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I am trying to create a process in which I can identify if a process is out of control.

My idea was to do something similar to 6 sigma, where when a point is outside of the mean by +-3 sigma, then its considered out of control.

However, my process is not normal which is throwing me off.

The process involves counting the number of times that a certain event happens on a day (60%+ of the time the count is 0).

So the process looks like an exponential distribution and I want to draw conclusions from the model However: 1) its not continuous, but discrete - I dont think this is an issue 2) the data is not from arrival times or something else but rather a count - I dont think this is an issue.

So, I can fit the data to an exponential model and obtain a lambda, but now what?

Questions: 1) In general, if the data fits a given distribution (assuming all assumptions are met) does it matter that the data is unconventional for that distribution in regards for using the model to draw conclusions? 2) How do I use this model to actually determine if new points are out of control? Is it even appropriate? Should I be looking elsewhere? Thanks.

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    $\begingroup$ It is not evident what you mean by an "exponential distribution," because such a distribution describes a continuous variable, which a count definitely is not. Moreover, since arrival times are not involved, no plausible mechanism to introduce exponential distributions appears to be present. Everything you describe suggests a Poisson model could be effective. $\endgroup$ – whuber♦ May 19 '16 at 19:58
  • $\begingroup$ Ok, so an exponential distribution has to be continuous by definition. Good to know. The reason I didnt go after poisson is because of the high number of 0 events. But I didnt realize poisson could capture this. So using poisson, how would I find the out of control points? $\endgroup$ – ggpatterson May 19 '16 at 20:01
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    $\begingroup$ Are lower amounts better than higher amounts? That is, typically in quality control we're implicitly measuring distance from the target, and higher distances than expected require attention. But if you're tracking, say, the number of sales closed, then a long drought requires attention in a way that a flood day does not. $\endgroup$ – Matthew Graves May 19 '16 at 20:20
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    $\begingroup$ High numbers are those out of control. $\endgroup$ – ggpatterson May 19 '16 at 20:23
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If you have a discrete exponential distribution you can look at the poisson distribution or geometric distribution for example. As you say you can estimate $\lambda $ from the data but also the variance (=$\sigma^2$). For the poisson distribution the variance is also simply estimated by $\lambda$. Then if a data point falls outside this mean $\pm 3 \sigma$ it is out of control.

As for your first question: I believe that the data you're describing could fit both a geometric and a poisson distribution but the terminology of events makes me inclined to believe you're really looking at a poission distribution with low $\lambda $. My intuition tells me you want the distribution -and model of your data- to fit the explanation you have but I can't say for sure that using a different, fitting distribution is or isn't allowed.

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  • $\begingroup$ Yeah, I think poisson is the way to go with a very small (less than 1 if thats allowed lambda). I saw geometric earlier, but my data violates the assumption "There are only two possible outcomes for each trial, often designated success or failure." $\endgroup$ – ggpatterson May 19 '16 at 20:16
  • $\begingroup$ The usual rules, such as setting a control limit to $3\sigma$, do not apply to these distributions. They were developed for Normal distributions. In particular, due to the preponderance of zeros this distribution must be very positively skewed, so we already know that a $+3\sigma$ limit will be too low and create too many invalid OOC conditions. $\endgroup$ – whuber♦ May 19 '16 at 21:26
  • $\begingroup$ ggpatterson, a Geometric distribution describes counts, not just binary results. There are many possible models for your data--Poisson, Geometric, Binomial, Negative Binomial, etc--and choosing one requires an analysis of periods of in control data. $\endgroup$ – whuber♦ May 19 '16 at 21:28
  • $\begingroup$ If you have a geometric distribution, you can use a G chart; if your data fits a Weibull distribution, you can use a T chart. blog.minitab.com/blog/michelle-paret/… $\endgroup$ – Tavrock Dec 13 '16 at 17:49
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Based upon your description of your data, it looks like a "rare events" control chart is what you are asking for. Because your data is discrete, the chart type would be a G control chart.

To construct this chart, you make a table of your time-stamped events (simply ignore the 60% of the time when nothing happens), and then calculate the time between events. Each of these times is then designated $g_1, g_2, \dots, g_i, \dots, g_k$.

The centerline of your chart is drawn as $\overline{g}$ where $\overline{g}=\frac{\sum g_i}{k}$ and the control limits are drawn at $UCL_g=\overline{g}+ 3 \sqrt{\overline{g}\left(\overline{g}+1\right)}$ and $LCL_g=\max \left\{ \overline{g}- 3 \sqrt{\overline{g}\left(\overline{g}+1\right)},0\right\}$. Your observations, $g_i$ are then plotted in relation to these control limits.

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