Distribution Identification – How to Determine What Distribution Data Follows

Question

Let us say that I have 1000 components and I have been collecting data on how many times these log a failure and each time they logged a failure, I am also keeping track of how long it took my team to fix the problem. In short, I have been recording the time to repair (in seconds) for each of these 1000 components. Data is given at the end of this question.

I took all these values and drew a Cullen and Frey graph in R using descdist from the fitdistrplus package. My hope was to understand if the time to repair follows a particular distribution. Here's the plot with boot=500 to get bootstrapped values:

I see that this plot is telling me that the observation falls into the beta distribution (or maybe not, in which case, what is it revealing?) Now, considering that I am a system architect and not a statistician, what is this plot revealing? (I am looking for a practical real-world intuition behind these results).

EDIT:

QQplot using the qqPlot function in package car. I first estimated the shape and scale parameters using the fitdistr function.

> fitdistr(Data$Duration, "weibull")
      shape          scale    
  3.783365e-01   5.273310e+03 
 (6.657644e-03) (3.396456e+02)

Then, I did this:

qqPlot(LB$Duration, distribution="weibull", shape=3.783365e-01, scale=5.273310e+03)

EDIT 2:

Updating with a lognormal QQplot.

Here's my data:

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196891L, 196634L, 2960L, 194319L, 194037L, 3008L, 3927L, 178762L, 
178567L, 403L, 178124L, 2590L, 177405L, 177179L, 301L, 328L, 
390685L, 390683L, 575L, 1049L, 819L, 367L, 289L, 277L, 390L, 
301L, 318L, 3806L, 3778L, 3699L, 3691L)

Answer 1

The thing is that real data doesn't necessarily follow any particular distribution you can name ... and indeed it would be surprising if it did.

So while I could name a dozen possibilities, the actual process generating these observations probably won't be anything that I could suggest either. As sample size increases, you will likely be able to reject any well-known distribution.

Parametric distributions are often a useful fiction, not a perfect description.

Let's at least look at the log-data, first in a normal qqplot and then as a kernel density estimate to see how it appears:

Note that in a Q-Q plot done this way around, the flattest sections of slope are where you tend to see peaks. This has a clear suggestion of a peak near 6 and another about 12.3. The kernel density estimate of the log shows the same thing:

In both cases, the indication is that the distribution of the log time is right skew, but it's not clearly unimodal. Clearly the main peak is somewhere around the 5 minute mark. It may be that there's a second small peak in the log-time density, that appears to be somewhere in the region of perhaps 60 hours. Perhaps there are two very qualitatively different "types" of repair, and your distribution is reflecting a mix of two types. Or just maybe once a repair hits a full day of work, it tends to just take a longer time (that is, rather than reflecting a peak at just over a week, it may reflect an anti-peak at just over a day - once you get longer than just under a day to repair, jobs tend to 'slow down').

Even the log of the log of the time is somewhat right skew. Let's look at a stronger transformation, where the second peak is quite clear - minus the inverse of the fourth root of time:

The marked lines are at 5 minutes (blue) and 60 hours (dashed green); as you see, there's a peak just below 5 minutes and another somewhere above 60 hours. Note that the upper "peak" is out at about the 95th percentile and won't necessarily be close to a peak in the untransformed distribution.

There's also a suggestion of another dip around 7.5 minutes with a broad peak between 10 and 20 minutes, which might suggest a very slight tendency to 'round up' in that region (not that there's necessarily anything untoward going on; even if there's no dip/peak in inherent job time there, it could even be something as simple as a function of human ability to focus in one unbroken period for more than a few minutes.)

It looks to me like a two-component (two peak) or maybe three component mixture of right-skew distributions would describe the process reasonably well but would not be a perfect description.

The package logspline seems to pick four peaks in log(time):

with peaks near 30, 270, 900 and 270K seconds (30s,4.5m,15m and 75h).

Using logspline with other transforms generally find 4 peaks but with slightly different centers (when translated to the original units); this is to be expected with transformations.