TargetFunctions option in FindDistribution yields error

Question

I have some data set that I am trying to find the maximums . I figured I could do this by fitting the many peaks of the dataset to Gaussian distributions and then finding the peaks of this data from these fit functions:

data = {{6, 2.1}, {6.25, 1.82394}, {6.5, 2.056}, {6.75, 2.48818}, {7, 
  5.73034}, {7.25, 11.3611}, {7.5, 11.5297}, {7.625, 10.6597}, {7.75, 
  14.5473}, {7.875, 13.7337}, {8, 14.291}, {8.125, 15.4141}, {8.25, 
  13.2849}, {8.375, 16.785}, {8.5, 14.6091}, {8.625, 17.0505}, {8.75, 
  17.9138}, {8.875, 17.662}, {9, 19.6648}, {9.125, 14.9366}, {9.25, 
  19.6298}, {9.375, 17.377}, {9.5, 19.4579}, {9.75, 17.3271}, {10, 
  19.0673}, {10.5, 14.7837}, {11, 15.2593}, {11.5, 14.2707}, {12, 
  16.7271}, {12.5, 15.0264}, {13, 12.8453}, {13.125, 11.8129}, {13.25,
   12.2551}, {13.375, 12.4764}, {13.5, 12.3054}, {13.625, 
  12.2718}, {13.75, 11.7467}, {13.875, 11.6148}, {14, 
  10.7611}, {14.125, 13.5223}, {14.25, 17.3761}, {14.375, 
  46.9357}, {14.5, 58.4376}, {14.625, 57.7595}, {14.75, 
  48.2346}, {14.875, 37.6358}, {15, 22.366}, {15.125, 
  14.6147}, {15.25, 13.7916}, {15.375, 13.6754}, {15.5, 
  13.483}, {15.625, 16.7153}, {15.75, 20.2869}, {15.875, 
  81.1927}, {16, 231.151}, {16.125, 243.424}, {16.25, 
  232.654}, {16.375, 181.878}, {16.5, 111.372}, {16.625, 
  64.9912}, {16.75, 32.0148}, {16.875, 13.9479}, {17, 13.0599}, {18, 
  8.65011}, {19, 5.44097}, {20, 4.66879}, {21, 6.56693}, {22, 
  5.12372}, {23, 4.34006}, {24, 2.44913}, {25, 4.56596}, {26, 
  3.78848}, {27, 4.33847}, {27.5, 2.89337}, {28, 3.12278}, {28.5, 
  5.02059}, {28.625, 3.68042}, {28.75, 3.90491}, {28.875, 
  3.68717}, {29, 3.57609}, {29.125, 3.35689}, {29.25, 
  3.46561}, {29.375, 3.91997}, {29.5, 6.82568}, {29.625, 
  8.61445}, {29.75, 11.7729}, {29.875, 13.0278}, {30, 
  11.555}, {30.125, 7.86307}, {30.25, 6.16789}, {30.375, 
  4.26774}, {30.5, 3.93371}, {31, 4.59788}, {31.5, 4.49401}, {32, 
  4.71784}, {32.5, 4.71958}, {32.625, 5.72564}, {32.75, 
  5.94579}, {32.875, 5.49908}, {33, 8.74719}, {33.125, 
  27.7096}, {33.25, 36.5386}, {33.375, 45.0817}, {33.5, 
  42.019}, {33.625, 29.3559}, {33.75, 20.2754}, {33.875, 
  14.0957}, {34, 6.77348}, {34.5, 5.07823}, {35, 4.40521}, {35.5, 
  4.40848}, {36, 4.86514}, {37, 4.18628}, {38, 4.98291}, {39, 
  3.62528}, {40, 5.21134}, {41, 4.4216}, {42, 4.42242}, {43, 
  5.32562}, {44, 4.54088}, {45, 6.24023}, {46, 5.45007}, {47, 
  5.11135}, {47.5, 7.20017}, {48, 7.94951}, {48.125, 8.18731}, {48.25,
   6.37029}, {48.375, 6.95074}, {48.5, 8.55079}, {48.625, 
  6.38578}, {48.75, 6.7367}, {48.875, 5.8178}, {49, 5.82325}, {50, 
  5.36753}, {50.5, 5.60118}, {51, 6.97421}, {51.5, 5.71228}, {52, 
  5.83199}, {52.5, 5.84295}, {53, 7.32958}, {53.5, 7.6847}, {54, 
  8.02426}, {54.125, 6.88053}, {54.25, 7.81263}, {54.375, 
  6.66246}, {54.5, 8.50358}, {54.625, 6.55995}, {54.75, 
  7.37252}, {54.875, 6.91174}, {55, 9.66547}, {55.125, 
  12.6715}, {55.25, 17.5031}, {55.375, 18.7592}, {55.5, 
  13.7057}, {55.625, 12.2038}, {55.75, 9.21217}, {55.875, 
  9.67643}, {56, 8.55033}, {56.125, 8.7981}, {56.25, 7.869}, {56.375, 
  8.46045}, {56.5, 8.34932}, {57, 7.9999}, {57.5, 7.07507}, {58.5, 
  6.97235}, {59.5, 8.48789}, {60.5, 8.03506}, {61.5, 8.74713}}

counts=data[[All,2]]

Then I attempt to find the fits to the peaks of this data, which should be a distribution made up of many NormalDistribution functions:

nd = 
  FindDistribution[counts, 
    TargetFunctions -> {NormalDistribution}]

However, This results in the following distribution:

0.000599601 E^(-0.0000772966 (-91.0024 + x)^2) + 
 0.0712972 E^(-0.0206633 (-9.10869 + x)^2)

The following plot of this distribution:

Show[Plot[
  0.0005996013554770311` E^(-0.00007729656053053849` \
(-91.00239606190587` + x)^2) + 
   0.07129715654128815` E^(-0.020663257993617158` \
(-9.108687446977836` + x)^2), {x, 0, 60}],
 ListPlot[data]]

Does not fit my data very well. Any suggestions on how to output the many Normal distributions that fit the peaks of my data set?

Ok. That fix was easy. Is there any way to Tell FindDistrobution to have a Minimum number of distrobutions. I know there is a MaxItems parameter; but I would need something like a MinItems parameter. — Daniel Schulze, Jul 06 '15 at 03:46

score 6 · Accepted Answer · edited Apr 13 '17 at 12:55

6

I am not sure FindDistributions is the approach you want. If I understand you wish to fit Gaussians (i.e. the probability density functions) to your data to estimate the peaks rather than estimate the Gaussian distribution parameters for a dataset you have some prior belief is normal. If the former is your aim there are obviously a variety of symmetric peak smooth functions you could use after piecewise processing your data.

I post this in case FindPeaks is either sufficient or useful for model fitting.

{pos, pl} = Transpose@FindPeaks[data[[All, 2]], 3];
pts = Part[data, pos];
Manipulate[
 Row[{ListPlot[{data, pts}, 
    PlotStyle -> {Blue, {Red, PointSize[0.02]}}, PlotRange -> {0, r}, 
    Frame -> True, ImageSize -> 300], Column[pts]}], {r, 50, 250}]

enter image description here

Important

See this very instructive and useful in relation to fitting multiple peaks: here.

edited Apr 13 '17 at 12:55

Community

1

answered Jul 06 '15 at 04:26

ubpdqn

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What does the second argument of the FindPeaks[] function do? I know it is Gaussian Blurring and changes what points are output by the function. – Daniel Schulze Jul 06 '15 at 04:35
@DanielSchulze I suggest you look through the documentation: https://reference.wolfram.com/language/ref/FindPeaks.html as well as play around. Essentially allows vary sharpness of peaks you are looking for. – ubpdqn Jul 06 '15 at 04:44
Neat as always, +1 – ciao Jul 06 '15 at 05:06

score 4 · Answer 2 · answered Jul 06 '15 at 04:34

Using some of the other parameters of FindPeaks used in answer by @ubpdqn

data = {{6, 2.1}, {6.25, 1.82394}, {6.5, 2.056}, {6.75, 2.48818}, {7, 
    5.73034}, {7.25, 11.3611}, {7.5, 11.5297}, {7.625, 10.6597}, {7.75, 
    14.5473}, {7.875, 13.7337}, {8, 14.291}, {8.125, 15.4141}, {8.25, 
    13.2849}, {8.375, 16.785}, {8.5, 14.6091}, {8.625, 17.0505}, {8.75, 
    17.9138}, {8.875, 17.662}, {9, 19.6648}, {9.125, 14.9366}, {9.25, 
    19.6298}, {9.375, 17.377}, {9.5, 19.4579}, {9.75, 17.3271}, {10, 
    19.0673}, {10.5, 14.7837}, {11, 15.2593}, {11.5, 14.2707}, {12, 
    16.7271}, {12.5, 15.0264}, {13, 12.8453}, {13.125, 11.8129}, {13.25, 
    12.2551}, {13.375, 12.4764}, {13.5, 12.3054}, {13.625, 12.2718}, {13.75, 
    11.7467}, {13.875, 11.6148}, {14, 10.7611}, {14.125, 13.5223}, {14.25, 
    17.3761}, {14.375, 46.9357}, {14.5, 58.4376}, {14.625, 57.7595}, {14.75, 
    48.2346}, {14.875, 37.6358}, {15, 22.366}, {15.125, 14.6147}, {15.25, 
    13.7916}, {15.375, 13.6754}, {15.5, 13.483}, {15.625, 16.7153}, {15.75, 
    20.2869}, {15.875, 81.1927}, {16, 231.151}, {16.125, 243.424}, {16.25, 
    232.654}, {16.375, 181.878}, {16.5, 111.372}, {16.625, 64.9912}, {16.75, 
    32.0148}, {16.875, 13.9479}, {17, 13.0599}, {18, 8.65011}, {19, 
    5.44097}, {20, 4.66879}, {21, 6.56693}, {22, 5.12372}, {23, 4.34006}, {24,
     2.44913}, {25, 4.56596}, {26, 3.78848}, {27, 4.33847}, {27.5, 
    2.89337}, {28, 3.12278}, {28.5, 5.02059}, {28.625, 3.68042}, {28.75, 
    3.90491}, {28.875, 3.68717}, {29, 3.57609}, {29.125, 3.35689}, {29.25, 
    3.46561}, {29.375, 3.91997}, {29.5, 6.82568}, {29.625, 8.61445}, {29.75, 
    11.7729}, {29.875, 13.0278}, {30, 11.555}, {30.125, 7.86307}, {30.25, 
    6.16789}, {30.375, 4.26774}, {30.5, 3.93371}, {31, 4.59788}, {31.5, 
    4.49401}, {32, 4.71784}, {32.5, 4.71958}, {32.625, 5.72564}, {32.75, 
    5.94579}, {32.875, 5.49908}, {33, 8.74719}, {33.125, 27.7096}, {33.25, 
    36.5386}, {33.375, 45.0817}, {33.5, 42.019}, {33.625, 29.3559}, {33.75, 
    20.2754}, {33.875, 14.0957}, {34, 6.77348}, {34.5, 5.07823}, {35, 
    4.40521}, {35.5, 4.40848}, {36, 4.86514}, {37, 4.18628}, {38, 
    4.98291}, {39, 3.62528}, {40, 5.21134}, {41, 4.4216}, {42, 4.42242}, {43, 
    5.32562}, {44, 4.54088}, {45, 6.24023}, {46, 5.45007}, {47, 
    5.11135}, {47.5, 7.20017}, {48, 7.94951}, {48.125, 8.18731}, {48.25, 
    6.37029}, {48.375, 6.95074}, {48.5, 8.55079}, {48.625, 6.38578}, {48.75, 
    6.7367}, {48.875, 5.8178}, {49, 5.82325}, {50, 5.36753}, {50.5, 
    5.60118}, {51, 6.97421}, {51.5, 5.71228}, {52, 5.83199}, {52.5, 
    5.84295}, {53, 7.32958}, {53.5, 7.6847}, {54, 8.02426}, {54.125, 
    6.88053}, {54.25, 7.81263}, {54.375, 6.66246}, {54.5, 8.50358}, {54.625, 
    6.55995}, {54.75, 7.37252}, {54.875, 6.91174}, {55, 9.66547}, {55.125, 
    12.6715}, {55.25, 17.5031}, {55.375, 18.7592}, {55.5, 13.7057}, {55.625, 
    12.2038}, {55.75, 9.21217}, {55.875, 9.67643}, {56, 8.55033}, {56.125, 
    8.7981}, {56.25, 7.869}, {56.375, 8.46045}, {56.5, 8.34932}, {57, 
    7.9999}, {57.5, 7.07507}, {58.5, 6.97235}, {59.5, 8.48789}, {60.5, 
    8.03506}, {61.5, 8.74713}};

Manipulate[
 Module[{peaks, pts},
  peaks = FindPeaks[data[[All, 2]], sigma, s, t];
  pts = data[[peaks[[All, 1]]]];
  Column[{
    ListLinePlot[data,
     PlotRange -> All,
     Epilog -> {Red, AbsolutePointSize[6], Point[pts]},
     ImageSize -> 360],
    pts}]],
 {{sigma, 3}, 2, 4, .1, Appearance -> "Labeled"},
 {{s, .1, "sharpness"}, .05, 2, .05, Appearance -> "Labeled"},
 {{t, 19, "threshold"}, 0, 60, 1, Appearance -> "Labeled"}]

enter image description here

beautiful and illustrative wrt other arguments of FindPeaks +1, I merely wished to point the way...bust this is much more instructive :) — ubpdqn, Jul 06 '15 at 05:10

TargetFunctions option in FindDistribution yields error

2 Answers2