Fit Gaussian function to histogram

[cubsfan334]

Hi,

I realize that I can generate a histogram from a data set using the
Histogram[{data},bin size] command, yet this only seems to create a
graphic. Is there anyway to fit a function (preferably Gaussian) to
the histogram Mathematica creates?

Thanks!

[Richard Hofler]

Hello,

Try this:

data=Table[Random[NormalDistribution[10,2]],{10000}];
histo=Histogram[data];
plt=Plot[2000 PDF[NormalDistribution[10,2]][x],{x,4,16}];
Show[histo,plt]

Richard

[Barrie Stokes]

Hi

The first thing I would do would be to find m=Mean[ data ] and sd=StandardDeviation[ data ], and then plot together the histogram and something like

Plot[ PDF[ NormalDistribution[ m, sd ], x ], {x, Min [data ], Max[ data] ],

and see if the Gaussian is at all a reasonable fit.

Fitting distributions to data is a big area, and there are tests for closeness of fit. Have a look through the Mathematica documentation.

There is

DistributionFitTest[ data, Automatic, "HypothesisTestData" ] in the documentation for DistributionFitTest.

(I found this by searching the online documentation for "Kolmogorov", because I know that the Kolmogorov-Smirnov test is such a test.)

Have a look also at AndersonDarlingTest[].

If you enter "fitting Normal distribution to data" into the search box for the online documentation, the second screen will lead you to

FindDistributionParameters[], and it will mention also EstimatedDistribution[].

Mathematica can give you a lot of power to attack a problem like this. There is a lot of example code in these documentation pages, too.

Cheers

Barrie

>>> On 29/03/2011 at 10:54 pm, in message <2011032911...@smc.vnet.net>,

[Bill Rowe]

On 3/29/11 at 6:54 AM, cubsf...@gmail.com (cubsfan334) wrote:

>I realize that I can generate a histogram from a data set using the
>Histogram[{data},bin size] command, yet this only seems to create a
>graphic. Is there anyway to fit a function (preferably Gaussian) to
>the histogram Mathematica creates?

Is this something like what you want to do?

data = RandomReal[NormalDistribution[], 1000];

Show[Histogram[data, Automatic, "PDF"],
Plot[PDF[NormalDistribution[], x], {x, -3, 3}]]

[Kevin J. McCann]

Histogram gives the plot, but if you want the actual bin counts, use
BinCounts. The mma help will give you the options to call it; I usually use

BinCounts[data,dx] in which dx is the bin width. You can also specify
all the bins with BinCounts[data,{xmin,xmax,dx}]. It is then easy to
convert the counts to probability densities.

Kevin

[Alexei Boulbitch]

Hi,

try this:

data = RandomVariate[NormalDistribution[], 1000];
hist = Histogram[data, "Wand", "PDF", Frame -> True];
dist = EstimatedDistribution[data,
NormalDistribution[\[Mu], \[Sigma]]];
distPlot = Plot[PDF[dist, x], {x, -4, 4}, PlotStyle -> {Thick, Red}];
Show[{hist, unimodalDistPlot1}, Epilog -> Inset[
Panel@Column[{
Row[{Style["\[Mu] \[TildeTilde]", 12, Blue], Spacer[5],
Style[NumberForm[dist[[1]], {4, 2}], 12, Blue]
}],

Row[{Style["\[Sigma] \[TildeTilde]", 12, Blue], Spacer[5],
Style[NumberForm[dist[[2]], {4, 2}], 12, Blue]
}],
Row[{Style["Test: ", 10, Blue], Spacer[5],

Style[NumberForm[DistributionFitTest[data, dist],
{4, 3}], 12, Blue]
}]
}], Scaled[{0.15, 0.7}]
]]

Fave fun, Alexei

-----Original Message-----
From: cubsfan334 [mailto:cubsfan334 at gmail.com]
Sent: Tuesday, March 29, 2011 7:55 AM
To: mathgroup at smc.vnet.net
Subject: Fit Gaussian function to histogram

Hi,

I realize that I can generate a histogram from a data set using the
Histogram[{data},bin size] command, yet this only seems to create a
graphic. Is there anyway to fit a function (preferably Gaussian) to
the histogram Mathematica creates?

Thanks!

--
Alexei Boulbitch, Dr. habil.
Senior Scientist
Material Development

IEE S.A.
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[dr DanW]

I find it easier to fit to the CDF. The shape of the histogram depends on bin widths, so some of the decisions you have to make to get a good quality fit are arbitrary. The cumulative frequency of your data, however, is unambiguous.

CumulativeFrequency[dat_] := With[{tsd = Tally[Sort[dat]]},
Transpose[{tsd[[All,1]], Accumulate[tsd[[All,2]]]/(Length[dat] + 1)}]]

FitCDF[d_] := Module[{mean, sd, cf, param}, mean = Mean[d]; sd = StandardDeviation[d];
cf = CumulativeFrequency[d]; param = FindFit[cf, CDF[NormalDistribution[\[Mu], \[Sigma]],
x], {{\[Mu], mean}, {\[Sigma], sd}}, x];
Print[Show[{ListPlot[cf], Plot[Evaluate[CDF[NormalDistribution[\[Mu], \[Sigma]], x] /.
param], Evaluate[{x, \[Mu] - 3*\[Sigma], \[Mu] + 3*\[Sigma]} /. param]]}]]; param]

Notice that this technique works with any distribution (with suitable modification of FitCDF).

Mathematica 8 has a lot of new tools for statistics and probability, many of which I do not understand. Perhaps you are looking for HypothesisTesting`?

Enjoy,
Daniel

[Mark Fisher]

Maybe you're interesting in something like this:

data = RandomReal[NormalDistribution[], 100];
histo = Histogram[data, 10, "PDF"];
hdata = Cases[histo,
RectangleBox[{x0_, _}, {x1_, y_}, _] :> {.5 (x0 + x1),
y}, \[Infinity]];
fun = PDF[NormalDistribution[m, s], x] /.
FindFit[hdata, PDF[NormalDistribution[m, s], x], {m, s}, {x}];
Show[histo, Plot[fun, {x, Min[data], Max[data]}]]

--Mark

[Ray Koopman]

There are several ways to do it, depending on how you use the data.
See the thread "Fitting a normal distribution to a histogram",
Jun 19-23, 2008, at
https://groups.google.com/group/sci.stat.math/browse_frm/thread/ca471d2c3a09a620/4581b219f3ddea2b?hl=en