Nonparametric and semiparametric profile monitoring via residuals

Profile monitoring tracks the functional relationship between a response and explanatory variables over time, in a Phase I stage that builds a model from historical data and a Phase II stage that monitors the ongoing process [4].

Parametric fits can be biased when the model is misspecified, while nonparametric fits can have inflated variances [2, 3]. When profiles are modelled with mixed effects, the random effects describe the differences between profiles and are a natural target for monitoring [1, 2].

Residuals from a parametric linear mixed model are modelled nonparametrically, and a semiparametric method combines the parametric and nonparametric estimates as a convex combination; Hotelling’s T² statistics based on the fitted values and on the estimated random effects are computed for each technique [1].

For Phase II, nonparametric (via residuals) and semiparametric multivariate EWMA (MEWMA) control charts are introduced, taking into account correlation between and within profiles [2].

Model robust regression technique 2 (MRR2), a semiparametric approach, is combined with a multivariate CUSUM (MCUSUM) chart to monitor the slope of linear mixed models in Phase II through the random effects [3].

A step-by-step guide applies parametric, nonparametric and semiparametric methods to profile monitoring, with comparative analyses for practitioners [4].

Residual-based nonparametric and semiparametric charts for linear mixed-effects profiles in Phase I [1] and Phase II, using MEWMA [2] and MCUSUM [3] statistics, together with a practitioners’ guide to choosing among parametric, nonparametric and semiparametric methods [4].

In simulations, the proposed methods were the most effective for monitoring autocorrelated profile data compared with existing approaches [1].

The MEWMA charts were compared with the parametric approach by average run length and average time to signal, for different sample and shift sizes, using simulations and real datasets [2].

Across correlated and uncorrelated profiles, profile and sample sizes and levels of model misspecification, the semiparametric MCUSUM chart had the best performance in detecting shifts, and in a real-data application it had the highest sensitivity to out-of-control scenarios [3].

Processes whose quality is characterised by a profile, a functional relationship between dependent and independent variables, illustrated with real datasets [2, 3].