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Notebook[{

Cell[CellGroupData[{
Cell["Loess local regression", "Section"],

Cell[CellGroupData[{

Cell["Acknowledgements", "Subsubsection"],

Cell[TextData[{
  "This code was jointly developed by Luci Ellis, RBA,  and David Burmaster, \
Alceon Corp, 1997\[Dash]98.  See \"Visualising Data\" for more information \
about the procedure.  The procedure requires the ",
  StyleBox["simpleLinRegress[] ",
    FontWeight->"Bold"],
  "function developed by Luci Ellis, from the econometric_functions.nb \
notebook."
}], "Text"]
}, Open  ]],

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Cell["Startup", "Subsubsection"],

Cell[TextData[{
  "The cell below just loads in some ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " add-ons."
}], "Text"],

Cell["\<\
Needs[\"Statistics`NormalDistribution`\"];
Needs[\"Statistics`ContinuousDistributions`\"];
Needs[\"Statistics`DataManipulation`\"];
Needs[\"Statistics`DescriptiveStatistics`\"];
Needs[\"Statistics`MultiDescriptiveStatistics`\"];
Needs[\"Statistics`LinearRegression`\"];
Needs[\"Statistics`DataSmoothing`\"];
Needs[\"Graphics`Graphics`\"];\
\>", "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(EGloess::usage\  = \ 
      "\<This function implements the fixed-window (variable sample size) \
version of the loess local regression procedure where there is a single \
explanator.  Loess is useful when there are few explanatory variables, a \
fairly large sample size, but the functional form is unknown and potentially \
complicated.\>"\)], "Input"],

Cell[BoxData[
    \(EGloessX::usage\  = \ 
      "\<This function implements the fixed-sample-size (nearest neighbour) \
version of the loess local regression procedure where there is a single \
explanator.  Loess is useful when there are few explanatory variables, a \
fairly large sample size, but the functional form is unknown and potentially \
complicated.\>"\)], "Input"],

Cell[TextData[{
  "EGloess[\txypairs:{..{_Real,_Real}},\n\t\t\tnwtfn_\t\t\t/; 1 <= nwtfn,\n\t\
\t\tnpoly_Integer\t/; 1  == npoly || 2 == npoly,\n\t\t\tndiv_Integer\t/; 10 \
<= ndiv,\n\t\t\tswin_\t\t\t/; 0  <  swin\t\t] :=\n\t\t\t",
  StyleBox["(*the conditions for the function to be valid*)",
    FontColor->GrayLevel[0.500008]],
  "\nModule[{xvec, yvec, xmin, xmax, wtfn, yloesspt,\n\t\txloessvec, \
yloessvec, wtvecvec,xypairssorted,regressdata},\n\t\t",
  StyleBox["(* these are the names for the local variables *)\n\t\t",
    FontColor->GrayLevel[0.500008]],
  "\nxypairssorted = Sort[xypairs];\n{xvec, yvec}\t=\t\
Transpose[xypairssorted];\n{xmin, xmax}\t=\t{First[xvec], Last[xvec]};\n",
  StyleBox["(* all pretty straight-forward *)",
    FontColor->GrayLevel[0.500008]],
  "\n\nxloessvec =\t\n(N[xmin + # ((xmax-xmin)/(ndiv-1))]&) /@ \
Range[0,ndiv-1];\n",
  StyleBox["(* these are the points at which the fit is evaluated *)",
    FontColor->GrayLevel[0.500008]],
  "\nwtvecvec  =\t\n\tTable[N[Exp[-Abs[((xvec\[LeftDoubleBracket]i\
\[RightDoubleBracket]-xloessvec\[LeftDoubleBracket]j\[RightDoubleBracket])/\
swin)^nwtfn]]],\n\t{j, 1, ndiv},{i, 1, Length[xypairs]}];\n",
  StyleBox[
  "(* a matrix where the rows are the vector of weights for each evaluation \
of the WLS fit *)\n",
    FontColor->GrayLevel[0.500008]],
  "\nregressdata = If[npoly==1,Reverse /@ \
xypairssorted,Transpose[Join[{yvec},{xvec},{xvec}^2]]];\n",
  StyleBox[
  "(* puts the data in a form appropriate for use in the simpleLinRegress[] \
auxiliary function *)\n",
    FontColor->GrayLevel[0.500008]],
  "\nyloessvec=\tTable[(Table[q^k,{k,0,npoly}]. \t\n\t\t\
Evaluate[simpleLinRegress[ regressdata,   \t\n\t\tGLSWeights->wtvecvec\
\[LeftDoubleBracket]jj\[RightDoubleBracket] ]]\[LeftDoubleBracket]1\
\[RightDoubleBracket]) /. q->xloessvec\[LeftDoubleBracket]jj\
\[RightDoubleBracket],\n\t\t\t\t {jj, 1, ndiv}];\n",
  StyleBox["(* finds the WLS-fitted values for y at gridpoints of x *)",
    FontColor->GrayLevel[0.500008]],
  "\n\nTranspose[{xloessvec, yloessvec}]\n",
  StyleBox["(* returns vector of ndiv {x,y} pairs for loess points *)",
    FontColor->GrayLevel[0.500008]],
  "\n]"
}], "Input"],

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  "EGloessX[\txypairs:{..{_Real,_Real}},\n\t\t\tnwtfn_\t\t\t/; 1 <= nwtfn,\n\t\
\t\tnpoly_Integer\t/; 1  == npoly || 2 == npoly,\n\t\t\tndiv_Integer    /; 10 \
<= ndiv,\n\t\t\tsamplesize_Integer\t/; 20 <= samplesize,\n\t\t\tswin_\t\t\t/; \
0  <  swin\t\t] :=\n\t\t\t",
  StyleBox["(*the conditions for the function to be valid*)",
    FontColor->GrayLevel[0.500008]],
  "\nModule[{xvec, yvec, xmin, xmax, wtfn, yloesspt,\n\t\txloessvec, \
yloessvec, wtvecvec,xypairssorted,baseregressdata,regressdata},\n\t\t",
  StyleBox["(* these are the names for the local variables *)\n\t\t",
    FontColor->GrayLevel[0.500008]],
  "\nxypairssorted = Sort[xypairs];\n{xvec, yvec}\t=\t\
Transpose[xypairssorted];\n{xmin, xmax}\t=\t{First[xvec], Last[xvec]};\n",
  StyleBox["(* all pretty straight-forward *)",
    FontColor->GrayLevel[0.500008]],
  "\n\nxloessvec =\t\n(N[xmin + # ((xmax-xmin)/(ndiv-1))]&) /@ \
Range[0,ndiv-1];\n",
  StyleBox["(* these are the points at which the fit is evaluated *)",
    FontColor->GrayLevel[0.500008]],
  "\n",
  StyleBox[
  "wtvecvec  =\t\n\tTable[N[Exp[-Abs[((xvec\[LeftDoubleBracket]i\
\[RightDoubleBracket]-xloessvec\[LeftDoubleBracket]j\[RightDoubleBracket])/\
swin)^nwtfn]]],\n\t{j, 1, ndiv},{i, 1, Length[xypairs]}];",
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  "(* a matrix where the rows are the vector of weights for each evaluation \
of the WLS fit *)\n",
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  "\nbaseregressdata = If[npoly==1,Reverse /@ xypairssorted, \
Transpose[Join[{yvec},{xvec},{xvec}^2]]];\n",
  StyleBox[
  "(* puts the data in a form appropriate for use in the simpleLinRegress[] \
auxiliary function *)",
    FontColor->GrayLevel[0.500008]],
  "\n\nregressdata = If[npoly==1, \n(Take[#,samplesize]&)/@  \
(RotateLeft[baseregressdata, Max[0,Floor[Position[xvec, \
Floor[Select[xvec,Min[Abs[xvec-#]] ]] ]-(samplesize/2)]] ] &/@xloessvec);\n",
  StyleBox[
  "(* Partitions the data into multiple samples of the desired size, \
adjusting for end-sample constraints  *)\n",
    FontColor->GrayLevel[0.500008]],
  "\nyloessvec=\tTable[(Table[q^k,{k,0,npoly}]. \t\n\t\t\
Evaluate[simpleLinRegress[ regressdata,   \t\n\t\tGLSWeights->wtvecvec\
\[LeftDoubleBracket]jj\[RightDoubleBracket] ]]\[LeftDoubleBracket]1\
\[RightDoubleBracket]) /. q->xloessvec\[LeftDoubleBracket]jj\
\[RightDoubleBracket],\n\t\t\t\t {jj, 1, ndiv}];\n",
  StyleBox["(* finds the WLS-fitted values for y at gridpoints of x *)",
    FontColor->GrayLevel[0.500008]],
  "\n\nTranspose[{xloessvec, yloessvec}]\n",
  StyleBox["(* returns vector of ndiv {x,y} pairs for loess points *)",
    FontColor->GrayLevel[0.500008]],
  "\n]"
}], "Input"]
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Cached data follows.  If you edit this Notebook file directly, not using
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***********************************************************************)

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End of Mathematica Notebook file.
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