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MathGroup Archive 2004

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Re: Integral of a bivariate function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48759] Re: Integral of a bivariate function
  • From: omid_rezayi at hotmail.com (Marc)
  • Date: Tue, 15 Jun 2004 02:50:05 -0400 (EDT)
  • References: <cadvmq$6mf$1@smc.vnet.net> <cagij5$i9n$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Many thanks for your reply Steve. Do you know if it is possible to get
an upper bound on the approximation error of this method in the
general case? For example if one is only interested in upper
p-quantiles of the distribution for small p.



"Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk> wrote in message news:<cagij5$i9n$1 at smc.vnet.net>...
> "Marc" <omid_rezayi at hotmail.com> wrote in message
> news:cadvmq$6mf$1 at smc.vnet.net...
> > For a given bivariate function I want  to calculate the integral of
> > the function over an arbitrary compact region A, for instance over
> > A={(x,y)| f(x,y)=c} for some constant c. The function is smooth and in
> > my application it is the joint density of two continuous random
> > variables. I wonder if this can be done in Mathematica and in that
> > case how. Otherwise I'd appreciate any pointer to other programs which
> > can be used for this.
> >
> 
> Here is a notebook that describes how I would solve this problem. Select
> from the first (*** to the last ****) and copy/paste anywhere in
> Mathematica; it will automatically detect that you are pasting a whole
> notebook.
> 
> Steve Luttrell
> 
> (************** Content-type: application/mathematica **************
>                      CreatedBy='Mathematica 5.0'
> 
>                     Mathematica-Compatible Notebook
> 
> This notebook can be used with any Mathematica-compatible
> application, such as Mathematica, MathReader or Publicon. The data
> for the notebook starts with the line containing stars above.
> 
> To get the notebook into a Mathematica-compatible application, do
> one of the following:
> 
> * Save the data starting with the line of stars above into a file
>   with a name ending in .nb, then open the file inside the
>   application;
> 
> * Copy the data starting with the line of stars above to the
>   clipboard, then use the Paste menu command inside the application.
> 
> Data for notebooks contains only printable 7-bit ASCII and can be
> sent directly in email or through ftp in text mode.  Newlines can be
> CR, LF or CRLF (Unix, Macintosh or MS-DOS style).
> 
> NOTE: If you modify the data for this notebook not in a Mathematica-
> compatible application, you must delete the line below containing
> the word CacheID, otherwise Mathematica-compatible applications may
> try to use invalid cache data.
> 
> For more information on notebooks and Mathematica-compatible
> applications, contact Wolfram Research:
>   web: http://www.wolfram.com
>   email: info at wolfram.com
>   phone: +1-217-398-0700 (U.S.)
> 
> Notebook reader applications are available free of charge from
> Wolfram Research.
> *******************************************************************)
> 
> (*CacheID: 232*)
> 
> 
> (*NotebookFileLineBreakTest
> NotebookFileLineBreakTest*)
> (*NotebookOptionsPosition[     86575,       2344]*)
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> (*  CellTagsIndexPosition[     87177,       2362]*)
> (*WindowFrame->Normal*)
> 
> 
> 
> Notebook[{
> 
> Cell[CellGroupData[{
> Cell["Mapping a Probability Density", "Title"],
> 
> Cell["\<\
> Thoughts on how to map a PDF using a Gaussian approximation to the \
> Dirac delta function
> 
> S P Luttrell
> 12 June 2004\
> \>", "Subtitle"],
> 
> Cell[TextData[{
>   "The basic relationship for mapping a PDF is\n\n",
>   Cell[BoxData[
>       FormBox[
>         RowBox[{\(Pr(a)\), "=",
>           RowBox[{"\[Integral]",
>             RowBox[{
>               StyleBox[
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>                     FontSlant->"Italic"]}]], " ",
>               StyleBox[
>                 RowBox[{"d",
>                   StyleBox["y",
>                     FontSlant->"Italic"]}]],
>               " ", \(Pr(x, y)\), \(\[Delta](a - f(x, y))\)}]}]}],
>         TraditionalForm]]],
>   "\n\nwhere ",
>   Cell[BoxData[
>       \(TraditionalForm\`Pr(x, y)\)]],
>   " is the joint PDF in ",
>   Cell[BoxData[
>       \(TraditionalForm\`x\)]],
>   " and ",
>   Cell[BoxData[
>       \(TraditionalForm\`y\)]],
>   ", ",
>   Cell[BoxData[
>       \(TraditionalForm\`f(x, y)\)]],
>   " maps to the variable whose PDF you wish to compute, and ",
>   Cell[BoxData[
>       \(TraditionalForm\`\[Delta](a - f(x, y))\)]],
>   " is a Dirac delta function that constrains the integral over ",
>   Cell[BoxData[
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>   " that contribute to ",
>   Cell[BoxData[
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>   "."
> }], "Text"],
> 
> Cell[TextData[{
>   "Define a Gaussian PDF ",
>   Cell[BoxData[
>       \(TraditionalForm\`Pr(x, y)\)]],
>   " to work with."
> }], "Text"],
> 
> Cell[BoxData[
>     \(\(p[x_,
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> 
> Cell["Check that it is correctly normalised.", "Text"],
> 
> Cell[CellGroupData[{
> 
> Cell[BoxData[
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> 
> Cell[BoxData[
>     \(\(f[x_, y_] := x\^2 + y\^2;\)\)], "Input"],
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> Cell[TextData[{
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> Cell["\<\
> Switch off warning messages that occur when integrating an almost \
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> \>", "Text"],
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>     \(Off[NIntegrate::"\<slwcon\>"]\)], "Input"],
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> Cell[TextData[{
>   "For concretness, fix ",
>   Cell[BoxData[
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>   ". Check how ",
>   Cell[BoxData[
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>   Cell[BoxData[
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>   Cell[BoxData[
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>   " this tends to a constant, as expected. This gives an idea of what \
> size ",
>   Cell[BoxData[
>       \(TraditionalForm\`\[Epsilon]\)]],
>   " should be to obtain a good estimate of ",
>   Cell[BoxData[
>       \(TraditionalForm\`Pr(a)\)]],
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> }], "Text"],
> 
> Cell[CellGroupData[{
> 
> Cell[BoxData[
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>   ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.0108859, \
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> }, Open  ]],
> 
> Cell[TextData[{
>   "Compare the above with the analytic result that can be computed in \
> this case. Here are the steps in a  quick derivation.\n\n",
>   Cell[BoxData[
>       FormBox[
>         RowBox[{\(Pr \((a)\)\), "=",
>           RowBox[{"\[Integral]",
>             RowBox[{
>               StyleBox[
>                 RowBox[{"d",
>                   StyleBox["x",
>                     FontSlant->"Italic"]}]], " ",
>               StyleBox[
>                 RowBox[{"d",
>                   StyleBox["y",
>                     FontSlant->"Italic"]}]],
>               " ", \(Pr(x, y)\), \(\[Delta](a - f(x, y))\)}]}]}],
>         TraditionalForm]]],
>   "\n\n",
>   Cell[BoxData[
>       FormBox[
>         RowBox[{\(Pr(a)\), "=",
>           RowBox[{\(\[Integral]\_0\%\[Infinity]\),
>             RowBox[{\(1\/2\), " ",
>               RowBox[{"d", "(",
>                 SuperscriptBox[
>                   StyleBox["r",
>                     FontSlant->"Italic"], "2"],
>                 StyleBox[")",
>                   FontSlant->"Italic"]}],
>               RowBox[{
>                 SubsuperscriptBox[
>                   StyleBox["\[Integral]",
>                     FontSlant->"Italic"], "0", \(2  \[Pi]\)],
>                 " ", \(d\[Theta]\ \ \(1\/\((\(\@\(2  \[Pi]\)\) \
> \[Sigma])\)\^2\)
>                   Exp[\(-\(r\^2\/\(2  \[Sigma]\^2\)\)\)] \(\[Delta](
>                     a - r\^2)\)\)}]}]}]}], TraditionalForm]]],
>   "\n\n",
>   Cell[BoxData[
>       \(TraditionalForm\`Pr(a) = \(1\/2\)
>           2  \[Pi] \( 1\/\((\(\@\(2  \[Pi]\)\) \[Sigma])\)\^2\)
>           Exp[\(-\(a\^2\/\(2  \[Sigma]\^2\)\)\)]\)]],
>   "\n\n",
>   Cell[BoxData[
>       \(TraditionalForm\`Pr(a) = \(1\/\(2  \[Sigma]\^2\)\)
>           Exp[\(-\(a\/\(2  \[Sigma]\^2\)\)\)]\)]]
> }], "Text"],
> 
> Cell[TextData[{
>   "Setting ",
>   Cell[BoxData[
>       \(TraditionalForm\`\[Sigma] = 1\)]],
>   ", plot this over the same range of ",
>   Cell[BoxData[
>       \(TraditionalForm\`a\)]],
>   " as the numerical approximation above."
> }], "Text"],
> 
> Cell[CellGroupData[{
> 
> Cell[BoxData[
>     \(\(g2 =
>         With[{\[Sigma] = 1},
>           Plot[\(1\/\(2  \[Sigma]\^2\)\)
>               Exp[\(-\(a\/\(2  \[Sigma]\^2\)\)\)], {a, 0,
>               0.1}]];\)\)], "Input"],
> 
> Cell[GraphicsData["PostScript", "\<\
> %!
> %%Creator: Mathematica
> %%AspectRatio: .61803
> MathPictureStart
> /Mabs {
> Mgmatrix idtransform
> Mtmatrix dtransform
> } bind def
> /Mabsadd { Mabs
> 3 -1 roll add
> 3 1 roll add
> exch } bind def
> %% Graphics
> %%IncludeResource: font Courier
> %%IncludeFont: Courier
> /Courier findfont 10  scalefont  setfont
> % Scaling calculations
> 0.0238095 9.52381 -11.4655 24.1377 [
> [.21429 .59082 -12 -9 ]
> [.21429 .59082 12 0 ]
> [.40476 .59082 -12 -9 ]
> [.40476 .59082 12 0 ]
> [.59524 .59082 -12 -9 ]
> [.59524 .59082 12 0 ]
> [.78571 .59082 -12 -9 ]
> [.78571 .59082 12 0 ]
> [.97619 .59082 -9 -9 ]
> [.97619 .59082 9 0 ]
> [.01131 .12057 -24 -4.5 ]
> [.01131 .12057 0 4.5 ]
> [.01131 .24125 -30 -4.5 ]
> [.01131 .24125 0 4.5 ]
> [.01131 .36194 -24 -4.5 ]
> [.01131 .36194 0 4.5 ]
> [.01131 .48263 -30 -4.5 ]
> [.01131 .48263 0 4.5 ]
> [ 0 0 0 0 ]
> [ 1 .61803 0 0 ]
> ] MathScale
> % Start of Graphics
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> newpath
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> closepath
> clip
> newpath
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> s
> % End of Graphics
> MathPictureEnd
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