butools.dph.DPHFromMG¶

butools.dph.DPHFromMG()¶

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

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

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

References

[R10]

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 = [-0.6,0.3,1.3];
>>> A = [0.1, 0.2, 0; 0.3, 0.1, 0.25; -0.3, 0.2, 0.77];
>>> flag = CheckMGRepresentation(a, A);
>>> disp(flag);
     1
>>> flag = CheckDPHRepresentation(a, A);
CheckProbMatrix: the matrix has negative element (precision: 1e-12)!
>>> disp(flag);
     0
>>> [b, B] = DPHFromMG(a, A);
>>> disp(b);
         0.05       0.1375       0.8125
>>> disp(B);
          0.1          0.2            0
        0.425      0.06875      0.15625
        0.141      0.01975      0.80125
>>> flag = CheckDPHRepresentation(b, B);
>>> disp(flag);
     1
>>> Cm = SimilarityMatrix(A, B);
>>> err1 = norm(A*Cm-Cm*B);
>>> err2 = norm(a*Cm-b);
>>> disp(max(err1, err2));
   4.0704e-16

For Mathematica:

>>> a = {-0.6,0.3,1.3};
>>> A = {{0.1, 0.2, 0},{0.3, 0.1, 0.25},{-0.3, 0.2, 0.77}};
>>> flag = CheckMGRepresentation[a, A];
>>> Print[flag];
True
>>> flag = CheckDPHRepresentation[a, A];
"CheckProbVector: The vector has negative element!"
>>> Print[flag];
False
>>> {b, B} = DPHFromMG[a, A];
>>> Print[b];
{0.050000000000000044, 0.13749999999999998, 0.8125}
>>> Print[B];
{{0.1, 0.2, 0.},
 {0.425, 0.06875, 0.15625},
 {0.14100000000000007, 0.019750000000000018, 0.8012500000000001}}
>>> flag = CheckDPHRepresentation[b, B];
>>> Print[flag];
True
>>> Cm = SimilarityMatrix[A, B];
>>> err1 = Norm[A.Cm-Cm.B];
>>> err2 = Norm[a.Cm-b];
>>> Print[Max[err1, err2]];
4.783309287441108*^-16

For Python/Numpy:

>>> a = ml.matrix([[-0.6,0.3,1.3]])
>>> A = ml.matrix([[0.1, 0.2, 0],[0.3, 0.1, 0.25],[-0.3, 0.2, 0.77]])
>>> flag = CheckMGRepresentation(a, A)
>>> print(flag)
True
>>> flag = CheckDPHRepresentation(a, A)
CheckProbMatrix: the matrix has negative element (precision: 1e-12)!
>>> print(flag)
False
>>> b, B = DPHFromMG(a, A)
>>> print(b)
[[ 0.05    0.1375  0.8125]]
>>> print(B)
[[ 0.1      0.2      0.     ]
 [ 0.425    0.06875  0.15625]
 [ 0.141    0.01975  0.80125]]
>>> flag = CheckDPHRepresentation(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))
3.34426942151e-16