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 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 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>
/// 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)));
}
