The detection of outliers is very essential because of
their responsibility for producing huge interpretative problem in
linear as well as in nonlinear regression analysis. Much work has
been accomplished on the identification of outlier in linear
regression, but not in nonlinear regression. In this article we propose
several outlier detection techniques for nonlinear regression. The
main idea is to use the linear approximation of a nonlinear model and
consider the gradient as the design matrix. Subsequently, the
detection techniques are formulated. Six detection measures are
developed that combined with three estimation techniques such as the
Least-Squares, M and MM-estimators. The study shows that among
the six measures, only the studentized residual and Cook Distance
which combined with the MM estimator, consistently capable of
identifying the correct outliers.
[1] Anskombe, F. J. and Tukey, J. w. (1963), The examination and analysis
of residuals. Technometrics, 5, 141-60.
[2] Atkinson, A.C., (1981), Two graphical displays for outlying and
influential observations in regression, Biometrika, 68, 1, 13-20.
[3] Atkinson, A.C., (1982), Regression Diagnostics, Transformations and
Constructed Variables, Journal od Royal Statistical Society, B, 44, 1, 1-
36.
[4] Atkinson, A.C., (1986). Masking unmasked, Biometrika, 73, 3, 533-541.
[5] Bates, D.M. Watts, D.G., (1980). Relative curvature measures of
nonlinearity, J. R. statist. Ser. B 42, 1-25.
[6] Belsley, D. A., Kuh, E., and Welsch, R. E. (1980), Regression
Diagnostics, John Wiley & Sons, New York.
[7] Cook, R. D., and Weisberg, S., (1982), Residuals and Influence in
Regression. CHAPMAN and HALL.
[8] Fox, T., Hinkley, D. and Larntz, K., (1980), Jackknifing in nonlinear
regression. Technometrics, 22, 29-33.
[9] Habshah, M., Noraznan, M. R., Imon, A. H. M. R. (2009). The
performance of diagnostic-robust generalized potential for the
identification of multiple high leverage points in linear regression,
Journal of Applied Statistics, 36(5):507-520.
[10] Hadi, A.H. (1992). A new measure of overall potential influence in
linear regression, Computational Statistics and Data Analysis 14 (1992)
1-27.
[11] Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J. and Stahel, W.A.
(1986), Robust Statistics: The Approach Based on InfluenceFunctions.
New York: John Wiley & Sons, Inc.
[12] Hoaglin, D. C., Mosteller, F., Tukey, J. W. (1983), Understanding
Robust and Exploratory Data Analysis, John Wiley and Sons.
[13] Hoaglin, D.C., & Wellsch, R. (1978). The hat Matrix in regression and
ANOVA. Ammerican Statistician 32, 17-22.
[14] Huber, P. J. (1981), Robust Statistics, Wiley, New York .
[15] Imon, A.H.M.R, (2002), Identifying multiple high leverage points in
linear regression, J. Stat. Stud. 3, 207-218.
[16] Kennedy. W. and Gentle, J. (1980). Statistical Computing. New
York:Dekker.
[17] Rousseeuw, P. J., and Leroy, A. M. (1987), Robust Regression and
outlier detection, New York: John Wiley.
[18] Riazoshams, H., Habshah, Midi, (2009), A Nonlinear regression model
for chickens- growth data. European Journal of Scientific Research, 35,
3, 393-404.
[19] Srikantan, K. S. (1961), Testing for a single outlier in a regression
model. Sankhya A, 23, 251-260.
[20] Stromberg, A. J., (1993), Computation of High Breakdown Nonlinear
Regression Parameters, Journal of American Statistical Association, 88
(421), 237-244.
[21] Seber, G., A. F. and Wild, C. J. (2003), Nonlinear Regression, John
Wiley and Sons.
[22] Yohai, V. J. (1987), High Breakdown point and high efficiency robust
estimates for regression, The Annals of Statistics, 15, 642-656.
[1] Anskombe, F. J. and Tukey, J. w. (1963), The examination and analysis
of residuals. Technometrics, 5, 141-60.
[2] Atkinson, A.C., (1981), Two graphical displays for outlying and
influential observations in regression, Biometrika, 68, 1, 13-20.
[3] Atkinson, A.C., (1982), Regression Diagnostics, Transformations and
Constructed Variables, Journal od Royal Statistical Society, B, 44, 1, 1-
36.
[4] Atkinson, A.C., (1986). Masking unmasked, Biometrika, 73, 3, 533-541.
[5] Bates, D.M. Watts, D.G., (1980). Relative curvature measures of
nonlinearity, J. R. statist. Ser. B 42, 1-25.
[6] Belsley, D. A., Kuh, E., and Welsch, R. E. (1980), Regression
Diagnostics, John Wiley & Sons, New York.
[7] Cook, R. D., and Weisberg, S., (1982), Residuals and Influence in
Regression. CHAPMAN and HALL.
[8] Fox, T., Hinkley, D. and Larntz, K., (1980), Jackknifing in nonlinear
regression. Technometrics, 22, 29-33.
[9] Habshah, M., Noraznan, M. R., Imon, A. H. M. R. (2009). The
performance of diagnostic-robust generalized potential for the
identification of multiple high leverage points in linear regression,
Journal of Applied Statistics, 36(5):507-520.
[10] Hadi, A.H. (1992). A new measure of overall potential influence in
linear regression, Computational Statistics and Data Analysis 14 (1992)
1-27.
[11] Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J. and Stahel, W.A.
(1986), Robust Statistics: The Approach Based on InfluenceFunctions.
New York: John Wiley & Sons, Inc.
[12] Hoaglin, D. C., Mosteller, F., Tukey, J. W. (1983), Understanding
Robust and Exploratory Data Analysis, John Wiley and Sons.
[13] Hoaglin, D.C., & Wellsch, R. (1978). The hat Matrix in regression and
ANOVA. Ammerican Statistician 32, 17-22.
[14] Huber, P. J. (1981), Robust Statistics, Wiley, New York .
[15] Imon, A.H.M.R, (2002), Identifying multiple high leverage points in
linear regression, J. Stat. Stud. 3, 207-218.
[16] Kennedy. W. and Gentle, J. (1980). Statistical Computing. New
York:Dekker.
[17] Rousseeuw, P. J., and Leroy, A. M. (1987), Robust Regression and
outlier detection, New York: John Wiley.
[18] Riazoshams, H., Habshah, Midi, (2009), A Nonlinear regression model
for chickens- growth data. European Journal of Scientific Research, 35,
3, 393-404.
[19] Srikantan, K. S. (1961), Testing for a single outlier in a regression
model. Sankhya A, 23, 251-260.
[20] Stromberg, A. J., (1993), Computation of High Breakdown Nonlinear
Regression Parameters, Journal of American Statistical Association, 88
(421), 237-244.
[21] Seber, G., A. F. and Wild, C. J. (2003), Nonlinear Regression, John
Wiley and Sons.
[22] Yohai, V. J. (1987), High Breakdown point and high efficiency robust
estimates for regression, The Annals of Statistics, 15, 642-656.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:58360", author = "Hossein Riazoshams and Midi Habshah and Jr. and Mohamad Bakri Adam", title = "On the outlier Detection in Nonlinear Regression", abstract = "The detection of outliers is very essential because of
their responsibility for producing huge interpretative problem in
linear as well as in nonlinear regression analysis. Much work has
been accomplished on the identification of outlier in linear
regression, but not in nonlinear regression. In this article we propose
several outlier detection techniques for nonlinear regression. The
main idea is to use the linear approximation of a nonlinear model and
consider the gradient as the design matrix. Subsequently, the
detection techniques are formulated. Six detection measures are
developed that combined with three estimation techniques such as the
Least-Squares, M and MM-estimators. The study shows that among
the six measures, only the studentized residual and Cook Distance
which combined with the MM estimator, consistently capable of
identifying the correct outliers.", keywords = "Nonlinear Regression, outliers, Gradient, LeastSquare, M-estimate, MM-estimate.", volume = "3", number = "12", pages = "1095-7", }