butools.ph.PHFromME¶

butools.ph.PHFromME()¶

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

Obtains a Markovian representation of a matrix exponential distribution of the same size, if possible.

Parameters:	alpha : vector, shape (1,M) The initial vector of the matrix-exponential distribution. A : matrix, shape (M,M) The matrix parameter of the matrix-exponential distribution. precision : double, optional A representation is considered to be a Markovian one if it is closer than the precision. The default value is 1e-14.
Returns:	beta : vector, 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

References

[R41]

G Horváth, M Telek, “A minimal representation of Markov arrival processes and a moments matching method,” Performance Evaluation 64:(9-12) pp. 1153-1168. (2007)

Examples

For Matlab:

>>> a = [-0.4,1.4];
>>> A = [-3.8, 2; 2, -9];
>>> flag = CheckMERepresentation(a, A);
>>> disp(flag);
     1
>>> flag = CheckPHRepresentation(a, A);
CheckProbVector: The vector has negative element (precision: 1e-12)!
>>> disp(flag);
     0
>>> [b, B] = PHFromME(a, A);
>>> disp(b);
     0.013037      0.98696
>>> disp(B);
      -3.2605       2.5924
      0.34843      -9.5395
>>> flag = CheckPHRepresentation(b, B);
>>> disp(flag);
     1
>>> Cm = SimilarityMatrix(A, B);
>>> err1 = norm(A*Cm-Cm*B);
>>> err2 = norm(a*Cm-b);
>>> disp(max(err1, err2));
   1.0162e-15
>>> a = [-0.5,1.5];
>>> A = [-3.8, 2; 2, -9];
>>> flag = CheckMERepresentation(a, A);
>>> disp(flag);
     1
>>> flag = CheckPHRepresentation(a, A);
CheckProbVector: The vector has negative element (precision: 1e-12)!
>>> disp(flag);
     0
>>> [b, B] = PHFromME(a, A);
>>> disp(b);
    0.0057038       0.9943
>>> disp(B);
      -3.1279       3.0636
     0.017405      -9.6721
>>> flag = CheckPHRepresentation(b, B);
>>> disp(flag);
     1
>>> Cm = SimilarityMatrix(A, B);
>>> err1 = norm(A*Cm-Cm*B);
>>> err2 = norm(a*Cm-b);
>>> disp(max(err1, err2));
   3.0445e-15

For Mathematica:

>>> a = {-0.4,1.4};
>>> A = {{-3.8, 2},{2, -9}};
>>> flag = CheckMERepresentation[a, A];
>>> Print[flag];
True
>>> flag = CheckPHRepresentation[a, A];
"CheckProbVector: The vector has negative element!"
>>> Print[flag];
False
>>> {b, B} = PHFromME[a, A];
>>> Print[b];
{0.013037109374999953, 0.9869628906249999}
>>> Print[B];
{{-3.2604571906887756, 2.5924299798044217},
 {0.3484263627325931, -9.539542809311223}}
>>> flag = CheckPHRepresentation[b, B];
>>> Print[flag];
True
>>> Cm = SimilarityMatrix[A, B];
>>> err1 = Norm[A.Cm-Cm.B];
>>> err2 = Norm[a.Cm-b];
>>> Print[Max[err1, err2]];
1.1393205654608455*^-15
>>> a = {-0.5,1.5};
>>> A = {{-3.8, 2},{2, -9}};
>>> flag = CheckMERepresentation[a, A];
>>> Print[flag];
True
>>> flag = CheckPHRepresentation[a, A];
"CheckProbVector: The vector has negative element!"
>>> Print[flag];
False
>>> {b, B} = PHFromME[a, A];
>>> Print[b];
{0.0057037812657654285, 0.9942962187342346}
>>> Print[B];
{{-3.1278937744575632, 3.0635853844348873},
 {0.017404720964309853, -9.672106225542432}}
>>> flag = CheckPHRepresentation[b, B];
>>> Print[flag];
True
>>> Cm = SimilarityMatrix[A, B];
>>> err1 = Norm[A.Cm-Cm.B];
>>> err2 = Norm[a.Cm-b];
>>> Print[Max[err1, err2]];
2.30940485129664*^-15

For Python/Numpy:

>>> a = ml.matrix([[-0.4,1.4]])
>>> A = ml.matrix([[-3.8, 2],[2, -9]])
>>> flag = CheckMERepresentation(a, A)
>>> print(flag)
True
>>> flag = CheckPHRepresentation(a, A)
CheckProbVector: The vector has negative element (precision: 1e-12)!
>>> print(flag)
False
>>> b, B = PHFromME(a, A)
>>> print(b)
[[ 0.01304  0.98696]]
>>> print(B)
[[-3.26046  2.59243]
 [ 0.34843 -9.53954]]
>>> flag = CheckPHRepresentation(b, B)
>>> print(flag)
True
>>> Cm = SimilarityMatrix(A, B)
>>> err1 = la.norm(A*Cm-Cm*B)
>>> err2 = la.norm(a*Cm-b)
>>> print(np.max(err1, err2))
1.18018326364e-15
>>> a = ml.matrix([[-0.5,1.5]])
>>> A = ml.matrix([[-3.8, 2],[2, -9]])
>>> flag = CheckMERepresentation(a, A)
>>> print(flag)
True
>>> flag = CheckPHRepresentation(a, A)
CheckProbVector: The vector has negative element (precision: 1e-12)!
>>> print(flag)
False
>>> b, B = PHFromME(a, A)
>>> print(b)
[[ 0.0057  0.9943]]
>>> print(B)
[[-3.12789  3.06359]
 [ 0.0174  -9.67211]]
>>> flag = CheckPHRepresentation(b, B)
>>> print(flag)
True
>>> Cm = SimilarityMatrix(A, B)
>>> err1 = la.norm(A*Cm-Cm*B)
>>> err2 = la.norm(a*Cm-b)
>>> print(np.max(err1, err2))
2.23152066184e-15