## Process Management in a Non-Stationary World

Since Walter Shewhart first codified ideas of statistical process control in the 1920’s, theory and practice in monitoring and managing industrial and commercial processes has generated good advice:

- Always plot process data in time order, “you can see a lot just by looking.”
- Apply control chart rules to distinguish signals of special (assignable) causes from patterns of common cause variation.
- Reduce variation by removing special causes and understanding common causes.
- Don’t adjust a process that is in statistical control, else you will increase variability.

To illustrate this last point, W.E. Deming described a simulated system with a funnel and marble, dropped to hit a target. (*Out of the Crisis*, MIT Press, pp. 327-330, “The funnel experiment.”)

The simulation presumes a process in statistical control. In the funnel simulation, with no adjustments to the position of the funnel, the deviations from target plotted in time order will show a state of statistical control. If you work through the details, you see that the variability increases if you try to adjust the funnel position.

But what if the funnel system had an inherent drift from target? What are appropriate actions?

Processes that drift from target, in the absence of adjustment, appear to be the rule rather than the exception in our world. That is, the funnel experiment is a useful special case but does not provide general guidance.

Box and Luceño in their 1996 book *Statistical Control by Monitoring and Adjustment* (New York: Wiley; second edition published 2009) outline discrete monitoring and control methods that help us manage processes when we face these conditions:

- Human engineered processes, left to their own devices, will drift away from target.
- Input factors that drive the drift of human processes are themselves drifting, not in states of statistical control.
- We often can’t technically or economically remove the drifts.

In particular, they make the case for the centrality of a particular non-stationary process, the integrated moving average process—called IMA(1,1). The process is defined by a first order difference equation with a single parameter θ: z_{t+1} – z_{t} = a_{t+1} – θa_{t}, where {a_{t}} is a white noise series. (By white noise, we mean that the a_{t }values are a realization of a series of random variables with zero mean, finite variance and zero serial correlation. Often, researchers use the more stringent conditions that the a_{t} are realizations from a series of independent and identically distributed normal (Gaussian) random variables.)

The IMA(1,1) process includes white noise process as one special case, θ = 1, and the random walk process, θ = 0, as another special case.

Starting with the IMA(1,1) process, Box and Luceño demonstrate that an exponentially weighted moving average (EWMA) yields optimal one-step ahead forecasts. In situations in which an adjustment factor exists, they then show how to use the EWMA to drive control actions. As a bonus, they link the discrete methods in their book to the “proportional-integral-derivative” continuous control methods used by control engineers for many years.

A quick simulation in R increased my belief that the proposed adjustment procedure is well-supported and merits a try in client projects. I generated 1000 values of an IMA(1,1) process with θ = 0.5 and then used the last 100 values for the test. Starting at the 921^{st} observation, I applied the adjustment procedure proposed by Box and Luceño. Here’s a graphical summary for adjustments using two values of the adjustment scheme parameter G, 0.2 and 0.5.

In the display, you see the series that would have occurred without adjustment (in red) and the adjusted series (in black). The big move between the first and second observations of the adjusted series arises from a start-up effect as I set the first value of the adjusted series to be equal to the value of the undadjusted series at index 921. The performance of the adjustment method is relatively insensitive to the choice of parameter G, as shown in the example. Given the structure of the simulatiion, the value 0.5 is the best choice; 0.2 is a parameter suggested by the authors as a reasonable value over a range of practical situations.

Tying back to the tools of statistical process control, if the adjustment method is appropriate for the data series, the forecast errors should demonstrate a state of statistical control, which can be checked by creating a control chart of the error series.

Here’s a link to more notes on this topic, part of a presentation I gave at the Southwest Quality Improvement Network annual meeting, October 2013.