An Asymmetry Coefficient for Multivariate Distributions

© Michel Petitjean, 2013
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 m3 of an univariate distribution is the centered third-order moment normalized to the cube of the standard deviation. The square of the skewness, M3, was introduced by Pearson in 1895 (see ref. [1], p. 351) to measure the degree of asymmetry of a distribution. The squared skewness has an upper bound: M3 ≤ m4 - 1, where m4 is the kurtosis of the distribution, i.e. the centered fourth-order moment normalized to the square of the variance. This inequality is a consequence of a more general one standing for multivariate distributions (see eq. 6 in [2]). Many asymmetry coefficients were proposed in the statistical literature (see section 4.2 in [3] 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 known 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 in [0;1/2] because Rmin cannot be positive.
As a consequence of the convergence theorem in [4], 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 [3])

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 [5] to lie in the interval [1/2;1]. When d=2, it is shown [6] to lie in the interval [1-1/π;1-1/2π].


REFERENCES


  1. 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.

  2. PETITJEAN M.
    The Chiral Index: Applications to Multivariate Distributions and to 3D molecular graphs.
    Proceedings of 12th International Symposium on Operations Research in Slovenia, SOR'13, pp.11-16,
    Dolenjske Toplice, Slovenia, 25-27 September 2013.
    L. Zadnik Stirn, J. Zerovnik, J. Povh, S. Drobne, A. Lisec, Eds.
    Slovenian Society Informatika (SDI), Section for Operations Research (SOR), Ljubljana, 2013.
    Download PDF paper in preprint form (posted with permission from Society Informatika, Section for Operations Research)
    Download PDF file of the lecture.

  3. 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)

  4. 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)

  5. 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)

  6. 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)