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Adaptive Exponential Power Depth with Application to Classification

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Abstract

Depth functions have many applications in multivariate data analysis, including discriminant analysis and classification. In this paper, we introduce a novel class of data depth: exponential power depth (EPD) functions. Under some conditions, we show that the EPD functions are a statistical depth function, and the sample EPD functions are consistent and asymptotically normal. Based on the proposed EPD functions, we construct a DD-plot (depth-versus-depth plot), which can be applied to the classification problem. Since the EPD functions contain the two tuning parameters, we provide a data-driven approach to select these tuning parameters. The simulation studies and two real data analysis are conducted to assess the finite sample performance of the proposed method.

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References

  • BARNETT, V. (1976), “The Ordering of Multivariate Data”, Journal of the Royal Statistical Society Series A, 139(3), 318–355.

    Article  MathSciNet  Google Scholar 

  • BUTLER, R., DAVIES, P., and JHUN, M. (1993), “Asymptotics for the Minimum Covariance Determinant Estimator”, Annals of Statististics, 21(3), 1385–1400.

    Article  MathSciNet  Google Scholar 

  • CATOR, E., and LOPUHAA, H. (2010), “Asymptotic Expansion of the Minimum Covariance Determinant Estimators”, Journal of Multivariate Analysis, 101(10), 2372–2388.

    Article  MathSciNet  Google Scholar 

  • CHAKRABORTY, B., CHAUDHURI, P., and OJA, H. (1998), “Operating Transformation Retransformation on Spatial Median and Angle Test”, Statistica Sinica, 8, 767–784.

    MathSciNet  MATH  Google Scholar 

  • CHEN, Y., DANG, X., PENG, H., and BART, H. (2009), “Outlier Detection with the Kernelized Spatial Depth Function”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(2), 288–305.

    Article  Google Scholar 

  • CROUX, C., and HAESBROECK, G. (1999), “Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator”, Journal of Multivariate Analysis, 71(2), 161–190.

    Article  MathSciNet  Google Scholar 

  • CUEVAS, A., FEBRERO, M., and FRAIMAN, R. (2007), “Robust Estimation and Classification for Functional Data Via Projection-Based Depth Notions”, Computational Statistics, 22(3), 481–496.

    Article  MathSciNet  Google Scholar 

  • DAVIES, P.L. (1987), “Asymptotic Behavior of S-Estimators of Multivariate Location Parameters and Dispersion Matrices”, Annals of Statistics, 15(3), 1269–1292.

    Article  MathSciNet  Google Scholar 

  • DONOHO, D. (1982), “Breakdown Properties of Multivariate Location Estimators”, unpublished Ph.D. qualifying paper, Department of Statistics, Harvard University, Boston.

  • DUTTA, S., and GHOSH, A. (2011), “On Classification Based on l p-Depth with an Adaptive Choice of p”, Technical Report Number R5/2011, Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, India.

  • DUTTA, S., and GHOSH, A. (2012), “On Robust Classification Using Projection Depth”, Annals of the Institute of Statistical Mathematics, 64(3), 657–676.

    Article  MathSciNet  Google Scholar 

  • FRAIMAN, R., MELOCHE, J., GARCÍA-ESCUDERO, L., GORDALIZA, A., HE, X., MARONNA, R., YOHAI, V., SHEATHER, S., MCKEAN, J., SMALL, C. et al. (1999), “Multivariate l-Estimation”, Test, 8(2), 255–317.

    Article  MathSciNet  Google Scholar 

  • GHOSH, A., and CHAUDHURI, P. (2005), “On Data Depth and Distribution-Free Discriminant Analysis Using Separating Surfaces”, Bernoulli, 11(1), 1–27.

    Article  MathSciNet  Google Scholar 

  • JIANG, Y.,WANG, S., GE, W., and WANG, X. (2011), “Depth-Based Weighted Empirical Likelihood and General Estimating Equations”, Journal of Nonparametric Statistics, 23(4), 1051–1062.

    Article  MathSciNet  Google Scholar 

  • LI, J., CUESTA-ALBERTOS, J.A., and LIU, R.Y. (2012), “DD-Classifier: Nonparametric Classification Procedure Based on DD-Plot”, Journal of the American Statistical Association, 107(498), 737–753.

    Article  MathSciNet  Google Scholar 

  • LI, J., and LIU, R. (2008), “Multivariate Spacings Based on Data Depth: I. Construction of Nonparametric Multivariate Tolerance Regions”, Annals of Statistics, 36(3), 1299–1323.

    Article  MathSciNet  Google Scholar 

  • LIU, R. (1990), “On a Notion of Data Depth Based Upon Random Simplices”, Annals of Statistics, 18(1), 405–414.

    Article  MathSciNet  Google Scholar 

  • LIU, R. (1995), “Control Charts for Multivariate Processes”, Journal of the American Statistical Association, 90(432), 1380–1387.

    Article  MathSciNet  Google Scholar 

  • LIU, X., and ZUO, Y. (2012), “Computing Projection Depth and Its Associated Estimators”, Statistics and Computing, 1–13.

  • LÓPEZ-PINTADO, S., and ROMO, J. (2006), “Depth-Based Classification for Functional Data”, DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, 72, 17–35.

    MathSciNet  MATH  Google Scholar 

  • MAHALANOBIS, P. (1936), “On the Generalized Distance in Statistics”, Proceedings of the National Academy of Sciences India, 2(1), 49–55.

    MathSciNet  MATH  Google Scholar 

  • MARONNA, R. (1976), “Robust m-Estimators of Multivariate Location and Scatter”, Annals of Statistics, 4(1), 51–67.

    Article  MathSciNet  Google Scholar 

  • ROUSSEEUW, P. (1985), “Multivariate Estimation with High Breakdown Point”, Mathematical Statistics and Applications, 8, 283–297.

    Article  MathSciNet  Google Scholar 

  • ROUSSEEUW, P., and DRIESSEN, K. (1999), “A Fast Algorithm for the Minimum Covariance Determinant Estimator”, Technometrics, 41(3), 212–223.

    Article  Google Scholar 

  • ROUSSEEUW, P., VAN AELST, S., and HUBERT, M. (1999), “Regression Depth: Rejoinder”, Journal of the American Statistical Association, 94(446), 419–433.

    Google Scholar 

  • SÁNCHEZ-MANZANO, E., GOMEZ-VILLEGAS, M., and MARÍN-DIAZARAQUE, J. (2002), “A Matrix Variate Generalization of the Power Exponential Family of Distributions”, Communications in Statistics -Theory and Methods, 31(12), 2167–2182.

    Article  MathSciNet  Google Scholar 

  • SERFLING, R. (1980), Approximation Theorems of Mathematical Statistics, Wiley Online Library.

  • SERFLING, R. (2002), “A Depth Function and a Scale Curve Based on Spatial Quantiles”, in Statistical Data Analysis Based on the L1-Norm and Related Methods, ed. Y. Dodge, pp. 25–38.

  • SINGH, K. (1991), “A Notion of Majority Depth”, unpublished document.

  • SMALL, C. (1990), “A Survey of Multidimensional Medians”, International Statistical Review, 58(3), 263–277.

    Article  Google Scholar 

  • STAHEL, W. (1981), “Robust Estimation: Infinitesimal Optimality and Covariance Matrix Estimators”, unpublished doctoral dissertation, ETH, Zurich, Switzerland.

  • TUKEY, J. (1975), “Mathematics and the Picturing of Data”, Proceedings of the International Congress of Mathematicians, 2, 523–531.

    MathSciNet  MATH  Google Scholar 

  • WHITTLE, P., KAY, J., HAND, D., TARASSENKO, L., BROWN, P., TITTERINGTON, D., TAYLOR, C., GILKS, W., CRITCHLEY, F., MAYNE, A., et al. (1994), “Neural Networks and Related Methods for Classification-Discussion”, Journal of the Royal Statistical Society Series B, 56(3), 437–456.

    Google Scholar 

  • YEH, I.-C., YANG, K.-J., and TING, T.-M. (2009), “Knowledge Discovery on rfm Model Using Bernoulli Sequence”, Expert Systems with Applications, 36(3), 5866–5871.

    Article  Google Scholar 

  • ZUO, Y. (2003), “Projection-Based Depth Functions and Associated Medians,” Annals of Statistics, 31(5), 1460–1490.

    Article  MathSciNet  Google Scholar 

  • ZUO, Y. (2006), “Robust Location and Scatter Estimators in Multivariate Analysis”, in Frontiers in Statistics. Dedicated to Peter John Bickel in Honor of his 65th Birthday, Imperial College Press, pp. 467–490.

  • ZUO, Y., and LAI, S. (2011), “Exact Computation of Bivariate Projection Depth and the Stahel–Donoho Estimator”, Computational Statistics and Data Analysis, 55(3), 1173–1179.

    Article  MathSciNet  Google Scholar 

  • ZUO, Y., and SERFLING, R. (2000a), “General Notions of Statistical Depth Function”, Annals of Statistics, 28(2), 461–482.

    Article  MathSciNet  Google Scholar 

  • ZUO, Y., and SERFLING, R. (2000b), “Structural Properties and Convergence Results for Contours of gns”, Annals of Statistisics, 28(2), 483–499.

    Article  Google Scholar 

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Authors and Affiliations

  1. Jinan University, Guangzhou, China

    Yunlu Jiang

  2. Sun Yat-Sen University, Guangzhou, China

    Canhong Wen & Xueqin Wang

  3. Department of Statistical Science, School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou, 510275, China

    Xueqin Wang

  4. Zhongshan School of Medicine, Sun Yat-Sen University, Guangzhou, 510080, China

    Xueqin Wang

  5. Xinhua College, Sun Yat-Sen University, Guangzhou, 510520, China

    Xueqin Wang

Authors
  1. Yunlu Jiang
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  2. Canhong Wen
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  3. Xueqin Wang
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Correspondence to Xueqin Wang.

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Jiang, Y., Wen, C. & Wang, X. Adaptive Exponential Power Depth with Application to Classification. J Classif 35, 466–480 (2018). https://doi.org/10.1007/s00357-018-9264-z

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