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MathGroup Archive 2004

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Re: Integral of a bivariate function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48753] Re: Integral of a bivariate function
  • From: "Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk>
  • Date: Sat, 12 Jun 2004 23:33:42 -0400 (EDT)
  • References: <cadvmq$6mf$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

"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
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*******************************************************************)

(*CacheID: 232*)


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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[
                RowBox[{"d",
                  StyleBox["x",
                    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[
      \(TraditionalForm\`x\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`y\)]],
  " to pick up only those parts of ",
  Cell[BoxData[
      \(TraditionalForm\`Pr(x, y)\)]],
  " that contribute to ",
  Cell[BoxData[
      \(TraditionalForm\`Pr(a)\)]],
  "."
}], "Text"],

Cell[TextData[{
  "Define a Gaussian PDF ",
  Cell[BoxData[
      \(TraditionalForm\`Pr(x, y)\)]],
  " to work with."
}], "Text"],

Cell[BoxData[
    \(\(p[x_,
          y_, \[Sigma]_] := \(1\/\((\(\@\(2  \[Pi]\)\) \[Sigma])\)\^2\
\) Exp[\(-\(\(x\^2 + y\^2\)\/\(2  \[Sigma]\^2\)\)\)];\)\)], "Input"],

Cell["Check that it is correctly normalised.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(NIntegrate[
      p[x, y, 1], {x, \(-\[Infinity]\), \[Infinity]}, {y, \(-\
\[Infinity]\), \[Infinity]}]\)], "Input"],

Cell[BoxData[
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Cell[TextData[{
  "Define an ",
  Cell[BoxData[
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  Cell[BoxData[
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compute is the probability density as a function of squared radius."
}], "Text"],

Cell[BoxData[
    \(\(f[x_, y_] := x\^2 + y\^2;\)\)], "Input"],

Cell[TextData[{
  "Define an approximation to the Dirac delta function. This is a \
Gaussian with standard devaiation ",
  Cell[BoxData[
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  ". As ",
  Cell[BoxData[
      \(TraditionalForm\`\[Epsilon]\[LongRightArrow]0\)]],
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}], "Text"],

Cell[BoxData[
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Cell["\<\
Switch off warning messages that occur when integrating an almost \
singular function. This is a dodgy procedure, so the quality of the \
numerical results must be verified. This is done below.\
\>", "Text"],

Cell[BoxData[
    \(Off[NIntegrate::"\<slwcon\>"]\)], "Input"],

Cell[TextData[{
  "For concretness, fix ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sigma] = 1\)]],
  ". Check how ",
  Cell[BoxData[
      \(TraditionalForm\`Pr(a = 0.1)\)]],
  " varies with the width ",
  Cell[BoxData[
      \(TraditionalForm\`\[Epsilon]\)]],
  " of the approximation to the Dirac delta function. As ",
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size ",
  Cell[BoxData[
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  " should be to obtain a good estimate of ",
  Cell[BoxData[
      \(TraditionalForm\`Pr(a)\)]],
  "."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(Plot[
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More generally, a better survey would need to be done to pick a good \
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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[{\(1\/2\), " ",
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                SuperscriptBox[
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                StyleBox[")",
                  FontSlant->"Italic"]}],
              RowBox[{
                SubsuperscriptBox[
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                    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[
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          Plot[\(1\/\(2  \[Sigma]\^2\)\)
              Exp[\(-\(a\/\(2  \[Sigma]\^2\)\)\)], {a, 0,
              0.1}]];\)\)], "Input"],

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