Statistics

## Process Behavior Charts and Chaos Theory

### What do predictable processes have in common with chaos?

Published: Monday, July 11, 2022 – 12:03

Walter Shewhart made a distinction between common causes and assignable causes based on the effects they have upon the process outcomes. While Shewhart’s distinction predated the arrival of chaos theory by 40 years, chaos theory provides a way to understand what Shewhart was talking about.

### Assignable causes

W. Edwards Deming described assignable causes as special causes. He then would say that the special causes are “not part of the system.” Thus, he considered a special cause as a temporary cause — fleeting and ephemeral. Shewhart saw things differently. He sometimes wrote of assignable causes as temporary, but at other times he had a different model in mind. For example, Shewhart defined “a state of statistical control” to exist when “the chance fluctuations in a phenomenon are produced by a constant system of a large number of chance causes in which no cause produces a predominating effect.” Here Shewhart clearly implies that when a chance cause begins to have a dominant effect it becomes an assignable cause.

Both of these conceptualizations are valid. Some assignable causes are indeed external to the system, but others are internal to the system. So while we might seek insurance against the effects of external assignable causes, we can actually improve our process when we identify an internal assignable cause and turn it into a control factor. The examples in Shewhart’s first book show this happening, and this phenomenon has been observed by many who have used Shewhart’s approach.

Shewhart set out to characterize the behavior of data from a process rather than to create a detailed mathematical model of the process itself. However, he did not ignore the mathematics involved. Shewhart visualized a process as the result of a system of differential equations. In his thinking, if we knew all of the differential equations, and if we could solve those equations, and if we knew the initial conditions, we could predict each new value exactly. (The sensitivity to initial conditions known as the butterfly effect would not be discovered until 1965.)

But since he did not know the equations, could not solve the equations, and did not know the initial conditions, Shewhart took an empirical approach to explain process behavior. He observed that a process that satisfied the “equation of control” would be predictable within limits. Shewhart’s empirical approach can be explained in various ways. Here we will discuss some aspects of the relationship between process behavior charts and chaos theory.

### Chaos theory

William L. Ditto and Louis M. Pecora, writing in the August 1993 issue of *Scientific American*, described chaos theory in the following terms: Chaos theory focuses on characterizations of complex phenomena. A system that combines several simple cause-and-effect relationships will often give rise to very complex dynamic behaviors. Consider, for example, the horizontal motion of the ball in the mechanism shown in Figure 1. The ball is attached to a spring that gets stiffer as it is stretched or compressed. As the board moves back and forth, the spring is pushed and pulled, which causes the ball to move.

The movement of the ball might be characterized by (1) the horizontal position of the ball and (2) its velocity at any point in time. If the force exerted by the board is weak, then the ball will move in a simple manner. Plotting the position over time might result in a time series like that shown in the left-hand portion of Figure 2, while plotting the velocity versus the position might result in the “state space” diagram shown on the right of the same figure.

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When the force from the board is weak, the ball moves in a simple trajectory that repeats with each cycle of the force from the board. Because the motion is periodic, the state-space path will retrace itself with each cycle. This simple orbit in the state space is said to be a “period-one attractor” because it repeats itself every 360 degrees.

As the driving force on the spring is increased beyond a certain point, the period-one attractor will become unstable and a two-period attractor will form. In this state, it will take two cycles before the motion of the ball repeats itself every 720 degrees. A period-two attractor is shown in Figure 3.

One theorem in chaos theory is that all attractors with period greater than three are chaotic. So, as the force on the spring increases past some threshold, the ball will begin to move chaotically.

When this happens no pattern will be apparent in the time-series in Figure 4, and the orbits in the state space will form a chaotic attractor where the orbits never retrace themselves exactly.

In this simple example we had one nonlinear cause system. In reality, most systems are subject to many different causes which all compete to create the variation. When a system can be described by a chaotic attractor, according to Ditto and Pecora, the chaotic system can be said to be “a collection of many orderly behaviors, none of which dominates under ordinary circumstances.”

The similarity between this description of chaotic behavior and Shewhart’s description of a “state of statistical control” is no coincidence. While Shewhart did not have the computational power to solve sets of differential equations, he discovered that we could characterize the overall behavior of the dynamic variables of any system as being either “predictable within limits” or not.

Since chaotic attractors are always bounded, a time-series plot of any one characteristic will also be bounded and will be “predictable within limits.” This is the basis for the similarity between the time-series in Figure 4 and the time-series for a process which displays a reasonable degree of predictability — both are characterized by bounded, quasi-random variation that is the result of many competing cause- and-effect relationships.

In a chaotic system, any attempt to control or manipulate the process by adjusting one of these competing causes will be frustrated by the fact that no one cause is dominant. If we control one of these common causes, the variation created by the other common causes will mask the effects of the adjustment. Therefore, when working with a chaotic system, it is futile to try to single out any one of the common causes as the path to process improvement.

On the other hand, the time-series in figures 2 and 3 show regular patterns. Such patterns arise when one cause-and-effect relationship dominates the others. Shewhart called these “assignable causes” because their dominant effects allow them to be identified in spite of the routine variation. Moreover, the dominant effects make it profitable to discover the nature of the assignable causes because such knowledge will provide you with the leverage to improve the process.

Thus, more than 90 years ago, Shewhart gave us a technique for detecting whether a system displays chaotic variation. Without getting lost in the theory, Shewhart examined the behavior of observable variables and came up with a major distinction which is useful as a guide for action:

### Case One: Predictable = Chaos

When a process displays chaotic variation, the time series will vary in a quasi-random manner within certain bounds. Such a process will therefore be predictable within these limits, and may be said to be operating predictably. Here it will be a waste of time to seek to improve the process by controlling any one of the many common causes of routine variation.

Since there will always be a large number of causes that we will never be able to control, a state of chaotic variation will represent the limit to which we may expect to go in removing variation from our process. Chaotic variation will be the same as predictable operation.

### Case Two: Unpredictable = Pre-chaos

However, as long as some causes have effects that dominate the effects of other causes, these dominant effects will break up the chaotic attractor and result in detectable changes or patterns in the time series. These changes in the time series will offer clues to the nature of the dominant causes. As these dominant causes are found and made part of the set of control factors, the process will move in the direction of minimum variance, a state that may be characterized as chaotic variation that is predictable within limits. Thus, when we have a pre-chaotic system, it will be profitable to look for the dominant, assignable causes of exceptional variation.

### Summary

Shewhart’s process behavior charts empirically characterize any time series as being either predictable or unpredictable. They allow us to characterize our process as operating at the limit known as chaotic variation, or as operating in a pre-chaotic state. Processes operated in a pre-chaotic state are not operating with minimum variance. Here, opportunities exist for process improvement and the process behavior chart helps us to identify these opportunities.

So do not be fooled by the apparent simplicity of process behavior charts. Their simplicity is the simplicity that arises out of a profound understanding of how the world works. With deep roots and a firm foundation, process behavior charts provide the needed insight with a minimum of effort.

Forty years before chaos theory was developed, Shewhart gave us a way to detect whether our process was displaying regular and predictable chaotic variation or was displaying irregular and unpredictable pre-chaotic variation. Since the latter state presents opportunities for improvement, this distinction is not academic. This fundamental dichotomy in process behavior is what makes the process behavior chart a technique for all seasons.

*This article is adapted from Dr. Wheeler’s *Advanced Topics in Statistical Process Control, Second Edition*, copyright 2004, SPC Press, and is used with permission. The example and figures are based on Ditto and Pecora’s article cited in the text, copyright 1993, *Scientific American*and are also used by permission.*