Geometric Brownian Motion-Based Time Series Modeling Methodology for Statistical Autocorrelated Process Control: Logarithmic Return Model

Abstract

Fitting a time series model to the process data before applying a control chart to the residuals is essential to fulfill the basic assumptions of statistical process control (SPC). Autoregressive integrated moving average (ARIMA) model has been one of the well-established time series modeling approaches that is extensively used for this purpose and is widely recognized for its accuracy and efficiency. Nevertheless, the research community commented that its iterative stages are laborious and time-consuming. In addressing this gap, a novel time series modeling technique with its conceptual assumptions of attributes that was derived from the geometric Brownian motion (GBM) law was developed in this study. It was termed as the logarithmic return (LR) model. Then, the model was employed and tested on a real-world autocorrelated data, whereby the results were assessed and benchmarked with the ARIMA model. The findings for LR model reported a mean average percentage error that ranged between 1.5851% and 3.3793% (less than 10%), which were as accurate as the ARIMA model. The running time (in second of CPU time) taken by the LR model was at least 96.2% faster than the ARIMA model. Interestingly, the corresponding multivariate control chart constructed from the LR model also portrayed a similar general conclusion as that of its counterpart. The LR model was obviously parsimonious and easier to compute and took a shorter running time than the ARIMA model. Therefore, it possessed the potential as an alternative time series modeling methodology for the ARIMA model in the procedures of SPC.

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