public void SolveTest5()
{
double[,] value =
{
{ 2.1, 3.1 },
{ 1.6, 4.2 },
{ 2.1, 5.1 },
};
double[] rhs = { 6.1, 4.3, 2.1 };
double[] expected = { 3.1839, -0.1891 };
LuDecomposition target = new LuDecomposition(value);
bool thrown = false;
try
{
double[] actual = target.Solve(rhs);
}
catch (InvalidOperationException)
{
thrown = true;
}
Assert.IsTrue(thrown);
}
public void InverseTest()
{
double[,] value =
{
{ 2, 3, 0 },
{ -1, 2, 1 },
{ 0, -1, 3 }
};
double[,] expectedInverse =
{
{ 0.3043, -0.3913, 0.1304 },
{ 0.1304, 0.2609, -0.0870 },
{ 0.0435, 0.0870, 0.3043 },
};
var target = new LuDecomposition(value);
double[,] actualInverse = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(expectedInverse, actualInverse, 0.001));
Assert.IsTrue(Matrix.IsEqual(value, target.Reverse()));
var target2 = new JaggedLuDecomposition(value.ToJagged());
actualInverse = target2.Inverse().ToMatrix();
Assert.IsTrue(Matrix.IsEqual(expectedInverse, actualInverse, 0.001));
Assert.IsTrue(Matrix.IsEqual(value, target2.Reverse()));
}
public void SolveTransposeTest()
{
double[,] a =
{
{ 2, 1, 4 },
{ 6, 2, 2 },
{ 0, 1, 6 },
};
double[,] b =
{
{ 1, 0, 7 },
{ 5, 2, 1 },
{ 1, 5, 2 },
};
double[,] expected =
{
{ 0.5062, 0.2813, 0.0875 },
{ 0.1375, 1.1875, -0.0750 },
{ 0.8063, -0.2188, 0.2875 },
};
double[,] actual = new LuDecomposition(b, true).SolveTranspose(a);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.001));
}
public void SolveTest()
{
double[,] value =
{
{ 2, 3, 0 },
{ -1, 2, 1 },
{ 0, -1, 3 }
};
double[,] rhs =
{
{ 1, 2, 3 },
{ 3, 2, 1 },
{ 5, 0, 1 },
};
double[,] expected =
{
{ -0.2174, -0.1739, 0.6522 },
{ 0.4783, 0.7826, 0.5652 },
{ 1.8261, 0.2609, 0.5217 },
};
LuDecomposition target = new LuDecomposition(value);
double[,] actual = target.Solve(rhs);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.001));
}
public void SolveTest3()
{
double[,] value =
{
{ 2.000, 3.000, 0.000 },
{ -1.000, 2.000, 1.000 },
};
LuDecomposition target = new LuDecomposition(value);
double[,] L = target.LowerTriangularFactor;
double[,] U = target.UpperTriangularFactor;
double[,] expectedL =
{
{ 1.000, 0.000 },
{ -0.500, 1.000 },
};
double[,] expectedU =
{
{ 2.000, 3.000, 0.000 },
{ 0.000, 3.500, 1.000 },
};
Assert.IsTrue(Matrix.IsEqual(expectedL, L, 0.001));
Assert.IsTrue(Matrix.IsEqual(expectedU, U, 0.001));
}
public List <double> GetAnswer()
{
var lu = new LuDecomposition(matrix);
vectorB = SwapVector(vectorB, lu.SwappedElements);
var vectorZ = new List <double>(vectorB);
for (int i = 0; i < dim - 1; i++)
{
for (int j = i + 1; j < dim; j++)
{
vectorZ[j] -= lu.L[j, i] * vectorZ[i];
}
}
for (int i = dim - 1; i >= 0; i--)
{
vectorZ[i] /= lu.U[i, i];
; for (int j = i - 1; j >= 0; j--)
{
vectorZ[j] -= lu.U[j, i] * vectorZ[i];
}
}
return(vectorZ);
}
public void SolveTest1()
{
double[,] value =
{
{ 2, 3, 0 },
{ -1, 2, 1 },
{ 0, -1, 3 }
};
double[] rhs = { 5, 0, 1 };
double[] expected =
{
1.6522,
0.5652,
0.5217,
};
var target = new LuDecomposition(value);
double[] actual = target.Solve(rhs);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 1e-3));
Assert.IsTrue(Matrix.IsEqual(value, target.Reverse()));
var target2 = new JaggedLuDecomposition(value.ToJagged());
actual = target2.Solve(rhs);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 1e-3));
Assert.IsTrue(Matrix.IsEqual(value, target2.Reverse()));
}
public void LogDeterminantTest()
{
LuDecomposition lu = new LuDecomposition(CholeskyDecompositionTest.bigmatrix);
Assert.AreEqual(0, lu.Determinant);
Assert.AreEqual(-2224.8931093738875, lu.LogDeterminant, 1e-12);
Assert.IsTrue(lu.Nonsingular);
}
public void DeterminantTest()
{
double[,] value =
{
{ 2, 3, 0 },
{ -1, 2, 1 },
{ 0, -1, 3 }
};
LuDecomposition lu = new LuDecomposition(value);
Assert.AreEqual(23, lu.Determinant);
Assert.IsTrue(lu.Nonsingular);
}
/// <summary>
/// 三维空间中两直线的交点
/// </summary>
/// <returns>如果这两条射线所对应的空间直线能够相交，则返回其交点，如果不能相交，则返回 null。</returns>
/// <remarks>
/// 空间三维直线的参数方程为：
/// x=x0 + m*t;
/// y=y0 + n*t;
/// z=z0 + p*t;
/// 其中{x0,y0,z0} 为直线中的某个点，{m,n,p}为直线的方向向量。</remarks>
private static XYZ GetIntersectPoint(Line3D line1, Line3D line2)
{
// 构造一个六个方程、五个未知数x,y,z,t,k的线性方程组
// 先用前五个方程解出对应的五个未知数
double[][] dataA = new double[][]
{
new double[] { 1, 0, 0, -line1.Direction.X, 0 },
new double[] { 0, 1, 0, -line1.Direction.Y, 0 },
new double[] { 0, 0, 1, -line1.Direction.Z, 0 },
new double[] { 1, 0, 0, 0, -line2.Direction.X },
new double[] { 0, 1, 0, 0, -line2.Direction.Y },
};
double[][] dataB = new double[][]
{
new double[] { line1.Origin.X },
new double[] { line1.Origin.Y },
new double[] { line1.Origin.Z },
new double[] { line2.Origin.X },
new double[] { line2.Origin.Y },
};
Matrix A = new Matrix(dataA);
Matrix b = new Matrix(dataB);
// 通过LU上下三角分解法 求解线性方程组 Ax = b
LuDecomposition d = new LuDecomposition(A);
Matrix x = d.Solve(b);
// 五个未知量 x,y,z,t,k 的值
var col = x.GetColumn(0);
// 将这五个变量代入第六个方程 z = z2 + p2 * k 中，
// 如果此方程左边与右边相等，则表示两条直线有交点，否则两条直线不相交。
var left = col[2];
var right = line2.Origin.Z + line2.Direction.Z * col[4];
// 计算左右两边的相对误差
double relative = Math.Abs((left - right) / Math.Max(left, right));
//
if (relative > 0.0001)
{
return(null);
}
return(relative < 0.0001 ? new XYZ(col[0], col[1], col[2]) : null);
}
public void SolveTest4()
{
double[,] value =
{
{ 2.1, 3.1 },
{ 1.6, 4.2 },
};
double[] rhs = { 6.1, 4.3 };
double[] expected = { 3.1839, -0.1891 };
var target1 = new LuDecomposition(value);
var target2 = new JaggedLuDecomposition(value.ToJagged());
Assert.IsTrue(Matrix.IsEqual(expected, target1.Solve(rhs), 1e-3));
Assert.IsTrue(Matrix.IsEqual(expected, target2.Solve(rhs), 1e-3));
}
public void LogDeterminantTest2()
{
double[,] value =
{
{ 2, 3, 0 },
{ -1, 2, 1 },
{ 0, -1, 3 }
};
LuDecomposition lu = new LuDecomposition(value);
Assert.AreEqual(23, lu.Determinant);
double expected = System.Math.Log(23);
double actual = lu.LogDeterminant;
Assert.AreEqual(expected, actual);
}
public void SolveTest4()
{
double[,] value =
{
{ 2.1, 3.1 },
{ 1.6, 4.2 },
};
double[] rhs = { 6.1, 4.3 };
double[] expected = { 3.1839, -0.1891 };
LuDecomposition target = new LuDecomposition(value);
double[] actual = target.Solve(rhs);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.001));
}
public void LuDecompositionConstructorTest()
{
double[,] value =
{
{ 2, -1, 0 },
{ -1, 2, -1 },
{ 0, -1, 2 }
};
double[,] expectedL =
{
{ 1.0000, 0, 0 },
{ -0.5000, 1.0000, 0 },
{ 0, -0.6667, 1.0000 },
};
double[,] expectedU =
{
{ 2.0000, -1.0000, 0 },
{ 0, 1.5000, -1.0000 },
{ 0, 0, 1.3333 },
};
LuDecomposition target = new LuDecomposition(value);
double[,] actualL = target.LowerTriangularFactor;
double[,] actualU = target.UpperTriangularFactor;
Assert.IsTrue(Matrix.IsEqual(expectedL, actualL, 0.001));
Assert.IsTrue(Matrix.IsEqual(expectedU, actualU, 0.001));
target = new LuDecomposition(value.Transpose(), true);
actualL = target.LowerTriangularFactor;
actualU = target.UpperTriangularFactor;
Assert.IsTrue(Matrix.IsEqual(expectedL, actualL, 0.001));
Assert.IsTrue(Matrix.IsEqual(expectedU, actualU, 0.001));
}
public static void m2()
{
double[][] dataA = new double[][]
{
new double[] { 0.03, 58.9 },
new double[] { 5.31, -6.10 },
};
double[][] dataB = new double[][]
{
new double[] { 59.2 },
new double[] { 47.0 },
};
Matrix A = new Matrix(dataA);
Matrix b = new Matrix(dataB);
// 通过LU上下三角分解法 求解线性方程组 Ax = b
LuDecomposition d = new LuDecomposition(A);
Matrix x = d.Solve(b);
}
public void InverseTestNaN()
{
int n = 5;
var I = Matrix.Identity(n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
double[,] value = Matrix.Magic(n);
value[i, j] = double.NaN;
var target = new LuDecomposition(value);
Assert.IsTrue(Matrix.IsEqual(target.Solve(I), target.Inverse()));
var target2 = new JaggedLuDecomposition(value.ToJagged());
Assert.IsTrue(Matrix.IsEqual(target2.Solve(I.ToJagged()), target2.Inverse()));
}
}
}
public void SolveTransposeTest()
{
double[,] a =
{
{ 2, 1, 4 },
{ 6, 2, 2 },
{ 0, 1, 6 },
};
double[,] b =
{
{ 1, 0, 7 },
{ 5, 2, 1 },
{ 1, 5, 2 },
};
double[,] expected =
{
{ 0.5062, 0.2813, 0.0875 },
{ 0.1375, 1.1875, -0.0750 },
{ 0.8063, -0.2188, 0.2875 },
};
Assert.IsTrue(Matrix.IsEqual(expected, new LuDecomposition(b, true).SolveTranspose(a), 1e-3));
Assert.IsTrue(Matrix.IsEqual(expected, new JaggedLuDecomposition(b.ToJagged(), true).SolveTranspose(a.ToJagged()), 1e-3));
var target = new LuDecomposition(b, true);
var p = target.PivotPermutationVector;
int[] idx = p.ArgSort();
var r = target.LowerTriangularFactor.Dot(target.UpperTriangularFactor)
.Submatrix(idx, null).Transpose();
Assert.IsTrue(Matrix.IsEqual(b, r, 1e-3));
Assert.IsTrue(Matrix.IsEqual(b.Transpose(), target.Reverse(), 1e-3));
Assert.IsTrue(Matrix.IsEqual(b.Transpose(), new JaggedLuDecomposition(b.ToJagged(), true).Reverse(), 1e-3));
}
public void InverseTestNaN()
{
int n = 5;
var I = Matrix.Identity(n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
double[,] value = Matrix.Magic(n);
value[i, j] = double.NaN;
var target = new LuDecomposition(value);
var solution = target.Solve(I);
var inverse = target.Inverse();
Assert.IsTrue(Matrix.IsEqual(solution, inverse));
}
}
}
/// <summary>
/// Finds the coefficients of a polynomial p(x) of degree n that fits the data,
/// p(x(i)) to y(i), in a least squares sense. The result p is a row vector of
/// length n+1 containing the polynomial coefficients in incremental powers.
/// </summary>
/// <param name="x">x axis values</param>
/// <param name="y">y axis values</param>
/// <param name="order">polynomial order including the constant</param>
internal PolyFit(double[] x, double[] y, int order)
{
// incrememnt the order to match matlab way
var matrixX = new double[x.Length, ++order];
var matrixY = new double[x.Length, 1];
if (x.Length != y.Length)
{
throw new ArgumentException("x and y array lengths do not match!");
}
// copy y matrix
for (int i = 0; i < y.Length; i++)
{
matrixY[i, 0] = y[i];
}
// create the X matrix
for (int row = 0; row < x.Length; row++)
{
var nVal = 1.0;
for (int col = 0; col < order; col++)
{
matrixX[row, col] = nVal;
nVal *= x[row];
}
}
var matrixXt = matrixX.Transpose();
var matrixXtX = matrixXt.Product(matrixX);
var matrixXtY = matrixXt.Product(matrixY);
var lu = new LuDecomposition(matrixXtX);
Coeff = lu.Solve(matrixXtY).GetColumn(0).ToArray();
}
public void SolveTest1()
{
double[,] value =
{
{ 2, 3, 0 },
{ -1, 2, 1 },
{ 0, -1, 3 }
};
double[] rhs = { 5, 0, 1 };
double[] expected =
{
1.6522,
0.5652,
0.5217,
};
LuDecomposition target = new LuDecomposition(value);
double[] actual = target.Solve(rhs);
Assert.IsTrue(Matrix.IsEqual(expected, actual, 0.001));
}
public static CubicBezier[] BuildC2Spline(Vector2[] pts, Vector2?m0, Vector2?mn, double tension)
{
if (pts.Length < 2)
{
throw new ArgumentException("There must be at least 2 points to construct cardinal spline", "pts");
}
// compute reasonable starting and ending tangents if not supplied
if (m0 == null)
{
m0 = tension * (pts[1] - pts[0]);
}
if (mn == null)
{
mn = tension * (pts[pts.Length - 1] - pts[pts.Length - 2]);
}
if (pts.Length == 2)
{
return(new CubicBezier[] { CubicBezier.FromCubicHermite(pts[0], m0.Value, pts[1], mn.Value) });
}
int n = pts.Length - 1;
// we have > 2 points, so we will have n-1 beziers
CubicBezier[] beziers = new CubicBezier[n];
// build up constraint matrix
Matrix A = new Matrix(2 * n, 2 * n, 0);
Matrix b = new Matrix(2 * n, 2);
// matrix row index value
int idx = 0;
// add starting/ending tangent constraint
Vector2 p01 = m0.Value / 3.0 + pts[0];
A[idx, 0] = 1;
b[idx, 0] = p01.X; b[idx, 1] = p01.Y;
idx++;
Vector2 pn2 = -mn.Value / 3.0 + pts[n];
A[idx, 2 * n - 1] = 1;
b[idx, 0] = pn2.X; b[idx, 1] = pn2.Y;
idx++;
// add C1 constraints
for (int i = 0; i < n - 1; i++)
{
A[idx, i * 2 + 1] = 1;
A[idx, (i + 1) * 2] = 1;
b[idx, 0] = pts[i + 1].X * 2; b[idx, 1] = pts[i + 1].Y * 2;
idx++;
}
// add C2 constraints
for (int i = 0; i < n - 1; i++)
{
A[idx, i * 2] = 1;
A[idx, i * 2 + 1] = -2;
A[idx, (i + 1) * 2] = 2;
A[idx, (i + 1) * 2 + 1] = -1;
b[idx, 0] = 0; b[idx, 1] = 0;
idx++;
}
// build the LUDecomposition of A to solve system A*P = b;
LuDecomposition lu = new LuDecomposition(A);
Matrix P = lu.Solve(b);
// work back the Parameters
for (int i = 0; i < n; i++)
{
beziers[i] = new CubicBezier(pts[i], new Vector2(P[2 * i, 0], P[2 * i, 1]), new Vector2(P[2 * i + 1, 0], P[2 * i + 1, 1]), pts[i + 1]);
}
return(beziers);
}
/// <summary>
/// Iterates one pass of the optimization algorithm trying to find
/// the best regression coefficients for the logistic model.
/// </summary>
/// <remarks>
/// An iterative Newton-Raphson algorithm is used to calculate
/// the maximum likelihood values of the parameters. This procedure
/// uses the partial second derivatives of the parameters in the
/// Hessian matrix to guide incremental parameter changes in an effort
/// to maximize the log likelihood value for the likelihood function.
/// </remarks>
/// <returns>
/// The absolute value of the largest parameter change.
/// </returns>
public double Regress(double[][] input, double[] output)
{
// Regress using Iterative Reweighted Least Squares estimation.
// Initial definitions and memory allocations
int N = input.Length;
int M = this.Coefficients.Length;
double[,] regression = new double[N, M];
double[,] hessian = new double[M, M];
double[,] inverse;
double[] gradient = new double[M];
double[] errors = new double[N];
double[] R = new double[N];
double[] deltas;
// Compute the regression matrix, errors and diagonal
for (int i = 0; i < N; i++)
{
double y = this.Compute(input[i]);
double o = output[i];
// Calculate error vector
errors[i] = y - o;
// Calculate R diagonal
R[i] = y * (1.0 - y);
// Compute the regression matrix
regression[i, 0] = 1;
for (int j = 1; j < M; j++)
{
regression[i, j] = input[i][j - 1];
}
}
// Compute error gradient and "Hessian" matrix (with diagonal R)
for (int i = 0; i < M; i++)
{
// Compute error gradient
for (int j = 0; j < N; j++)
{
gradient[i] += regression[j, i] * errors[j];
}
// Compute "Hessian" matrix (regression'*R*regression)
for (int j = 0; j < M; j++)
{
for (int k = 0; k < N; k++)
{
hessian[j, i] += regression[k, i] * (R[k] * regression[k, j]);
}
}
}
// Decompose to solve the linear system. Usually the hessian will
// be invertible and LU will succeed. However, sometimes the hessian
// may be singular and a Singular Value Decomposition may be needed.
LuDecomposition lu = new LuDecomposition(hessian);
// The SVD is very stable, but is quite expensive, being on average
// about 10-15 times more expensive than LU decomposition. There are
// other ways to avoid a singular Hessian. For a very interesting
// reading on the subject, please see:
//
// - Jeff Gill & Gary King, "What to Do When Your Hessian Is Not Invertible",
// Sociological Methods & Research, Vol 33, No. 1, August 2004, 54-87.
// Available in: http://gking.harvard.edu/files/help.pdf
//
// Moreover, the computation of the inverse is optional, as it will
// be used only to compute the standard errors of the regression.
if (lu.Nonsingular)
{
// Solve using LU decomposition
deltas = lu.Solve(gradient);
inverse = lu.Inverse(); // optional
}
else
{
// Hessian Matrix is singular, try pseudo-inverse solution
SingularValueDecomposition svd = new SingularValueDecomposition(hessian);
deltas = svd.Solve(gradient);
inverse = svd.Inverse(); // optional
}
// Update coefficients using the calculated deltas
for (int i = 0; i < coefficients.Length; i++)
{
this.coefficients[i] -= deltas[i];
}
// Calculate Coefficients standard errors (optional)
for (int i = 0; i < standardErrors.Length; i++)
{
standardErrors[i] = System.Math.Sqrt(inverse[i, i]);
}
// Return the absolute value of the largest parameter change
return(Matrix.Max(Matrix.Abs(deltas)));
}
public void LuDecompositionConstructorTest()
{
#region doc_ctor
// Let's say we would like to compute the
// LU decomposition of the following matrix:
double[,] matrix =
{
{ 2, -1, 0 },
{ -1, 2, -1 },
{ 0, -1, 2 }
};
// Compute the LU decomposition with:
var lu = new LuDecomposition(matrix);
// Retrieve the lower triangular factor L:
double[,] L = lu.LowerTriangularFactor;
// Should be equal to
double[,] expectedL =
{
{ 1.0000, 0, 0 },
{ -0.5000, 1.0000, 0 },
{ 0, -0.6667, 1.0000 },
};
// Retrieve the upper triangular factor U:
double[,] U = lu.UpperTriangularFactor;
// Should be equal to
double[,] expectedU =
{
{ 2.0000, -1.0000, 0 },
{ 0, 1.5000, -1.0000 },
{ 0, 0, 1.3333 },
};
// Certify that the decomposition has worked as expected by
// trying to reconstruct the original matrix with R = L * U:
double[,] reconstruction = L.Dot(U);
// reconstruction should be equal to
// {
// { 2, -1, 0 },
// { -1, 2, -1 },
// { 0, -1, 2 }
// };
#endregion
Assert.IsTrue(Matrix.IsEqual(matrix, reconstruction, 1e-4));
Assert.IsTrue(Matrix.IsEqual(expectedL, L, 1e-4));
Assert.IsTrue(Matrix.IsEqual(expectedU, U, 1e-4));
lu = new LuDecomposition(matrix.Transpose(), true);
L = lu.LowerTriangularFactor;
U = lu.UpperTriangularFactor;
Assert.IsTrue(Matrix.IsEqual(expectedL, L, 0.001));
Assert.IsTrue(Matrix.IsEqual(expectedU, U, 0.001));
}
/// <summary>
/// Runs one iteration of the Reweighted Least Squares algorithm.
/// </summary>
/// <param name="inputs">The input data.</param>
/// <param name="outputs">The outputs associated with each input vector.</param>
/// <returns>The maximum relative change in the parameters after the iteration.</returns>
///
public double Run(double[][] inputs, double[] outputs)
{
// Regress using Iteratively Reweighted Least Squares estimation.
// References:
// - Bishop, Christopher M.; Pattern Recognition
// and Machine Learning. Springer; 1st ed. 2006.
// Initial definitions and memory allocations
int N = inputs.Length;
double[][] design = new double[N][];
double[] errors = new double[N];
double[] weights = new double[N];
double[] coefficients = this.regression.Coefficients;
double[] deltas;
// Compute the regression matrix
for (int i = 0; i < inputs.Length; i++)
{
double[] row = design[i] = new double[parameterCount];
row[0] = 1; // for intercept
for (int j = 0; j < inputs[i].Length; j++)
{
row[j + 1] = inputs[i][j];
}
}
// Compute errors and weighing matrix
for (int i = 0; i < inputs.Length; i++)
{
double y = regression.Compute(inputs[i]);
// Calculate error vector
errors[i] = y - outputs[i];
// Calculate weighting matrix
weights[i] = y * (1.0 - y);
}
// Reset Hessian matrix and gradient
for (int i = 0; i < gradient.Length; i++)
{
gradient[i] = 0;
for (int j = 0; j < gradient.Length; j++)
{
hessian[i, j] = 0;
}
}
// (Re-) Compute error gradient
for (int j = 0; j < design.Length; j++)
{
for (int i = 0; i < gradient.Length; i++)
{
gradient[i] += design[j][i] * errors[j];
}
}
// (Re-) Compute weighted "Hessian" matrix
for (int k = 0; k < weights.Length; k++)
{
double[] rk = design[k];
for (int j = 0; j < rk.Length; j++)
{
for (int i = 0; i < rk.Length; i++)
{
hessian[j, i] += rk[i] * rk[j] * weights[k];
}
}
}
// Decompose to solve the linear system. Usually the hessian will
// be invertible and LU will succeed. However, sometimes the hessian
// may be singular and a Singular Value Decomposition may be needed.
LuDecomposition lu = new LuDecomposition(hessian);
// The SVD is very stable, but is quite expensive, being on average
// about 10-15 times more expensive than LU decomposition. There are
// other ways to avoid a singular Hessian. For a very interesting
// reading on the subject, please see:
//
// - Jeff Gill & Gary King, "What to Do When Your Hessian Is Not Invertible",
// Sociological Methods & Research, Vol 33, No. 1, August 2004, 54-87.
// Available in: http://gking.harvard.edu/files/help.pdf
//
// Moreover, the computation of the inverse is optional, as it will
// be used only to compute the standard errors of the regression.
if (lu.Nonsingular)
{
// Solve using LU decomposition
deltas = lu.Solve(gradient);
decomposition = lu;
}
else
{
// Hessian Matrix is singular, try pseudo-inverse solution
decomposition = new SingularValueDecomposition(hessian);
deltas = decomposition.Solve(gradient);
}
previous = (double[])coefficients.Clone();
// Update coefficients using the calculated deltas
for (int i = 0; i < coefficients.Length; i++)
{
coefficients[i] -= deltas[i];
}
if (computeStandardErrors)
{
// Grab the regression information matrix
double[,] inverse = decomposition.Inverse();
// Calculate coefficients' standard errors
double[] standardErrors = regression.StandardErrors;
for (int i = 0; i < standardErrors.Length; i++)
{
standardErrors[i] = Math.Sqrt(inverse[i, i]);
}
}
// Return the relative maximum parameter change
for (int i = 0; i < deltas.Length; i++)
{
deltas[i] = Math.Abs(deltas[i]) / Math.Abs(previous[i]);
}
return(Matrix.Max(deltas));
}
/// <summary>
/// Compute the Thin Plate Spline of the image, return a 2D tab
/// </summary>
/// <param name="control_points">Control points </param>
/// <param name="input">Input image to get the dim xy</param>
public double[,] calc_tps(List<cPoint3D> control_points, cDRC_Region AssociatedRegion, double Regularization)
{
int p = control_points.Count;
if (p < 3) return null;
double[,] grid = new double[AssociatedRegion.SizeX, AssociatedRegion.SizeY];
Matrix mtx_l = new Matrix(p + 3, p + 3);
Matrix mtx_v = new Matrix(p + 3, 1);
Matrix mtx_orig_k = new Matrix(p, p);
double a = 0.0;
for (int i = 0; i < p; ++i)
{
for (int j = i + 1; j < p; ++j)
{
cPoint3D pt_i = new cPoint3D(control_points[i].X, control_points[i].Y, control_points[i].Z);
cPoint3D pt_j = new cPoint3D(control_points[j].X, control_points[j].Y, control_points[j].Z);
pt_i.Y = pt_j.Y = 0;
//double elen = Math.Sqrt((pt_i.X - pt_j.X) * (pt_i.X - pt_j.X) + (pt_i.Z - pt_j.Z) * (pt_i.Z - pt_j.Z));
double elen = pt_i.DistTo(pt_j);
mtx_l[i, j] = mtx_l[j, i] = mtx_orig_k[i, j] = mtx_orig_k[j, i] = tps_base_func(elen);
a += elen * 2; // same for upper & lower tri
}
}
a /= (double)(p * p);
//regularization = 0.3f;
//Fill the rest of L
for (int i = 0; i < p; ++i)
{
//diagonal: reqularization parameters (lambda * a^2)
mtx_l[i, i] = mtx_orig_k[i, i] = Regularization * (a * a);
// P (p x 3, upper right)
mtx_l[i, p + 0] = 1.0;
mtx_l[i, p + 1] = control_points[i].X;
mtx_l[i, p + 2] = control_points[i].Z;
// P transposed (3 x p, bottom left)
mtx_l[p + 0, i] = 1.0;
mtx_l[p + 1, i] = control_points[i].X;
mtx_l[p + 2, i] = control_points[i].Z;
}
// O (3 x 3, lower right)
for (int i = p; i < p + 3; ++i)
for (int j = p; j < p + 3; ++j)
mtx_l[i, j] = 0.0;
// Fill the right hand vector V
for (int i = 0; i < p; ++i)
mtx_v[i, 0] = control_points[i].Y;
mtx_v[p + 0, 0] = mtx_v[p + 1, 0] = mtx_v[p + 2, 0] = 0.0;
// Solve the linear system "inplace"
Matrix mtx_v_res = new Matrix(p + 3, 1);
LuDecomposition ty = new LuDecomposition(mtx_l);
mtx_v_res = ty.Solve(mtx_v);
if (mtx_v_res == null)
{
return null;
}
// Interpolate grid heights
for (int x = 0; x < AssociatedRegion.SizeX; ++x)
{
for (int z = 0; z < AssociatedRegion.SizeY; ++z)
{
//float x = 0f; float z = 0.5f;
double h = mtx_v_res[p + 0, 0] + mtx_v_res[p + 1, 0] * (float)x / (float)AssociatedRegion.SizeX + mtx_v_res[p + 2, 0] * (float)z / (float)AssociatedRegion.SizeY;
//double h = mtx_v[p + 0, 0] + mtx_v[p + 1, 0] * (float)x + mtx_v[p + 2, 0] * (float)z ;
cPoint3D pt_ia;
cPoint3D pt_cur = new cPoint3D((float)x / (float)AssociatedRegion.SizeX, 0, (float)z / (float)AssociatedRegion.SizeY);
//Vector3 pt_cur = new Vector3((float)x , 0, (float)z);
for (int i = 0; i < p; ++i)
{
pt_ia = control_points[i];
pt_ia.Y = 0;
h += mtx_v_res[i, 0] * tps_base_func(pt_ia.DistTo(pt_cur));
}
grid[x, z] = h;
}
}
// Calc bending energy
Matrix w = new Matrix(p, 1);
for (int i = 0; i < p; ++i)
w[i, 0] = mtx_v_res[i, 0];
Matrix be;
be = Matrix.Multiply(Matrix.Multiply(w.Transpose(), mtx_orig_k), w);
bending_energy = be[0, 0];
Console.WriteLine("be= " + be[0, 0]);
return grid;
}
private double run(double[][] inputs, double[][] outputs)
{
// Regress using Lower-Bound Newton-Raphson estimation
//
// The main idea is to replace the Hessian matrix with a
// suitable lower bound. Indeed, the Hessian is lower
// bounded by a negative definite matrix that does not
// even depend on w [Krishnapuram et al].
//
// - http://www.lx.it.pt/~mtf/Krishnapuram_Carin_Figueiredo_Hartemink_2005.pdf
//
// Initial definitions and memory allocations
int N = inputs.Length;
double[][] design = new double[N][];
double[][] coefficients = this.regression.Coefficients;
// Compute the regression matrix
for (int i = 0; i < inputs.Length; i++)
{
double[] row = design[i] = new double[M];
row[0] = 1; // for intercept
for (int j = 0; j < inputs[i].Length; j++)
{
row[j + 1] = inputs[i][j];
}
}
// Reset Hessian matrix and gradient
for (int i = 0; i < gradient.Length; i++)
{
gradient[i] = 0;
}
if (UpdateLowerBound)
{
for (int i = 0; i < gradient.Length; i++)
{
for (int j = 0; j < gradient.Length; j++)
{
lowerBound[i, j] = 0;
}
}
}
// In the multinomial logistic regression, the objective
// function is the log-likelihood function l(w). As given
// by Krishnapuram et al and Böhning, this is a concave
// function with Hessian given by:
//
// H(w) = -sum(P(w) - p(w)p(w)') (x) xx'
// (see referenced paper for proper indices)
//
// In which (x) denotes the Kronocker product. By using
// the lower bound principle, Krishnapuram has shown that
// we can replace H(w) with a lower bound approximation B
// which does not depend on w (eq. 8 on aforementined paper):
//
// B = -(1/2) [I - 11/M] (x) sum(xx')
//
// Thus we can compute and invert this matrix only once.
//
// For each input sample in the dataset
for (int i = 0; i < inputs.Length; i++)
{
// Grab variables related to the sample
double[] x = design[i];
double[] y = outputs[i];
// Compute and estimate outputs
this.compute(inputs[i], output);
// Compute errors for the sample
for (int j = 0; j < errors.Length; j++)
{
errors[j] = y[j + 1] - output[j];
}
// Compute current gradient and Hessian
// We can take advantage of the block structure of the
// Hessian matrix and gradient vector by employing the
// Kronocker product. See [Böhning, 1992]
// (Re-) Compute error gradient
double[] g = Matrix.KroneckerProduct(errors, x);
for (int j = 0; j < g.Length; j++)
{
gradient[j] += g[j];
}
if (UpdateLowerBound)
{
// Compute xxt matrix
for (int k = 0; k < x.Length; k++)
{
for (int j = 0; j < x.Length; j++)
{
xxt[k, j] = x[k] * x[j];
}
}
// (Re-) Compute weighted "Hessian" matrix
double[,] h = Matrix.KroneckerProduct(weights, xxt);
for (int j = 0; j < parameterCount; j++)
{
for (int k = 0; k < parameterCount; k++)
{
lowerBound[j, k] += h[j, k];
}
}
}
}
if (UpdateLowerBound)
{
UpdateLowerBound = false;
// Decompose to solve the linear system. Usually the hessian will
// be invertible and LU will succeed. However, sometimes the hessian
// may be singular and a Singular Value Decomposition may be needed.
LuDecomposition lu = new LuDecomposition(lowerBound);
// The SVD is very stable, but is quite expensive, being on average
// about 10-15 times more expensive than LU decomposition. There are
// other ways to avoid a singular Hessian. For a very interesting
// reading on the subject, please see:
//
// - Jeff Gill & Gary King, "What to Do When Your Hessian Is Not Invertible",
// Sociological Methods & Research, Vol 33, No. 1, August 2004, 54-87.
// Available in: http://gking.harvard.edu/files/help.pdf
//
// Moreover, the computation of the inverse is optional, as it will
// be used only to compute the standard errors of the regression.
if (lu.Nonsingular)
{
// Solve using LU decomposition
deltas = lu.Solve(gradient);
decomposition = lu;
}
else
{
// Hessian Matrix is singular, try pseudo-inverse solution
decomposition = new SingularValueDecomposition(lowerBound);
deltas = decomposition.Solve(gradient);
}
}
else
{
deltas = decomposition.Solve(gradient);
}
previous = coefficients.Reshape(1);
// Update coefficients using the calculated deltas
for (int i = 0, k = 0; i < coefficients.Length; i++)
{
for (int j = 0; j < coefficients[i].Length; j++)
{
coefficients[i][j] -= deltas[k++];
}
}
solution = coefficients.Reshape(1);
if (computeStandardErrors)
{
// Grab the regression information matrix
double[,] inverse = decomposition.Inverse();
// Calculate coefficients' standard errors
double[][] standardErrors = regression.StandardErrors;
for (int i = 0, k = 0; i < standardErrors.Length; i++)
{
for (int j = 0; j < standardErrors[i].Length; j++, k++)
{
standardErrors[i][j] = Math.Sqrt(Math.Abs(inverse[k, k]));
}
}
}
// Return the relative maximum parameter change
for (int i = 0; i < deltas.Length; i++)
{
deltas[i] = Math.Abs(deltas[i]) / Math.Abs(previous[i]);
}
return(Matrix.Max(deltas));
}
static void Main(string[] args)
{
#region 1. Declaring matrices
// 1.1 Using standard .NET declaration
double[,] A =
{
{ 1, 2, 3 },
{ 6, 2, 0 },
{ 0, 0, 1 }
};
double[,] B =
{
{ 2, 0, 0 },
{ 0, 2, 0 },
{ 0, 0, 2 }
};
{
// 1.2 Using Accord extension methods
double[,] Bi = Matrix.Identity(3).Multiply(2);
double[,] Bj = Matrix.Diagonal(3, 2.0); // both are equal to B
// 1.2 Using Accord extension methods with implicit typing
var I = Matrix.Identity(3);
}
#endregion
#region 2. Matrix Operations
{
// 2.1 Addition
var C = A.Add(B);
// 2.2 Subtraction
var D = A.Subtract(B);
// 2.3 Multiplication
{
// 2.3.1 By a scalar
var halfM = A.Multiply(0.5);
// 2.3.2 By a vector
double[] m = A.Multiply(new double[] { 1, 2, 3 });
// 2.3.3 By a matrix
var M = A.Multiply(B);
// 2.4 Transposing
var At = A.Transpose();
}
}
// 2.5 Elementwise operations
// 2.5.1 Elementwise multiplication
A.ElementwiseMultiply(B); // A.*B
// 2.5.1 Elementwise division
A.ElementwiseDivide(B); // A./B
#endregion
#region 3. Matrix characteristics
{
// 3.1 Calculating the determinant
double det = A.Determinant();
// 3.2 Calculating the trace
double tr = A.Trace();
// 3.3 Computing the sum vector
{
double[] sumVector = A.Sum();
// 3.3.1 Computing the total sum of elements
double sum = sumVector.Sum();
// 3.3.2 Computing the sum along the rows
sumVector = A.Sum(0); // Equivalent to Octave's sum(A, 1)
// 3.3.2 Computing the sum along the columns
sumVector = A.Sum(1); // Equivalent to Octave's sum(A, 2)
}
}
#endregion
#region 4. Linear Algebra
{
// 4.1 Computing the inverse
var invA = A.Inverse();
// 4.2 Computing the pseudo-inverse
var pinvA = A.PseudoInverse();
// 4.3 Solving a linear system (Ax = B)
var x = A.Solve(B);
}
#endregion
#region 5. Special operators
{
// 5.1 Finding the indices of elements
double[] v = { 5, 2, 2, 7, 1, 0 };
int[] idx = v.Find(e => e > 2); // finding the index of every element in v higher than 2.
// 5.2 Selecting elements by index
double[] u = v.Submatrix(idx); // u is { 5, 7 }
// 5.3 Converting between different matrix representations
double[][] jaggedA = A.ToArray(); // from multidimensional to jagged array
// 5.4 Extracting a column or row from the matrix
double[] a = A.GetColumn(0); // retrieves the first column
double[] b = B.GetRow(1); // retrieves the second row
// 5.5 Taking the absolute of a matrix
var absA = A.Abs();
// 5.6 Applying some function to every element
var newv = v.Apply(e => e + 1);
}
#endregion
#region 7. Vector operations
{
double[] u = { 1, 2, 3 };
double[] v = { 4, 5, 6 };
var w1 = u.InnerProduct(v);
var w2 = u.OuterProduct(v);
var w3 = u.CartesianProduct(v);
double[] m = { 1, 2, 3, 4 };
double[,] M = Matrix.Reshape(m, 2, 2);
}
#endregion
#region Decompositions
{
// Singular value decomposition
{
SingularValueDecomposition svd = new SingularValueDecomposition(A);
var U = svd.LeftSingularVectors;
var S = svd.Diagonal;
var V = svd.RightSingularVectors;
}
// or (please see documentation for details)
{
SingularValueDecomposition svd = new SingularValueDecomposition(A.Transpose());
var U = svd.RightSingularVectors;
var S = svd.Diagonal;
var V = svd.LeftSingularVectors;
}
// Eigenvalue decomposition
{
EigenvalueDecomposition eig = new EigenvalueDecomposition(A);
var V = eig.Eigenvectors;
var D = eig.DiagonalMatrix;
}
// QR decomposition
{
QrDecomposition qr = new QrDecomposition(A);
var Q = qr.OrthogonalFactor;
var R = qr.UpperTriangularFactor;
}
// Cholesky decomposition
{
CholeskyDecomposition chol = new CholeskyDecomposition(A);
var R = chol.LeftTriangularFactor;
}
// LU decomposition
{
LuDecomposition lu = new LuDecomposition(A);
var L = lu.LowerTriangularFactor;
var U = lu.UpperTriangularFactor;
}
}
#endregion
}
public LuDecompositionWrapper(LuDecomposition lu)
{
_lu = lu;
}
