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Notebook[{

Cell[CellGroupData[{
Cell["Business cycle tools", "Subtitle"],

Cell[TextData[{
  "Luci Ellis\nReserve Bank of Australia and Australian National University\n \
",
  ButtonBox["elisha@dot.net.au",
    ButtonData:>{
      URL[ "mailto:elisha@dot.net.au"], None},
    Active->True,
    ButtonStyle->"Hyperlink",
    ButtonNote->"Send mail to Luci Ellis at home"]
}], "Author"],

Cell[CellGroupData[{

Cell["Finding peaks in data", "Section"],

Cell[BoxData[
    \(Clear[findPeaks]\)], "Input",
  CellLabel->"In[3]:="],

Cell[CellGroupData[{

Cell[BoxData[
    \(findPeaks::usage = "\<findPeaks[data,consec] finds the points in data \
that are an increase from the previous point, and following which there are \
consec consecutive declines. This function can be used to identify recessions \
in macroeconomic data. The parameter consec is optional. If omitted, it takes \
the value 2. findPeaks[data,{consecups,consecdowns}] finds the points in data \
that complete a sequence of consecups consecutive rises and following which \
there are consecdowns consecutive falls.\>"\)], "Input",
  CellLabel->"In[4]:="],

Cell[BoxData[
    \("findPeaks[data,consec] finds the points in data that are an increase \
from the previous point, and following which there are consec consecutive \
declines. This function can be used to identify recessions in macroeconomic \
data. The parameter consec is optional. If omitted, it takes the value 2. \
findPeaks[data,{consecups,consecdowns}] finds the points in data that \
complete a sequence of consecups consecutive rises and following which there \
are consecdowns consecutive falls."\)], "Output",
  CellLabel->"Out[4]="]
}, Open  ]],

Cell[BoxData[
    \(\(Options[findPeaks] = {DataInLevels \[Rule] True};\)\)], "Input",
  CellLabel->"In[5]:="],

Cell[BoxData[
    \(findPeaks[data : {__?NumericQ}, consec_:  2, 
          opts___Rule] /; \ \((Head[consec] \[Equal] Integer && 
            Length[data] > 4\  && \ consec > 0\  && 
            Length[data] > 
              consec + 2)\) := \[IndentingNewLine]Module[{difffirst, ddata, \ 
          nested, \ negtest, 
          posns}, \ \[IndentingNewLine]difffirst\  = \ \(DataInLevels /. \
{opts}\) /. Options[findPeaks]; \[IndentingNewLine]ddata = 
          If[difffirst, ListCorrelate[{\(-1\), 1}, data], data]; \  (*\ 
          differencing\ data\ if\ required\ *) \[IndentingNewLine]nested = 
          Transpose[\ \(Drop[#, \(-consec\)] &\)\  /@ \ \((\(RotateRight[
                      ddata, #] &\)\  /@ \ 
                  Range[consec - 
                      1, \(-1\), \(-1\)])\)]; \[IndentingNewLine]negtest = 
          Join[{_?Positive}, 
            Table[_?Negative, {consec}]]; \[IndentingNewLine]posns = 
          Cases[Flatten[
                Position[nested, negtest, {1}, 
                  Heads \[Rule] False]] - \((consec - 
                  If[difffirst, 2, 1])\), _?
              Positive]; \[IndentingNewLine]Transpose[{data\
\[LeftDoubleBracket]posns\[RightDoubleBracket], posns}]]\)], "Input",
  CellLabel->"In[6]:="],

Cell[BoxData[
    \(findPeaks[
          data : {__?NumericQ}, {consecups_Integer?Positive, 
            consecdowns_Integer?Positive}, 
          opts___Rule] /; \ \((Length[data] > 4\ \  && 
            Length[data] > 
              consecups + consecdowns + 
                2)\) := \[IndentingNewLine]Module[{difffirst, ddata, \ 
          nested, \ negtest, 
          posns}, \ \[IndentingNewLine]difffirst\  = \ \(DataInLevels /. \
{opts}\) /. Options[findPeaks]; \[IndentingNewLine]ddata = 
          If[difffirst, ListCorrelate[{\(-1\), 1}, data], data]; \  (*\ 
          differencing\ data\ if\ required\ *) \[IndentingNewLine]nested = 
          Transpose[\ \(Drop[#, \(-consecups\) - 
                    consecdowns] &\)\  /@ \ \((\(RotateRight[
                      ddata, #] &\)\  /@ \ 
                  Range[consecdowns - 
                      1, \(-consecups\), \(-1\)])\)]; \
\[IndentingNewLine]negtest = 
          Join[Table[_?Positive, {consecups}], 
            Table[_?Negative, {consecdowns}]]; \[IndentingNewLine]posns = 
          Cases[Flatten[
                Position[nested, negtest, {1}, Heads \[Rule] False]] + 
              consecups - 1 - \((\(+consecdowns\) - If[difffirst, 2, 1])\), _?
              Positive]; \[IndentingNewLine]Transpose[{data\
\[LeftDoubleBracket]posns\[RightDoubleBracket], posns}]]\)], "Input",
  CellLabel->"In[49]:="],

Cell[CellGroupData[{

Cell["Error cases", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(findPeaks::negcon = "\<The second parameter to this function must be \
positive.\>"\)], "Input",
  CellLabel->"In[8]:="],

Cell[BoxData[
    \("The second parameter to this function must be positive."\)], "Output",
  CellLabel->"Out[8]="]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(findPeaks::conlen\  = \ "\<The number of consecutive falls to be found \
`1` is larger than the length of the data series, `2`.\>"\)], "Input",
  CellLabel->"In[9]:="],

Cell[BoxData[
    \("The number of consecutive falls to be found `1` is larger than the \
length of the data series, `2`."\)], "Output",
  CellLabel->"Out[9]="]
}, Open  ]],

Cell[BoxData[
    \(findPeaks[data : {__?NumericQ}, consec_, opts___Rule] /; consec < 1 := 
      Message[findPeaks::negcon]\)], "Input",
  CellLabel->"In[10]:="],

Cell[BoxData[
    \(findPeaks[data : {__?NumericQ}, consec_, 
          opts___Rule] /; \((consec + 2)\) > Length[data] := 
      Message[findPeaks::conlen, consec, Length[data]]\)], "Input",
  CellLabel->"In[11]:="]
}, Closed]],

Cell[CellGroupData[{

Cell["Demonstration", "Subsubsection"],

Cell[BoxData[
    \(\(testdata = 
        NestList[#\ 0.95\  - 0.5\  + Random[] &, 0.3, 200];\)\)], "Input",
  CellLabel->"In[12]:="],

Cell[BoxData[
    \(Show[
      ListPlot[Reverse /@ \ findPeaks[testdata, 3], Frame \[Rule] True, 
        PlotStyle \[Rule] {Hue[0], AbsolutePointSize[6]}, 
        DisplayFunction \[Rule] Identity], 
      ListPlot[testdata, PlotJoined \[Rule] True, 
        DisplayFunction \[Rule] Identity], 
      DisplayFunction \[Rule] $DisplayFunction]\)], "Input",
  CellLabel->"In[57]:="]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Assymmetric smoothing", "Section"],

Cell[TextData[{
  "Reference: Julian Allwood and  ",
  ButtonBox["David Shepherd",
    ButtonData:>{
      URL[ "mailto:d.shepherd@ecomfac.unimelb.edu.au"], None},
    Active->True,
    ButtonStyle->"Hyperlink",
    ButtonNote->"Send mail to David Shepherd"],
  " (1999), \"Alternative Detrending Procedures for Macroeconomic Time Series\
\", University of Melbourne Department of Economics Research Paper Number \
698, June 1999"
}], "Text"],

Cell[BoxData[
    \(Needs["\<NumericalMath`SplineFit`\>"]\)], "Input",
  CellLabel->"In[59]:="],

Cell[BoxData[
    \(assymetricDetrending[data : {__?NumericQ}, 
          consec_Integer?Positive] /; \((\ 
          Head[consec] \[Equal] Integer && Length[data] > 4\  && \ 
            consec > 0\  && 
            Length[data] > consec + 2)\) := \[IndentingNewLine]With[{points = 
            findPeaks[data, consec]}, SplineFit[points, Cubic]]\)], "Input",
  CellLabel->"In[70]:="],

Cell[BoxData[
    \(assymetricDetrending[
          data : {__?NumericQ}, {consecups_Integer?Positive, 
            consecdowns_Integer?Positive}]\  /; \((\ 
          Length[data] > 4\  && 
            Length[data] > 
              consecups\  + consecdowns + 
                2)\) := \[IndentingNewLine]With[{points = 
            findPeaks[data, {consecups, consecdowns}]}, 
        SplineFit[points, Cubic]]\)], "Input",
  CellLabel->"In[77]:="],

Cell[CellGroupData[{

Cell["Demonstration", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(assDTtest = assymetricDetrending[testdata, 2]\)], "Input",
  CellLabel->"In[72]:="],

Cell[BoxData[
    InterpretationBox[\("SplineFunction["\[InvisibleSpace]Cubic\
\[InvisibleSpace]", "\[InvisibleSpace]{0.`, 21.`}\[InvisibleSpace]", <>]"\),
      SequenceForm[ "SplineFunction[", Cubic, ", ", {0.0, 21.0}, ", <>]"],
      Editable->False]], "Output",
  CellLabel->"Out[72]="]
}, Open  ]],

Cell["\<\
Note that you different time series result in SplineFunctions with \
different ranges, so you may have to tweak the number in blue in the next \
cell.\
\>", "Text"],

Cell[BoxData[
    RowBox[{"Show", "[", 
      RowBox[{\(ListPlot[Reverse /@ \ findPeaks[testdata, 2], 
          PlotStyle \[Rule] {Hue[0], AbsolutePointSize[6]}, 
          DisplayFunction \[Rule] Identity]\), 
        ",", \(ListPlot[testdata, PlotJoined \[Rule] True, 
          DisplayFunction \[Rule] Identity]\), ",", 
        RowBox[{"ListPlot", "[", 
          RowBox[{
            RowBox[{"Reverse", "/@", 
              RowBox[{"Table", "[", 
                RowBox[{\(assDTtest[x]\), ",", 
                  RowBox[{"{", 
                    RowBox[{"x", ",", "0", ",", 
                      StyleBox["21",
                        FontColor->RGBColor[0, 0, 1]], ",", "0.2"}], "}"}]}], 
                "]"}]}], ",", \(PlotJoined \[Rule] True\), 
            ",", \(PlotStyle \[Rule] Hue[0.6]\), 
            ",", \(DisplayFunction \[Rule] Identity\)}], "]"}], 
        ",", \(DisplayFunction \[Rule] $DisplayFunction\)}], "]"}]], "Input",
  CellLabel->"In[73]:="],

Cell["This suggests a further possible function:", "Text"],

Cell[BoxData[
    \(plotAssymetricDetrendedData[data : {__?NumericQ}, 
          consec_:  Integer] /; \((\ 
          Head[consec] \[Equal] Integer && Length[data] > 4\  && \ 
            consec > 0\  && 
            Length[data] > 
              consec + 2)\) := \[IndentingNewLine]With[{spliner = 
            assymetricDetrending[data, consec], 
          dots = Reverse /@ \ findPeaks[data, consec], 
          T = Length[data]}, \[IndentingNewLine]With[{splinedots = 
              Reverse /@ 
                Table[spliner[x], 
                  Evaluate[
                    Join[{x}, 
                      spliner\[LeftDoubleBracket]2\[RightDoubleBracket], \
{spliner\[LeftDoubleBracket]2, 2\[RightDoubleBracket]/
                          T}]]]}, \[IndentingNewLine]Show[
            ListPlot[dots, PlotStyle \[Rule] {Hue[0], AbsolutePointSize[6]}, 
              Frame \[Rule] True, DisplayFunction \[Rule] Identity], 
            ListPlot[data, PlotJoined \[Rule] True, Frame \[Rule] True, 
              DisplayFunction \[Rule] Identity], 
            ListPlot[splinedots, PlotJoined \[Rule] True, 
              PlotStyle \[Rule] Hue[0.6], Frame \[Rule] True, 
              DisplayFunction \[Rule] Identity], 
            DisplayFunction \[Rule] $DisplayFunction]\ ]]\)], "Input",
  CellLabel->"In[83]:="],

Cell[BoxData[
    \(plotAssymetricDetrendedData[
          data : {__?NumericQ}, {consecups_Integer?Positive, 
            consecdowns_Integer?Positive}] /; \((\ 
          Length[data] > 4\  && \ 
            Length[data] > 
              consecups + consecdowns + 
                2)\) := \[IndentingNewLine]With[{spliner = 
            assymetricDetrending[data, {consecups, consecdowns}], 
          dots = Reverse /@ \ findPeaks[data, {consecups, consecdowns}], 
          T = Length[data]}, \[IndentingNewLine]With[{splinedots = 
              Reverse /@ 
                Table[spliner[x], 
                  Evaluate[
                    Join[{x}, 
                      spliner\[LeftDoubleBracket]2\[RightDoubleBracket], \
{spliner\[LeftDoubleBracket]2, 2\[RightDoubleBracket]/
                          T}]]]}, \[IndentingNewLine]Show[
            ListPlot[dots, PlotStyle \[Rule] {Hue[0], AbsolutePointSize[6]}, 
              Frame \[Rule] True, DisplayFunction \[Rule] Identity], 
            ListPlot[data, PlotJoined \[Rule] True, Frame \[Rule] True, 
              DisplayFunction \[Rule] Identity], 
            ListPlot[splinedots, PlotJoined \[Rule] True, 
              PlotStyle \[Rule] Hue[0.6], Frame \[Rule] True, 
              DisplayFunction \[Rule] Identity], 
            DisplayFunction \[Rule] $DisplayFunction]\ ]]\)], "Input",
  CellLabel->"In[84]:="],

Cell[CellGroupData[{

Cell[BoxData[
    \(plotAssymetricDetrendedData[testdata, {2, 2}] // Timing\)], "Input",
  CellLabel->"In[87]:="],

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