FactorialMoment

FactorialMoment[data,r]

gives the order r factorial moment of data.

FactorialMoment[data,{r1,,rm}]

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

FactorialMoment[dist,]

gives the factorial moment of the distribution dist.

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 » columnwise sum of r factorial powers »
• FactorialMoment[x,r] is equivalent to ArrayReduce[FactorialMoment[#,r]&,x,1].
• For vector order {r1,,rm} and data being array :
•  sum the rj factorial power in the j column sum the rj factorial power in the j column »
• FactorialMoment[x,{r1,,rm}] is equivalent to ArrayReduce[FactorialMoment[#,{r1,,rm}]&,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),xdist]. »
• For a multivariate distribution dist, the {r1,,rm} factorial moment is given by Expectation[x1(r1) xm(rm),{x1,,xm}dist]. »
• 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(22)

Basic Uses(6)

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:

Find a factorial moment for data involving quantities:

Array Data(5)

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:

Find the factorial moment of a QuantityArray:

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)

Factorial moments of univariate distributions:

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)

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 :

MomentConvert produces the same result:

Moment can be computed from FactorialMoment through :

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:

Wolfram Research (2010), FactorialMoment, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorialMoment.html (updated 2023).

Text

Wolfram Research (2010), FactorialMoment, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorialMoment.html (updated 2023).

CMS

Wolfram Language. 2010. "FactorialMoment." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/FactorialMoment.html.

APA

Wolfram Language. (2010). FactorialMoment. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FactorialMoment.html

BibTeX

@misc{reference.wolfram_2024_factorialmoment, author="Wolfram Research", title="{FactorialMoment}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/FactorialMoment.html}", note=[Accessed: 16-June-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_factorialmoment, organization={Wolfram Research}, title={FactorialMoment}, year={2023}, url={https://reference.wolfram.com/language/ref/FactorialMoment.html}, note=[Accessed: 16-June-2024 ]}