Re: Need help NIntegrating stepwise 3D probability density functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg114095] Re: Need help NIntegrating stepwise 3D probability density functions*From*: Barrie Stokes <Barrie.Stokes at newcastle.edu.au>*Date*: Tue, 23 Nov 2010 06:00:47 -0500 (EST)

Hi Jason I admit I tried a few things before I tried the following: thresholdDensity = 5/100; NIntegrate[ PDF[ testDistribution, {x, y, z} ]* UnitStep[ PDF[ testDistribution, {x, y, z} ] - thresholdDensity ] , {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, Infinity} , Method -> {"LocalAdaptive"} ] thresholdDensity = 4/100; NIntegrate[ PDF[ testDistribution, {x, y, z} ]* UnitStep[ PDF[ testDistribution, {x, y, z} ] - thresholdDensity ] , {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, Infinity} , Method -> {"LocalAdaptive"} ] thresholdDensity = 6/100; NIntegrate[ PDF[ testDistribution, {x, y, z} ]* UnitStep[ PDF[ testDistribution, {x, y, z} ] - thresholdDensity ] , {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, Infinity} , Method -> {"LocalAdaptive"} ] thresholdDensity = 0/100; NIntegrate[ PDF[ testDistribution, {x, y, z} ]* UnitStep[ PDF[ testDistribution, {x, y, z} ] - thresholdDensity ] , {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, Infinity} , Method -> {"LocalAdaptive"} ] The answers look like they are behaving correctly. I too read the documentation in UnitStep about "SymbolicProcessing" -> 0, and I only tried removing it on a whim. Go figure. Maybe Daniel Lichtblau can explain it for us? Cheers Barrie >>> On 22/11/2010 at 11:40 pm, in message <201011221240.HAA06705 at smc.vnet.net>, Jason Neuswanger <jrneuswanger at alaska.edu> wrote: > I need to integrate 3D probability density functions over the region > in which the probability density exceeds some threshold (the end goal > being to calculate isopleths in 3D animal territories fit by kernel > distributions). > > The simplest test case I could come up with is this: > > testDistribution = > MultinormalDistribution[{1, 2, > 4}, {{2, -1/4, 1/3}, {-1/4, 2/3, 1/5}, {1/3, 1/5, 1/2}}]; > > thresholdDensity = 0.05; > > NIntegrate[ > PDF[testDistribution, {x, y, z}]* > UnitStep[PDF[testDistribution, {x, y, z}] - thresholdDensity] > , {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, > Infinity} > , Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0} > ] > > I've tried quite a few things and just can't get remotely sensible > results. The analogous case in 1 dimension works just fine, shown > below: > > testDistribution2 = NormalDistribution[]; > > thresholdDensity2 = 0.35; > > NIntegrate[ > PDF[testDistribution2, {x}]* > UnitStep[PDF[testDistribution2, {x}] - thresholdDensity2] > , {x, -Infinity, Infinity} > , Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0} > ] > > (* check answer using symbolic methods *) > > thresholdRoot = > x /. FindRoot[ > PDF[testDistribution2, {x}] - thresholdDensity2, {x, 1}]; > Integrate[ > PDF[testDistribution2, {x}], {x, -thresholdRoot, thresholdRoot}] > > Any help here would be VERY much appreciated!