Re: MultinormalDistribution Question

*To*: mathgroup at smc.vnet.net*Subject*: [mg120367] Re: MultinormalDistribution Question*From*: Ray Koopman <koopman at sfu.ca>*Date*: Wed, 20 Jul 2011 06:32:45 -0400 (EDT)*References*: <201107100901.FAA24634@smc.vnet.net> <ivel76$89d$1@smc.vnet.net> <ivuc25$grq$1@smc.vnet.net>

On Jul 17, 3:03 am, Steve <s... at epix.net> wrote: > [...] > What I really need to do is perform this analysis on test data for > which I have only a few data points, hence the Student T distribution > would be more appropriate than the Normal distribution. Secondly, > values for the "independent" and "dependent" variables have no > physical meaning below zero. So this implies that I need truncated > distributions. I'm hoping that the solution Andrzej provided can be > generalized for these added complications. > Here are my 9 {F,t} data points where "F" is considered "independent" > and t considered "dependent". > {{1.01041, 0.3152}, {10.455, 0.3386}, {17.9032, 0.2534}, {24.9581, > 0.5412}, {26.4688, 0.3251}, {27.4651, 0.4428}, {30.1682, > 0.3402}, {36.6174, 0.2106}, {45.6129, 0.2154}} > Would someone be so kind as to plop this data into their notebook to > confirm a solution or two for me ? My results are below which are > based on truncating the Student T distribution, 8 degrees of freedom > and a calculated rho of -0.2327. > [...] Another approach is to regress u = log[t] on f linearly. This solves the problem of keeping the conditional distributions of t non-negative, but makes the regression of t on f nonlinear. For your data, switching to u increases the correlation, but the conditional s.d. is still bigger than the marginal s.d., so there is still room for questioning the whole exercise. FWTW, here are the numbers I got: {mf, sf} = {Mean@f, StandardDeviation@f} {24.5177, 13.3704} {mu, su} = {Mean@u, StandardDeviation@u} {-1.1482, .311145} r = Correlation[f,u] -.322776 b = r*su/sf (* slope *) -.00751142 a = mu - b*mf (* intercept *) -.964034 se = su*Sqrt[(1-r^2)(n-1)/(n-2)] (* conditional s.d. *) .314825 Table[{x, CDF[StudentTDistribution[n-2], (a+b*x-Log[.5])/se]}, {x,0,50,5}] (* f, Pr[(t|f) > .5] *) {{0, .209020}, { 5, .179928}, {10, .154056}, {15, .131283}, {20, .111422}, {25, .0942435}, {30, .0794909}, {35, .0669001}, {40, .0562108}, {45, .0471758}, {50, .0395667}}

**References**:**MultinormalDistribution Question***From:*Steve <s123@epix.net>