Abstract:
Under normality, Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] investigated the asymptotic properties of the quadratic discrimination procedure under hierarchical models for the scatter matrices, that is: (i) arbitrary scatter matrices, (ii) common principal components, (iii) proportional scatter matrices and (iv) identical matrices. In this paper, we study the properties of robust quadratic discrimination rules based on robust estimates of the involved parameters. Our analysis is based on the partial influence functions of the functionals related to these parameters and allows to derive the asymptotic variances of the estimated coefficients under models (i)-(iv). From them, we conclude that the asymptotic variances verify the same order relations as those obtained by Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] for the classical estimators. We also perform a Monte Carlo study for different sample sizes and different hierarchies which shows the advantage of using robust procedures over classical ones, when anomalous data are present. It also confirms that better rates of misclassification can be achieved if a more parsimonious model among all the correct ones is used instead of the standard quadratic discrimination. © 2007 Elsevier Inc. All rights reserved.
Registro:
Documento: |
Artículo
|
Título: | Robust discrimination under a hierarchy on the scatter matrices |
Autor: | Bianco, A.; Boente, G.; Pires, A.M.; Rodrigues, I.M. |
Filiación: | Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, CONICET, Ciudad Universitaria, Pabellon 2, Buenos Aires, C1428EHA, Argentina Departamento de Matemática, CEMAT, Instituto Superior Técnico, Lisboa, Portugal
|
Palabras clave: | Common principal components; Outliers; Partial influence functions; Plug-in methods; Proportional scatter matrices; Quadratic discrimination; Robust estimation |
Año: | 2008
|
Volumen: | 99
|
Número: | 6
|
Página de inicio: | 1332
|
Página de fin: | 1357
|
DOI: |
http://dx.doi.org/10.1016/j.jmva.2007.08.008 |
Título revista: | Journal of Multivariate Analysis
|
Título revista abreviado: | J. Multivariate Anal.
|
ISSN: | 0047259X
|
CODEN: | JMVAA
|
PDF: | https://bibliotecadigital.exactas.uba.ar/download/paper/paper_0047259X_v99_n6_p1332_Bianco.pdf |
Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0047259X_v99_n6_p1332_Bianco |
Referencias:
- A. Bianco, G. Boente, A.M. Pires, I.M. Rodrigues, Robust discrimination under a hierarchy on the scatter matrices, Working Paper, 2007, Universidad de Buenos Aires. Available at 〈http://www.ic.fcen.uba.ar/preprints/biancoboentepiresrodrigues.pdf〉; Boente, G., Critchley, F., Orellana, L., Influence functions for robust estimators under proportional scatter matrices (2007) Statist. Methods Appl., 15, pp. 295-327
- Boente, G., Orellana, L., A robust approach to common principal components (2001) Statistics in Genetics and in the Environmental Sciences, pp. 117-147. , Fernholz L.T., Morgenthaler S., and Stahel W. (Eds), Birkhauser, Basel
- Boente, G., Orellana, L., Robust plug-in estimators in proportional scatter models (2004) J. Statist. Plann. Inference, 122, pp. 95-110
- Boente, G., Pires, A.M., Rodrigues, I.M., Influence functions and outlier detection under the common principal components model: a robust approach (2002) Biometrika, 89, pp. 861-875
- G. Boente, A.M. Pires, I.M. Rodrigues, Robust tests for the common principal components model, Working Paper, 2005, Universidad de Buenos Aires. Available at 〈http://www.ic.fcen.uba.ar/preprints/boentepiresrodrigues2.pdf〉; Boente, G., Pires, A.M., Rodrigues, I.M., General projection-pursuit estimators for the common principal components model: influence functions and Monte Carlo study (2006) J. Multivariate Anal., 97, pp. 124-147
- Campbell, N.A., The influence function as an aid in outlier detection in discriminant analysis (1978) Appl. Statist., 27, pp. 251-258
- Critchley, F., Vitiello, C., The influence of observations on misclassification probability estimates in linear discriminant analysis (1991) Biometrika, 78, pp. 677-690
- Croux, C., Dehon, C., Robust linear discriminant analysis using S-estimators (2001) Canad. J. Statist., 29, pp. 473-492
- C. Croux, P. Filzmoser, K. Joossens, Classification efficiencies for robust linear discriminant analysis, Statist. Sinica (2007), to appear; Croux, C., Haesbroeck, G., Principal component analysis based on robust estimators of the covariance or correlation matrix: influence functions and efficiencies (2000) Biometrika, 87, pp. 603-618
- Croux, C., Joossens, K., Influence of observations on the misclassification probability in quadratic discriminant analysis (2005) J. Multivariate Anal., 96, pp. 384-403
- D.L. Donoho, Breakdown properties of multivariate location estimators, Ph.D. Qualifying Paper, Harvard University, 1982; Efron, B., The efficiency of logistic regression compared to normal discrimination analysis (1975) J. Amer. Statist. Assoc., 70, pp. 892-898
- Flury, B.K., (1988) Common Principal Components and Related Multivariate Models, , Wiley, New York
- Flury, B.K., Schmid, M.J., Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results (1992) J. Multivariate Anal., 40, pp. 244-261
- Flury, B.K., Schmid, M.J., Narayanan, A., Error rates in quadratic discrimination with constraints on the covariance matrices (1994) J. Classification, 11, pp. 101-120
- Fung, W.K., Diagnostics in linear discriminant analysis (1995) J. Amer. Statist. Assoc., 90, pp. 952-956
- Fung, W.K., Diagnosing influential observations in quadratic discriminant analysis (1996) Biometrics, 52, pp. 1235-1241
- Gervini, D., The influence function of the Donoho-Stahel estimator of multivariate location and scale (2002) Statist. Probab. Lett., 60, pp. 425-435
- Lachenbruch, P.A., Note on initial misclassification effects on the quadratic discriminant function (1979) Technometrics, 21, pp. 129-132
- Lopuhaä, H.P., Breakdown points and asymptotic properties of multivariate S-estimators and τ-estimators: a summary (1991) Directions in Robust Statistics and Diagnostics, Part I, pp. 167-182. , Stahel W., and Weisberg S. (Eds), Springer, New York
- Maronna, R., Martin, D., Yohai, V., (2006) Robust Statistics: Theory and Methods, , Wiley, New York
- T.J. O'Neill, A theoretical method of comparing classification rules under non-optimal conditions with application to the estimates of Fisher's linear and quadratic discriminant rules under unequal covariances matrices, Technical Report No. 217, 1984, Department of Statistics, Stanford University; O'Neill, T.J., Error rates of non-Bayes classification rules and robustness of Fisher's linear discriminant function (1992) Biometrika, 79, pp. 177-184
- Pires, A.M., Branco, J., Partial influence functions (2002) J. Multivariate Anal., 83, pp. 451-468
- W. Stahel, Robust estimation: infinitesimal optimality and covariance matrix estimation, Thesis, ETH, Zurich, 1981, (in German); Tyler, D.E., Radial estimates and the test for sphericity (1982) Biometrika, 69, pp. 429-436
- H.P. Lopuhaä (1989). On the relation between S-estimators and M-estimators of multivariate location and covariance. Annals of Statistics, 17, 1662-1683
Citas:
---------- APA ----------
Bianco, A., Boente, G., Pires, A.M. & Rodrigues, I.M.
(2008)
. Robust discrimination under a hierarchy on the scatter matrices. Journal of Multivariate Analysis, 99(6), 1332-1357.
http://dx.doi.org/10.1016/j.jmva.2007.08.008---------- CHICAGO ----------
Bianco, A., Boente, G., Pires, A.M., Rodrigues, I.M.
"Robust discrimination under a hierarchy on the scatter matrices"
. Journal of Multivariate Analysis 99, no. 6
(2008) : 1332-1357.
http://dx.doi.org/10.1016/j.jmva.2007.08.008---------- MLA ----------
Bianco, A., Boente, G., Pires, A.M., Rodrigues, I.M.
"Robust discrimination under a hierarchy on the scatter matrices"
. Journal of Multivariate Analysis, vol. 99, no. 6, 2008, pp. 1332-1357.
http://dx.doi.org/10.1016/j.jmva.2007.08.008---------- VANCOUVER ----------
Bianco, A., Boente, G., Pires, A.M., Rodrigues, I.M. Robust discrimination under a hierarchy on the scatter matrices. J. Multivariate Anal. 2008;99(6):1332-1357.
http://dx.doi.org/10.1016/j.jmva.2007.08.008