Extreme value theory in software engineering

As its name suggests, extreme value theory deals with extreme deviations from the average, e.g., how often will rainfall be heavy enough to cause a river to overflow its banks.

The initial list of statistical topics to I thought ought to be covered in my evidence-based software engineering book included extreme value theory. At the time, and even today, there were/are no books covering “Statistics for software engineering”, so I had no prior work to guide my selection of topics. I was keen to cover all the important topics, had heard of it in several (non-software) contexts and jumped to the conclusion that it must be applicable to software engineering.

Years pass: the draft accumulate a wide variety of analysis techniques applied to software engineering data, but, no use of extreme value theory.

Something else does not happen: I don’t find any ‘Using extreme value theory to analyse data’ books. Yes, there are some really heavy-duty maths books available, but nothing of a practical persuasion.

The book’s Extreme value section becomes a subsection, then a subsubsection, and ended up inside a comment (I cannot bring myself to delete it).

It appears that extreme value theory is more talked about than used. I can understand why. Extreme events are newsworthy; rivers that don’t overflow their banks are not news.

Just over a month ago a discussion cropped up on the UK’s C++ standards’ panel mailing list: was email traffic down because of COVID-19? The panel’s convenor, Roger Orr, posted some data on monthly volumes. Oh, data :-)

Monthly data is a bit too granular for detailed analysis over relatively short periods. After some poking around Roger was able to send me the date&time of every post to the WG21‘s Core and Lib reflectors, since February 2016 (there have been various changes of hosts and configurations over the years, and date of posts since 2016 was straightforward to obtain).

During our email exchanges, Roger had mentioned that every now and again a huge discussion thread comes out of nowhere. Woah, sounds like WG21 could do with some extreme value theory. How often are huge discussion threads likely to occur, and how huge is a once in 10-years thread that they might have to deal with?

There are two techniques for analysing the distribution of extreme values present in a sample (both based around the generalized extreme value distribution):

  • Generalized Extreme Value (GEV) uses block maxima, e.g., maximum number of daily emails sent in each month,
  • Generalized Pareto (GP) uses peak over threshold: pick a threshold and extract day values for when more than this threshold number of emails was sent.

The plots below show the maximum number of monthly emails that are expected to occur (y-axis) within a given number of months (x-axis), for WG21’s Core and Lib email lists. The circles are actual occurrences, and dashed lines 95% confidence intervals; GEP was used for these fits (code+data):

Expected maximum for emails appearing on C++'s core and lib reflectors within a given period

The 10-year return value for Core is around a daily maximum of 70 +-30, and closer to 200 +-100 for Lib.

The model used is very simplistic, and fails to take into account the growth in members joining these lists and traffic lost when a new mailing list is created for a new committee subgroup.

If any readers have suggests for uses of extreme value theory in software engineering, please let me know.

Postlude. This discussion has reordered events. My original interest in the mailing list data was the desire to find some evidence for the hypothesis that the volume of email increased as the date of the next WG21 meeting approached. For both Core and Lib, the volume actually decreases slightly as the date of the next meeting approaches; see code for details. Also, the volume of email at the weekend is around 60% lower than during weekdays.

Statistical techniques not needed to analyze software engineering data

One of the methods I used to try to work out what statistical techniques were likely to be useful to software developers, was to try to apply techniques that were useful in other areas. Of course, applying techniques requires the appropriate data to apply them to.

Extreme value statistics are used to spot patterns in rare events, e.g., frequency of rivers over spilling their banks and causing extensive flooding. I have tried and failed to find any data where Extreme value theory might be applicable. There probably is some such data, somewhere.

The fact that I have spent a lot of time looking for data and failed to find particular kinds of data, suggests that occurrences are rare. If data needing a particular kind of analysis technique is rare, there is no point including a discussion of the technique in a book aimed at providing general coverage of material.

I have spent some time looking for data drawn from a zero-inflated Poisson distribution. Readers are unlikely to have ever heard of this and might well ask why I would be interested in such an obscure distribution. Well, zero-truncated Poisson distributions crop up regularly (the Poisson distribution applies to count data that starts at zero, when count data starts at one the zeroes are said to be truncated and the Poisson distribution has to be offset to adjust for this). There is a certain symmetry to zero-truncated/inflated (although the mathematics involved is completely different), plus there is probably a sunk cost effect (i.e., I have spent time learning about them, I am going to find the data).

I spotted a plot in a paper investigating record data structure usage in Racket, that looked like it might be well fitted by a zero-inflated Poisson distribution. Tobias Pape kindly sent me the data (number of record data structures having a given size), which I then failed miserably to fit to any kind of Poisson related distribution; see plot below; data points along red line through the plus symbols (code+data):

Number of Racket record data structures having a given size.

I can only imagine what the authors thought of my reason for wanting the data (I made data requests to a few other researchers for similar reasons; and again I failed to fit the desired distribution).

I had expected to make more use of time series analysis; but, it has just not been that applicable.

Machine learning is useful for publishing papers, but understanding what is going on is the subject of my book, not building black boxes to make predictions.

It is possible that researchers are not publishing work relating to data that requires statistical techniques I have not used, because they don’t know how to analyze the data or the data is too hard to collect. Inability to use the correct techniques to analyze data is rarely a reason for not publishing a paper. Data being too hard to collect is very believable, as-is the data rarely occurring in software engineering related work.

There are statistical tests I have intentionally ignored, the Mann–Whitney U test (aka, the Wilcoxon rank-sum test) and the t-test probably being the most well-known. These tests became obsolete once computers became generally available. If you are ever stuck on a desert island without a computer, these are the statistical tests you will have to use.