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Linear discrimination with equicorrelated training vectors. (English) Zbl 1112.62057

Summary: Fisher’s linear discrimination rule requires uncorrelated training vectors. In this paper a linear discrimination method is developed to be used when the training vectors are equicorrelated. Also, maximum likelihood ratio tests are proposed to decide whether the training samples are uncorrelated or equicorrelated.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
68T10 Pattern recognition, speech recognition
62H15 Hypothesis testing in multivariate analysis
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[1] Anderson, T.W., An introduction to multivariate statistical analysis, (1958), Wiley New York · Zbl 0083.14601
[2] Barghava, R.P.; Srivastava, M.S., On Tukey’s confidence intervals for the contrasts in the means of the intraclass correlation model, J. roy. statist. soc. B, 34, 147-152, (1973) · Zbl 0259.62033
[3] Bartlett, M.S., An inverse matrix adjustment arising in discriminant analysis, Ann. math. statist., 22, 107-111, (1951) · Zbl 0042.38203
[4] Basu, J.P.; Odell, P.L., Effect of intraclass correlation among training samples on the misclassification probabilities of bayes’ procedure, Pattern recognition, 6, 13-16, (1974)
[5] Bhandary, M.; Alam, M.K., Test for the equality of intraclass correlation coefficients under unequal family sizes for several populations, Comm. statist.—theory methods, 29, 755-768, (2000) · Zbl 0992.62052
[6] Bhattacharyya, A., On a measure of divergence between two statistical populations defined by probability distributions, Bull. Calcutta math. soc., 35, 99-109, (1943) · Zbl 0063.00364
[7] Chaudhuri, G.; Borwankar, J.D.; Rao, P.R.K., Bhattacharyya distance based linear discriminant function for stationary time series, Comm. statist.—theory methods, 20, 2195-2205, (1991) · Zbl 0900.62461
[8] Donner, A.; Bull, S., Inferences concerning a common intraclass correlation coefficient, Biometrics, 39, 771-775, (1983) · Zbl 0526.62052
[9] Donner, A.; Zou, G., Testing the equality of dependent intraclass correlation coefficients, Statistician, 51, 367-379, (2002)
[10] Fisher, R.A., The use of multiple measurements in taxonomic problems, Ann. eugenics, 7, 179-188, (1936)
[11] Gupta, A.K.; Nagar, D.K., Nonnull distribution of the likelihood ratio criterion for testing equality of covariance matrices under intraclass correlation model, Comm. statist.—theory methods, 16, 3323-3341, (1987) · Zbl 0631.62060
[12] Harville, D.A., D.A. matrix algebra from a Statistician’s perspective, (1997), Springer Berlin, ISBN 0-387-94978-X · Zbl 0881.15001
[13] M. Herrera, Analisis discriminate para poblaciones con matrices de covarianza distintas, Master Thesis, Facultad de Ciencias Físico Matemáticas y Naturales, Universidad Nacional de San Luis, Argentina, 2000.
[14] Khan, S.; Bhatti, M.I., Predictive inference on equicorrelated linear regression models, Appl. math. comput., 95, 205-217, (1998) · Zbl 0943.62065
[15] Khatri, C.G.; Pukkila, T.M.; Radhakrishna Rao, C., Testing intraclass correlation coefficient, Comm. statist.—simulation, 18, 755-768, (1989) · Zbl 0695.62135
[16] Konishi, S.; Gupta, A.K., Testing the equality of several intraclass correlation coefficients, J. statist. plann. inference, 21, 93-105, (1989) · Zbl 0666.62057
[17] Leiva, R.A.; Herrera, M., Generalización de la distancia de Mahalanobis para el análisis discriminante lineal en poblaciones con matrices de covarianza desiguales, Rev. sociedad Argentina estadística, 3, 64-85, (1999)
[18] Mahalanobis, P.C., On the generalized distance in statistics, Calcutta statist. assoc. bull., 14, 9, (1936) · Zbl 0015.03302
[19] Matushita, K., A distance and related statistics in multivariate analysis, (), 187-202
[20] McLachlan, G.J., Discriminant analysis and statistical pattern recognition, (1992), Wiley New York
[21] Mentz, R.P., Likelihood ratio test of equality of several variances in the intraclass correlation model, Comm. statist.—theory methods, 30, 75-91, (2001) · Zbl 1008.62609
[22] Okamoto, M., An asymptotic expansion for the distribution of the linear discriminant function, Ann. math. statist., 34, 1286-1301, (1963) · Zbl 0117.37101
[23] Paranjpe, S.A.; Gore, A.P., Selecting variables for discrimination when covariance matrices are unequal, Statist. probab. lett., 21, 5, (1994) · Zbl 0825.62545
[24] Park, P.S.; Kshirsagar, A.M., Distances between normal populations when covariance matrices are unequal, Comm. statist.—theory methods, 23, 12, 3549-3556, (1994) · Zbl 0825.62240
[25] Paul, S.R., Maximum likelihood estimation of intraclass correlation in the analysis of familial data: estimating equation approach, Biometrika, 77, 3, 549-555, (1990)
[26] Paul, S.R.; Barnwal, R.K., Maximum likelihood estimation and a \(C(\alpha)\) test for a common intraclass correlation, Statistician, 39, 15-30, (1990)
[27] Rao, C.R.; Varadarajan, V.S., Discrimination of Gaussian processes, Sankhya A, 25, 303-330, (1963) · Zbl 0122.14801
[28] Richards, J.A.; Jia, X., Remote sensing and digital image analysis: an introduction, (1999), Springer Berlin
[29] Shoukri, M.; Ward, R., On the estimation of the intraclass correlation, Comm. statist.—theory methods, 13, 1239-1255, (1984)
[30] Smith, J.R.; Lewis, T.O., Determining the effects of the intraclass correlation on factorial experiments, Comm. statist.—theory methods, 9, 1253-1364, (1980) · Zbl 0446.62081
[31] Viana, M.A.G., Combined estimators for the equicorrelation coefficient, Comm. statist.—theory methods, 11, 1483-1504, (1982) · Zbl 0533.62049
[32] Viana, M.A.G., Combined maximum likelihood estimates for the equicorrelation coefficient, Biometrics, 50, 813-820, (1994) · Zbl 0821.62026
[33] Young, D.J.; Bhandary, M., Test for equality of intraclass correlation coefficients under unequal family sizes, Biometrics, 54, 1363-1373, (1998) · Zbl 1058.62673
[34] Zerbe, G.O.; Goldgar, D.E., Comparison of intraclass correlation coefficients with the ratio of two independent F statistics, Comm. statist.—theory methods A, 9, 15, 1641-1655, (1980) · Zbl 0459.62057
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