CentralMoment

CentralMoment[data,r]

gives the order r central moment of data.

CentralMoment[data,{r₁,…,r_m}]

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

CentralMoment[dist,…]

gives the central moment of the distribution dist.

CentralMoment[r]

represents the order r formal central moment.

Details

CentralMoment is also known as a moment about the mean.
For scalar order r and data being an array , with mean (first raw moment) :
sum of r powers »

columnwise sum of r central powers »

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

$x in TemplateBox[{Arrays, paclet:ref/Arrays}, RefLink, BaseStyle -> {3ColumnTableMod}][{n_(1),...,n_(k)}]$ sum the r_j central power in the j column »
CentralMoment[x,{r₁,…,r_m}] is equivalent to ArrayReduce[CentralMoment[#, ${r_1,r_2,...,r_m}$ ]&,x,{{1},{2}}].
CentralMoment 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 channels' values or grayscale intensity value »
	Audio	amplitude values of all channels »
	DateObject, TimeObject	list of dates or list of times »

For a distribution dist, the rcentral moment is given by Expectation[(x-Mean[dist])^r,xdist]. »
For a multivariate distribution dist, the {r₁,…,r_m} central moment is given by Expectation[(x₁-μ₁)^r₁⋯(x₂-μ_m)^r_m,{x₁,…,x_m}dist] and {μ₁,…,μ_m}=Mean[dist]. »
For a random process proc, the central moment function can be computed for slice distribution at time t, SliceDistribution[proc,t], as [t]=CentralMoment[SliceDistribution[proc,t],r]. »
CentralMoment[r] can be used in functions such as MomentConvert and MomentEvaluate, etc. »

Examples

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Basic Examples (3)

Compute central moments from data:

Use symbolic data:

Central moment of a list of dates:

Compute the second central moment of a univariate distribution:

The central moment of a multivariate distribution:

Scope (26)

Basic Uses (6)

Exact input yields exact output:

Approximate input yields approximate output:

Find central moments of WeightedData:

Find a central moment of EventData:

Find a central moment of TimeSeries:

Central moment depends only on the values:

Find a central moment for data involving quantities:

Array Data (5)

For a matrix, CentralMoment gives columnwise moments:

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

Multivariate CentralMoment for an array:

Works with large arrays:

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

SparseArray data can be used just like dense arrays:

Find the central moment of a QuantityArray:

Image and Audio Data (2)

Channelwise central moment of an RGB image:

Central moment intensity value of a grayscale image:

On audio objects, CentralMoment works channelwise:

Date and Time (4)

Compute central moment of dates:

Compute the weighted central moment of dates:

Compute the central moment of dates given in different calendars:

Compare with variance:

Compute the central moment of times:

List of times with different time zone specifications:

Distribution and Process Moments (5)

Scalar central moment for univariate distributions:

Scalar central moment for multivariate distributions:

Joint central moment for multivariate distributions:

Compute a central moment for a symbolic order r:

A central moment may only evaluate for specific orders:

A central moment may only evaluate numerically:

Central moments for derived distributions:

Data distribution:

Central moment function for a random process:

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

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

Formal Moments (4)

TraditionalForm formatting for formal moments:

Convert combinations of formal moments to an expression involving CentralMoment:

Evaluate an expression involving formal moments for a distribution:

Evaluate for data:

Find a sample estimator for an expression involving CentralMoment:

Evaluate the resulting estimator for data:

Applications (11)

The first central moment is always 0:

The second central moment is a measure of dispersion:

The third central moment is a measure of skewness:

Estimate parameters of a distribution using the method of moments:

Compare data and the estimated parametric distribution:

Find a normal approximation to GammaDistribution using the method of moments:

Show how and depend on and :

Compare an original and an approximated distribution:

Construct a sample estimator of the second central moment:

Find its sample distribution expectation, assuming sample size :

Find sample distribution variance of the estimator:

Variance of the estimator for uniformly distributed sample:

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

Edgeworth expansion for near-normal data correcting for third and fourth central moments:

Function computing sample Jarque–Bera statistics [link]:

Accumulate statistics on samples of normal random variates:

Compare the statistics histogram with an asymptotic distribution:

Compute a moving central moment for some data:

Use the window of length .1:

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

Choose a few slice times:

Plot central moments over these paths:

Properties & Relations (11)

Central moments are translation invariant:

The second central moment is a scaled Variance:

The odd moments of 2×2 matrices vanish:

For a multivariate order, the total order must be odd:

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

Sqrt of the second central moment is RootMeanSquare of deviations from the Mean:

Skewness is a ratio of powers of third and second central moments:

Kurtosis is a ratio of powers of fourth and second central moments:

CentralMoment is equivalent to an Expectation of a power of a random variable around its mean:

CentralMoment of order is equivalent to when both exist:

Use CentralMoment directly:

Find the central moment–generating function by using GeneratingFunction:

Compare with direct evaluation of CentralMomentGeneratingFunction:

CentralMoment can be expressed in terms of Moment, Cumulant, or FactorialMoment:

Possible Issues (2)

Central moments of higher order are undefined for a heavy-tailed distribution:

Compute central moments on 5 independent samples of the distribution:

Sample central moments of higher order exhibit wild fluctuations:

Sample estimators of central moments are biased:

Find sampling population expectation assuming a sample of size :

The estimator is asymptotically unbiased:

Construct an unbiased estimator:

The expected value of the estimator is the central moment for all sample sizes:

Neat Examples (1)

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

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