butools.queues.MAPMAP1¶
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butools.queues.MAPMAP1()¶ Matlab: Ret = MAPMAP1(D0, D1, S0, S1, ...)Mathematica: Ret = MAPMAP1[D0, D1, S0, S1, ...]Python/Numpy: Ret = MAPMAP1(D0, D1, S0, S1, ...)Returns various performane measures of a continuous time MAP/MAP/1 queue.
In a MAP/MAP/1 queue both the arrival and the service processes are characterized by Markovian arrival processes.
Parameters: D0 : matrix, shape(N,N)
The transitions of the arrival MAP not accompanied by job arrivals
D1 : matrix, shape(N,N)
The transitions of the arrival MAP accompanied by job arrivals
S0 : matrix, shape(N,N)
The transitions of the service MAP not accompanied by job service
S1 : matrix, shape(N,N)
The transitions of the service MAP accompanied by job service
further parameters :
The rest of the function parameters specify the options and the performance measures to be computed.
The supported performance measures and options in this function are:
Parameter name Input parameters Output “ncMoms” Number of moments The moments of the number of customers “ncDistr” Upper limit K The distribution of the number of customers from level 0 to level K-1 “ncDistrMG” None The vector-matrix parameters of the matrix-geometric distribution of the number of customers in the system “ncDistrDPH” None The vector-matrix parameters of the matrix-geometric distribution of the number of customers in the system, converted to a discrete PH representation “stMoms” Number of moments The sojourn time moments “stDistr” A vector of points The sojourn time distribution at the requested points (cummulative, cdf) “stDistrME” None The vector-matrix parameters of the matrix-exponentially distributed sojourn time distribution “stDistrPH” None The vector-matrix parameters of the matrix-exponentially distributed sojourn time distribution, converted to a continuous PH representation “prec” The precision Numerical precision used as a stopping condition when solving the matrix-quadratic equation (The quantities related to the number of customers in the system include the customer in the server, and the sojourn time related quantities include the service times as well)
Returns: Ret : list of the performance measures
Each entry of the list corresponds to a performance measure requested. If there is just a single item, then it is not put into a list.
Notes
“ncDistrMG” and “stDistrME” behave much better numerically than “ncDistrDPH” and “stDistrPH”.
Examples
For Matlab:
>>> D0 = [-8., 1., 2.; 0., -6., 4.; 3., 0., -3.]; >>> D1 = [4., 1., 0.; 0., 2., 0.; 0., 0., 0.]; >>> S0 = [-10., 4.; 0., -7.]; >>> S1 = [5., 1.; 4., 3.]; >>> [ncd, ncm] = MAPMAP1(D0, D1, S0, S1, 'ncDistr', 11, 'ncMoms', 5); >>> disp(ncd); Columns 1 through 8 0.67697 0.18891 0.07951 0.032563 0.013182 0.0053087 0.0021328 0.00085583 Columns 9 through 11 0.00034322 0.0001376 5.5158e-05 >>> disp(ncm); 0.54864 1.306 4.357 19.193 105.39 >>> [alphap, Ap] = MAPMAP1(D0, D1, S0, S1, 'ncDistrDPH'); >>> disp(alphap); 0.067133 0.075328 0.0407 0.047054 0.035736 0.057079 >>> disp(Ap); 0.2813 0.046292 0.0076126 0.0034456 0 0 0.10552 0.33506 0.0083548 0.01223 0 0 0.17544 0.045811 0.13666 0.010563 0 0 0.079843 0.20353 0.042265 0.16552 0 0 0.20818 0.091826 0.05939 0.013516 0 0 0.1325 0.20874 0.034792 0.065764 0 0 >>> [alpha, A] = MAPMAP1(D0, D1, S0, S1, 'ncDistrMG'); >>> disp(alpha); 0.1111 0.010481 0.10662 -0.80772 0.48646 0.41608 >>> disp(A); 0.16291 0.023017 0.1119 -0.13707 -0.0046024 0.28152 -0.06844 0.32235 -0.06034 0.037524 -0.15214 0.35383 -0.060271 0.1733 0.035807 0.028632 -0.12729 0.35975 -0.059415 0.16196 -0.10712 0.18069 -0.1826 0.39939 -0.059415 0.16196 -0.10712 0.18069 -0.1826 0.39939 -0.059415 0.16196 -0.10712 0.18069 -0.1826 0.39939 >>> ncdFromDPH = PmfFromDPH(alphap, Ap, (0:1:10)); >>> disp(ncdFromDPH); Columns 1 through 8 0.67697 0.18891 0.07951 0.032563 0.013182 0.0053087 0.0021328 0.00085583 Columns 9 through 11 0.00034322 0.0001376 5.5158e-05 >>> ncmFromMG = MomentsFromMG(alpha, A, 5); >>> disp(ncmFromMG); 0.54864 1.306 4.357 19.193 105.39 >>> [std, stm] = MAPMAP1(D0, D1, S0, S1, 'stDistr', (0.:0.1:1.), 'stMoms', 5); >>> disp(std); Columns 1 through 8 1.1102e-16 0.3122 0.53197 0.68345 0.78667 0.85655 0.90367 0.93538 Columns 9 through 11 0.95667 0.97097 0.98055 >>> disp(stm); 0.25908 0.13145 0.09911 0.099178 0.12376 >>> [betap, Bp] = MAPMAP1(D0, D1, S0, S1, 'stDistrPH'); >>> disp(betap); 0.27119 0.39548 0.13559 0.19774 >>> disp(Bp); -8.1822 1.8689 0.12464 0.12862 3.6856 -5.9854 0.047157 0.063652 1.2252 1.2825 -9.0589 0.94899 0.34891 0.66063 3.4502 -6.4625 >>> [beta, B] = MAPMAP1(D0, D1, S0, S1, 'stDistrME'); >>> disp(beta); 0.40071 0.26596 0.19675 0.13659 0 0 >>> disp(B); -8.1822 4.607 0.61259 0.21807 0 0 1.4951 -5.9854 0.51299 0.33032 0 0 0.24927 0.11789 -9.0589 4.3127 0 0 0.20579 0.1273 0.75919 -6.4625 0 0 0.5575 0.23374 0.18872 0.080814 -10 4 0.44853 0.30439 0.1539 0.099095 0 -7 >>> stdFromPH = CdfFromPH(betap, Bp, (0.:0.1:1.)); >>> disp(stdFromPH); Columns 1 through 8 0 0.3122 0.53197 0.68345 0.78667 0.85655 0.90367 0.93538 Columns 9 through 11 0.95667 0.97097 0.98055 >>> stmFromME = MomentsFromME(beta, B, 5); >>> disp(stmFromME); 0.25908 0.13145 0.09911 0.099178 0.12376 >>> delta = [0.5,0.1,0.4]; >>> Dm = [-8., 1., 2.; 0., -6., 4.; 3., 0., -3.]; >>> sigma = [0.2,0.7,0.1]; >>> S = [-10., 4., 0.; 5., -7., 2.; 1., 2., -8.]; >>> D0 = Dm; >>> D1 = sum(-Dm,2)*delta; >>> S0 = S; >>> S1 = sum(-S,2)*sigma; >>> [ncd, ncm] = MAPMAP1(D0, D1, S0, S1, 'ncDistr', 11, 'ncMoms', 5); >>> disp(ncd); Columns 1 through 8 0.33826 0.21299 0.1455 0.098491 0.066529 0.044917 0.030323 0.02047 Columns 9 through 11 0.013818 0.0093279 0.0062968 >>> disp(ncm); 2.0439 10.554 80.619 820.69 10443 >>> [alphap, Ap] = MAPMAP1(D0, D1, S0, S1, 'ncDistrDPH'); >>> disp(alphap); Columns 1 through 8 0.074729 0.11816 0.037058 0.019693 0.03096 0.0097648 0.12199 0.18879 Column 9 0.0606 >>> disp(Ap); Columns 1 through 8 0.26782 0.31102 0.050454 0.028231 0.032596 0.005318 0 0 0.12042 0.4268 0.051941 0.012693 0.04473 0.0054746 0 0 0.093612 0.29656 0.24254 0.0098678 0.03108 0.025565 0 0 0.25805 0.31793 0.051505 0.027202 0.03332 0.0054287 0 0 0.12382 0.42297 0.053367 0.013052 0.044328 0.005625 0 0 0.095477 0.30306 0.2325 0.010064 0.031762 0.024506 0 0 0.22985 0.33605 0.056865 0.024229 0.035219 0.0059937 0 0 0.13243 0.412 0.058567 0.01396 0.043179 0.0061731 0 0 0.10836 0.32375 0.19362 0.011422 0.03393 0.020408 0 0 Column 9 0 0 0 0 0 0 0 0 0 >>> [alpha, A] = MAPMAP1(D0, D1, S0, S1, 'ncDistrMG'); >>> disp(alpha); Columns 1 through 8 0.33539 0.11124 -0.11565 0.22684 0.049774 -0.83292 -2.2177 2.6612 Column 9 0.44353 >>> disp(A); Columns 1 through 8 0.23011 0.17196 -0.041192 0.15685 0.08314 -1.0418 -1.0077 1.5932 0.092072 0.32509 -0.031637 0.067255 0.14976 -1.3362 -0.52827 1.4421 0.093673 0.13219 0.17527 0.068436 0.063802 -1.1347 -0.53458 1.0089 0.1138 0.13806 0.14334 0.081476 0.066654 -1.121 -0.60436 1.0951 0.093316 0.1752 0.12914 0.068173 0.082966 -1.1796 -0.53317 1.1055 0.093673 0.13219 0.17527 0.068436 0.063802 -1.1347 -0.53458 1.0089 0.093673 0.13219 0.17527 0.068436 0.063802 -1.1347 -0.53458 1.0089 0.093673 0.13219 0.17527 0.068436 0.063802 -1.1347 -0.53458 1.0089 0.093673 0.13219 0.17527 0.068436 0.063802 -1.1347 -0.53458 1.0089 Column 9 0.53741 0.49554 0.79874 0.76019 0.73114 0.79874 0.79874 0.79874 0.79874 >>> ncdFromDPH = PmfFromDPH(alphap, Ap, (0:1:10)); >>> disp(ncdFromDPH); Columns 1 through 8 0.33826 0.21299 0.1455 0.098491 0.066529 0.044917 0.030323 0.02047 Columns 9 through 11 0.013818 0.0093279 0.0062968 >>> ncmFromMG = MomentsFromMG(alpha, A, 5); >>> disp(ncmFromMG); 2.0439 10.554 80.619 820.69 10443 >>> [std, stm] = MAPMAP1(D0, D1, S0, S1, 'stDistr', (0.:0.1:1.), 'stMoms', 5); >>> disp(std); Columns 1 through 8 2.2204e-16 0.067321 0.14411 0.21822 0.28688 0.34979 0.40723 0.45962 Columns 9 through 11 0.50738 0.55093 0.59062 >>> disp(stm); 1.1135 2.4113 7.8173 33.786 182.52 >>> [betap, Bp] = MAPMAP1(D0, D1, S0, S1, 'stDistrPH'); >>> disp(betap); Columns 1 through 8 0.35369 0 0.14631 0.070739 0 0.029261 0.28295 0 Column 9 0.11705 >>> disp(Bp); Columns 1 through 8 -9.7151 8.1485 0.63533 0.032747 0.049378 0.016146 0.087771 0.17848 3.2004 -6.235 0.94726 0.073403 0.11068 0.036192 0.19674 0.40007 0.28699 6.635 -7.86 0.032986 0.049738 0.016264 0.088411 0.17978 0.28492 0.34127 0.13895 -9.9673 7.8566 0.51253 0.087771 0.17848 0.63864 0.76496 0.31145 2.6351 -6.8893 0.67199 0.19674 0.40007 0.28699 0.34375 0.13996 0.032986 6.341 -7.9837 0.088411 0.17978 0.28492 0.34127 0.13895 0.032747 0.049378 0.016146 -9.9122 7.9857 0.63864 0.76496 0.31145 0.073403 0.11068 0.036192 2.7585 -6.5999 0.28699 0.34375 0.13996 0.032986 0.049738 0.016264 0.088411 6.471 Column 9 0.044593 0.099956 0.044918 0.044593 0.099956 0.044918 0.54098 0.73576 -7.9551 >>> [beta, B] = MAPMAP1(D0, D1, S0, S1, 'stDistrME'); >>> disp(beta); Columns 1 through 8 0.14329 0.28553 0.071181 0.028658 0.057106 0.014236 0.11463 0.22842 Column 9 0.056945 >>> disp(B); Columns 1 through 8 -9.7151 4.9972 0.14246 0.056983 0.19944 0.028492 0.22793 0.79777 5.2186 -6.235 2.1093 0.043712 0.15299 0.021856 0.17485 0.61196 1.2799 2.9797 -7.86 0.055984 0.19594 0.027992 0.22394 0.78377 0.16374 0.57307 0.081868 -9.9673 4.1146 0.016374 0.13099 0.45846 0.15812 0.55341 0.079058 5.0316 -6.8893 2.0158 0.12649 0.44273 0.16264 0.56923 0.081319 1.0325 2.1138 -7.9837 0.13011 0.45539 0.10971 0.384 0.054857 0.021943 0.0768 0.010971 -9.9122 4.3072 0.14288 0.50009 0.071442 0.028577 0.10002 0.014288 5.1143 -6.5999 0.11229 0.39303 0.056147 0.022459 0.078606 0.011229 1.0898 2.3144 Column 9 0.11397 0.087423 0.11197 0.065494 0.063247 0.065055 0.043886 2.0572 -7.9551 >>> stdFromPH = CdfFromPH(betap, Bp, (0.:0.1:1.)); >>> disp(stdFromPH); Columns 1 through 8 0 0.067321 0.14411 0.21822 0.28688 0.34979 0.40723 0.45962 Columns 9 through 11 0.50738 0.55093 0.59062 >>> stmFromME = MomentsFromME(beta, B, 5); >>> disp(stmFromME); 1.1135 2.4113 7.8173 33.786 182.52
For Mathematica:
>>> D0 = {{-8., 1., 2.},{0., -6., 4.},{3., 0., -3.}}; >>> D1 = {{4., 1., 0.},{0., 2., 0.},{0., 0., 0.}}; >>> S0 = {{-10., 4.},{0., -7.}}; >>> S1 = {{5., 1.},{4., 3.}}; >>> {ncd, ncm} = MAPMAP1[D0, D1, S0, S1, "ncDistr", 11, "ncMoms", 5]; "Final Residual Error for G: "1.0408340855860843*^-16 "Final Residual Error for R: "1.3877787807814457*^-16 >>> Print[ncd]; {0.6769690927218346, 0.18890601544446484, 0.07950988455548241, 0.032562815367099616, 0.013182008065251377, 0.005308701544215693, 0.00213277061566337, 0.0008558335516777699, 0.0003432229641681273, 0.00013760305702500175, 0.00005515776673456938} >>> Print[ncm]; {0.5486395318641362, 1.3060488095463447, 4.357004424902784, 19.19295358204142, 105.38633755592997} >>> {alphap, Ap} = MAPMAP1[D0, D1, S0, S1, "ncDistrDPH"]; "Final Residual Error for G: "1.0408340855860843*^-16 "Final Residual Error for R: "1.3877787807814457*^-16 >>> Print[alphap]; {0.06713336638603387, 0.07532832075280335, 0.040700005311250634, 0.04705391014775885, 0.03573592042634463, 0.05707938425397416} >>> Print[Ap]; {{0.28130240531969064, 0.04629222581204726, 0.007612641693714353, 0.003445567933621212, 0., 0.}, {0.10551669990408467, 0.33506431281604, 0.008354813144417458, 0.012229921483571509, 0., 0.}, {0.17544233493101746, 0.04581068687974456, 0.13665561446203264, 0.010562750164590453, 0., 0.}, {0.07984336109470196, 0.20353241432916766, 0.042265324532051944, 0.16551629424385653, 0., 0.}, {0.20817723425870532, 0.0918264144940291, 0.059390327559432174, 0.013516066113380536, 0., 0.}, {0.13250054324765906, 0.2087351339587931, 0.034792155582504015, 0.0657638702586556, 0., 0.}} >>> {alpha, A} = MAPMAP1[D0, D1, S0, S1, "ncDistrMG"]; "Final Residual Error for G: "1.0408340855860843*^-16 "Final Residual Error for R: "1.3877787807814457*^-16 >>> Print[alpha]; {0.11109802951004608, 0.010480702136161618, 0.10662329054401752, -0.8077182308946177, 0.48646319457458154, 0.41608392140797645} >>> Print[A]; {{0.1629053064844125, 0.023017401892469813, 0.11189525710750912, -0.13707103216123398, -0.00460237048747536, 0.2815167630916619}, {-0.06843965017167865, 0.3223468624823151, -0.060340481045871086, 0.03752370826293791, -0.15213536522543497, 0.35382777946490074}, {-0.060271277810733125, 0.17329866481605158, 0.03580746865942097, 0.02863218049808125, -0.12729406912309077, 0.35975047082003087}, {-0.05941546558542623, 0.1619589748583506, -0.10712188703259878, 0.18068938099433576, -0.18260444681358848, 0.3993940550347238}, {-0.05941546558542623, 0.1619589748583506, -0.10712188703259878, 0.18068938099433576, -0.18260444681358848, 0.3993940550347238}, {-0.05941546558542623, 0.1619589748583506, -0.10712188703259878, 0.18068938099433576, -0.18260444681358848, 0.3993940550347238}} >>> ncdFromDPH = PmfFromDPH[alphap, Ap, Range[0,10,1]]; >>> Print[ncdFromDPH]; {0.6769690927218345, 0.18890601544446484, 0.0795098845554824, 0.03256281536709962, 0.01318200806525138, 0.005308701544215695, 0.0021327706156633714, 0.0008558335516777703, 0.00034322296416812746, 0.0001376030570250018, 0.000055157766734569404} >>> ncmFromMG = MomentsFromMG[alpha, A, 5]; >>> Print[ncmFromMG]; {0.5486395318641364, 1.3060488095463452, 4.357004424902785, 19.192953582041415, 105.38633755592993} >>> {std, stm} = MAPMAP1[D0, D1, S0, S1, "stDistr", Range[0.,1.,0.1], "stMoms", 5]; "Final Residual Error for G: "1.0408340855860843*^-16 "Final Residual Error for R: "1.3877787807814457*^-16 >>> Print[std]; {1.1102230246251565*^-16, 0.31220200485532823, 0.5319693886055359, 0.683448492874903, 0.786668497914873, 0.8565469353128969, 0.9036716256123, 0.9353763920782243, 0.9566744696604379, 0.9709671432122016, 0.9805517895663979} >>> Print[stm]; {0.25907977893584216, 0.1314530565761404, 0.09910977975365252, 0.0991778409591816, 0.12375826425492883} >>> {betap, Bp} = MAPMAP1[D0, D1, S0, S1, "stDistrPH"]; "Final Residual Error for G: "1.0408340855860843*^-16 "Final Residual Error for R: "1.3877787807814457*^-16 >>> Print[betap]; {0.27118644067796616, 0.39548022598870064, 0.13559322033898305, 0.1977401129943503} >>> Print[Bp]; {{-8.18221808955257, 1.868862984211661, 0.12463702298427055, 0.12861844665139152}, {3.6856163463839864, -5.985377122566605, 0.04715723015286752, 0.06365232793573995}, {1.2251890316565601, 1.282470479592513, -9.058949015866476, 0.9489932799691041}, {0.3489096698540739, 0.6606305461695843, 3.450175746844795, -6.4624885396327}} >>> {beta, B} = MAPMAP1[D0, D1, S0, S1, "stDistrME"]; "Final Residual Error for G: "1.0408340855860843*^-16 "Final Residual Error for R: "1.3877787807814457*^-16 >>> Print[beta]; {0.4007102782337436, 0.26595638843292296, 0.19674632576397322, 0.13658700756936012, 0., 0.} >>> Print[B]; {{-8.18221808955257, 4.607020432979983, 0.61259451582828, 0.21806854365879616, 0., 0.}, {1.4950903873693289, -5.985377122566605, 0.5129881918370052, 0.3303152730847921, 0., 0.}, {0.24927404596854114, 0.11789307538216881, -9.058949015866476, 4.3127196835559936, 0., 0.}, {0.20578951464222647, 0.12730465587147993, 0.7591946239752834, -6.4624885396327, 0., 0.}, {0.5574964766296526, 0.23373913475846325, 0.18872072136071122, 0.08081415066754914, -10., 4.}, {0.44852711621079855, 0.30438686323001857, 0.15389645755110154, 0.0990945819254376, 0., -7.}} >>> stdFromPH = CdfFromPH[betap, Bp, Range[0.,1.,0.1]]; >>> Print[stdFromPH]; {0., 0.3122020048553279, 0.5319693886055358, 0.6834484928749028, 0.786668497914873, 0.856546935312897, 0.9036716256123, 0.9353763920782242, 0.9566744696604379, 0.9709671432122016, 0.9805517895663979} >>> stmFromME = MomentsFromME[beta, B, 5]; >>> Print[stmFromME]; {0.25907977893584216, 0.1314530565761404, 0.09910977975365252, 0.0991778409591816, 0.12375826425492883} >>> delta = {0.5,0.1,0.4}; >>> Dm = {{-8., 1., 2.},{0., -6., 4.},{3., 0., -3.}}; >>> sigma = {0.2,0.7,0.1}; >>> S = {{-10., 4., 0.},{5., -7., 2.},{1., 2., -8.}}; >>> D0 = Dm; >>> D1 = Transpose[{Total[-Dm,{2}]}].{delta}; >>> S0 = S; >>> S1 = Transpose[{Total[-S,{2}]}].{sigma}; >>> {ncd, ncm} = MAPMAP1[D0, D1, S0, S1, "ncDistr", 11, "ncMoms", 5]; "Final Residual Error for G: "2.8796409701215*^-16 "Final Residual Error for R: "2.5673907444456745*^-16 >>> Print[ncd]; {0.3382583000173821, 0.21298529276749126, 0.1455036749660385, 0.09849079999116112, 0.06652869034717646, 0.044917143499797356, 0.030322557234957335, 0.020469519523922355, 0.013818043581100023, 0.009327918288020327, 0.006296840863138073} >>> Print[ncm]; {2.043920190427152, 10.553791196138821, 80.61928537723435, 820.6894786219923, 10442.52887895297} >>> {alphap, Ap} = MAPMAP1[D0, D1, S0, S1, "ncDistrDPH"]; "Final Residual Error for G: "2.8796409701215*^-16 "Final Residual Error for R: "2.5673907444456745*^-16 >>> Print[alphap]; {0.0747292656813806, 0.1181643446209385, 0.03705768376723692, 0.019693372356747885, 0.030959986518879018, 0.009764840891516689, 0.12198619213743217, 0.1887863241704555, 0.060599689838030685} >>> Print[Ap]; {{0.26781773085286764, 0.31101976305978485, 0.05045448201370804, 0.028231158164525778, 0.03259584844252456, 0.005317979312709587, 0., 0., 0.}, {0.1204151193930397, 0.426803260037457, 0.051940881424856866, 0.012693178566480858, 0.04473032273605763, 0.0054746480763838545, 0., 0., 0.}, {0.09361178537818889, 0.29655828531121786, 0.24254481404417771, 0.009867789972901992, 0.031080240134197804, 0.025564593114670967, 0., 0., 0.}, {0.25805347211788343, 0.3179330582067931, 0.05150455357229029, 0.027201889744437005, 0.03332038349661052, 0.005428658455622758, 0., 0., 0.}, {0.12382168736754393, 0.4229664164893307, 0.05336708312117147, 0.013052271144034176, 0.04432820947623766, 0.005624971909154385, 0., 0., 0.}, {0.095476911834167, 0.3030612398902468, 0.23250227823873204, 0.010064396373112615, 0.03176177020740381, 0.024506094532800775, 0., 0., 0.}, {0.22985295491631016, 0.33604933463865083, 0.05686493792626865, 0.024229221508828593, 0.03521902619090838, 0.005993651137440084, 0., 0., 0.}, {0.13242835135132583, 0.41200326098416273, 0.05856727781254947, 0.01395951537846673, 0.04317923633130734, 0.006173080356354317, 0., 0., 0.}, {0.10835583923850761, 0.32374817692321156, 0.19362199117845982, 0.011421987729680155, 0.03392982621012461, 0.020408053010888768, 0., 0., 0.}} >>> {alpha, A} = MAPMAP1[D0, D1, S0, S1, "ncDistrMG"]; "Final Residual Error for G: "2.8796409701215*^-16 "Final Residual Error for R: "2.5673907444456745*^-16 >>> Print[alpha]; {0.335388097798228, 0.11124374834176529, -0.11565482233389898, 0.22684247560717574, 0.049774394541475475, -0.8329212637990596, -2.217672674567333, 2.661207209480798, 0.4435345349134663} >>> Print[A]; {{0.23010874461138647, 0.17196369830044164, -0.04119187617645898, 0.1568451647402447, 0.08314023718352927, -1.0418469110562227, -1.0076930929970267, 1.5931894863967555, 0.5374143415183332}, {0.09207203502871297, 0.3250868076906807, -0.031637329628947004, 0.06725527006540137, 0.1497556640191295, -1.3361636344419388, -0.5282671320547201, 1.4421155298051003, 0.49554465753990445}, {0.09367269488934829, 0.1321911062922013, 0.17526524304297714, 0.06843629532664251, 0.06380192761345282, -1.1347373274317223, -0.5345801680271869, 1.0088804369648803, 0.798736724637723}, {0.11379592294037874, 0.13805724607558675, 0.14333954611613192, 0.08147589772328324, 0.06665417388115764, -1.1210367326381352, -0.6043605472044968, 1.0950613567287801, 0.760193759482519}, {0.09331582169637492, 0.17519793526434796, 0.12913552894210745, 0.06817298126078337, 0.08296563947595605, -1.1796460873121937, -0.533172652689134, 1.1054718457284263, 0.7311389021831358}, {0.09367269488934832, 0.1321911062922014, 0.17526524304297716, 0.06843629532664249, 0.06380192761345285, -1.1347373274317225, -0.5345801680271869, 1.0088804369648805, 0.798736724637723}, {0.09367269488934832, 0.1321911062922014, 0.17526524304297716, 0.06843629532664249, 0.06380192761345285, -1.1347373274317225, -0.5345801680271869, 1.0088804369648805, 0.798736724637723}, {0.09367269488934832, 0.1321911062922014, 0.17526524304297716, 0.06843629532664249, 0.06380192761345285, -1.1347373274317225, -0.5345801680271869, 1.0088804369648805, 0.798736724637723}, {0.09367269488934832, 0.1321911062922014, 0.17526524304297716, 0.06843629532664249, 0.06380192761345285, -1.1347373274317225, -0.5345801680271869, 1.0088804369648805, 0.798736724637723}} >>> ncdFromDPH = PmfFromDPH[alphap, Ap, Range[0,10,1]]; >>> Print[ncdFromDPH]; {0.3382583000173821, 0.21298529276749126, 0.1455036749660385, 0.09849079999116114, 0.06652869034717646, 0.044917143499797356, 0.030322557234957335, 0.02046951952392236, 0.013818043581100023, 0.009327918288020327, 0.006296840863138071} >>> ncmFromMG = MomentsFromMG[alpha, A, 5]; >>> Print[ncmFromMG]; {2.0439201904271664, 10.553791196138974, 80.61928537723583, 820.6894786220133, 10442.528878953299} >>> {std, stm} = MAPMAP1[D0, D1, S0, S1, "stDistr", Range[0.,1.,0.1], "stMoms", 5]; "Final Residual Error for G: "2.8796409701215*^-16 "Final Residual Error for R: "2.5673907444456745*^-16 >>> Print[std]; {-2.220446049250313*^-16, 0.06732092429975123, 0.14410990918817657, 0.21821569553423958, 0.2868837551361466, 0.34979180437976287, 0.4072286119979047, 0.45961534708806473, 0.5073799327606132, 0.5509251846596359, 0.590622172200536} >>> Print[stm]; {1.1135106870764584, 2.411324091748759, 7.8172702796512095, 33.78557510760976, 182.52100725089815} >>> {betap, Bp} = MAPMAP1[D0, D1, S0, S1, "stDistrPH"]; "Final Residual Error for G: "2.8796409701215*^-16 "Final Residual Error for R: "2.5673907444456745*^-16 >>> Print[betap]; {0.35369318181818193, 0., 0.14630681818181818, 0.07073863636363638, 0., 0.029261363636363637, 0.2829545454545455, 0., 0.11704545454545455} >>> Print[Bp]; {{-9.715083156753732, 8.148496507481427, 0.6353335255307762, 0.03274706510215011, 0.049378103925532595, 0.01614619488986971, 0.08777139112801251, 0.17848405075047077, 0.044593109982684585}, {3.2003730166901305, -6.235044479758062, 0.9472560892491555, 0.07340295071122381, 0.11068162955530263, 0.03619189518138234, 0.19674065681086136, 0.4000742032639796, 0.09995600655840951}, {0.28699148045437195, 6.635014685933965, -7.860040259477494, 0.03298551460532111, 0.04973765322596125, 0.016263764270135758, 0.08841050319933302, 0.17978369189671212, 0.04491781709420901}, {0.284916843246268, 0.3412675918187785, 0.1389479833621017, -9.96725293489785, 7.856607019588182, 0.5125317370585444, 0.08777139112801254, 0.17848405075047077, 0.044593109982684585}, {0.6386446216284014, 0.7649555202419378, 0.31145362011335304, 2.6351313457729524, -6.889318370444697, 0.6719943643171847, 0.19674065681086134, 0.4000742032639796, 0.09995600655840951}, {0.28699148045437195, 0.3437525500116338, 0.1399597405225054, 0.03298551460532111, 6.3409997891482925, -7.983736235729865, 0.08841050319933302, 0.1797836918967121, 0.04491781709420901}, {0.284916843246268, 0.3412675918187785, 0.1389479833621017, 0.032747065102150114, 0.0493781039255326, 0.016146194889869712, -9.912228608871988, 7.98571296641312, 0.5409786521513593}, {0.6386446216284014, 0.7649555202419378, 0.31145362011335304, 0.07340295071122381, 0.1106816295553026, 0.03619189518138234, 2.75846905187259, -6.59992579673602, 0.7357584756942119}, {0.28699148045437195, 0.3437525500116338, 0.1399597405225054, 0.03298551460532111, 0.049737653225961256, 0.016263764270135758, 0.08841050319933302, 6.471045827819043, -7.955082182905792}} >>> {beta, B} = MAPMAP1[D0, D1, S0, S1, "stDistrME"]; "Final Residual Error for G: "2.8796409701215*^-16 "Final Residual Error for R: "2.5673907444456745*^-16 >>> Print[beta]; {0.14328764742307518, 0.28553117573889425, 0.07118117683803069, 0.028657529484615033, 0.057106235147778836, 0.01423623536760614, 0.11463011793846013, 0.22842494059111534, 0.056944941470424545} >>> Print[B]; {{-9.715083156753732, 4.997208951361937, 0.14245842162313396, 0.05698336864925359, 0.1994417902723875, 0.028491684324626794, 0.22793347459701435, 0.79776716108955, 0.11396673729850718}, {5.218558720069125, -6.235044479758062, 2.1092793600345625, 0.043711744013825035, 0.15299110404838756, 0.021855872006912518, 0.17484697605530014, 0.6119644161935502, 0.08742348802765007}, {1.2799194810450107, 2.9797181836575377, -7.860040259477494, 0.05598389620900216, 0.19594363673150755, 0.02799194810450108, 0.22393558483600864, 0.7837745469260302, 0.11196779241800432}, {0.1637353255107505, 0.5730736392876267, 0.08186766275537526, -9.96725293489785, 4.114614727857525, 0.016373532551075053, 0.13098826040860043, 0.45845891143010137, 0.06549413020430021}, {0.15811661365043234, 0.5534081477765131, 0.07905830682521617, 5.031623322730087, -6.889318370444697, 2.0158116613650434, 0.1264932909203459, 0.44272651822121045, 0.06324664546017295}, {0.16263764270135755, 0.5692317494547514, 0.08131882135067878, 1.0325275285402715, 2.11384634989095, -7.9837362357298645, 0.13011011416108606, 0.4553853995638012, 0.06505505708054303}, {0.10971423891001564, 0.3839998361850546, 0.05485711945500782, 0.02194284778200313, 0.07679996723701092, 0.010971423891001565, -9.912228608871988, 4.307199868948044, 0.04388569556400626}, {0.1428836440228499, 0.5000927540799744, 0.07144182201142495, 0.028576728804569976, 0.1000185508159949, 0.014288364402284988, 5.11430691521828, -6.59992579673602, 2.05715345760914}, {0.1122945427355225, 0.3930308995743287, 0.05614727136776125, 0.0224589085471045, 0.07860617991486574, 0.01122945427355225, 1.0898356341884181, 2.314424719659463, -7.955082182905791}} >>> stdFromPH = CdfFromPH[betap, Bp, Range[0.,1.,0.1]]; >>> Print[stdFromPH]; {-2.220446049250313*^-16, 0.06732092429975112, 0.14410990918817634, 0.2182156955342397, 0.2868837551361467, 0.3497918043797634, 0.4072286119979047, 0.4596153470880646, 0.5073799327606132, 0.5509251846596357, 0.5906221722005369} >>> stmFromME = MomentsFromME[beta, B, 5]; >>> Print[stmFromME]; {1.1135106870764584, 2.411324091748759, 7.8172702796512095, 33.78557510760976, 182.52100725089815}
For Python/Numpy:
>>> D0 = ml.matrix([[-8., 1., 2.],[0., -6., 4.],[3., 0., -3.]]) >>> D1 = ml.matrix([[4., 1., 0.],[0., 2., 0.],[0., 0., 0.]]) >>> S0 = ml.matrix([[-10., 4.],[0., -7.]]) >>> S1 = ml.matrix([[5., 1.],[4., 3.]]) >>> ncd, ncm = MAPMAP1(D0, D1, S0, S1, "ncDistr", 11, "ncMoms", 5) Final Residual Error for G: 2.35922392733e-16 Final Residual Error for R: 1.31838984174e-16 >>> print(ncd) [ 6.76969e-01 1.88906e-01 7.95099e-02 3.25628e-02 1.31820e-02 5.30870e-03 2.13277e-03 8.55834e-04 3.43223e-04 1.37603e-04 5.51578e-05] >>> print(ncm) [0.54863953186413617, 1.3060488095463447, 4.3570044249027839, 19.192953582041419, 105.38633755592997] >>> alphap, Ap = MAPMAP1(D0, D1, S0, S1, "ncDistrDPH") Final Residual Error for G: 2.35922392733e-16 Final Residual Error for R: 1.31838984174e-16 >>> print(alphap) [[ 0.06713 0.07533 0.0407 0.04705 0.03574 0.05708]] >>> print(Ap) [[ 0.2813 0.04629 0.00761 0.00345 0. 0. ] [ 0.10552 0.33506 0.00835 0.01223 0. 0. ] [ 0.17544 0.04581 0.13666 0.01056 0. 0. ] [ 0.07984 0.20353 0.04227 0.16552 0. 0. ] [ 0.20818 0.09183 0.05939 0.01352 0. 0. ] [ 0.1325 0.20874 0.03479 0.06576 0. 0. ]] >>> alpha, A = MAPMAP1(D0, D1, S0, S1, "ncDistrMG") Final Residual Error for G: 2.35922392733e-16 Final Residual Error for R: 1.31838984174e-16 >>> print(alpha) [[ 0.1111 0.01048 0.10662 -0.80772 0.48646 0.41608]] >>> print(A) [[ 0.16291 0.02302 0.1119 -0.13707 -0.0046 0.28152] [-0.06844 0.32235 -0.06034 0.03752 -0.15214 0.35383] [-0.06027 0.1733 0.03581 0.02863 -0.12729 0.35975] [-0.05942 0.16196 -0.10712 0.18069 -0.1826 0.39939] [-0.05942 0.16196 -0.10712 0.18069 -0.1826 0.39939] [-0.05942 0.16196 -0.10712 0.18069 -0.1826 0.39939]] >>> ncdFromDPH = PmfFromDPH(alphap, Ap, np.arange(0,11.0,1)) >>> print(ncdFromDPH) [ 6.76969e-01 1.88906e-01 7.95099e-02 3.25628e-02 1.31820e-02 5.30870e-03 2.13277e-03 8.55834e-04 3.43223e-04 1.37603e-04 5.51578e-05] >>> ncmFromMG = MomentsFromMG(alpha, A, 5) >>> print(ncmFromMG) [0.54863953186413594, 1.3060488095463438, 4.3570044249027813, 19.192953582041405, 105.38633755592987] >>> std, stm = MAPMAP1(D0, D1, S0, S1, "stDistr", np.arange(0.,1.1,0.1), "stMoms", 5) Final Residual Error for G: 2.35922392733e-16 Final Residual Error for R: 1.31838984174e-16 >>> print(std) [ 1.11022e-16 3.12202e-01 5.31969e-01 6.83448e-01 7.86668e-01 8.56547e-01 9.03672e-01 9.35376e-01 9.56674e-01 9.70967e-01 9.80552e-01] >>> print(stm) [0.25907977893584216, 0.1314530565761404, 0.099109779753652538, 0.099177840959181612, 0.12375826425492886] >>> betap, Bp = MAPMAP1(D0, D1, S0, S1, "stDistrPH") Final Residual Error for G: 2.35922392733e-16 Final Residual Error for R: 1.31838984174e-16 >>> print(betap) [[ 0.27119 0.39548 0.13559 0.19774]] >>> print(Bp) [[-8.18222 1.86886 0.12464 0.12862] [ 3.68562 -5.98538 0.04716 0.06365] [ 1.22519 1.28247 -9.05895 0.94899] [ 0.34891 0.66063 3.45018 -6.46249]] >>> beta, B = MAPMAP1(D0, D1, S0, S1, "stDistrME") Final Residual Error for G: 2.35922392733e-16 Final Residual Error for R: 1.31838984174e-16 >>> print(beta) [[ 0.40071 0.26596 0.19675 0.13659 0. 0. ]] >>> print(B) [[ -8.18222 4.60702 0.61259 0.21807 0. 0. ] [ 1.49509 -5.98538 0.51299 0.33032 0. 0. ] [ 0.24927 0.11789 -9.05895 4.31272 0. 0. ] [ 0.20579 0.1273 0.75919 -6.46249 0. 0. ] [ 0.5575 0.23374 0.18872 0.08081 -10. 4. ] [ 0.44853 0.30439 0.1539 0.09909 0. -7. ]] >>> stdFromPH = CdfFromPH(betap, Bp, np.arange(0.,1.1,0.1)) >>> print(stdFromPH) [ 0. 0.3122 0.53197 0.68345 0.78667 0.85655 0.90367 0.93538 0.95667 0.97097 0.98055] >>> stmFromME = MomentsFromME(beta, B, 5) >>> print(stmFromME) [0.25907977893584216, 0.1314530565761404, 0.099109779753652524, 0.099177840959181612, 0.12375826425492886] >>> delta = ml.matrix([[0.5,0.1,0.4]]) >>> Dm = ml.matrix([[-8., 1., 2.],[0., -6., 4.],[3., 0., -3.]]) >>> sigma = ml.matrix([[0.2,0.7,0.1]]) >>> S = ml.matrix([[-10., 4., 0.],[5., -7., 2.],[1., 2., -8.]]) >>> D0 = Dm >>> D1 = np.sum(-Dm,1)*delta >>> S0 = S >>> S1 = np.sum(-S,1)*sigma >>> ncd, ncm = MAPMAP1(D0, D1, S0, S1, "ncDistr", 11, "ncMoms", 5) Final Residual Error for G: 1.80411241502e-16 Final Residual Error for R: 4.02455846427e-16 >>> print(ncd) [ 0.33826 0.21299 0.1455 0.09849 0.06653 0.04492 0.03032 0.02047 0.01382 0.00933 0.0063 ] >>> print(ncm) [2.0439201904271491, 10.553791196138791, 80.619285377234021, 820.68947862198797, 10442.5288789529] >>> alphap, Ap = MAPMAP1(D0, D1, S0, S1, "ncDistrDPH") Final Residual Error for G: 1.80411241502e-16 Final Residual Error for R: 4.02455846427e-16 >>> print(alphap) [[ 0.07473 0.11816 0.03706 0.01969 0.03096 0.00976 0.12199 0.18879 0.0606 ]] >>> print(Ap) [[ 0.26782 0.31102 0.05045 0.02823 0.0326 0.00532 0. 0. 0. ] [ 0.12042 0.4268 0.05194 0.01269 0.04473 0.00547 0. 0. 0. ] [ 0.09361 0.29656 0.24254 0.00987 0.03108 0.02556 0. 0. 0. ] [ 0.25805 0.31793 0.0515 0.0272 0.03332 0.00543 0. 0. 0. ] [ 0.12382 0.42297 0.05337 0.01305 0.04433 0.00562 0. 0. 0. ] [ 0.09548 0.30306 0.2325 0.01006 0.03176 0.02451 0. 0. 0. ] [ 0.22985 0.33605 0.05686 0.02423 0.03522 0.00599 0. 0. 0. ] [ 0.13243 0.412 0.05857 0.01396 0.04318 0.00617 0. 0. 0. ] [ 0.10836 0.32375 0.19362 0.01142 0.03393 0.02041 0. 0. 0. ]] >>> alpha, A = MAPMAP1(D0, D1, S0, S1, "ncDistrMG") Final Residual Error for G: 1.80411241502e-16 Final Residual Error for R: 4.02455846427e-16 >>> print(alpha) [[ 0.33539 0.11124 -0.11565 0.22684 0.04977 -0.83292 -2.21767 2.66121 0.44353]] >>> print(A) [[ 0.23011 0.17196 -0.04119 0.15685 0.08314 -1.04185 -1.00769 1.59319 0.53741] [ 0.09207 0.32509 -0.03164 0.06726 0.14976 -1.33616 -0.52827 1.44212 0.49554] [ 0.09367 0.13219 0.17527 0.06844 0.0638 -1.13474 -0.53458 1.00888 0.79874] [ 0.1138 0.13806 0.14334 0.08148 0.06665 -1.12104 -0.60436 1.09506 0.76019] [ 0.09332 0.1752 0.12914 0.06817 0.08297 -1.17965 -0.53317 1.10547 0.73114] [ 0.09367 0.13219 0.17527 0.06844 0.0638 -1.13474 -0.53458 1.00888 0.79874] [ 0.09367 0.13219 0.17527 0.06844 0.0638 -1.13474 -0.53458 1.00888 0.79874] [ 0.09367 0.13219 0.17527 0.06844 0.0638 -1.13474 -0.53458 1.00888 0.79874] [ 0.09367 0.13219 0.17527 0.06844 0.0638 -1.13474 -0.53458 1.00888 0.79874]] >>> ncdFromDPH = PmfFromDPH(alphap, Ap, np.arange(0,11.0,1)) >>> print(ncdFromDPH) [ 0.33826 0.21299 0.1455 0.09849 0.06653 0.04492 0.03032 0.02047 0.01382 0.00933 0.0063 ] >>> ncmFromMG = MomentsFromMG(alpha, A, 5) >>> print(ncmFromMG) [2.0439201904271451, 10.553791196138775, 80.61928537723395, 820.6894786219882, 10442.528878952886] >>> std, stm = MAPMAP1(D0, D1, S0, S1, "stDistr", np.arange(0.,1.1,0.1), "stMoms", 5) Final Residual Error for G: 1.80411241502e-16 Final Residual Error for R: 4.02455846427e-16 >>> print(std) [ 0. 0.06732 0.14411 0.21822 0.28688 0.34979 0.40723 0.45962 0.50738 0.55093 0.59062] >>> print(stm) [1.1135106870764573, 2.411324091748754, 7.817270279651181, 33.785575107609588, 182.52100725089704] >>> betap, Bp = MAPMAP1(D0, D1, S0, S1, "stDistrPH") Final Residual Error for G: 1.80411241502e-16 Final Residual Error for R: 4.02455846427e-16 >>> print(betap) [[ 0.35369 0. 0.14631 0.07074 0. 0.02926 0.28295 0. 0.11705]] >>> print(Bp) [[-9.71508 8.1485 0.63533 0.03275 0.04938 0.01615 0.08777 0.17848 0.04459] [ 3.20037 -6.23504 0.94726 0.0734 0.11068 0.03619 0.19674 0.40007 0.09996] [ 0.28699 6.63501 -7.86004 0.03299 0.04974 0.01626 0.08841 0.17978 0.04492] [ 0.28492 0.34127 0.13895 -9.96725 7.85661 0.51253 0.08777 0.17848 0.04459] [ 0.63864 0.76496 0.31145 2.63513 -6.88932 0.67199 0.19674 0.40007 0.09996] [ 0.28699 0.34375 0.13996 0.03299 6.341 -7.98374 0.08841 0.17978 0.04492] [ 0.28492 0.34127 0.13895 0.03275 0.04938 0.01615 -9.91223 7.98571 0.54098] [ 0.63864 0.76496 0.31145 0.0734 0.11068 0.03619 2.75847 -6.59993 0.73576] [ 0.28699 0.34375 0.13996 0.03299 0.04974 0.01626 0.08841 6.47105 -7.95508]] >>> beta, B = MAPMAP1(D0, D1, S0, S1, "stDistrME") Final Residual Error for G: 1.80411241502e-16 Final Residual Error for R: 4.02455846427e-16 >>> print(beta) [[ 0.14329 0.28553 0.07118 0.02866 0.05711 0.01424 0.11463 0.22842 0.05694]] >>> print(B) [[-9.71508 4.99721 0.14246 0.05698 0.19944 0.02849 0.22793 0.79777 0.11397] [ 5.21856 -6.23504 2.10928 0.04371 0.15299 0.02186 0.17485 0.61196 0.08742] [ 1.27992 2.97972 -7.86004 0.05598 0.19594 0.02799 0.22394 0.78377 0.11197] [ 0.16374 0.57307 0.08187 -9.96725 4.11461 0.01637 0.13099 0.45846 0.06549] [ 0.15812 0.55341 0.07906 5.03162 -6.88932 2.01581 0.12649 0.44273 0.06325] [ 0.16264 0.56923 0.08132 1.03253 2.11385 -7.98374 0.13011 0.45539 0.06506] [ 0.10971 0.384 0.05486 0.02194 0.0768 0.01097 -9.91223 4.3072 0.04389] [ 0.14288 0.50009 0.07144 0.02858 0.10002 0.01429 5.11431 -6.59993 2.05715] [ 0.11229 0.39303 0.05615 0.02246 0.07861 0.01123 1.08984 2.31442 -7.95508]] >>> stdFromPH = CdfFromPH(betap, Bp, np.arange(0.,1.1,0.1)) >>> print(stdFromPH) [ 0. 0.06732 0.14411 0.21822 0.28688 0.34979 0.40723 0.45962 0.50738 0.55093 0.59062] >>> stmFromME = MomentsFromME(beta, B, 5) >>> print(stmFromME) [1.1135106870764573, 2.411324091748754, 7.8172702796511846, 33.78557510760961, 182.52100725089716]