Adaptive Reweighted Minimum Vector Variance Estimator of Covariance Used for as a New Robust Approach to Partial Least Squares Regression

Abstract

Partial Least Squares Regression (PLSR), which is developed as partial type of the least squares estimator of regression in case of multicollinearity existence among independent variables, is a linear regression method. If there are outliers in data set, robust methods can be applied for diminishing or getting rid of the negative impacts of them. Past studies have shown that if the covariance matrix is appropriately robustified, PLS1 algorithm (PLSR for one dependent variable) becomes robust against outliers. In this study, an adaptive reweighted estimator of covariance based on Minimum Vector Variance as the first estimator is used and a new robust PLSR method: “PLS-ARWMVV“ is introduced. PLS-ARWMVV is compared with ordinary PLSR and four popular robust PLSR methods. The simulation and real data application are revealed that if there are contaminated observations, proposed robust PLS-ARWMVV is robust and efficient. It generally performs better than robust PRM and good alternative for other robust PLS-KurSD, RSIMPLS and PLS-SD methods.

Keywords

• Minimum Vector Variance (MVV)
• Partial least squares regression
• Robust covariance

References

1. Hubert, M., Vanden Branden, K., ”Robust methods for Partial Least Squares Regression”, Journal of Chemometrics, 17: 537-549, (2003).
2. Liebmann, B., Filzmoser, P., Varmuza, K., “Robust and classical PLS Regression compared”, Journal of Chemometrics, 24(3-4): 111-120, (2010).
3. González, J., Peña, D., Romera, R, “A robust partial least squares regression method with applications”, Journal of Chemometrics, 23, 78–90, (2009).
4. Wakeling, I.N., Macfie, H.J.H., “ A Robust PLS Procedure”, Journal of Chemometrics, 6: 189–198, (1992).
5. Griep, M.I., Wakeling, I.N., Vankeerberghen, P., Massart, D.L., “Comparison of semirobust and robust partial least squares procedures”, Chemometrics and Intelligent Laboratory Systems, 29: 37-50, (1995).
6. Gil, J.A., Romera, R., “On robust partial least squares (PLS) methods”, Journal of Chemometrics, 12: 365-378, (1998).
7. Engelen, S., Hubert, M., Vanden Branden, K., Verboven, S., “Robust PCR and Robust PLSR: a comparative study”, In: Hubert, M., Pison, G., Struyf, A., Aelst, S.V. (Eds.), Theory and Applications of Recent Robust Methods, Birkhäuser, Basel, 105–117, (2004).
8. Serneels, S., Croux, C., Filzmoser, P., Van Espen, P.J., “Partial Robust M-regression”, Chemometrics and Intelligent Laboratory Systems, 79: 55-64, (2005).

9. Herwindiati, D.E., “A new criterion in robust estimation for location and covariance matrix, and its application for outlier labeling”. Phd. Thesis, Institut Teknologi Bandung, Bandung, (2006).
10. Djauhari, M.A., Mashuri, M., Herwindiati, D.E., “Multivariate process variability monitoring”, Communications in Statistics - Theory and Methods, 37(11): 1742-1754, (2008).
11. Herwindiati, D.E., Djauhari, M.A., Mashuri, M., “Robust multivariate outlier labeling”, Communication in Statistics–Computation and Simulation, 36: 1287-1294, (2007).
12. Ali, H., Syed-Yahaya, S.S., “On robust mahalonobis distance issued from minimum vector variance”, Far East Journal of Mathematical Sciences (FJMS), 74(2): 249-268, (2013).
13. Yahaya, S.S., Ali, H., Omar, Z., “An alternative Hotelling control chart based on Minimum Vector Variance (MVV)”, Modern Applied Science, 5(4): 132-151, (2011).
14. Rousseeuw, P.J., Van Zomeren, B.C., “Unmasking multivariate outliers and leverage points”, Journal of the American Statistical Association, 85: 633–639, (1990).
15. Gervini, D., “A robust and efficient adaptive reweighted estimator of multivariate location and scatter”, Journal of Multivariate Analysis, 84: 116–144, (2003).
16. Gervini, D., Yohai, V.J., “A class of robust and fully efficient regression estimators”, The Annals of Statistics, 30 (2): 583–616, (2002).
17. Rousseeuw, P.J., Van Driessen, K., “A fast algorithm for the minimum covariance determinant estimator”, Technometrics, 41: 212–224, (1999).
18. Naes, T., “Multivariate calibration when the error covariance matrix is structured”, Technometrics, 27(3): 301-311, (1985).
19. Hardy, A.J., MacLaurin, P., Haswell, S.J., De Jong, S., Vandeginste, B.G.M., “Double-case diagnostic for outliers identification”, Chemometrics and Intelligent Laboratory Systems, 34: 117-129, (1996).
20. Polat, E., Gunay, S., “A New Robust Partial Least Squares Regression Method Based on Multivariate MM-Estimators”, International Journal of Mathematics and Statistics, 18(3): 82-99, (2017).