butools.ph.AcyclicPHFromME¶

butools.ph.AcyclicPHFromME()¶

Matlab:	[beta, B] = AcyclicPHFromME(alpha, A, maxSize, precision)
Mathematica:	{beta, B} = AcyclicPHFromME[alpha, A, maxSize, precision]
Python/Numpy:	beta, B = AcyclicPHFromME(alpha, A, maxSize, precision)

Transforms an arbitrary matrix-exponential representation to an acyclic phase-type representation. (see [R5]).

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 acyclic representation B : matrix, shape (M,M) Transient generator matrix of the Markovian acyclic representation

Notes

Raises an error if no Markovian acyclic representation has been found.

References

[R5]	(1, 2) Mocanu, S., Commault, C.: “Sparse representations of phase-type distributions,” Stoch. Models 15, 759-778 (1999)

Examples

For Matlab:

>>> a = [-0.4,1.4,0.];
>>> A = [-4., 1., 1.; 0., -2., 1.; 1., 0., -8.];
>>> [b, B] = AcyclicPHFromME(a, A);
>>> disp(b);
      0.55273       0.3741     0.073173
>>> disp(B);
      -1.9145       1.9145            0
            0      -3.8858       3.8858
            0            0      -8.1997
>>> ma = MomentsFromME(a, A, 5);
>>> disp(ma);
      0.64918      0.73131       1.1825       2.5062       6.5898
>>> mb = MomentsFromME(b, B, 5);
>>> disp(mb);
      0.64918      0.73131       1.1825       2.5062       6.5898

For Mathematica:

>>> a = {-0.4,1.4,0.};
>>> A = {{-4., 1., 1.},{0., -2., 1.},{1., 0., -8.}};
>>> {b, B} = AcyclicPHFromME[a, A];
>>> Print[b];
{0.5527262576934738, 0.3741003779708655, 0.07317336433566028}
>>> Print[B];
{{-1.914468283493477, 1.914468283493477, 0},
 {0, -3.8858267357749496, 3.8858267357749496},
 {0, 0, -8.199704980731571}}
>>> ma = MomentsFromME[a, A, 5];
>>> Print[ma];
{0.6491803278688524, 0.7313087879602256, 1.1825377454500594, 2.50620910640242, 6.589761685446926}
>>> mb = MomentsFromME[b, B, 5];
>>> Print[mb];
{0.6491803278688524, 0.7313087879602258, 1.18253774545006, 2.506209106402422, 6.5897616854469305}

For Python/Numpy:

>>> a = ml.matrix([[-0.4,1.4,0.]])
>>> A = ml.matrix([[-4., 1., 1.],[0., -2., 1.],[1., 0., -8.]])
>>> b, B = AcyclicPHFromME(a, A)
>>> print(b)
[[ 0.55273  0.3741   0.07317]]
>>> print(B)
[[-1.91447  1.91447  0.     ]
 [ 0.      -3.88583  3.88583]
 [ 0.       0.      -8.1997 ]]
>>> ma = MomentsFromME(a, A, 5)
>>> print(ma)
[0.64918032786885238, 0.73130878796022558, 1.1825377454500594, 2.5062091064024199, 6.5897616854469261]
>>> mb = MomentsFromME(b, B, 5)
>>> print(mb)
[0.64918032649380331, 0.73130878479925954, 1.182537737569604, 2.5062090835980859, 6.5897616090589066]