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Notebook[{

Cell[CellGroupData[{
Cell["Say No to Loops!", "Title"],

Cell[TextData[{
  "One of the reasons people who are used to C, GAUSS, FORTRAN, etc, find ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " a bit strange to begin with is that they are used to \"procedural\" \
programming styles instead of \"functional\" programming. So when they start \
to program in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " and use the loops and nested If[] statements and what-not, they think ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is slow.\n\nThis notebook shows just how much you can speed up your \
computations in ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " by using functional programming instead of a loop.  The only problem is \
you have to be a bit more lateral in your approach to the problem."
}], "Text"],

Cell[CellGroupData[{

Cell["The Problem: generating artificial ARMA time series.", "Subsection"],

Cell[TextData[{
  "Anyone who's done econometrics or time series work would be familiar with \
this sort of process:\n\n\t",
  Cell[BoxData[
      \(TraditionalForm
      \`y\_t = \[Sum]\+\(i = 1\)\%p\( \[Phi]\_i\) y\_\(t - 1\)\  + 
          \[Sum]\+\(j = 0\)\%q\( \[Theta]\_j\) \[Epsilon]\_\(t - j\)\)]],
  "\n\nThe \[Epsilon]'s are usually normally distributed, or at least iid. \
(should that be \"normally normally\").\n\nSuppose you want to create a \
function that generates artificial ARMA processes.  You'd probably do \
something like this:"
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Version 1: Use a loop.", "Subsection"],

Cell[BoxData[
    RowBox[{
    \(ARMAlist[length_Integer, sv_, sd_Real, pcoefs_List, qcoefs_List]\), 
      ":=", 
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        RowBox[{
        \(Length[Dimensions[pcoefs]] == 1 && 
            Length[Dimensions[qcoefs]] == 1\), ",", 
          RowBox[{"Module", "[", 
            RowBox[{
            \({shocklist, maparts, q, \n\t\t\t\tp, qplus, thelist}\), ",", 
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              RowBox[{
                RowBox[{
                  StyleBox["p",
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                  StyleBox["shocklist",
                    FontColor->GrayLevel[0.333333]], "=", 
                  \(RandomArray[NormalDistribution[0, sd], 
                    length + Max[p, q]\ ]\)}], ";", "\n", "\t\t\t", 
                RowBox[{
                  StyleBox["maparts",
                    FontColor->GrayLevel[0.333333]], "=", 
                  \(Expand/@\ \n\t\t\t\t\t
                    Table[qplus.shocklist
                          \[LeftDoubleBracket]Range[t - q, t]
                            \[RightDoubleBracket], {t, q + 1, length + q}]
                      \)}], ";", "\n", "\t\t\t", 
                RowBox[{
                  StyleBox["thelist",
                    FontColor->GrayLevel[0.333333]], "=", "\n", "\t\t\t\t", 
                  \(Join[Table[sv, {p}], Table[0, {length}]]\)}], ";", 
                \(Do[thelist = 
                    ReplacePart[thelist, \n\t\t\t
                      pcoefs.Reverse[
                            thelist
                              \[LeftDoubleBracket]Range[i - p, i - 1]
                                \[RightDoubleBracket]] + 
                        maparts
                          \[LeftDoubleBracket]i - p\[RightDoubleBracket], 
                      i], {i, p + 1, length + p}]\), ";", 
                StyleBox[\(Drop[thelist, p]\),
                  FontColor->RGBColor[0, 0, 1]]}]}], " ", "]"}], ",", 
          \(Message[ARMAlist::"\<badcf\>"]\)}], "]"}]}]], "Input",
  PageWidth->PaperWidth,
  AspectRatioFixed->True],

Cell["\<\
Here are the timings for generating 1000 to 12 000 artificial data \
points on a Power Mac G3 (233 MHz):\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(timingtesttable1 = 
      Table[Timing[
          ARMAlist[1000\ i, 0, 0.5, {0.9, 0.09}, {0.8, 0.8, 0.5}]; ], {i, 1, 
          8}]\)], "Input"],

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        10.0833333333339392`\ Second, Null}, {15.4666666666662422`\ Second, 
        Null}, {22.4833333333335749`\ Second, Null}, {
        30.03333333333285`\ Second, Null}, {38.5`\ Second, Null}}\)], "Output"]
}, Open  ]],

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Cell[BoxData[
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Cell["\<\
Ok, so this first version looks OK, runs reasonably well.  But \
there is a better way.  Functional programming is quicker than that \
loop.\
\>", "Text"]
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Cell["Version 2: Functional Programming", "Subsection"],

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          macoefs = Flatten[Reverse[Join[{1}, qcoefs]]], 
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              Expand\ /@
                \((Take[
                    \((macoefs.
                        \((\(RotateLeft[rawshocks, #]&\)\ /@\ Range[0, q])\))
                      \), n])\)}, \n\t\t\t
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          30}]\)], "Input"],

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There!  Heaps faster and approximately linear in the number of data \
points to be generated.  The previous version would have taken AGES to \
generate 10,000 data points, especially on a slower machine, but with the \
functional programming style you can do it in less than two seconds.  \
Consider that the 8000-point case is nearly TWENTY times faster!!  And there \
are fewer lines of code. All right!\
\>", "Text"],

Cell["\<\
One final thing: you might have noticed that I've defined pcoefs in \
the new version to match {_,__}, that is, a vector with two or more elements. \
 That's because the AR(1) case admits a simpler function and I decided to \
code that as a special case.\
\>", "Text"],

Cell["\<\
Luci Ellis
8 July 1998\
\>", "Text"]
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