Profile Monitoring via Nonlinear Mixed Models

Abstract

Profile monitoring is a relatively new technique in quality control best used where the process data follows a profile (or curve) at each time period. Little work has been done on the monitoring on nonlinear profiles. Previous work has assumed that the measurements within a profile are uncorrelated. To relax this restriction we propose the use of nonlinear mixed models to monitor the nonlinear profiles in order to account for the correlation structure. We evaluate the effectiveness of fitting separate nonlinear regression models to each profile in Phase I control chart applications for data with uncorrelated errors and no random effects. For data with random effects, we compare the effectiveness of charts based on a separate nonlinear regression approach versus those based on a nonlinear mixed model approach. Our proposed approach uses the separate nonlinear regression model fits to obtain a nonlinear mixed model fit. The nonlinear mixed model approach results in charts with good abilities to detect changes in Phase I data and has a simple to calculate control limit.

On monitoring of multiple non-linear profiles

“…Equation (10) shows the operated function in this simulation study. Note that the coefficients in the equation (10), A=5, B=8, C=0.6, and D=0, are as same as the setup in Jensen and Birch's (2009) study. Further, to simulate the correlated multiple non-linear profiles, it is assumed that the parameters between profiles are generated from independently and identically multivariate normal distribution from one observation to another.…”

“…The simulation study conducts two correlated four-parameter logistic curves. The curve equation is adapted from Jensen and Birch (2009). They used this curve in their simulation study for generating ARL property and then applied this property to the real world case.…”

“…Kim et al (2003) showed their method of three univariate EWMA charts monitoring slope, intercept, and the variance of deviation between samples and regression line performs better than EWMA/R chart in terms of average run length (ARL). Studies in nonlinear profile analysis can be categorized into four typesapplying multiple and polynomial regression (Zou et al, 2007;Kazemzadeh et al 2008;Mahmoud 2008), applying nonlinear regression models (Ding et al, 2006;Williams et al, 2007;Shiau et al, 2009;Chang and Yadama 2010;Chen and Nembhard 2011;), use of mixed models (Jensen et al, 2008;Jensen and Birch, 2009;Qiu et al, 2010;Paynabar and Jin, 2011 ), and use of wavelets (Reis and Saraiva, 2006;Zhou et al, 2007;Chicken et al, 2009). For more detail of those methods to monitor the process stability can be found in Woodall (2007) and Noorossana et al (2011).…”

Scalable Statistical Monitoring of Fleet Data

“…Profile monitoring involves fitting parametric or nonparametric models to longitudinal or profile data and monitoring these models. Some applications are discussed in Wang and Tsung [2005], Shiau et al [2009], Jensen et al [2006], and Mosesova et al [2006]. In Wang and Tsung [2005], the authors propose an SPC method to monitor processes with large amounts of data.…”

Identifying the time of a step change in AR(1) auto-correlated simple linear profiles

“…Although in some applications the error terms in successive profiles are auto-correlated [see for example Jensen et al (2008); Kazemzadeh et al (2010); Noorossana et al (2008);Zhang et al (2014); Keramatpour et al (2014); Niaki et al (2014); Jensen and Birch (2009);Amiri et al (2010);and Khedmati and Niaki (2015)], to the best of authors' knowledge there has not been any research work on estimating the time of a step change in auto-correlated simple linear profiles in Phase II. Therefore, in this paper, a maximum likelihood estimator of a step change in the parameters of auto-correlated simple linear regression profiles is first proposed in which the auto-correlation structure between observations in each profile is assumed to follow a first-order auto-regressive, AR(1), model.…”