //Vector<double> resultX = matrixA.LU().Solve(vectorB); Vector<double> resultX = matrixA.QR().Solve(vectorB); //matrixA.GramSchmidt().Solve(vectorB, resultX); List<double> w2 = new List<double>(resultX.ToArray()); int i = 0; w2.ForEach((double weight) => { if (i < CenterNode.Centers.Count) CenterNode[i].w = weight; // set the center's weight else Poly[i - CenterNode.Centers.Count] = weight;//store the polynomial coefficients ++i; }); return 0; }
/// <summary> /// Train. Single iteration. /// </summary> public void Iteration() { int rowCount = _trainingData.Count; int inputColCount = _trainingData[0].Input.Length; Matrix<double> xMatrix = new DenseMatrix(rowCount, inputColCount + 1); Matrix<double> yMatrix = new DenseMatrix(rowCount, 1); for (int row = 0; row < _trainingData.Count; row++) { BasicData dataRow = _trainingData[row]; int colSize = dataRow.Input.Count(); xMatrix[row, 0] = 1; for (int col = 0; col < colSize; col++) { xMatrix[row, col + 1] = dataRow.Input[col]; } yMatrix[row, 0] = dataRow.Ideal[0]; } // Calculate the least squares solution QR qr = xMatrix.QR(); Matrix<double> beta = qr.Solve(yMatrix); double sum = 0.0; for (int i = 0; i < inputColCount; i++) sum += yMatrix[i, 0]; double mean = sum/inputColCount; for (int i = 0; i < inputColCount; i++) { double dev = yMatrix[i, 0] - mean; _sst += dev*dev; } Matrix<double> residuals = xMatrix.Multiply(beta).Subtract(yMatrix); _sse = residuals.L2Norm()*residuals.L2Norm(); for (int i = 0; i < _algorithm.LongTermMemory.Length; i++) { _algorithm.LongTermMemory[i] = beta[i, 0]; } // calculate error _errorCalculation.Clear(); foreach (BasicData dataRow in _trainingData) { double[] output = _algorithm.ComputeRegression(dataRow.Input); _errorCalculation.UpdateError(output, dataRow.Ideal, 1.0); } _error = _errorCalculation.Calculate(); }
private double[] Polyfit(double[] x, double[] y, int degree) { // Vandermonde matrix var v = new DenseMatrix(x.Length, degree + 1); for (int i = 0; i < v.RowCount; i++) for (int j = 0; j <= degree; j++) v[i, j] = Math.Pow(x[i], j); var yv = new DenseVector(y).ToColumnMatrix(); QR<double> qr = v.QR(); // Math.Net doesn't have an "economy" QR, so: // cut R short to square upper triangle, then recompute Q var r = qr.R.SubMatrix(0, degree + 1, 0, degree + 1); var q = v.Multiply(r.Inverse()); var p = r.Inverse().Multiply(q.TransposeThisAndMultiply(yv)); return p.Column(0).ToArray(); }
public CurveFit(List<CurvePoint> data) { var x = new DenseMatrix(data.Count, data.Count); for(int i = 0; i < data.Count; i++) { for (int j = 0; j < data.Count; j++) { x[i, j] = Pow(data[i].x, data.Count - 1 - j); } } var y = new DenseVector(data.Select(p => (double)p.y).ToArray()); var coefficents = x.QR().Solve(y); Coefficents = coefficents.Select(c => Round(c)).ToList(); }