CovarianceMatrix¶
- class CovarianceMatrix(*args)¶
Covariance (real symmetric positive definite) matrix.
- Parameters:
- sizeint, , optional
Matrix size. Default is 1.
- valuessequence of float with size , optional
Values. OpenTURNS uses column-major ordering (like Fortran) for reshaping the flat list of values. Default creates an identity matrix.
- Raises:
- TypeErrorIf the matrix is not symmetric.
Examples
Create a matrix
>>> import openturns as ot >>> C = ot.CovarianceMatrix(2, [1.0, 0.5, 0.5, 1.0]) >>> print(C) [[ 1 0.5 ] [ 0.5 1 ]]
Get or set terms
>>> print(C[0, 1]) 0.5 >>> C[0, 1] = 0.6 >>> print(C[0, 1]) 0.6 >>> print(C[:, 0]) [[ 1 ] [ 0.6 ]]
Create an openturns matrix from a symmetric numpy 2d-array (or matrix, or 2d-list)…
>>> import numpy as np >>> np_2d_array = np.array([[1.0, 0.5], [0.5, 1.0]]) >>> ot_matrix = ot.CovarianceMatrix(np_2d_array)
and back
>>> np_matrix = np.matrix(ot_matrix)
Methods
Check if the internal representation is really symmetric.
clean
(threshold)Set elements smaller than a threshold to zero.
computeCholesky
([keepIntact])Compute the Cholesky factor.
computeDeterminant
([keepIntact])Compute the determinant.
computeEV
([keepIntact])Compute the eigenvalues decomposition (EVD).
computeEigenValues
([keepIntact])Compute eigenvalues.
computeGram
([transpose])Compute the associated Gram matrix.
computeHadamardProduct
(other)Compute the Hadamard product matrix.
Compute the largest eigenvalue module.
computeLogAbsoluteDeterminant
([keepIntact])Compute the logarithm of the absolute value of the determinant.
computeQR
([fullQR, keepIntact])Compute the QR factorization.
Compute the regularized Cholesky factor.
computeSVD
([fullSVD, keepIntact])Compute the singular values decomposition (SVD).
computeSingularValues
([keepIntact])Compute the singular values.
Compute the sum of the matrix elements.
Compute the trace of the matrix.
Accessor to the object's name.
getDiagonal
([k])Get the k-th diagonal of the matrix.
getDiagonalAsPoint
([k])Get the k-th diagonal of the matrix.
Accessor to the dimension (the number of rows).
getId
()Accessor to the object's id.
Accessor to the underlying implementation.
getName
()Accessor to the object's name.
Accessor to the number of columns.
Accessor to the number of rows.
Test whether the matrix is diagonal or not.
isEmpty
()Tell if the matrix is empty.
Test whether the matrix is positive definite or not.
reshape
(newRowDim, newColDim)Reshape the matrix.
reshapeInPlace
(newRowDim, newColDim)Reshape the matrix, in place.
setDiagonal
(*args)Set the k-th diagonal of the matrix.
setName
(name)Accessor to the object's name.
solveLinearSystem
(*args)Solve a square linear system whose the present matrix is the operator.
solveLinearSystemInPlace
(*args)Solve a rectangular linear system whose the present matrix is the operator.
Square the Matrix, ie each element of the matrix is squared.
Transpose the matrix.
- __init__(*args)¶
- checkSymmetry()¶
Check if the internal representation is really symmetric.
- clean(threshold)¶
Set elements smaller than a threshold to zero.
- Parameters:
- thresholdfloat
Threshold for zeroing elements.
- Returns:
- cleaned_matrix
Matrix
Input matrix with elements smaller than the threshold set to zero.
- cleaned_matrix
- computeCholesky(keepIntact=True)¶
Compute the Cholesky factor.
The Cholesky factor of a covariance (real symmetric positive definite) matrix is the lower triangular matrix such that:
- Parameters:
- keep_intactbool, optional
A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.
- Returns:
- cholesky_factor
TriangularMatrix
The left (lower) Cholesky factor.
- cholesky_factor
Notes
This uses LAPACK’s DPOTRF.
- computeDeterminant(keepIntact=True)¶
Compute the determinant.
- Parameters:
- keep_intactbool, optional
A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.
- Returns:
- determinantfloat
The square matrix determinant.
Examples
>>> import openturns as ot >>> A = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]]) >>> A.computeDeterminant() -2.0
- computeEV(keepIntact=True)¶
Compute the eigenvalues decomposition (EVD).
The eigenvalues decomposition of a square matrix with size reads:
where is an diagonal matrix and is an orthogonal matrix.
- Parameters:
- keep_intactbool, optional
A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.
- Returns:
- eigenvalues
Point
The vector of eigenvalues with size that form the diagonal of the matrix of the EVD.
- Phi
SquareComplexMatrix
The left matrix of the EVD.
- eigenvalues
Notes
This uses LAPACK’S DSYEV.
Examples
>>> import openturns as ot >>> import numpy as np >>> M = ot.SymmetricMatrix([[1.0, 2.0], [2.0, -4.0]]) >>> eigen_values, Phi = M.computeEV() >>> Lambda = ot.SquareMatrix(M.getDimension()) >>> for i in range(eigen_values.getSize()): ... Lambda[i, i] = eigen_values[i] >>> np.testing.assert_array_almost_equal(Phi * Lambda * Phi.transpose(), M)
- computeEigenValues(keepIntact=True)¶
Compute eigenvalues.
- Parameters:
- keep_intactbool, optional
A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.
- Returns:
- eigenvalues
Point
Eigenvalues.
- eigenvalues
See also
Examples
>>> import openturns as ot >>> M = ot.SymmetricMatrix([[1.0, 2.0], [2.0, -4.0]]) >>> print(M.computeEigenValues()) [-4.70156,1.70156]
- computeGram(transpose=True)¶
Compute the associated Gram matrix.
- Parameters:
- transposedbool
Tells if matrix is to be transposed or not. Default value is True
- Returns:
- MMT
Matrix
The Gram matrix.
- MMT
Notes
When transposed is set to True, the method computes . Otherwise it computes
Examples
>>> import openturns as ot >>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]) >>> MtM = M.computeGram() >>> print(MtM) [[ 35 44 ] [ 44 56 ]] >>> MMt = M.computeGram(False) >>> print(MMt) [[ 5 11 17 ] [ 11 25 39 ] [ 17 39 61 ]]
- computeHadamardProduct(other)¶
Compute the Hadamard product matrix.
Notes
The matrix resulting from the Hadamard product of the matrices and :cB is as follows:
Examples
>>> import openturns as ot >>> A = ot.Matrix([[1.0, 2.0], [3.0, 4.0]]) >>> B = ot.Matrix([[1.0, 2.0], [3.0, 4.0]]) >>> C = A.computeHadamardProduct(B) >>> print(C) [[ 1 4 ] [ 9 16 ]] >>> print(B.computeHadamardProduct(A)) [[ 1 4 ] [ 9 16 ]]
- computeLargestEigenValueModule(*args)¶
Compute the largest eigenvalue module.
- Parameters:
- maximumIterationsint, optional
The maximum number of power iterations to perform to get the approximation. Default is given by the ‘Matrix-LargestEigenValueIterations’ key in the
ResourceMap
.- epsilonfloat, optional
The target relative error. Default is given by the ‘Matrix-LargestEigenValueRelativeError’ key in the
ResourceMap
.
- Returns:
- largestEigenvalueModulefloat
The largest eigenvalue module.
See also
Examples
>>> import openturns as ot >>> M = ot.SymmetricMatrix([[1.0, 3.0], [3.0, 4.0]]) >>> M.computeLargestEigenValueModule() 5.8541...
- computeLogAbsoluteDeterminant(keepIntact=True)¶
Compute the logarithm of the absolute value of the determinant.
- Parameters:
- keep_intactbool, optional
A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.
- Returns:
- determinantfloat
The logarithm of the absolute value of the square matrix determinant.
- signfloat
The sign of the determinant.
Examples
>>> import openturns as ot >>> A = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]]) >>> A.computeLogAbsoluteDeterminant() [0.693147..., -1.0]
- computeQR(fullQR=False, keepIntact=True)¶
Compute the QR factorization. By default, it is the economic decomposition which is computed.
The economic QR factorization of a rectangular matrix with (more rows than columns) is defined as follows:
where is an upper triangular matrix, is , is , and and both have orthogonal columns.
- Parameters:
- full_qrbool, optional
A flag telling whether Q, R or Q1, R1 are returned. Default is False and returns Q1, R1.
- keep_intactbool, optional
A flag telling whether the present matrix is preserved or not in the computation of the decomposition. Default is True and leaves the present matrix unchanged.
- Returns:
- Q1
Matrix
The orthogonal matrix of the economic QR factorization.
- R1
TriangularMatrix
The right (upper) triangular matrix of the economic QR factorization.
- Q
Matrix
The orthogonal matrix of the full QR factorization.
- R
TriangularMatrix
The right (upper) triangular matrix of the full QR factorization.
- Q1
Notes
The economic QR factorization is often used for solving overdetermined linear systems (where the operator has ) in the least-square sense because it implies solving a (simple) triangular system:
This uses LAPACK’s DGEQRF and DORGQR.
Examples
>>> import openturns as ot >>> import numpy as np >>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]) >>> Q1, R1 = M.computeQR() >>> np.testing.assert_array_almost_equal(Q1 * R1, M)
- computeRegularizedCholesky()¶
Compute the regularized Cholesky factor.
Similar to
computeCholesky()
but with a regularization loop according to the largest eigenvalue and keys Matrix-StartingScaling and Matrix-MaximalScaling.- Returns:
- cholesky_factor
TriangularMatrix
The left (lower) Cholesky factor.
- cholesky_factor
- computeSVD(fullSVD=False, keepIntact=True)¶
Compute the singular values decomposition (SVD).
The singular values decomposition of a rectangular matrix with size reads:
where is an orthogonal matrix, is an diagonal matrix and is an orthogonal matrix.
- Parameters:
- fullSVDbool, optional
Whether the null parts of the orthogonal factors are explicitly stored or not. Default is False and computes a reduced SVD.
- keep_intactbool, optional
A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.
- Returns:
- singular_values
Point
The vector of singular values with size that form the diagonal of the matrix of the SVD.
- U
SquareMatrix
The left orthogonal matrix of the SVD.
- VT
SquareMatrix
The transposed right orthogonal matrix of the SVD.
- singular_values
Notes
This uses LAPACK’s DGESDD.
Examples
>>> import openturns as ot >>> import numpy as np >>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]) >>> singular_values, U, VT = M.computeSVD(True) >>> Sigma = ot.Matrix(M.getNbRows(), M.getNbColumns()) >>> for i in range(singular_values.getSize()): ... Sigma[i, i] = singular_values[i] >>> np.testing.assert_array_almost_equal(U * Sigma * VT, M)
- computeSingularValues(keepIntact=True)¶
Compute the singular values.
- Parameters:
- fullSVDbool, optional
Whether the null parts of the orthogonal factors are explicitly stored or not. Default is False and computes a reduced SVD.
- keep_intactbool, optional
A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.
- Returns:
- singular_values
Point
The vector of singular values with size that form the diagonal of the matrix of the SVD decomposition.
- singular_values
See also
Examples
>>> import openturns as ot >>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]) >>> print(M.computeSingularValues(True)) [9.52552,0.514301]
- computeSumElements()¶
Compute the sum of the matrix elements.
- Returns:
- suma float
The sum of the elements.
Notes
We compute here the sum of elements of the matrix , that defines as:
Examples
>>> import openturns as ot >>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]) >>> s = M.computeSumElements() >>> print(s) 21.0
- computeTrace()¶
Compute the trace of the matrix.
- Returns:
- tracefloat
The trace of the matrix.
Examples
>>> import openturns as ot >>> M = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]]) >>> M.computeTrace() 5.0
- getClassName()¶
Accessor to the object’s name.
- Returns:
- class_namestr
The object class name (object.__class__.__name__).
- getDiagonal(k=0)¶
Get the k-th diagonal of the matrix.
- Parameters:
- kint
The k-th diagonal to extract Default value is 0
- Returns:
- D:
Matrix
The k-th diagonal.
- D:
Examples
>>> import openturns as ot >>> M = ot.Matrix([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]) >>> diag = M.getDiagonal() >>> print(diag) [[ 1 ] [ 5 ] [ 9 ]] >>> print(M.getDiagonal(1)) [[ 2 ] [ 6 ]]
- getDiagonalAsPoint(k=0)¶
Get the k-th diagonal of the matrix.
- Parameters:
- kint
The k-th diagonal to extract Default value is 0
- Returns:
- pt
Point
The k-th digonal.
- pt
Examples
>>> import openturns as ot >>> M = ot.Matrix([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]) >>> pt = M.getDiagonalAsPoint() >>> print(pt) [1,5,9]
- getDimension()¶
Accessor to the dimension (the number of rows).
- Returns:
- dimensionint
- getId()¶
Accessor to the object’s id.
- Returns:
- idint
Internal unique identifier.
- getImplementation()¶
Accessor to the underlying implementation.
- Returns:
- implImplementation
A copy of the underlying implementation object.
- getName()¶
Accessor to the object’s name.
- Returns:
- namestr
The name of the object.
- getNbColumns()¶
Accessor to the number of columns.
- Returns:
- n_columnsint
- getNbRows()¶
Accessor to the number of rows.
- Returns:
- n_rowsint
- isDiagonal()¶
Test whether the matrix is diagonal or not.
- Returns:
- testbool
Answer.
- isEmpty()¶
Tell if the matrix is empty.
- Returns:
- is_emptybool
True if the matrix contains no element.
Examples
>>> import openturns as ot >>> M = ot.Matrix([[]]) >>> M.isEmpty() True
- isPositiveDefinite()¶
Test whether the matrix is positive definite or not.
A matrix is positive definite if is positive for every compatible non-zero column vector .
- Returns:
- testbool
Answer.
Notes
This uses LAPACK’s DPOTRF.
- reshape(newRowDim, newColDim)¶
Reshape the matrix.
- Parameters:
- newRowDimint
The row dimension of the reshaped matrix.
- newColDimint
The column dimension of the reshaped matrix.
- Returns:
- MT
Matrix
The reshaped matrix.
- MT
Notes
If the size of the reshaped matrix is smaller than the size of the matrix to be reshaped, only the first elements are kept (in a column-major storage sense). If the size is greater, the new elements are set to zero.
Examples
>>> import openturns as ot >>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]) >>> print(M) [[ 1 2 ] [ 3 4 ] [ 5 6 ]] >>> print(M.reshape(1, 6)) 1x6 [[ 1 3 5 2 4 6 ]] >>> print(M.reshape(2, 2)) [[ 1 5 ] [ 3 2 ]] >>> print(M.reshape(2, 6)) 2x6 [[ 1 5 4 0 0 0 ] [ 3 2 6 0 0 0 ]]
- reshapeInPlace(newRowDim, newColDim)¶
Reshape the matrix, in place.
- Parameters:
- newRowDimint
The row dimension of the reshaped matrix.
- newColDimint
The column dimension of the reshaped matrix.
Notes
If the size of the reshaped matrix is smaller than the size of the matrix to be reshaped, only the first elements are kept (in a column-major storage sense). If the size is greater, the new elements are set to zero. If the size is unchanged, no copy of data is done.
Examples
>>> import openturns as ot >>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]) >>> print(M) [[ 1 2 ] [ 3 4 ] [ 5 6 ]] >>> M.reshapeInPlace(1, 6) >>> print(M) 1x6 [[ 1 3 5 2 4 6 ]] >>> M.reshapeInPlace(2, 2) >>> print(M) [[ 1 5 ] [ 3 2 ]] >>> M.reshapeInPlace(2, 6) >>> print(M) 2x6 [[ 1 5 0 0 0 0 ] [ 3 2 0 0 0 0 ]]
- setDiagonal(*args)¶
Set the k-th diagonal of the matrix.
- Parameters:
Examples
>>> import openturns as ot >>> M = ot.Matrix([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]) >>> M.setDiagonal([-1, 23, 9]) >>> print(M) [[ -1 2 3 ] [ 4 23 6 ] [ 7 8 9 ]] >>> M.setDiagonal(1.0) >>> print(M) [[ 1 2 3 ] [ 4 1 6 ] [ 7 8 1 ]] >>> M.setDiagonal([2, 6, 9]) >>> print(M) [[ 2 2 3 ] [ 4 6 6 ] [ 7 8 9 ]]
- setName(name)¶
Accessor to the object’s name.
- Parameters:
- namestr
The name of the object.
- solveLinearSystem(*args)¶
Solve a square linear system whose the present matrix is the operator.
- Parameters:
- rhssequence of float or
Matrix
with values or rows, respectively The right hand side member of the linear system.
- rhssequence of float or
- Returns:
Notes
This will handle both matrices and vectors. Note that you’d better type explicitly the matrix if it has some properties that could simplify the resolution (see
TriangularMatrix
).This uses LAPACK’S DGESV for matrices and DGELSY for vectors.
Examples
>>> import openturns as ot >>> import numpy as np >>> M = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]]) >>> b = [1.0] * 2 >>> x = M.solveLinearSystem(b) >>> np.testing.assert_array_almost_equal(M * x, b)
- solveLinearSystemInPlace(*args)¶
Solve a rectangular linear system whose the present matrix is the operator.
Similar to
solveLinearSystem()
except the matrix is modified in-place during the resolution avoiding the need to allocate an extra copy if the original copy is not re-used.
- squareElements()¶
Square the Matrix, ie each element of the matrix is squared.
Examples
>>> import openturns as ot >>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]) >>> M.squareElements() >>> print(M) [[ 1 4 ] [ 9 16 ] [ 25 36 ]]
- transpose()¶
Transpose the matrix.
- Returns:
- MT
SquareMatrix
The transposed matrix.
- MT
Examples
>>> import openturns as ot >>> M = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]]) >>> print(M) [[ 1 2 ] [ 3 4 ]] >>> print(M.transpose()) [[ 1 3 ] [ 2 4 ]]
Examples using the class¶
Estimate Wilks and empirical quantile
Estimate a multivariate ARMA process
Estimate a non stationary covariance function
Create and draw multivariate distributions
Quick start guide to distributions
Draw minimum volume level sets
Extract the copula from a distribution
Create a functional basis process
Create a parametric spectral density function
Create a stationary covariance model
Create a custom covariance model
Sample trajectories from a Gaussian Process with correlated outputs
Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos
Kriging :configure the optimization solver
Analyse the central tendency of a cantilever beam
Estimate moments from Taylor expansions
Use the post-analytical importance sampling algorithm
Time variant system reliability problem
Create unions or intersections of events
Create an event based on a process
Estimate Sobol indices on a field to point function
Parallel coordinates graph as sensitivity tool
Create mixed deterministic and probabilistic designs of experiments
Calibration without observed inputs
Calibration of the deflection of a tube
Calibration of the flooding model
Calibration of the Chaboche mechanical model
Bayesian calibration of a computer code
Bayesian calibration of the flooding model
Linear Regression with interval-censored observations
Compute leave-one-out error of a polynomial chaos expansion