Abstract
Invariant coordinate selection (ICS), introduced by Tyler et al. (J. Roy. Stat. Soc. B 71(3):549–592, 2009), is a powerful tool to find potentially interesting projections of multivariate data. In some cases, some of the projections proposed by ICS come close to really interesting ones, but little deviations can result in a blurred view which does not reveal the feature (e.g., a clustering), which would otherwise be clearly visible. To remedy this problem, we propose an automated and localized version of projection pursuit (PP), cf. Huber (Ann. Stat. 13(2):435–525, 1985). Precisely, our local search is based on gradient descent applied to estimated differential entropy as a function of the projection matrix.
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References
Diaconis, P., & Freedman, D. (1984). Asymptotics of graphical projection pursuit. Annals of Statistics, 12(3), 793–815.
Dümbgen, L., & Del Conte-Zerial, P. (2013). On low-dimensional projections of high-dimensional distributions. In M. Banerjee, F. Bunea, J. Huang, V. Koltchinskii, & M. H. Maathuis (Eds.) From probability to statistics and back: High-dimensional models and processes. A Festschrift in Honor of Jon Wellner, vol. 9 of IMS collections (pp. 91–104). Hayward, California: Institute of Mathematical Statistics.
Friedman, J. H., & Tukey, J. W. (1974). A projection pursuit algorithm for exploratory data analysis. IEEE Transactions on Computers, C-23, 881–890.
Huber, P. J. (1985). Projection pursuit. Annals of Statistics, 13(2), 435–525. With discussion.
Jones, M. C., & Sibson, R. (1987). What is projection pursuit? Journal of the Royal Statistical Society: Series A, 150(1), 1–36. With discussion.
Kent, J. T., & Tyler, D. E. (1991). Redescending M-estimates of multivariate location and scatter. Annals of Statistics, 19(4), 2102–2119.
Nocedal, J., & Wright, S. J. (2006). Numerical optimization (2nd ed.). Springer Series in Operations Research and Financial Engineering. Springer, New York.
Tyler, D. E. (1987). A distribution-free M-estimator of multivariate scatter. Annals of Statistics, 15(1), 234–251.
Tyler, D. E., Critchley, F., Dümbgen, L., & Oja, H. (2009). Invariant coordinate selection (with discussion). Journal of the Royal Statistical Society: Series B, 71(3), 549–592.
Acknowledgements
Part of this research was supported by Swiss National Science Foundation. The authors are grateful to two referees for constructive comments.
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Dümbgen, L., Gysel, K., Perler, F. (2023). Refining Invariant Coordinate Selection via Local Projection Pursuit. In: Yi, M., Nordhausen, K. (eds) Robust and Multivariate Statistical Methods. Springer, Cham. https://doi.org/10.1007/978-3-031-22687-8_6
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DOI: https://doi.org/10.1007/978-3-031-22687-8_6
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