butools.ph.PH3From5Moments¶

butools.ph.PH3From5Moments()¶

Matlab:	`[alpha, A] = PH3From5Moments(moms)`
Mathematica:	`{alpha, A} = PH3From5Moments[moms]`
Python/Numpy:	`alpha, A = PH3From5Moments(moms)`

Returns a PH(3) which has the same 5 moments as given.

Parameters:

moms : vector of doubles, length(5)

The moments to match

Returns:

alpha : vector, shape (1,3)

The initial probability vector of the PH(3)

A : matrix, shape (3,3)

Transient generator matrix of the PH(3)

Notes

Raises an error if the moments are not feasible with a PH(3). Also note that the numerical behavior of the procedure can be poor if the moments are close to the boundary of the feasible region.

References

[R40]

G. Horvath and M. Telek, “On the canonical representation of phase type distributions,” Performance Evaluation, vol. 66, no. 8, pp. 396 - 409, 2009.

Examples

For Matlab:

>>> a = [0.1,0.9,0];
>>> A = [-6.2, 2, 0; 2, -9, 1; 1, 0, -3];
>>> moms = MomentsFromPH(a, A);
>>> disp(moms);
      0.20939      0.10449     0.089092      0.11027      0.17953
>>> [a, A] = PH3From5Moments(moms);
>>> disp(a);
      0.58305      0.32736     0.089589
>>> disp(A);
      -9.9819            0            0
       5.3405      -5.3405            0
            0       2.8776      -2.8776
>>> phmoms = MomentsFromME(a, A, 5);
>>> disp(phmoms);
      0.20939      0.10449     0.089092      0.11027      0.17953
>>> a = [0.2,0.3,0.5];
>>> A = [-1, 0, 0; 0, -3, 0.5; 0, -0.5, -3];
>>> moms = MomentsFromME(a, A);
>>> disp(moms);
      0.44865       0.5496       1.3298       4.9428       24.182
>>> [a, A] = PH3From5Moments(moms);
>>> disp(a);
      0.94865     0.036778     0.014574
>>> disp(A);
           -3            0      0.15385
        2.866       -2.866            0
            0        1.134       -1.134
>>> phmoms = MomentsFromME(a, A, 5);
>>> disp(phmoms);
      0.44865       0.5496       1.3298       4.9428       24.182

For Mathematica:

>>> a = {0.1,0.9,0};
>>> A = {{-6.2, 2, 0},{2, -9, 1},{1, 0, -3}};
>>> moms = MomentsFromPH[a, A];
>>> Print[moms];
{0.20938722294654497, 0.10448912014333092, 0.08909150039190732, 0.11026674096545433, 0.179530273247209}
>>> {a, A} = PH3From5Moments[moms];
>>> Print[a];
{0.5830541253440302, 0.3273566132692404, 0.08958926138672949}
>>> Print[A];
{{-9.981920626264277, 0., 0.},
 {5.340471031780809, -5.340471031780809, 0.},
 {0., 2.8776083419564285, -2.8776083419564285}}
>>> phmoms = MomentsFromME[a, A, 5];
>>> Print[phmoms];
{0.2093872229465451, 0.10448912014333109, 0.08909150039190755, 0.11026674096545479, 0.17953027324721005}
>>> a = {0.2,0.3,0.5};
>>> A = {{-1, 0, 0},{0, -3, 0.5},{0, -0.5, -3}};
>>> moms = MomentsFromME[a, A];
>>> Print[moms];
{0.4486486486486486, 0.5495982468955442, 1.3298244921327464, 4.942768524155609, 24.182331446704147}
>>> {a, A} = PH3From5Moments[moms];
>>> Print[a];
{0.9690995849472338, 0.014984267780593008, 0.015916147272173234}
>>> Print[A];
{{-2.9342585459110544, 0., 0.1481671658935929},
 {2.9342585459110566, -2.9342585459110566, 0.},
 {0., 1.1314829081787512, -1.1314829081787512}}
>>> phmoms = MomentsFromME[a, A, 5];
>>> Print[phmoms];
{0.4486486486486484, 0.5495982468955433, 1.329824492132743, 4.942768524155592, 24.182331446704055}

For Python/Numpy:

>>> a = ml.matrix([[0.1,0.9,0]])
>>> A = ml.matrix([[-6.2, 2, 0],[2, -9, 1],[1, 0, -3]])
>>> moms = MomentsFromPH(a, A)
>>> print(moms)
[0.20938722294654497, 0.10448912014333091, 0.089091500391907288, 0.11026674096545433, 0.17953027324720897]
>>> a, A = PH3From5Moments(moms)
>>> print(a)
[[ 0.58305  0.32736  0.08959]]
>>> print(A)
[[-9.98192  0.       0.     ]
 [ 5.34047 -5.34047  0.     ]
 [ 0.       2.87761 -2.87761]]
>>> phmoms = MomentsFromME(a, A, 5)
>>> print(phmoms)
[0.20938722294654719, 0.10448912014333331, 0.089091500391910647, 0.11026674096546003, 0.17953027324722048]
>>> a = ml.matrix([[0.2,0.3,0.5]])
>>> A = ml.matrix([[-1, 0, 0],[0, -3, 0.5],[0, -0.5, -3]])
>>> moms = MomentsFromME(a, A)
>>> print(moms)
[0.44864864864864862, 0.54959824689554426, 1.3298244921327464, 4.9427685241556087, 24.182331446704147]
>>> a, A = PH3From5Moments(moms)
>>> print(a)
[[ 0.94865  0.03678  0.01457]]
>>> print(A)
[[-3.       0.       0.15385]
 [ 2.86603 -2.86603  0.     ]
 [ 0.       1.13397 -1.13397]]
>>> phmoms = MomentsFromME(a, A, 5)
>>> print(phmoms)
[0.44864864864864928, 0.54959824689554582, 1.3298244921327509, 4.9427685241556283, 24.18233144670425]