Cumulant

Cumulant[data,r]

gives the r cumulant of data.

Cumulant[data,{r₁,…,r_m} ]

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

Cumulant[,r]

gives the r cumulant of the distribution dist.

Cumulant[r]

represents the r formal cumulant.

Details

Formally, the r cumulant is defined as the coefficient of the Taylor series of the CumulantGeneratingFunction.
The first few cumulants expressed in terms of moments:
In general, MomentConvert[Cumulant[r],"Moment"] gives the in terms of moments.
Cumulant[data,r] effectively uses MomentConvert to compute it in terms of other moments for data.
For x∈Arrays[{n₁,n₂,… ,n_k}], Cumulant[x,r] is equivalent to ArrayReduce[Cumulant[#,r]&,data,1]. »
For x∈Arrays[{n₁,n₂,… ,n_k}], Cumulant[x,{r₁,…,r_m}] is equivalent to ArrayReduce[Cumulant[#,{r₁,…,r_m}]&,x,{{1},{2}}]. »
Cumulant 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 »
	DateObject, TimeObject	list of dates or list of times »

For a distribution dist with G=CumulantGeneratingFunction[,…]:
Cumulant[,r] »

Cumulant[dist,{r₁,…,r_m}] »
For a random process proc, the cumulant function can be computed for slice distribution at time t, SliceDistribution[proc,t], as [t]=Cumulant[SliceDistribution[proc,t],r]. »
Cumulant[r] can be used in such functions as MomentConvert and MomentEvaluate, etc. »

Examples

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Basic Examples (3)

Compute cumulants from data:

For symbolic data:

Cumulant of a list of dates:

Compute the second cumulant of a continuous univariate distribution:

The cumulant for a multivariate distribution:

Scope (26)

Basic Uses (6)

Exact input yields exact output:

Approximate input yields approximate output:

Find cumulants of WeightedData:

Find a cumulant of EventData:

Find a cumulant of TimeSeries:

The cumulant depends only on the values:

Find a cumulant for data involving quantities:

Array Data (5)

For a matrix, Cumulant gives columnwise cumulants:

For an array, Cumulant gives columnwise cumulants at the first level:

Works with large arrays:

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

SparseArray data can be used just like dense arrays:

Compute the multivariate cumulant of an array in terms of its raw moments:

Image and Audio Data (2)

Channelwise cumulant of an RGB image:

Cumulant intensity value of a grayscale image:

On audio objects, Cumulant works channelwise:

Date and Time (4)

Compute cumulant of dates:

Compute the weighted cumulant of dates:

Compare with simple cumulant:

Compute the cumulant of dates given in different calendars:

Compute the cumulant of times:

List of times with different time zone specifications:

Distribution and Process Cumulants (5)

Scalar cumulant for univariate distributions:

Scalar cumulant for multivariate distributions:

Joint cumulant for multivariate distributions:

Compute a cumulant for a symbolic order r:

A cumulant may only evaluate for specific orders:

A cumulant may only evaluate numerically:

Cumulants for derived distributions:

Data distribution:

Cumulant function for a random process:

Find a cumulant of TemporalData at time t=0.5:

Find the corresponding cumulant function together with all the simulations:

Formal Cumulants (4)

TraditionalForm formatting for formal cumulants:

Convert combinations of formal moments to an expression involving Cumulant:

Evaluate an expression involving formal cumulants for a distribution:

Evaluate for data:

Find a sample estimator for an expression involving Cumulant:

Evaluate the resulting estimator for data:

Applications (5)

Estimate parameters of a distribution using the method of cumulants:

The law of large numbers states that a sample moment approaches the population moment as the sample size increases. Use Histogram to show the probability distribution of sample cumulant of standard normal random variates for different sample sizes: