استفاده از برآوردكننده‌هاي استوار به‌منظور پايش پروفايل خطي ساده چندمتغيره

Monitoring Multivariate Simple Linear Profiles Using Robust Regression M-estimators

در برخي كاربردهاي كنترل فرايند آماري، كيفيت يك محصول يا فرايند را مي توان به‌صورت رابطه‌ي بين يك يا چند متغير وابسته‌/ مستقل بيان ‌كرد كه آن را «پروفايل» مي نامند. در اين نوشتار به بررسي پروفايل‌هاي خطي ساده‌ي چندمتغيره مي‌پردازيم كه در آن چندين مشخصه‌ي كيفي مرتبط به‌صورت مجموعه‌يي از توابع خطي برحسب يك متغير مستقل مدل‌بندي مي‌شود. در حالي كه حضور داده‌هاي دورافتاده ممكن است در برآورد پارامترهاي مدل با استفاده از برآوردكننده‌هاي كلاسيك اختلال ايجاد ‌كند. به‌منظور برآورد پارامترهاي پروفايل دو رويكرد استوار مبتني بر برآوردكننده‌ي M پيشنهاد مي‌شود كه در دو وضعيت تك‌مرحله‌يي و دو‌مرحله‌يي كاربرد داشته است. عملكرد برآوردكننده‌هاي پيشنهادي از دو نقطه‌نظر استواري و كارايي مورد بررسي قرار مي گيرد. نتايج نشان مي دهد در فاز 1 كنترل فرايند آماري، براي مقادير متوسط و بزرگ از آلودگي، عملكرد برآوردكننده‌هاي استوار بسيار بهتر از رويكرد كلاسيك است.

In some quality control applications, quality of a product or process can be characterized by a relationship between two or more variables that is typically referred to as profile. Research on profile monitoring commenced with studying simple linear profiles nearly fifteen years ago and then was pursued by investigating other types of profiles including multiple linear profiles, polynomial profiles, generalized linear profiles, and recently multivariate linear profiles. This paper has focused on the last case where several correlated quality characteristics can be modeled as a set of linear functions of one or more explanatory variables. Since, for the sake of simplicity, the model structure of our study consists of only one predictor, this type of profile has been referred to as multivariate simple linear profile. This paper deals with statistical process control in phase one where parameters of the model are to be estimated in order to develop a control chart for process monitoring. While outliers may hamper the expected performance of the classical regression estimators leading to erroneous interpretation of the process, this study resorts to robust regression estimators as the remedy of the estimation problem in the presence of outliers. Two robust approaches are proposed in our study to estimate parameters of multivariate simple linear profiles on the basis of robust regression M-estimates with bi-square weight function. In the first approach, the aggregation of estimates in the sample level is done by averaging while the second approach employs a second M-estimate to aggregate parameter estimates. Extensive simulation studies are conducted to investigate and compare the performance of the proposed estimators in terms of robustness and efficiency. The results show that in the absence of outliers or for small amount of contamination, the robust methods perform as well as the classical method, while for medium and large amounts of contamination the proposed estimators perform considerably better than the classical ones.