An Asymmetry Coefficient for Multivariate Distributions
© Michel Petitjean, 2010
Author's professional address:
MTi, INSERM UMR-S 973, Université Paris 7
35 rue Hélène Brion, 75205 Paris Cedex 13, France.
petitjean.chiral@gmail.com
Formerly: CEA/DSV/iBiTec-S/SB2SM (CNRS URA 2096), Saclay, France
(and formerly: ITODYS, UMR 7086, CNRS, Université Paris 7).
Related topics:
The skewness of an univariate distribution is the centered third-order moment
normalized to the cube of the standard deviation. The square of this quantity
was introduced by Pearson in 1895 (see ref. [1], p. 351)
to measure the degree of asymmetry of a distribution. Many asymmetry coefficients
were proposed in the statistical literature (see section 4.2 in [2]
for an historical survey). The centered third-order moment and its multivariate
analogs vanish for a symmetric distribution, although there are non-symmetric
distributions with a null third-order moment. It should be noted here that
the term "symmetry" denotes in this context an indirect symmetry (chirality)
rather a direct symmetry (this latter is undefined in the univariate case).
Despite its major drawback, the third-order moment has been widely used,
probably due to its simplicity and to the fact that most knwon
asymmetry coefficients offer the same drawback.
The chiral index is an asymmetry coefficient
which is null IF and ONLY IF the distribution is symmetric (i.e. achiral).
In the univariate case, the chiral index is expressed from
the lower bound Rmin of the correlation coefficient between the distribution and itself:
CHI = ( 1 + Rmin ) / 2
The mean m and the variance s2 are assumed to exist. CHI takes values over [0..1/2]
because Rmin cannot be positive.
As a consequence of the convergence theorem in [3],
the chiral index
of a sample of n observations of a random vector in Rd
converges to the chiral index of its parent distribution.
The chiral index CHI of a set of n observations xi (i=1..n),
sorted in increasing order is calculated like this:
2 CHI - 1 = Rmin = [ (x1-m)(xn-m) + (x2-m)(xn-1-m)
+ ... +
(xn-1-m)(x2-m) + (xn-m)(x1-m)
] / ns2
CHI is thus easily computable with a pocket calculator: (a) sort the set
with increasing values and then with decreasing values, (b) compute
the correlation coefficients between the sorted sets, (c) add 1 and
then divide by 2. Note that the correlation coefficient cannot be positive.
CHI may be expressed with the squared midranges or with the squared range lengths:
(see equations 2.9.4 and 2.9.5 in [2])
The chiral index is defined
for multivariate distributions. It is derived from a probability metric
and has formal relations with the Monge-Kantorovitch transportation problem:
see The Mathematical Theory of Chirality.
The upper bound of the chiral index of a d-variate distribution is unknown when d>1.
For any d value, this upper bound is shown [4] to lie in the interval [1/2;1].
When d=2, it is shown [5] to lie in the interval [1-1/π;1-1/2π].
REFERENCES
- PEARSON K.
Contributions to the Mathematical Theory of Evolution,-II.
Skew Variation in Homogeneous Material.
Phil. Trans. Roy. Soc. London (A.) 1895,186,343-414.
- PETITJEAN M.
Chirality and Symmetry Measures: A Transdisciplinary Review.
Entropy 2003,5[3],271-312.
(Zbl 1078.00503)
(DOI 10.3390/e5030271)
Download PDF paper
(open access paper posted with permission from MDPI)
- PETITJEAN M.
Chiral Mixtures.
J. Math. Phys. 2002,43[8],4147-4157.
(DOI 10.1063/1.1484559)
Copyright AIP (American Institute of Physics), 2002.
Download PDF paper (for personal use only.
Other uses: see copyright information)
- PETITJEAN M.
About the Upper Bound of the Chiral Index of Multivariate Distributions.
AIP Conf. Proc. 2008,1073,61-66.
Copyright AIP (American Institute of Physics), 2008.
Download PDF paper (for personal use only.
Other uses: see copyright information)
- COPPERSMITH D., PETITJEAN M.
About the Optimal Density Associated to the Chiral Index of a Sample from a Bivariate Distribution.
Compt. Rend. Acad. Sci. Paris, série I, 2005,340[8],599-604.
(DOI 10.1016/j.crma.2005.03.011)