/// <summary>Creates a 'n x r' matrix B such that B * B' is a correlation matrix and 'near' to the specified symmetric, normalized matrix of dimension n. A rank reduction will apply if r is strict less than n.
/// </summary>
/// <param name="rawCorrelationMatrix">The symmetric, normalized matrix where to find the 'nearest' correlation matrix.</param>
/// <param name="state">The state of the operation in its <see cref="PseudoSqrtMatrixDecomposer.State"/> representation (output).</param>
/// <param name="triangularMatrixType">A value indicating which part of <paramref name="rawCorrelationMatrix"/> to take into account.</param>
/// <param name="outputEntries">This argument will be used to store the matrix entries of the resulting matrix B, i.e. the return value array points to this array if != <c>null</c>; otherwise a memory allocation will be done.</param>
/// <param name="worksspaceContainer">A specific <see cref="PseudoSqrtMatrixDecomposer.WorkspaceContainer"/> object to reduce memory allocation; ignored if <c>null</c>.</param>
/// <returns>A <see cref="DenseMatrix"/> object that represents a matrix B such that B * B' is the 'nearest' correlation matrix with respect to <paramref name="rawCorrelationMatrix"/>.</returns>
/// <remarks>In general the return object does <b>not</b> represents the pseudo-root of <paramref name="rawCorrelationMatrix"/>, i.e. output of the Cholesky decomposition.
/// <para>The parameters <paramref name="outputEntries"/>, <paramref name="worksspaceContainer"/> allows to avoid memory allocation and to re-use arrays if the calculation of correlation matrices will be done often.</para></remarks>
public override DenseMatrix Create(DenseMatrix rawCorrelationMatrix, out State state, double[] outputEntries = null, PseudoSqrtMatrixDecomposer.WorkspaceContainer worksspaceContainer = null, BLAS.TriangularMatrixType triangularMatrixType = BLAS.TriangularMatrixType.LowerTriangularMatrix)
{
if (rawCorrelationMatrix.IsQuadratic == false)
{
throw new ArgumentException("rawCorrelationMatrix");
}
int n = rawCorrelationMatrix.RowCount;
if ((outputEntries == null) || (outputEntries.Length < n * n))
{
outputEntries = new double[n * n];
}
var ws = worksspaceContainer as Workspace;
if ((ws == null) || (ws.Dimension < n))
{
ws = new Workspace(n);
}
int m;
BLAS.Level1.dcopy(n * n, rawCorrelationMatrix.Data, ws.data);
LAPACK.EigenValues.Symmetric.driver_dsyevr(LapackEigenvalues.SymmetricGeneralJob.All, n, ws.data, out m, ws.eigenValues, outputEntries, ws.isuppz, ws.work, ws.iwork);
var originalEigenValueDataTable = InfoOutputDetailLevel.IsAtLeastAsComprehensiveAs(InfoOutputDetailLevel.High) ? CreateDataTableWithEigenvalues("Eigenvalues.Original", m, ws.eigenValues) : null;
int rank = n;
int minNumberOfEigenvaluesToSetZero = n - Math.Min(MaximalRank ?? n, n);
int i = 0;
while ((i < minNumberOfEigenvaluesToSetZero) || (ws.eigenValues[i] < 0.0))
{
ws.eigenValues[i] = 0.0;
i++;
rank--;
}
var adjustedEigenValueDataTable = InfoOutputDetailLevel.IsAtLeastAsComprehensiveAs(InfoOutputDetailLevel.High) ? CreateDataTableWithEigenvalues("Eigenvalues.Adjusted", m, ws.eigenValues) : null;
VectorUnit.Basics.Sqrt(n, ws.eigenValues); // calculate sqrt of eigenvalues only once, i.e. the array 'eigenValues' contains the sqrt of the eigenvalues!
for (i = 0; i < n; i++)
{
var t_i = 0.0;
for (int j = n - 1; j >= n - rank; j--)
{
t_i += outputEntries[i + j * n] * outputEntries[i + j * n] * ws.eigenValues[j] * ws.eigenValues[j];
outputEntries[i + j * n] *= ws.eigenValues[j];
}
BLAS.Level1.dscal(rank, 1.0 / Math.Sqrt(t_i), outputEntries, -n, i + (n - 1) * n); // [i+j*n] *= 1/Math.Sqrt(tempValue) for j = n-1, ..., n-rank
}
/* The eigenvalues are in ascending order. Thefore the first (and not last) 'rank' columns of the eigenvectors are not required. Therefore we swap the relevant part */
BLAS.Level1.dscal(n * (n - rank), 0.0, outputEntries);
BLAS.Level1.dswap(n * rank, outputEntries, 1, outputEntries, 1, n * (n - rank), 0);
state = State.Create(rank, detailProperties: new[] { InfoOutputProperty.Create("Eigenvalues set to 0.0", n - rank) }, detailDataTables: new[] { originalEigenValueDataTable, adjustedEigenValueDataTable }, iterationsNeeded: 1, infoOutputDetailLevel: InfoOutputDetailLevel);
return(new DenseMatrix(n, rank, outputEntries));
}
/// <summary>Creates a 'n x r' matrix B such that B * B' is a correlation matrix and 'near' to the specified symmetric, normalized matrix of dimension n. A rank reduction will apply if r is strict less than n.
/// </summary>
/// <param name="rawCorrelationMatrix">The symmetric, normalized matrix where to find the 'nearest' correlation matrix.</param>
/// <param name="state">The state of the operation in its <see cref="PseudoSqrtMatrixDecomposer.State"/> representation (output).</param>
/// <param name="triangularMatrixType">A value indicating which part of <paramref name="rawCorrelationMatrix"/> to take into account.</param>
/// <param name="outputEntries">This argument will be used to store the matrix entries of the resulting matrix B, i.e. the return value array points to this array if != <c>null</c>; otherwise a memory allocation will be done.</param>
/// <param name="worksspaceContainer">A specific <see cref="PseudoSqrtMatrixDecomposer.WorkspaceContainer"/> object to reduce memory allocation; ignored if <c>null</c>.</param>
/// <returns>A <see cref="DenseMatrix"/> object that represents a matrix B such that B * B' is the 'nearest' correlation matrix with respect to <paramref name="rawCorrelationMatrix"/>.</returns>
/// <remarks>In general the return object does <b>not</b> represents the pseudo-root of <paramref name="rawCorrelationMatrix"/>, i.e. output of the Cholesky decomposition.
/// <para>The parameters <paramref name="outputEntries"/>, <paramref name="worksspaceContainer"/> allows to avoid memory allocation and to re-use arrays if the calculation of correlation matrices will be done often.</para></remarks>
public override DenseMatrix Create(DenseMatrix rawCorrelationMatrix, out State state, double[] outputEntries = null, PseudoSqrtMatrixDecomposer.WorkspaceContainer worksspaceContainer = null, BLAS.TriangularMatrixType triangularMatrixType = BLAS.TriangularMatrixType.LowerTriangularMatrix)
{
if (rawCorrelationMatrix.IsQuadratic == false)
{
throw new ArgumentException("rawCorrelationMatrix");
}
int n = rawCorrelationMatrix.RowCount;
var ws = worksspaceContainer as Workspace;
if ((ws == null) || (ws.Dimension < n))
{
ws = new Workspace(n, this);
}
/* calculate an initial value for the optimizer:
* (i) Apply the EZN algorithm for the calculation of a matrix B_0 such that B_0 * B_0^t is near to the (raw) correlation matrix
* (ii) calculate angle parameters \theta such that B_0 = B(\theta).
* */
State initState;
var initialAngleParameterMatrix = GetAngleParameter(m_InitialDecomposer.Create(rawCorrelationMatrix, out initState, outputEntries, ws.InitalDecomposerWorkspace, triangularMatrixType), ws.ArgMinData);
/* prepare and apply optimization algorithm: */
int rank = initState.Rank;
if ((outputEntries == null) || (outputEntries.Length < n * n))
{
outputEntries = new double[n * rank];
}
var B = new DenseMatrix(n, rank, outputEntries, createDeepCopyOfArgument: false);
var C = new DenseMatrix(n, n, ws.CorrelationMatrixData, createDeepCopyOfArgument: false);
var optAlgorithm = m_MultiDimOptimizer.Create(n * (rank - 1));
optAlgorithm.SetFunction(theta =>
{
GetParametricMatrix(theta, B);
C.AddAssignment(B, B.T, beta: 0.0); // C = B * B^t
VectorUnit.Basics.Sub(n * n, C.Data, rawCorrelationMatrix.Data, ws.TempDifferences);
return(BLAS.Level1.dnrm2sq(n * n, ws.TempDifferences));
});
double minimum;
var optState = optAlgorithm.FindMinimum(ws.ArgMinData, out minimum);
state = State.Create(rank, optState.IterationsNeeded, InfoOutputDetailLevel,
Tuple.Create <string, IInfoOutputQueriable>("Initial.State", initState),
Tuple.Create <string, IInfoOutputQueriable>("Final.Optimizer", optState),
Tuple.Create <string, IInfoOutputQueriable>("Initial.Parameters", initialAngleParameterMatrix),
InfoOutputDetailLevel.IsAtLeastAsComprehensiveAs(InfoOutputDetailLevel.High) ? Tuple.Create <string, IInfoOutputQueriable>("Final.Parameters", new DenseMatrix(n, rank, ws.ArgMinData, createDeepCopyOfArgument: false)) : null
);
GetParametricMatrix(ws.ArgMinData, B); // B should be already set to B(\theta^*), we just want to be sure that B is correct on exit
return(B);
}
