Re: Serious problem with CDF[MultinormalDistribution...

*To*: mathgroup at smc.vnet.net*Subject*: [mg113925] Re: Serious problem with CDF[MultinormalDistribution...*From*: Myeong Ae Kang <makang at purdue.edu>*Date*: Thu, 18 Nov 2010 07:05:27 -0500 (EST)

With version 6.0 and 7.0.1, the difference is quite large. With other input values, the results are much worse even than this. In[1326]:= Needs["MultivariateStatistics`"] (1.0-CDF[MultinormalDistribution[{0,0,0},Rationalize[{{1,0.559887,0.289988}, {0.559887,1,0.880157},{0.289988,0.880157,1}}]],SetPrecision [{4.5,4.5,6.9958},50]]) (1.0-CDF[MultinormalDistribution[{0,0},{{1,0.559887},{0.559887,1}}],{4.5,4.5}]) Out[1327]= 5.00832*10^-6 Out[1328]= 6.75155*10^-6 And, similarly to this, with version 6.0, I did the calculation of CDF with 6 RV's in different orders of RV's. I just changed the orders of the sigma matrix and range vector with zero mean. But, the results(which are suppossed to equal) are quite different like this following: In[1432]:= ClearAll; Print["R,A,W ======================="] TT={{1,0.576363,0.300013,0.888477,0.106961,0.215919}, {0.576363,1,0.888477,0.300013,0.215919,0.106961},{0.300013,0.888477,1,- 9.72249*10^-6,0.466809,-0.191847},{0.888477,0.300013,-9.72249*10^-6,1,- 0.191847,0.466809},{0.106961,0.215919,0.466809,-0.191847,1,-0.862483}, {0.215919,0.106961,-0.191847,0.466809,-0.862483,1}}; TTbnd={7.14659,7.14659,7.83599,7.83599,5.55663,5.55663}; TTp={{0,0,0,0,0,1},{0,0,0,0,1,0},{0,0,0,1,0,0},{0,0,1,0,0,0},{0,1,0,0,0,0}, {1,0,0,0,0,0}}.TT.{{0,0,0,0,0,1},{0,0,0,0,1,0},{0,0,0,1,0,0},{0,0,1,0,0,0}, {0,1,0,0,0,0},{1,0,0,0,0,0}} Tbnd=TTbnd.Transpose[{{0,0,0,0,0,1},{0,0,0,0,1,0},{0,0,0,1,0,0},{0,0,1,0,0,0}, {0,1,0,0,0,0},{1,0,0,0,0,0}}] 1.0-CDF[MultinormalDistribution[{0,0,0,0,0,0},TTp],Tbnd] Print["A,R,W ======================="] TTp2={{0,0,1,0,0,0},{0,0,0,1,0,0},{0,0,0,0,1,0},{0,0,0,0,0,1},{1,0,0,0,0,0}, {0,1,0,0,0,0}}.TT.Transpose[{{0,0,1,0,0,0},{0,0,0,1,0,0},{0,0,0,0,1,0}, {0,0,0,0,0,1},{1,0,0,0,0,0},{0,1,0,0,0,0}}] Tbnd2=TTbnd.Transpose[{{0,0,1,0,0,0},{0,0,0,1,0,0},{0,0,0,0,1,0},{0,0,0,0,0,1}, {1,0,0,0,0,0},{0,1,0,0,0,0}}] 1.0-CDF[MultinormalDistribution[{0,0,0,0,0,0},TTp2],Tbnd2] Print["A,W,R ======================="] TTp3={{0,0,1,0,0,0},{0,0,0,1,0,0},{1,0,0,0,0,0},{0,1,0,0,0,0},{0,0,0,0,1,0}, {0,0,0,0,0,1}}.TT.Transpose[{{0,0,1,0,0,0},{0,0,0,1,0,0},{1,0,0,0,0,0}, {0,1,0,0,0,0},{0,0,0,0,1,0},{0,0,0,0,0,1}}] Tbnd3=TTbnd.Transpose[{{0,0,1,0,0,0},{0,0,0,1,0,0},{1,0,0,0,0,0},{0,1,0,0,0,0}, {0,0,0,0,1,0},{0,0,0,0,0,1}}] 1.0-CDF[MultinormalDistribution[{0,0,0,0,0,0},TTp3],Tbnd3] During evaluation of In[1432]:= R,A,W ======================= Out[1436]= {{1.,-0.862483,0.466809,-0.191847,0.106961,0.215919},{-0.862483,1.,- 0.191847,0.466809,0.215919,0.106961},{0.466809,-0.191847,1.,-9.72249*10^- 6,0.300013,0.888477},{-0.191847,0.466809,-9.72249*10^-6,1.,0.888477,0.300013}, {0.106961,0.215919,0.300013,0.888477,1.,0.576363}, {0.215919,0.106961,0.888477,0.300013,0.576363,1.}} Out[1437]= {5.55663,5.55663,7.83599,7.83599,7.14659,7.14659} Out[1438]= 1.37516*10^-8 During evaluation of In[1432]:= A,R,W ======================= Out[1440]= {{1.,-9.72249*10^-6,0.466809,-0.191847,0.300013,0.888477},{- 9.72249*10^-6,1.,-0.191847,0.466809,0.888477,0.300013},{0.466809,-0.191847,1.,- 0.862483,0.106961,0.215919},{-0.191847,0.466809,-0.862483,1.,0.215919,0.106961}, {0.300013,0.888477,0.106961,0.215919,1.,0.576363}, {0.888477,0.300013,0.215919,0.106961,0.576363,1.}} Out[1441]= {7.83599,7.83599,5.55663,5.55663,7.14659,7.14659} Out[1442]= 6.50208*10^-9 During evaluation of In[1432]:= A,W,R ======================= Out[1444]= {{1.,-9.72249*10^-6,0.300013,0.888477,0.466809,-0.191847},{- 9.72249*10^-6,1.,0.888477,0.300013,-0.191847,0.466809}, {0.300013,0.888477,1.,0.576363,0.106961,0.215919}, {0.888477,0.300013,0.576363,1.,0.215919,0.106961},{0.466809,- 0.191847,0.106961,0.215919,1.,-0.862483},{-0.191847,0.466809,0.215919,0.106961,- 0.862483,1.}} Out[1445]= {7.83599,7.83599,7.14659,7.14659,5.55663,5.55663} Out[1446]= 1.17684*10^-14 --------------- Now, I have two questions. 1) Does mathematica version 8.0 give three similar results of the second part? I don't have version 8.0. Before upgrading it (if possible), I wonder that. 2) Is there any other way to get the proper results with version 6.0 or 7.01? Thank you. Regards, Myeong Ae Kang ================================= Darren Glosemeyer <darreng at wolfram.com> wrote: > On 11/17/2010 4:29 AM, Myeong Ae Kang wrote: > > Hello. > > > > I have a problem, serious and urgent. > > > > I got wrong results of CDF of MultinormalDistribution. > > > > I'm calculating (1.0 -CDF[MultinormalDistribution[{0, 0,0}, {{1, 0.559887, > > 0.289988}, {0.559887, 1,0.880157}, {0.289988, 0.880157, 1}}], > > {4.5,4.5,6.9958}]). > > The results is smaller than (1.0 -CDF[MultinormalDistribution[{0,0}, {{1, > > 0.559887}, {0.559887, 1}}], {4.5,4.5}]). It should be larger. > > > > Actually, I have to calculate more than 6 to 12 random variables (12 x 12 > sigma > > matrix.). > > > > Mathematica 6.0 and 7.0.1 gave me the same wrong result. > > > > I compared this with the result of another system. > > That system gave the expected value. > > > > To fix this problem, what should I do? > > > > > > Regards, > > Myeong Ae Kang > > > > > > > > > This is addressed in version 8. Here are the results: > > > > In[1]:= threeD = (1.0 - > CDF[MultinormalDistribution[{0, 0, > 0}, {{1, 0.559887, 0.289988}, {0.559887, 1, > 0.880157}, {0.289988, 0.880157, 1}}], {4.5, 4.5, 6.9958}]) > > -6 > Out[1]= 6.75155 10 > > In[2]:= twoD = (1.0 - > CDF[MultinormalDistribution[{0, > 0}, {{1, 0.559887}, {0.559887, 1}}], {4.5, 4.5}]) > > -6 > Out[2]= 6.75155 10 > > In[3]:= threeD > twoD > > Out[3]= True > > > The difference is at the level of machine precision numerical error: > > > In[4]:= threeD - twoD > > -16 > Out[4]= 1.11022 10 > > > > Note that if we artificially increase the precision as in the example > below, we get a difference at the same order of magnitude, so the > precision in the difference above is as good as could be expected (and > in fact probably better than should be expected) at machine precision. > > > In[5]:= CDF[MultinormalDistribution[{0, 0, 0}, > Rationalize[{{1, 0.559887, 0.289988}, {0.559887, 1, > 0.880157}, {0.289988, 0.880157, 1}}, 0]], > SetPrecision[{4.5, 4.5, 6.9958}, 50]] > > Out[5]= 0.999993248446123977600817893853727681422304494350 > > In[6]:= CDF[MultinormalDistribution[{0, 0}, > Rationalize[{{1, 0.559887}, {0.559887, 1}}, 0]], > SetPrecision[{4.5, 4.5}, 50]] > > Out[6]= 0.999993248446124136306526964076086153533229885505 > > In[7]:= % - %% > > -16 > Out[7]= 1.5870570907022235847211092539116 10 > > > Darren Glosemeyer > Wolfram Research >