butools.ph.MonocyclicPHFromME¶
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butools.ph.MonocyclicPHFromME()¶ Matlab: [beta, B] = MonocyclicPHFromME(alpha, A, maxSize, precision)Mathematica: {beta, B} = MonocyclicPHFromME[alpha, A, maxSize, precision]Python/Numpy: beta, B = MonocyclicPHFromME(alpha, A, maxSize, precision)Transforms an arbitrary matrix-exponential representation to a Markovian monocyclic representation.
Parameters: alpha : matrix, shape (1,N)
Initial vector of the distribution
A : matrix, shape (N,N)
Matrix parameter of the distribution
maxSize : int, optional
The maximum number of phases for the result. The default value is 100.
precision : double, optional
Vector and matrix entries smaller than the precision are considered to be zeros. The default value is 1e-14.
Returns: beta : matrix, shape (1,M)
The initial probability vector of the Markovian monocyclic representation
B : matrix, shape (M,M)
Transient generator matrix of the Markovian monocyclic representation
Notes
Raises an error if no Markovian monocyclic representation has been found.
References
[R38] Mocanu, S., Commault, C.: “Sparse representations of phase-type distributions,” Stoch. Models 15, 759-778 (1999) Examples
For Matlab:
>>> a = [0.2,0.3,0.5]; >>> A = [-1., 0., 0.; 0., -3., 2.; 0., -2., -3.]; >>> [b, B] = MonocyclicPHFromME(a, A); >>> disp(b); Columns 1 through 6 -8.1021e-17 0.010077 0.019799 0.0077544 0.0010922 4.0852e-05 Columns 7 through 12 1.3538e-05 0.00011267 0.00039486 0.00093038 0.0018049 0.0031208 Columns 13 through 18 0.0049985 0.0075762 0.01101 0.015473 0.021149 0.028228 Columns 19 through 24 0.036901 0.047341 0.059692 0.074045 0.090409 0.10868 Columns 25 through 27 0.12858 0.14965 0.17112 >>> disp(B); Columns 1 through 6 -1 1 0 0 0 0 0 -3 3 0 0 0 0 0 -3 3 0 0 0 0 0 -3 3 0 0 0.59259 0 0 -3 2.4074 0 0 0 0 0 -17.531 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 7 through 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 13 through 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 19 through 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 17.531 0 0 0 0 0 -17.531 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 25 through 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17.531 0 0 -17.531 17.531 0 0 -17.531 17.531 0 0 -17.531 >>> ma = MomentsFromME(a, A, 5); >>> disp(ma); 0.35385 0.41893 1.1552 4.6998 23.838 >>> mb = MomentsFromME(b, B, 5); >>> disp(mb); 0.35385 0.41893 1.1552 4.6998 23.838
For Mathematica:
>>> a = {0.2,0.3,0.5}; >>> A = {{-1., 0., 0.},{0., -3., 2.},{0., -2., -3.}}; >>> {b, B} = MonocyclicPHFromME[a, A]; >>> Print[b]; {-2.5397600778722706*^-17, 0.010076564579360592, 0.019799026572787613, 0.007754395700638792, 0.0010921892175280797, 0.00004085157730567139, 0.000013537575173626456, 0.00011267376891308164, 0.0003948571244651649, 0.0009303831689188193, 0.001804898763447438, 0.003120837336642398, 0.004998456760755956, 0.007576247022797484, 0.01101041315944734, 0.01547306860911693, 0.02114869573602008, 0.028228345073631006, 0.036900955049397284, 0.04734108305260393, 0.059692251707772884, 0.07404503804493777, 0.09040897715773615, 0.10867732789998653, 0.1285837713807389, 0.1496502023195005, 0.1711249516403759} >>> Print[B]; {{-1., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., -3., 3., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., -3., 3., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., -3., 3., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0.5925925925925928, 0., 0., -3., 2.4074074074074074, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363, 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363, 17.531049512315363}, {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -17.531049512315363}} >>> ma = MomentsFromME[a, A, 5]; >>> Print[ma]; {0.3538461538461538, 0.41893491124260357, 1.1552116522530724, 4.699835439935577, 23.837756165615836} >>> mb = MomentsFromME[b, B, 5]; >>> Print[mb]; {0.3538461538461526, 0.4189349112425999, 1.1552116522530596, 4.69983543993552, 23.83775616561553}
For Python/Numpy:
>>> a = ml.matrix([[0.2,0.3,0.5]]) >>> A = ml.matrix([[-1., 0., 0.],[0., -3., 2.],[0., -2., -3.]]) >>> b, B = MonocyclicPHFromME(a, A) >>> print(b) [[ -4.29548e-17 1.00766e-02 1.97990e-02 7.75440e-03 1.09219e-03 4.08516e-05 1.35376e-05 1.12674e-04 3.94857e-04 9.30383e-04 1.80490e-03 3.12084e-03 4.99846e-03 7.57625e-03 1.10104e-02 1.54731e-02 2.11487e-02 2.82283e-02 3.69010e-02 4.73411e-02 5.96923e-02 7.40450e-02 9.04090e-02 1.08677e-01 1.28584e-01 1.49650e-01 1.71125e-01]] >>> print(B) [[ -1. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. -3. 3. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. -3. 3. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. -3. 3. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0.59259 0. 0. -3. 2.40741 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105 0. ] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105 17.53105] [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17.53105]] >>> ma = MomentsFromME(a, A, 5) >>> print(ma) [0.35384615384615381, 0.41893491124260357, 1.1552116522530724, 4.6998354399355771, 23.837756165615836] >>> mb = MomentsFromME(b, B, 5) >>> print(mb) [0.35384615384615531, 0.41893491124260573, 1.155211652253076, 4.699835439935578, 23.83775616561579]