Monitoring quadratic behavior with non-constant standard deviation

The control chart (CC) was originally developed to monitor a quality characteristic aiming a target value. However, new processes have developed with many components, which have resulted in more complex systems, thus requiring a different handling due to the trended behavior commonly encountered. When the trend is linear, approaches such as the regression control chart and even adapted traditional approaches (such as Shewhart's CC or EWMA CC) have been employed. However, under a non-linear trend (such as in many econometric series) not much research has been performed. A non-linear trend provokes non-constant variability, which complicates the direct use of traditional parametric procedures. This paper proposes a theoretical model for the monitoring of complex systems represented by quadratic behavior with non-constant variability. The proposition is done considering Shewhart's foundations, time series analysis, and probability theory. Areas of current and future research are also provided.