Gram–Schmidt Orthonormalization based Projection Depth

Muthukrishnan R; Vadivel M; Ramkumar N

doi:https://doi.org/10.14445/22315373/IJMTT-V52P560

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 52 | Number 6 | Year 2017 | Article Id. IJMTT-V52P560 | DOI : https://doi.org/10.14445/22315373/IJMTT-V52P560

Gram–Schmidt Orthonormalization based Projection Depth

Muthukrishnan R, Vadivel M, Ramkumar N

Citation :

Muthukrishnan R, Vadivel M, Ramkumar N, "Gram–Schmidt Orthonormalization based Projection Depth," International Journal of Mathematics Trends and Technology (IJMTT), vol. 52, no. 6, pp. 430-434, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V52P560

Abstract

Gram-Schmidt Orthonormalization (GSO) Euclidean vectors based depth function is proposed to compute projection depth. The performance of GSO algorithm has been studied with exact and approximate algorithms, used in the associated estimator namelyStahel-Donoho (S-D) location and scatter estimators, for bivariate data.The efficiency of GSO algorithm is checked out by computing average misclassification error in discriminant analysis under real and stimulatingenvironment. The study concludesthat GSO algorithm based projection depth estimators performs well when compared with exact and approximate algorithms.

Keywords

Gram-Schmidt process, Projection depth, Robust Discriminant analysis.

References

[1] Ben Noble and James Daniel.: Applied Linear Algebra, 3rdedition. Prentice Hall, Englewood Cliffs, N.J., (1988).
[2] Donoho, D. L., and Gasko, M.: Breakdown properties of location estimates based on halfspace depth and projected outlyingness. The Annals of Statistics. 20, 1803–1827, (1992).
[3] Dyckerhoff, R,.:Data depthssatistiying the projection property. All emeinestatistischesArchiv, 88, 163-190, (2004).
[4] Hubert, M. and Van Driessen, K.: Fast and Robust Discriminant Analysis. Computaional Statistics and Data Analysis, 45,301,320, (2004).
[5] Johnson, R.A., and Wichern, D.W.: Applied multivariate analysis,3rd ed. Prentice Hall, Englewood Cliffs, New Jersey, (2007).
[6] Johnson, R.A., and Wichern, D.W.: Applied multivariate analysis, 9th ed. Prentice Hall, Englewood Cliffs, New Jersey, (2009).
[7] Liu, R.Y.: On a notion of data depth based on random simplices. The Annals of Statistics, 18, 191-219, (1990).
[8] Liu, R.Y.: Data depth and multivariate rank test. In: L1- Statistical Analysis and Related Methods, 279-294. North-Holland, Amsterdam, (1992).
[9] Liu, X.H., Zuo, Y.J., Wang, Z.Z.: Exactly computing bivariate projection depth contours and median. Preprint, (2011).
[10] Liu, X., and Zuo, Y.: Computing projection depth and its associated estimators. Stat.Comput., 24, 51-63, (2014).
[11] Rousseeuw, P.J., and Hubert, M.: Regression depth. Journal of the American Statistical Association, 94, 388-433, (1999).
[12] Stahel, W.A.: Breakdown of covariance estimators. Research Report 31, 1029-1036, (1981).
[13] Tukey, J.W.: Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematicians, pp. 523-531. Canadian Mathematical Congress, Montreal, (1975).
[14] Zuo, Y. Projection-based depth functions and associated medians. The Annals of Statistics, 31, 1460-1490, (2003).
[15] Zuo, Y.J.: Multidimensional trimming based on projection depth. The Annals of Statistics, 34, 2211-2251, (2006).
[16] Zuo, Y.J., Cui, H.J., He, X.M.: On the Stahel-Donoho estimators and depth – weighted means for multivariate data. The Annals of Statistics, 32, 189-218, (2004).
[17] Zuo, Y.J., and Lai, S.Y.: Exact computation of bivariate projection depth and Stahel-Donoho estimator. Computational Statistics & Data Analysis, 55, 1173-1179, (2011).
[18] Zuo, Y., and Serfling, R.: General notions of statistical depth function. The Annals of Statistics, 28, 461–482, (2000).