Identify important minima and maxima in the w / Mathematica time series

Question

I need a way to define local minima and maxima in time series data using Mathematica. It seems like it should be easy to do, but it is difficult. I posted this on MathForum, but thought that I could get extra eyes on it here.

You can find an article discussing the issue: http://www.cs.cmu.edu/~eugene/research/full/compress-series.pdf

I have tried this so far ...

Get and format some data:

data = FinancialData["SPY", {"May 1, 2006", "Jan. 21, 2011"}][[All, 2]];
data = data/First@data;
data = Transpose[{Range[Length@data], data}];

Define 2 functions:

First method:

findMinimaMaxima[data_, window_] := With[{k = window},
  data[[k + Flatten@Position[Partition[data[[All, 2]], 2 k + 1, 1],  x_List /;  x[[k + 1]] < Min[Delete[x, k + 1]] || x[[k + 1]] > Max[Delete[x, k + 1]]]]]]

Now a different approach, although not as flexible:

findMinimaMaxima2[data_] := data[[Accumulate@(Length[#] & /@ Split[Prepend[Sign[Rest@data[[All, 2]] - Most@data[[All, 2]]], 0]])]]

See what each function does. The first findMinimaMaxima2 []:

minmax = findMinimaMaxima2[data];
{Length@data, Length@minmax}
ListLinePlot@minmax

( ) 49% , . . 2, , , :

minmax2 = findMinimaMaxima[data, 2];
{Length@data, Length@minmax2}
ListLinePlot@minmax2

, , 60:

minmax2 = findMinimaMaxima[data, 60];
ListLinePlot[{data, minmax2}]

. findMinimaMaxima2 [] findMinimaMaxima [] ...

minmax3 = findMinimaMaxima2[minmax2];
ListLinePlot[{data, minmax2, minmax3}]

.

, , . , R (, ), , . :

findMinimaMaxima3[data_, R_] := Module[{d, n, positions},
  d = data[[All, 2]];
  n = Transpose[{data[[All, 1]], Rest@FoldList[If[(#2 <= #1 + #1*R && #2 >= #1) || (#2 >= #1 - #1* R && #2 <= #1), #1, #2] &, d[[1]], d]}];
  n = Sign[Rest@n[[All, 2]] - Most@n[[All, 2]]];
  positions = Flatten@Rest[Most[Position[n, Except[0]]]];
  data[[positions]]
  ]

minmax4 = findMinimaMaxima3[data, 0.1];
ListLinePlot[{data, minmax4}]

- findMinimaMaxima2 []

ListLinePlot[{data, findMinimaMaxima2[minmax4]}]

, , , R - , . R , :

minmax4 = findMinimaMaxima3[data, 0.15];
ListLinePlot[{data, minmax4}]

, . . . / R , ( , ).

- - ?

. , - .

, Jagra

Answer 1

. , , , :

localMinPositionsC = 
 Compile[{{pts, _Real, 1}}, 
   Module[{result = Table[0, {Length[pts]}], i = 1, ctr = 0}, 
    For[i = 2, i < Length[pts], i++, 
     If[pts[[i - 1]] > pts[[i]] && pts[[i + 1]] > pts[[i]], 
      result[[++ctr]] = i]];
    Take[result, ctr]]];

localMaxPositionsC = 
  Compile[{{pts, _Real, 1}}, 
    Module[{result = Table[0, {Length[pts]}], i = 1, ctr = 0}, 
      For[i = 2, i < Length[pts], i++, 
        If[pts[[i - 1]] < pts[[i]] && pts[[i + 1]] < pts[[i]], 
          result[[++ctr]] = i]];
       Take[result, ctr]]];

:

dplot = ListLinePlot[data]

mins, 3 :

mins = ListPlot[Nest[#[[localMinPositionsC[#[[All, 2]]]]] &, data, 3],
   PlotStyle -> Directive[PointSize[0.015], Red]]

:

maxs = ListPlot[Nest[#[[localMaxPositionsC[#[[All, 2]]]]] &, data, 3],
   PlotStyle -> Directive[PointSize[0.015], Green]]

:

Show[{dplot, mins, maxs}]

, /.

Edit:

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