butools.map.MAPFromFewMomentsAndCorrelations¶
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butools.map.MAPFromFewMomentsAndCorrelations()¶ Matlab: [D0, D1] = MAPFromFewMomentsAndCorrelations(moms, corr1, r)Mathematica: {D0, D1} = MAPFromFewMomentsAndCorrelations[moms, corr1, r]Python/Numpy: D0, D1 = MAPFromFewMomentsAndCorrelations(moms, corr1, r)Creates a Markovian arrival process that has the given 2 or 3 marginal moments and lag-1 autocorrelation. The decay of the autocorrelation function can be optionally adjusted as well. The lag-k autocorrelation function \(\rho_k\) of the resulting MAP is \(\rho_k=r(corr_1/r)^k\).
Parameters: moms : vector of doubles, length 2 or 3
The list of marginal moments to match.
corr1 : double
The lag-1 autocorrelation coefficient to match.
r : double, optional
The decay of the autocorrelation function.
Returns: D0 : matrix, shape (M,M)
The D0 matrix of the Markovian arrival process
D1 : matrix, shape (M,M)
The D1 matrix of the Markovian arrival process
Notes
With 2 marginal moments, or with 3 marginal moments and positive autocorrelation the procedure always returns a valid Markovian representation.
References
[R19] G Horvath, “Matching marginal moments and lag autocorrelations with MAPs,” ValueTools 2013, Torino, Italy (2013). Examples
For Matlab:
>>> moms = [1.1, 6.05]; >>> corr1 = -0.17; >>> [D0, D1] = MAPFromFewMomentsAndCorrelations(moms, corr1); >>> disp(D0); -0.28494 0.28494 0 0 0 -18.134 0 0 0 0 -0.28494 0.28494 0 0 0 -0.95707 >>> disp(D1); 0 0 0 0 0.022795 1.4279 4.9669 11.716 0 0 0 0 0.013835 0.86667 0.022795 0.053771 >>> rmoms = MarginalMomentsFromMAP(D0, D1, 2); >>> disp(rmoms); 1.1 6.05 >>> rcorr1 = LagCorrelationsFromMAP(D0, D1, 1); >>> disp(rcorr1); -0.17 >>> moms = [1.2, 4.32, 20.]; >>> corr1 = -0.4; >>> [D0, D1] = MAPFromFewMomentsAndCorrelations(moms, corr1); >>> disp(D0); -0.33604 0.33604 0 0 0 0 0 -36.027 0 0 0 0 0 0 -1.2829 1.2829 0 0 0 0 0 -1.2829 1.2829 0 0 0 0 0 -1.2829 1.2829 0 0 0 0 0 -1.3747 >>> disp(D1); 0 0 0 0 0 0 0.024741 1.7766 23.475 0 0 10.751 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.017937 1.288 0.047145 0 0 0.021591 >>> BuToolsCheckPrecision = 10.^-12; >>> rmoms = MarginalMomentsFromMAP(D0, D1, 3); >>> disp(rmoms); 1.2 4.32 20 >>> rcorr1 = LagCorrelationsFromMAP(D0, D1, 1); >>> disp(rcorr1); -0.4 >>> moms = [1.2, 4.32, 20.]; >>> corr1 = 0.4; >>> [D0, D1] = MAPFromFewMomentsAndCorrelations(moms, corr1); >>> disp(D0); Columns 1 through 6 -0.54184 0.54184 0 0 0 0 0 -116.34 0 0 0 0 0 0 -0.24417 0.24417 0 0 0 0 0 -2.014 2.014 0 0 0 0 0 -2.014 2.014 0 0 0 0 0 -2.014 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 7 through 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.014 0 0 -2.014 2.014 0 0 -2.014 2.014 0 0 -2.014 >>> disp(D1); Columns 1 through 6 0 0 0 0 0 0 16.128 88.575 0.099774 11.534 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.062048 0.34076 0.013818 1.5974 0 0 Columns 7 through 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 >>> rmoms = MarginalMomentsFromMAP(D0, D1, 3); >>> disp(rmoms); 1.2 4.32 20 >>> rcorr1 = LagCorrelationsFromMAP(D0, D1, 1); >>> disp(rcorr1); 0.4
For Mathematica:
>>> moms = {1.1, 6.05}; >>> corr1 = -0.17; >>> {D0, D1} = MAPFromFewMomentsAndCorrelations[moms, corr1]; >>> Print[D0]; {{-0.28493894165535966, 0.28493894165535966, 0, 0}, {0., -18.13383801258688, 0, 0}, {0, 0, -0.2849389416553596, 0.2849389416553596}, {0, 0, 0., -0.9570710783221891}} >>> Print[D1]; {{0., 0., 0., 0.}, {0.022795115332428794, 1.4279119256745232, 4.966897224470778, 11.71623374710915}, {0., 0., 0., 0.}, {0.013835475664900425, 0.8666699163915135, 0.022795115332428787, 0.053770570933346404}} >>> rmoms = MarginalMomentsFromMAP[D0, D1, 2]; >>> Print[rmoms]; {1.0999999999999996, 6.049999999999997} >>> rcorr1 = LagCorrelationsFromMAP[D0, D1, 1][[1]]; >>> Print[rcorr1]; -0.16999999999999998 >>> moms = {1.2, 4.32, 20.}; >>> corr1 = -0.4; >>> {D0, D1} = MAPFromFewMomentsAndCorrelations[moms, corr1]; >>> Print[D0]; {{-0.3360418695910829, 0.3360418695910829, 0, 0, 0, 0}, {0, -36.0266728289002, 0, 0, 0, 0}, {0, 0, -1.2828628105692446, 1.2828628105692446, 0, 0}, {0, 0, 0, -1.2828628105692446, 1.2828628105692446, 0}, {0, 0, 0, 0, -1.2828628105692446, 1.2828628105692446}, {0, 0, 0, 0, 0, -1.3747185152915538}} >>> Print[D1]; {{0., 0., 0., 0., 0., 0.}, {0.024740745552158373, 1.7765928958928536, 23.47455308878542, 0., 0., 10.750786098669774}, {0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0.}, {0.017937256152678223, 1.2880453333742978, 0.04714475230266865, 0., 0., 0.021591173461909096}} >>> BuTools`CheckPrecision = 10.^-12; >>> rmoms = MarginalMomentsFromMAP[D0, D1, 3]; >>> Print[rmoms]; {1.2, 4.319999999999999, 20.00000000000002} >>> rcorr1 = LagCorrelationsFromMAP[D0, D1, 1][[1]]; >>> Print[rcorr1]; -0.4 >>> moms = {1.2, 4.32, 20.}; >>> corr1 = 0.4; >>> {D0, D1} = MAPFromFewMomentsAndCorrelations[moms, corr1]; >>> Print[D0]; {{-0.5418383137500514, 0.5418383137500514, 0, 0, 0, 0, 0, 0, 0}, {0, -116.33647358652203, 0, 0, 0, 0, 0, 0, 0}, {0, 0, -0.24417076816625743, 0.24417076816625743, 0, 0, 0, 0, 0}, {0, 0, 0, -2.0140275715489198, 2.0140275715489198, 0, 0, 0, 0}, {0, 0, 0, 0, -2.0140275715489198, 2.0140275715489198, 0, 0, 0}, {0, 0, 0, 0, 0, -2.0140275715489198, 2.0140275715489198, 0, 0}, {0, 0, 0, 0, 0, 0, -2.0140275715489198, 2.0140275715489198, 0}, {0, 0, 0, 0, 0, 0, 0, -2.0140275715489198, 2.0140275715489198}, {0, 0, 0, 0, 0, 0, 0, 0, -2.0140275715489198}} >>> Print[D1]; {{0., 0., 0., 0., 0., 0., 0., 0., 0.}, {16.128313181474883, 88.57451304639496, 0.0997738996285687, 11.533873459023635, 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0., 0., 0., 0., 0., 0., 0., 0., 0.}, {0.062047737583266864, 0.3407577767265171, 0.013818358324120445, 1.5974036989150155, 0., 0., 0., 0., 0.}} >>> rmoms = MarginalMomentsFromMAP[D0, D1, 3]; >>> Print[rmoms]; {1.2000000000000002, 4.320000000000002, 19.999999999999577} >>> rcorr1 = LagCorrelationsFromMAP[D0, D1, 1][[1]]; >>> Print[rcorr1]; 0.40000000000000063For Python/Numpy:
>>> moms = [1.1, 6.05] >>> corr1 = -0.17 >>> D0, D1 = MAPFromFewMomentsAndCorrelations(moms, corr1) >>> print(D0) [[ -0.28494 0.28494 0. 0. ] [ -0. -18.13384 0. 0. ] [ 0. 0. -0.28494 0.28494] [ 0. 0. -0. -0.95707]] >>> print(D1) [[ 0. 0. 0. 0. ] [ 0.0228 1.42791 4.9669 11.71623] [ 0. 0. 0. 0. ] [ 0.01384 0.86667 0.0228 0.05377]] >>> rmoms = MarginalMomentsFromMAP(D0, D1, 2) >>> print(rmoms) [1.0999999999999996, 6.0499999999999963] >>> rcorr1 = LagCorrelationsFromMAP(D0, D1, 1)[0] >>> print(rcorr1) -0.17 >>> moms = [1.2, 4.32, 20.] >>> corr1 = -0.4 >>> D0, D1 = MAPFromFewMomentsAndCorrelations(moms, corr1) >>> print(D0) [[ -0.33604 0.33604 0. 0. 0. 0. ] [ 0. -36.02667 0. 0. 0. 0. ] [ 0. 0. -1.28286 1.28286 0. 0. ] [ 0. 0. 0. -1.28286 1.28286 0. ] [ 0. 0. 0. 0. -1.28286 1.28286] [ 0. 0. 0. 0. 0. -1.37472]] >>> print(D1) [[ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 2.47407e-02 1.77659e+00 2.34746e+01 0.00000e+00 0.00000e+00 1.07508e+01] [ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 1.79373e-02 1.28805e+00 4.71448e-02 0.00000e+00 0.00000e+00 2.15912e-02]] >>> butools.checkPrecision = 10.**-12 >>> rmoms = MarginalMomentsFromMAP(D0, D1, 3) >>> print(rmoms) [1.2000000000000002, 4.3200000000000021, 19.999999999999915] >>> rcorr1 = LagCorrelationsFromMAP(D0, D1, 1)[0] >>> print(rcorr1) -0.4 >>> moms = [1.2, 4.32, 20.] >>> corr1 = 0.4 >>> D0, D1 = MAPFromFewMomentsAndCorrelations(moms, corr1) >>> print(D0) [[ -0.54184 0.54184 0. 0. 0. 0. 0. 0. 0. ] [ 0. -116.33647 0. 0. 0. 0. 0. 0. 0. ] [ 0. 0. -0.24417 0.24417 0. 0. 0. 0. 0. ] [ 0. 0. 0. -2.01403 2.01403 0. 0. 0. 0. ] [ 0. 0. 0. 0. -2.01403 2.01403 0. 0. 0. ] [ 0. 0. 0. 0. 0. -2.01403 2.01403 0. 0. ] [ 0. 0. 0. 0. 0. 0. -2.01403 2.01403 0. ] [ 0. 0. 0. 0. 0. 0. 0. -2.01403 2.01403] [ 0. 0. 0. 0. 0. 0. 0. 0. -2.01403]] >>> print(D1) [[ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 1.61283e+01 8.85745e+01 9.97739e-02 1.15339e+01 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00] [ 6.20477e-02 3.40758e-01 1.38184e-02 1.59740e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00]] >>> rmoms = MarginalMomentsFromMAP(D0, D1, 3) >>> print(rmoms) [1.2, 4.3199999999999985, 19.999999999999851] >>> rcorr1 = LagCorrelationsFromMAP(D0, D1, 1)[0] >>> print(rcorr1) 0.4