Abstract
Projection pursuit is a multivariate statistical technique aimed at finding interesting low-dimensional data projections. A projection pursuit index is a function which associates a data projection to a real value measuring its interestingness: the higher the index, the more interesting the projection. Consequently, projection pursuit looks for the data projection which maximizes the projection pursuit index. The absolute value of the fourth standardized cumulant is a prominent projection pursuit index. In the general case, a projection achieving either minimal or maximal kurtosis poses computational difficulties. We address them by an algorithm which converges to the global optimum, whose computational advantages are illustrated with air pollution data.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Adcock, C.J., Shutes, K.: On the multivariate extended skew-normal, normal-exponential, and normal-gamma distributions. J. Stat. Theory Pract. 6, 636–664 (2012)
Bartoletti, S., Loperfido, N.: Modelling air pollution data by the skew-normal distribution. Stoch. Environ. Res. Risk Assess. 24, 513–517 (2010)
Caussinus, H., Ruiz-Gazen, A.: Exploratory Projection Pursuit. In: Gerard Govaert, G. (ed.) Data Analysis, pp. 76–92. Wiley (2009)
de Lathauwer, L., Comon, P., De Moor, B., Vandewalle, J.: Higher-order power method Application in independent component analysis. In: Proceedings of the International Symposium on Nonlinear Theory and Its Applications (NOLTA’95), Las Vegas, NV, pp. 91–96 (1995)
de Lathauwer L., de Moor, B., Vandewalle, J.: On the best rank-1 and rank-(\(R_{1}\),\(R_{2}\),...\(R_{N}\)) approximation of high-order tensors. SIAM J. Matrix Ana. Appl. 21, 1324–1342 (2000)
Hou, S., Wentzell, P.D.: Fast and simple methods for the optimization of kurtosis used as a projection pursuit index. Anal. Chim. Acta 704, 1–15 (2011)
Huber, P.J.: Projection pursuit (with discussion). Ann. Stat. 13, 435–525 (1985)
Hyvarinen, A., Oja, E.: A fast fixed point-algorithm for independent component analysis. Neural Comput. 9, 1483–1492 (1997)
Hyvarinen, A., Karhunen, J., Oja, E.: Independent Compon. Anal. Wiley, New York (2001)
Jondeau, E., Rockinger, M.: Optimal portfolio allocation under higher moments. Eur. Finan. Manag. 12, 29–55 (2006)
Kent, J.T.: Discussion on the paper by Tyler, Critchley, Dumbgen & Oja: invariant co-ordinate selection. J. R. Stat. Soc. Ser. B 71, 575–576 (2009)
Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23, 863–884 (2002)
Loperfido, N.: Spectral analysis of the fourth moment matrix. Linear Algebra Appl. 435, 1837–1844 (2011)
Loperfido, N.: A new Kurtosis matrix, with statistical applications. Linear Algebra Appl. 512, 1–17 (2017)
Paajarvi, P., Leblanc, J.P.: Skewness maximization for impulsive sources in blind deconvolution. In: Proceedings of the 6th Nordic Signal Processing Symposium—NORSIG 2004, 9–11 June, Espoo, Finland (2004)
Peña, D., Prieto, F.J.: Multivariate outlier detection and robust covariance estimation (with discussion). Technometrics 43, 286–310 (2001)
Peña, D., Prieto, F.J.: Cluster identification using projections. J. Am. Stat. Assoc. 96, 1433–1445 (2001)
Peña, D., Prieto, F.J., Viladomat, J.: Eigenvectors of a kurtosis matrix as interesting directions to reveal cluster structure. J. Multiv. Anal. 101, 1995–2007 (2010)
Qi, L.: Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines. J. Symb. Comput. 41, 1309–1327 (2006)
Ruiz-Gazen A., Marie-Sainte S.L., Berro, A.: Detecting multivariate outliers using projection pursuit with Particle Swarm Optimization. In: Proceedings of COMPSTAT, pp. 89–98 (2010)
Zarzoso, V., Comon, P.: Robust independent component analysis by iterative maximization of the Kurtosis contrast with algebraic optimal step size. IEEE Trans. Neural Netw. 21, 248–261 (2010)
Acknowledgements
The authors would like to thank an anonymous referee for her/his comments which greatly helped in improving the present paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Franceschini, C., Loperfido, N. (2018). An Algorithm for Finding Projections with Extreme Kurtosis. In: Perna, C., Pratesi, M., Ruiz-Gazen, A. (eds) Studies in Theoretical and Applied Statistics. SIS 2016. Springer Proceedings in Mathematics & Statistics, vol 227. Springer, Cham. https://doi.org/10.1007/978-3-319-73906-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-73906-9_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-73905-2
Online ISBN: 978-3-319-73906-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)