Sequential monitoring of linear profiles using mixed memory recursions and stochastic approximation

Many modern data sets contain sequences of short regression profiles collected over time, where departures from stability are small, persistent, and difficult to separate from routine variation. This paper proposes a mixed memory procedure for monitoring linear profiles, denoted, that jointly tracks the intercept, the slope, and the error variance in a model with a coded explanatory variable. The method combines exponential smoothing with simple one sided accumulation recursions, which improves sensitivity to gradual changes while keeping componentwise interpretation. The main contribution is a simulation-based design strategy that calibrates control limits to a target in control average run length using projected Robbins Monro updates. This calibration step is a stochastic approximation routine that works with Monte Carlo estimates and adapts the limits through stable, low-cost iterations, which makes the design practical when closed-form limits are unavailable. A unified scaling places all charting statistics on a common unit control limit, which simplifies comparison across components and across competing designs. Performance is studied using run length summaries and domain averaged efficiency criteria over ranges of location and scale shifts, with comparisons to standard profile charts based on Shewhart, exponentially weighted moving average, and cumulative sum ideas under matched false alarm protection. An interactive Shiny application (Application available at https://linearprofiles.shinyapps.io/MEC3/) implements the full workflow, including calibration, simulation experiments, and exportable reports for reproducible use. A data example with 225 profiles illustrates earlier signaling under small sustained departures while maintaining the intended false alarm control.