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Central Moment
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A moment mu_n of a univariate probability function P(x) taken about the mean mu=mu_1^',

mu_n = <(x-<x>)^n>
(1)
= int(x-mu)^nP(x)dx,
(2)

where <X> denotes the expectation value. The central moments mu_n can be expressed as terms of the raw moments mu_n^' (i.e., those taken about zero) using the binomial transform

 mu_n=sum_(k=0)^n(n; k)(-1)^(n-k)mu_k^'mu_1^('n-k),
(3)

with mu_0^'=1 (Papoulis 1984, p. 146). The first few central moments expressed in terms of the raw moments are therefore

mu_1 = 0
(4)
mu_2 = -mu_1^('2)+mu_2^'
(5)
mu_3 = 2mu_1^('3)-3mu_1^'mu_2^'+mu_3^'
(6)
mu_4 = -3mu_1^('4)+6mu_1^('2)mu_2^'-4mu_1^'mu_3^'+mu_4^'
(7)
mu_5 = 4mu_1^('5)-10mu_1^('3)mu_2^'+10mu_1^('2)mu_3^'-5mu_1^'mu_4^'+mu_5^'.
(8)

These transformations can be obtained using CentralToRaw[n] in the Mathematica application package mathStatica.

The central moments mu_n can also be expressed in terms of the cumulants kappa_n, with the first few cases given by

mu_2 = kappa_2
(9)
mu_3 = kappa_3
(10)
mu_4 = 3kappa_2^2+kappa_4
(11)
mu_5 = 10kappa_2kappa_3+kappa_5.
(12)

These transformations can be obtained using CentralToCumulant[n] in the Mathematica application package mathStatica.

The central moment of a multivariate probability function P(x_1,x_2,...) can be similarly defined as

 mu_(m,n,...)=<(x_1-<x_1>)^m(x_2-<x_2>)^n...>.
(13)

Therefore,

 mu_(n,0,...,0)=mu_n.
(14)

For example,

mu_(1,1) = -mu_(0,1)^'mu_(1,0)^'+mu_(1,1)^'
(15)
mu_(2,1) = 2mu_(0,1)^'mu_(1,0)^'^2-2mu_(1,0)^'mu_(1,1)^'-mu_(0,1)^'mu_(2,0)^'+mu_(2,1)^'.
(16)

Similarly, the multivariate central moments can be expressed in terms of the multivariate cumulants. For example,

mu_(1,1) = kappa_(1,1)
(17)
mu_(2,1) = kappa_(2,1)
(18)
mu_(3,1) = 3kappa_(1,1)kappa_(2,0)+kappa_(3,1)
(19)
mu_(4,1) = 6kappa_(2,0)kappa_(2,1)+4kappa_(1,1)kappa_(3,0)+kappa_(4,1)
(20)
mu_(5,1) = 15kappa_(1,1)kappa_(2,0)^2+10kappa_(2,1)kappa_(3,0)+10kappa_(2,0)kappa_(3,1)+5kappa_(1,1)kappa_(4,0)]+kappa_(5,1).
(21)

These transformations can be obtained using CentralToRaw[{m, n, ...}] in the Mathematica application package mathStatica and CentralToCumulant[{m, n, ...}], respectively.

SEE ALSO: Absolute Moment, Cumulant, Kurtosis, Moment, Raw Moment, Sample Central Moment, Skewness

REFERENCES:

Kendall, M. G. "The Derivation of Multivariate Sampling Formulae from Univariate Formulae by Symbolic Operation." Ann. Eugenics 10, 392-402, 1940.

Kenney, J. F. and Keeping, E. S. "Moments About the Mean." §7.3 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 92-93, 1962.

Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 146, 1984.

Smith, P. J. "A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa." Amer. Stat. 49, 217-218, 1995.




CITE THIS AS:

Weisstein, Eric W. "Central Moment." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CentralMoment.html

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