# Cumulant

Cumulant[data,r]

gives the r cumulant of data.

Cumulant[data,{r1,,rm} ]

gives the multivariate cumulant of order {r1,,rm} 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 xArrays[{n1,n2, ,nk}], Cumulant[x,r] is equivalent to ArrayReduce[Cumulant[#,r]&,data,1]. »
• For xArrays[{n1,n2, ,nk}], Cumulant[x,{r1,,rm}] is equivalent to ArrayReduce[Cumulant[#,{r1,,rm}]&,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 »
• For a distribution dist with :
•  Cumulant[,r] Cumulant[dist,{r1,…,rm}]
• Cumulant[r] can be used in such functions as MomentConvert and MomentEvaluate etc. »

# Examples

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## Basic Examples(2)

Compute cumulants from data:

For symbolic data:

Compute the second cumulant of a continuous univariate distribution:

The cumulant for a multivariate distribution:

## Scope(22)

### 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:

### Distribution and Process Cumulants(5)

Find the cumulants of univariate distributions:

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:

Edgeworth's expansion of order :

Approximate SechDistribution:

Compute a moving cumulant for some data:

Use the window of length .1:

Compute cumulants for slices of a collection of paths of a random process:

Choose a few slice times:

Plot cumulants over these paths:

## Properties & Relations(5)

First cumulant is equivalent to the first moment :

Second cumulant is equivalent to the second central moment :

Third cumulant is equivalent to the third central moment :

Cumulant is equal to the derivative of the cumulant-generating function at zero :

Use Cumulant directly:

Find the cumulant-generating function using GeneratingFunction:

Check using CumulantGeneratingFunction:

Formally, cumulants can be computed using the fact that CumulantGeneratingFunction[dist,t] is given by Log[MomentGeneratingFunction[dist,t]]:

Plug in the definition for the moments in terms of their generating function:

Sample estimator of Cumulant on data is biased:

Find a sampling population expectation, assuming size :

Construct an unbiased sample estimator using PowerSymmetricPolynomial:

Verify unbiasedness on a small sample size:

The sample estimator is biased:

Compare with the sampling population expectation of the sample estimator:

## Possible Issues(1)

For some distributions with long tails, cumulants of only several low orders are defined:

## Neat Examples(2)

Find an unbiased estimator for a product of cumulants:

Check the sampling population expectation:

The distribution of Cumulant estimates for 20, 100, and 300 samples:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_cumulant, author="Wolfram Research", title="{Cumulant}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/Cumulant.html}", note=[Accessed: 17-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_cumulant, organization={Wolfram Research}, title={Cumulant}, year={2023}, url={https://reference.wolfram.com/language/ref/Cumulant.html}, note=[Accessed: 17-July-2024 ]}