BUILT-IN MATHEMATICA SYMBOL

HistogramDistribution

represents the probability distribution corresponding to a histogram of the data values

.

represents a multivariate histogram distribution based on data values

.

HistogramDistribution

represents a histogram distribution with bins specified by bspec.

MORE INFORMATION

HistogramDistribution returns a DataDistribution object that can be used like any other probability distribution.

The probability density function for HistogramDistribution for a value is given by where is the number of data points in bin , is the width of bin , are bin delimiters, and is the total number of data points.

The width of each bin is computed according to the values , the width according to the , etc.

The following bin specifications bpsec can be given:

	n	use n bins »
	{w}	use bins of width w »
	{min,max,w}	use bins of width w from min to max »
	{{b₁,b₂,...}}	use bins »
	Automatic	determine bin widths automatically
	"name"	use a named binning method »
	fw	apply fw to get an explicit bin specification
	{xspec,yspec,...}	give different x, y, etc. specifications

Possible named binning methods include:

	"FreedmanDiaconis"	twice the interquartile range divided by the cube root of sample size
	"Knuth"	balance likelihood and prior probability of a piecewise uniform model
	"Scott"	asymptotically minimize the mean square error
	"Sturges"	compute the number of bins based on the length of data
	"Wand"	one-level recursive approximate Wand binning

The probability density for value in a histogram distribution is a piecewise constant function.

HistogramDistribution can be used with such functions as Mean, CDF, and RandomVariate.

EXAMPLES

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Basic Examples (2)

Create a histogram distribution of univariate data:

Use the resulting distribution to perform analysis, including visualizing distribution functions:

Compute moments and quantiles:

Create a histogram distribution of bivariate data:

Visualize the PDF and CDF:

Compute covariance and general moments:

Create a histogram distribution of univariate data:

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Use the resulting distribution to perform analysis, including visualizing distribution functions:

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Compute moments and quantiles:

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Create a histogram distribution of bivariate data:

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Visualize the PDF and CDF:

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Compute covariance and general moments:

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Scope (28)

Create a distribution from a histogram of some data:

Compute probabilities from the distribution:

Decrease the number of bins to decrease local sensitivity:

Increase the bin width to decrease local sensitivity:

Create distributions from histograms in higher dimensions:

Plot the univariate marginal PDFs:

Plot the bivariate marginal PDFs:

Estimate distribution functions:

Compute moments of the distribution:

Special moments:

General moments:

Quantile function:

Special quantile values:

Generate random numbers:

Compare with HistogramDistribution:

Compute probabilities and expectations:

Generating functions:

Estimate distribution functions for bivariate data:

Compute moments of a bivariate distribution:

Special moments:

General moments:

Generate random numbers:

Show the point distribution:

Having fewer bins yields a coarser approximation to the underlying distribution:

Generating functions:

Automatically compute the number of bins:

More data yields smaller bins:

Explicitly specify the number of bins to use:

Specify 5 and 50 bins, respectively:

Explicitly specify bin width:

Use bin widths 1. and 0.1:

Specify bin range and bin width:

Use bin widths of

and

respectively over the interval

to

:

Provide explicit bin delimiters:

Use different automatic binning methods:

Delimit bins on integer boundaries using a binning function:

Automatically compute the number of bins for bivariate data:

More data yields smaller bins:

Explicitly specify the number of bins to use:

Specify 5 and 10 bins in each dimension:

Explicitly specify bin width:

Use bin widths 1.0 or 2.0 for each dimension:

Specify bin range and bin width:

Use the range from

to

with bin widths

and

:

Explicitly give bin delimiters:

Use the bin delimiters

and

:

Use different automatic binning methods:

Use different bin specifications in each dimension:

Specify 3 bins in the row dimension and bin width 0.5 in the column dimension:

Applications (5)

Compare an estimated density to a theoretical model:

Distribution of lengths of human chromosomes:

Compute the probability that the sequence length is greater than 15:

Compare the distributions of word length for some of the parts of speech:

The expected number of characters for a randomly chosen English noun:

Estimate the distribution of day-to-day point changes in the S&P 500 index:

Compute the probability of a 1% point change or more on a given day:

Determine the number of bins to use for bimodal data by Knuth's Bayesian method:

The optimal number of bins maximizes the log of the posterior density:

Density estimates using Knuth's method, Scott's rule, and the Freedman-Diaconis rule:

Knuth's method outperforms the other two in terms of LogLikelihood:

Properties & Relations (8)

The PDF of HistogramDistribution is equivalent to a probability density Histogram:

The resulting density estimate integrates to unity:

The precision of the output matches the precision of the data:

The PDF is piecewise constant:

The CDF and SurvivalFunction are piecewise linear:

The HazardFunction is linear fractional:

HistogramDistribution is a MixtureDistribution of uniform distributions:

HistogramDistribution is a consistent estimator of the underlying distribution:

Possible Issues (1)

It is possible to drop data from the estimation by specifying a binning range:

Specifying a width alone uses all the data:

Neat Examples (1)

Random pop art with HistogramDistribution: