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FactorialMoment

FactorialMoment[data,r]

gives the order r factorial moment of data.

FactorialMoment[data,{r₁,…,r_m}]

gives the order {r₁,…,r_m} multivariate factorial moment of data.

FactorialMoment[dist,…]

gives the factorial moment of the distribution dist.

FactorialMoment[r]

represents the order r formal factorial moment.

Details

Factorial moments are defined using FactorialPower[x,r] given by .
For scalar order r and data being an array :
sum of r factorial powers »

columnwise sum of r factorial powers »

$x in TemplateBox[{Arrays, paclet:ref/Arrays}, RefLink, BaseStyle -> {3ColumnTableMod}][{n_(1),...,n_(k)}]$ columnwise sum of r factorial powers »
FactorialMoment[x,r] is equivalent to ArrayReduce[FactorialMoment[#,r]&,x,1].
For vector order {r₁,…,r_m} and data being array :
sum the r_j factorial power in the j column

$x in TemplateBox[{Arrays, paclet:ref/Arrays}, RefLink, BaseStyle -> {3ColumnTableMod}][{n_(1),...,n_(k)}]$ sum the r_j factorial power in the j column »
FactorialMoment[x,{r₁,…,r_m}] is equivalent to ArrayReduce[FactorialMoment[#,{r₁,…,r_m}]&,x,{{1},{2}}].
FactorialMoment handles both numerical and symbolic data.
The data can have the following additional forms and interpretations:

	Association	the values (the keys are ignored) »
	WeightedData	weighted mean, based on the underlying EmpiricalDistribution »
	EventData	based on the underlying SurvivalDistribution »
	TimeSeries, TemporalData, …	vector or array of values (the time stamps ignored) »
	Image,Image3D	RGB channel's values or grayscale intensity value »
	Audio	amplitude values of all channels »

For a distribution dist, the r factorial moment is given by Expectation[x^(r),xdist]. »
For a multivariate distribution dist, the {r₁,…,r_m} factorial moment is given by Expectation[x₁^(r₁)⋯ x_m^(r_m),{x₁,…,x_m}dist]. »
For a random process proc, the factorial moment function can be computed for slice distribution at time t, SliceDistribution[proc,t], as [t]=FactorialMoment[SliceDistribution[proc,t],r]. »
FactorialMoment[r] can be used in such functions as MomentConvert and MomentEvaluate, etc. »

Examples

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Basic Examples (2)

Compute factorial moment from data:

Use symbolic data:

Compute the second factorial moment of a discrete univariate distribution:

The factorial moment for a multivariate distribution:

Scope (20)

Basic Uses (5)

Exact input yields exact output:

Approximate input yields approximate output:

Find factorial moments of WeightedData:

Find a factorial moment of EventData:

Find a factorial moment of TimeSeries:

Factorial moment depends only on the values:

Array Data (4)

For a matrix, FactorialMoment gives columnwise moments:

For an array, FactorialMoment gives columnwise moments at the first level:

Multivariate FactorialMoment for an array:

Works with large arrays:

When the input is an Association, FactorialMoment works on its values:

SparseArray data can be used just like dense arrays:

Image and Audio Data (2)

Channelwise factorial moment of an RGB image:

Factorial moment intensity value of a grayscale image:

On audio objects, FactorialMoment works channelwise:

Distribution and Process Moments (5)

Scalar factorial moment for univariate distributions:

Scalar factorial moment for multivariate distributions:

Joint factorial moment for multivariate distributions:

Compute a factorial moment for a symbolic order r:

A factorial moment may only evaluate for specific orders:

A factorial moment may only evaluate numerically:

Factorial moments for derived distributions:

Data distribution:

Factorial moment function for a random process:

Find a factorial moment of TemporalData at some time t=0.5:

Find the corresponding factorial moment function together with all the simulations:

Formal Moments (4)

TraditionalForm formatting for formal moments:

Convert combinations of formal moments to an expression involving FactorialMoment:

Evaluate an expression involving formal moments for a distribution:

Evaluate for data:

Find a sample estimator for an expression involving FactorialMoment:

Evaluate the resulting estimator for data:

Applications (4)

Estimate parameters of a distribution using the method of factorial moments:

Compare data and the estimated parametric distribution:

Reconstruct probability mass function from the sequence of factorial moments:

Find the factorial moment-generating function (FMGF):

Use equivalence of the FMGF and the probability generating function:

Verify that factorial moments of the found distribution match the originals:

Compute a moving factorial moment for some data:

Use the window of length .1:

Compute factorial moments for slices of a collection of paths of a random process:

Choose a few slice times:

Plot factorial moments over these paths:

Properties & Relations (5)

Factorial moment is equivalent to an expectation of FactorialPower:

First factorial moment is equivalent to Mean:

FactorialMoment can be computed from Moment through $mu^__r=sum_(k=1)^rTemplateBox[{r, k}, StirlingS1]mu_k$ :

MomentConvert produces the same result:

Moment can be computed from FactorialMoment through $mu_r=sum_(k=0)^rmu^__k TemplateBox[{r, k}, StirlingS2]$ :

MomentConvert produces the same result:

The multivariate factorial moment of an array of depth has depth :

Neat Examples (1)

The distribution of FactorialMoment estimates for 30, 100, and 300 samples:

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FactorialMoment

Details

Examples

Basic Examples (2)