butools.ph.APHFrom3Moments¶
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butools.ph.APHFrom3Moments()¶ Matlab: [alpha, A] = APHFrom3Moments(moms, maxSize)Mathematica: {alpha, A} = APHFrom3Moments[moms, maxSize]Python/Numpy: alpha, A = APHFrom3Moments(moms, maxSize)Returns an acyclic PH which has the same 3 moments as given. If detects the order and the structure automatically to match the given moments.
Parameters: moms : vector of doubles, length(3)
The moments to match
maxSize : int, optional
The maximal size of the resulting APH. The default value is 100.
Returns: alpha : vector, shape (1,M)
The initial probability vector of the APH
A : matrix, shape (M,M)
Transient generator matrix of the APH
Raises an error if the moments are not feasible with an
APH of size “maxSize”.
References
[R4] A. Bobbio, A. Horvath, M. Telek, “Matching three moments with minimal acyclic phase type distributions,” Stochastic models, pp. 303-326, 2005. Examples
For Matlab:
>>> moms = [10.0, 125.0, 8400.0]; >>> [a, A] = APHFrom3Moments(moms); >>> disp(a); 1.3212e-05 0.99999 0 0 0 0 >>> disp(A); -0.0022936 0.0022936 0 0 0 0 0 -0.50029 0.50029 0 0 0 0 0 -0.50029 0.50029 0 0 0 0 0 -0.50029 0.50029 0 0 0 0 0 -0.50029 0.50029 0 0 0 0 0 -0.50029 >>> phmoms = MomentsFromPH(a, A, 3); >>> disp(phmoms); 10 125 8400 >>> moms = [10.0, 525.0, 31400.0]; >>> [a, A] = APHFrom3Moments(moms); >>> disp(a); Columns 1 through 6 0.21179 0 0 0 0 0 Columns 7 through 8 0 0.78821 >>> disp(A); Columns 1 through 6 -0.15079 0.15079 0 0 0 0 0 -0.15079 0.15079 0 0 0 0 0 -0.15079 0.15079 0 0 0 0 0 -0.15079 0.15079 0 0 0 0 0 -0.15079 0.15079 0 0 0 0 0 -0.15079 0 0 0 0 0 0 0 0 0 0 0 0 Columns 7 through 8 0 0 0 0 0 0 0 0 0 0 0.15079 0 -0.15079 0.15079 0 -5.9502 >>> phmoms = MomentsFromPH(a, A, 3); CheckMERepresentation warning: There are more than one eigenvalue with the same absolute value as the largest eigenvalue! >>> disp(phmoms); 10 525 31400
For Mathematica:
>>> moms = {10.0, 125.0, 8400.0}; >>> {a, A} = APHFrom3Moments[moms]; >>> Print[a]; {0.000013211714931595383, 0.9999867882850684, 0, 0, 0, 0} >>> Print[A]; {{-0.002293559800424418, 0.002293559800424418, 0, 0, 0, 0}, {0, -0.5002881836726082, 0.5002881836726082, 0, 0, 0}, {0, 0, -0.5002881836726082, 0.5002881836726082, 0, 0}, {0, 0, 0, -0.5002881836726082, 0.5002881836726082, 0}, {0, 0, 0, 0, -0.5002881836726082, 0.5002881836726082}, {0, 0, 0, 0, 0, -0.5002881836726082}} >>> phmoms = MomentsFromPH[a, A, 3]; >>> Print[phmoms]; {10., 125.00000000000003, 8400.000000000162} >>> moms = {10.0, 525.0, 31400.0}; >>> {a, A} = APHFrom3Moments[moms]; >>> Print[a]; {0.21178752772283896, 0, 0, 0, 0, 0, 0, 0.7882124722771611} >>> Print[A]; {{-0.15078539360511473, 0.15078539360511473, 0, 0, 0, 0, 0, 0}, {0, -0.15078539360511473, 0.15078539360511473, 0, 0, 0, 0, 0}, {0, 0, -0.15078539360511473, 0.15078539360511473, 0, 0, 0, 0}, {0, 0, 0, -0.15078539360511473, 0.15078539360511473, 0, 0, 0}, {0, 0, 0, 0, -0.15078539360511473, 0.15078539360511473, 0, 0}, {0, 0, 0, 0, 0, -0.15078539360511473, 0.15078539360511473, 0}, {0, 0, 0, 0, 0, 0, -0.15078539360511473, 0.15078539360511473}, {0, 0, 0, 0, 0, 0, 0, -5.950197455082666}} >>> phmoms = MomentsFromPH[a, A, 3]; >>> Print[phmoms]; {9.999999999999998, 524.9999999999998, 31399.99999999998}
For Python/Numpy:
>>> moms = [10.0, 125.0, 8400.0] >>> a, A = APHFrom3Moments(moms) >>> print(a) [[ 1.32117e-05 9.99987e-01 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00]] >>> print(A) [[-0.00229 0.00229 0. 0. 0. 0. ] [ 0. -0.50029 0.50029 0. 0. 0. ] [ 0. 0. -0.50029 0.50029 0. 0. ] [ 0. 0. 0. -0.50029 0.50029 0. ] [ 0. 0. 0. 0. -0.50029 0.50029] [ 0. 0. 0. 0. 0. -0.50029]] >>> phmoms = MomentsFromPH(a, A, 3) >>> print(phmoms) [9.9999999999999964, 124.99999999999994, 8400.0000000001164] >>> moms = [10.0, 525.0, 31400.0] >>> a, A = APHFrom3Moments(moms) >>> print(a) [[ 0.21179 0. 0. 0. 0. 0. 0. 0.78821]] >>> print(A) [[-0.15079 0.15079 0. 0. 0. 0. 0. 0. ] [ 0. -0.15079 0.15079 0. 0. 0. 0. 0. ] [ 0. 0. -0.15079 0.15079 0. 0. 0. 0. ] [ 0. 0. 0. -0.15079 0.15079 0. 0. 0. ] [ 0. 0. 0. 0. -0.15079 0.15079 0. 0. ] [ 0. 0. 0. 0. 0. -0.15079 0.15079 0. ] [ 0. 0. 0. 0. 0. 0. -0.15079 0.15079] [ 0. 0. 0. 0. 0. 0. 0. -5.9502 ]] >>> phmoms = MomentsFromPH(a, A, 3) CheckMERepresentation warning: There are more than one eigenvalue with the same absolute value as the largest eigenvalue! >>> print(phmoms) [10.0, 525.0, 31399.999999999993]