Testing rounded data for a circular uniform distribution

Derek Jones from The Shape of Code

Circular statistics deals with analysis of measurements made using a circular scale, e.g., minutes past the hour, days of the week. Wikipedia uses the term directional statistics, the traditional use being measurements of angles, e.g., wind direction.

Package support for circular statistics is rather thin on the ground. R’s circular package is one of the best, and the book “Circular Statistics in R” provides the only best introduction to the subject.

Circular statistics has a few surprises for those new to the subject (apart from a few name changes, e.g., the von Mises distribution is effectively the ‘circular Normal distribution’), including:

  • the mean value contains two components, a direction and a length, e.g., mean wind direction and strength,
  • there are several definitions of variance, with angular variance having a value between 0 and 2, and circular variance having a value between 0 and 1. The circular standard deviation is not the square root of variance, but rather: sqrt{-2 log R}, where R is the mean length.

The basic techniques used in circular statistics are still relatively new, compared to the more well known basic statistical techniques. For instance, it was recently discovered that having more measurements may reduce the reliability of the Rao spacing test (used to test whether a sample has a uniform circular distribution); generally, having more measurements improves the reliability of a statistical test.

The plot below shows Rose diagrams for the number of commits in each 3-hour period of a day for Linux and FreeBSD (mean direction and length in green; code+data):

Project actual/estimate ratio against percent complete.

The Linux kernel source has far fewer commits at the weekend, compared to working days. Given the number of people whose job is to work on the Linux kernel, compared to the number of people doing it out of interest, this difference is not surprising. The percentage of people working on OpenBSD as a job is small, and there does not appear to be a big difference between weekends and workdays. There is a lot of variation in the number of commits during each 3-hour period of a day, but the number of commits per day does not vary so much; the number of OpenBSD commits per day of week is:

            Mon   Tue   Wed   Thu   Fri   Sat   Sun 
          26909 26144 25705 25104 24765 22812 24304 

Does this distribution of commits per day have a uniform distribution (to some confidence level)?

Like all measurements, those made on a circular scale are rounded to some number of digits. Measurements may also be rounded, or binned, to particular units of the scale, e.g., measured to the nearest degree, or nearest minute.

A recent paper, by Landler, Ruxton and Malkemper, found that for samples containing around five hundred or more measurements, rounding to the nearest degree was sufficient to cause the Rao spacing test to almost always report non-uniformity, i.e., for non-trivial samples the rounding was sufficient to cause the test to detect non-uniformity (things worked as expected for samples containing fewer than 100 measurements).

Landler et al found that adding a small amount of noise (drawn from a von Mises distribution) to the rounded measurements appeared to ‘fix’ the incorrect behavior, i.e., rejecting the hypothesis of a uniform distribution, when a uniform distribution may be present.

The rao.spacing.test function, in the circular package, rejected that null hypothesis that the OpenBSD daily data has a uniform distribution. However, when noise is added to each day value (i.e., adding a random fraction to the day values, using rvonmises(length(c_per_day), circular(0), 2.0)), the call to rao.spacing.test failed to reject the null hypothesis of uniformity at the 0.05 level (i.e., the daily distribution is probably uniform).

How many research results are affected by this discovery?

I very rarely encounter the use of circular statistics (even though they should probably have been used in places), but then I spend my time reading software engineering papers, whose use of statistics tends to be primitive. I plan to include a brief mention of the use of the Rao spacing test with binned data in the addendum to my Evidence-based software engineering book (which includes the above example).

Testing rounded data for a circular uniform distribution

Derek Jones from The Shape of Code

Circular statistics deals with analysis of measurements made using a circular scale, e.g., minutes past the hour, days of the week. Wikipedia uses the term directional statistics, the traditional use being measurements of angles, e.g., wind direction.

Package support for circular statistics is rather thin on the ground. R’s circular package is one of the best, and the book “Circular Statistics in R” provides the only best introduction to the subject.

Circular statistics has a few surprises for those new to the subject (apart from a few name changes, e.g., the von Mises distribution is effectively the ‘circular Normal distribution’), including:

  • the mean value contains two components, a direction and a length, e.g., mean wind direction and strength,
  • there are several definitions of variance, with angular variance having a value between 0 and 2, and circular variance having a value between 0 and 1. The circular standard deviation is not the square root of variance, but rather: sqrt{-2 log R}, where R is the mean length.

The basic techniques used in circular statistics are still relatively new, compared to the more well known basic statistical techniques. For instance, it was recently discovered that having more measurements may reduce the reliability of the Rao spacing test (used to test whether a sample has a uniform circular distribution); generally, having more measurements improves the reliability of a statistical test.

The plot below shows Rose diagrams for the number of commits in each 3-hour period of a day for Linux and FreeBSD (mean direction and length in green; code+data):

Project actual/estimate ratio against percent complete.

The Linux kernel source has far fewer commits at the weekend, compared to working days. Given the number of people whose job is to work on the Linux kernel, compared to the number of people doing it out of interest, this difference is not surprising. The percentage of people working on OpenBSD as a job is small, and there does not appear to be a big difference between weekends and workdays. There is a lot of variation in the number of commits during each 3-hour period of a day, but the number of commits per day does not vary so much; the number of OpenBSD commits per day of week is:

            Mon   Tue   Wed   Thu   Fri   Sat   Sun 
          26909 26144 25705 25104 24765 22812 24304 

Does this distribution of commits per day have a uniform distribution (to some confidence level)?

Like all measurements, those made on a circular scale are rounded to some number of digits. Measurements may also be rounded, or binned, to particular units of the scale, e.g., measured to the nearest degree, or nearest minute.

A recent paper, by Landler, Ruxton and Malkemper, found that for samples containing around five hundred or more measurements, rounding to the nearest degree was sufficient to cause the Rao spacing test to almost always report non-uniformity, i.e., for non-trivial samples the rounding was sufficient to cause the test to detect non-uniformity (things worked as expected for samples containing fewer than 100 measurements).

Landler et al found that adding a small amount of noise (drawn from a von Mises distribution) to the rounded measurements appeared to ‘fix’ the incorrect behavior, i.e., rejecting the hypothesis of a uniform distribution, when a uniform distribution may be present.

The rao.spacing.test function, in the circular package, rejected that null hypothesis that the OpenBSD daily data has a uniform distribution. However, when noise is added to each day value (i.e., adding a random fraction to the day values, using rvonmises(length(c_per_day), circular(0), 2.0), although runif(length(c_per_day)) is probably more appropriate {and produces essentially the same result}), the call to rao.spacing.test failed to reject the null hypothesis of uniformity at the 0.05 level (i.e., the daily distribution is probably uniform).

How many research results are affected by this discovery?

I very rarely encounter the use of circular statistics (even though they should probably have been used in places), but then I spend my time reading software engineering papers, whose use of statistics tends to be primitive. I plan to include a brief mention of the use of the Rao spacing test with binned data in the addendum to my Evidence-based software engineering book (which includes the above example).

Where are the industrial strength R compilers?

Derek Jones from The Shape of Code

Why don’t compiler projects for the R language make it into production use? The few that have been written have remained individual experimental products, e.g., RLLVMCompile.

Most popular languages attract many compiler implementations. I’m not saying that any of these implementations have more than a handful of users, that they implement the full language (a full implementation is not common), or that they fulfil any need other than their implementers desire to build something.

A commonly heard reason for the lack of production R compilers is that it is not worth the time and effort, because most of an R program’s time is spent in the library code which is written in a compiled language (e.g., C or Fortran). The fact that it is probably not worth the time and effort has not stopped people writing compilers for other languages, but then I think that the kind of people who use R tend not to be the kind of people who want to spend their time writing compilers. On the whole, they are the kind of people who are into statistics and data analysis.

Is it true that that most R programs spend most of their time executing library code? It’s certainly true for me. But I have noticed that a lot of the library functions executed by my code are written in R. Also, if somebody uses R for all their programming needs (it might be the only language they know), then their code might not be heavily library dependent.

I was surprised to read about Tierney’s byte code compiler, because his implementation is how I thought the R-core’s existing implementation worked (it does now). The internals of R is based on 1980s textbook functional techniques, and like many book implementations of the day, performance is dependent on the escape hatch of compiled code. R’s implementers wisely spent their time addressing user concerns, which revolved around statistics and visual presentation, i.e., not internal implementation technicalities.

Building an R compiler is easy, the much harder and time-consuming part is the runtime system.

Threaded code is a quick and simple approach to compiler implementation. R source gets mapped to a sequence of C function calls, with these functions proving a wrapper to library functions implementing the appropriate basic functionality, e.g., add two vectors. This approach has been the subject of at least one Master’s thesis. Thesis implementations rarely reach production use because those involved significantly underestimate the work that remains to be done, which is usually a lot more than the original implementation.

A simple threaded code approach provides a base for subsequent optimization, with the base having a similar performance to an interpreter. Optimizing requires figuring out details of the operations performed and replacing generic function calls with ones designed to be fast for specific cases, or even better replacing calls with inline code, e.g., adding short vectors of integers. There is a lot of existing work for scripting languages and a few PhD thesis researching R (e.g., Wang). The key technique is static analysis of R source.

Jan Vitek is running what appears to be the most active R compiler research group, at the moment e.g., the Ř project. Research can be good for uncovering language usage and trying out different techniques, but it is not intended to produce industry strength code. Lots of the fancy optimizations in early versions of the gcc C compiler started life as a PhD thesis, with the respective individual sometimes going on to spend a few years creating a production quality version for the released compiler.

The essential ingredient for building a production compiler is persistence. There are an awful lot of details that need to be sorted out (this is why research project code does not directly translate to production code, they ignore ‘minor’ details in order to concentrate on the ‘interesting’ research problem). Is there a small group of people currently beavering away on a production quality compiler for R? If there is, I can understand being discrete, on long-term projects it can be very annoying to have people regularly asking when the software is going to be released.

To have a life, once released, a production compiler needs to attract users, who are often loyal to their current compiler (because they know that their code works for this compiler); there needs to be a substantial benefit to entice people to switch. The benefit of compiling R to machine code, rather than interpreting, is performance. What performance improvement is needed to attract a viable community of users (there is always a tiny subset of users who will pay lots for even small performance improvements)?

My R code is rarely cpu bound, so I am not in the target audience, no matter what the speed-up. I don’t have any insight in the performance problems experienced by the R community, and have no idea whether a factor of two, five, ten or more would be enough.

Evidence-based software engineering: book released

Derek Jones from The Shape of Code

My book, Evidence-based software engineering, is now available; the pdf can be downloaded here, here and here, plus all the code+data. Report any issues here. I’m investigating the possibility of a printed version.

The original goals of the book, from 10-years ago, have been met, i.e., discuss what is currently known about software engineering based on an analysis of all the publicly available software engineering data, and having the pdf+data+code freely available for download. The definition of “all the public data” started out as being “all”, but as larger and higher quality data was discovered the corresponding were ignored.

The intended audience has always been software developers and their managers. Some experience of building software systems is assumed.

How much data is there? The data directory contains 1,142 csv files and 985 R files, the book cites 895 papers that have data available of which 556 are cited in figure captions; there are 628 figures. I am currently quoting the figure of 600+ for the ‘amount of data’.


Cover image of book Evidence-based software engineering.

Things that might be learned from the analysis has been discussed in previous posts on the chapters: Human cognition, Cognitive capitalism, Ecosystems, Projects and Reliability.

The analysis of the available data is like a join-the-dots puzzle, except that the 600+ dots are not numbered, some of them are actually specs of dust, and many dots are likely to be missing. The future of software engineering research is joining the dots to build an understanding of the processes involved in building and maintaining software systems; work is also needed to replicate some of the dots to confirm that they are not specs of dust, and to discover missing dots.

Some missing dots are very important. For instance, there is almost no data on software use, but there can be lots of data on fault experiences. Without software usage data it is not possible to estimate whether the software is very reliable (i.e., few faults experienced per amount of use), or very unreliable (i.e., many faults experienced per amount of use).

The book treats the creation of software systems as an economically motivated cognitive activity occurring within one or more ecosystems. Algorithms are now commodities and are not discussed. The labour of the cognitariate is the means of production of software systems, and this is the focus of the discussion.

Existing books treat the creation of software as a craft activity, with developers applying the skills and know-how acquired through personal practical experience. The craft approach has survived because building software systems has been a sellers market, customers have paid what it takes because the potential benefits have been so much greater than the costs.

Is software development shifting from being a sellers market to a buyers market? In a competitive market for development work and staff, paying people to learn from mistakes that have already been made by many others is an unaffordable luxury; an engineering approach, derived from evidence, is a lot more cost-effective than craft development.

As always, if you know of any interesting software engineering data, please let me know.

beta: Evidence-based Software Engineering – book

Derek Jones from The Shape of Code

My book, Evidence-based software engineering: based on the publicly available data is now out on beta release (pdf, and code+data). The plan is for a three-month review, with the final version available in the shops in time for Christmas (I plan to get a few hundred printed, and made available on Amazon).

The next few months will be spent responding to reader comments, and adding material from the remaining 20 odd datasets I have waiting to be analysed.

You can either email me with any comments, or add an issue to the book’s Github page.

While the content is very different from my original thoughts, 10-years ago, the original aim of discussing all the publicly available software engineering data has been carried through (in some cases more detailed data, in greater quantity, has supplanted earlier less detailed/smaller datasets).

The aim of only discussing a topic if public data is available, has been slightly bent in places (because I thought data would turn up, and it didn’t, or I wanted to connect two datasets, or I have not yet deleted what has been written).

The outcome of these two aims is that the flow of discussion is very disjoint, even disconnected. Another reason might be that I have not yet figured out how to connect the material in a sensible way. I’m the first person to go through this exercise, so I have no idea where it’s going.

The roughly 620+ datasets is three to four times larger than I thought was publicly available. More data is good news, but required more time to analyse and discuss.

Depending on the quantity of issues raised, updates of the beta release will happen.

As always, if you know of any interesting software engineering data, please tell me.

Beta release of data analysis chapters: Evidence-based software engineering

Derek Jones from The Shape of Code

When I started my evidence-based software engineering book, nobody had written a data analysis book for software developers, so I had to write one (in fact, a book on this topic has still to be written). When I say “I had to write one”, what I mean is that the 200 pages in the second half of my evidence-based software engineering book contains a concentrated form of such a book.

This 200 pages is now on beta release (it’s 186 pages, if the bibliography is excluded); chapters 8 to 15 of the draft pdf. Originally I was going to wait until all the material was ready, before making a beta release; the Coronavirus changed my plans.

Here is your chance to learn a new skill during the lockdown (yes, these are starting to end; my schedule has not changed, I’m just moving with the times).

All the code+data is available for you to try out any ideas you might have.

The software engineering material, the first half of the book, is also part of the current draft pdf, and the polished form should be available on beta release in about 6 weeks.

If you have a comment or find a problem, either email me or raise an issue on the book’s Github page.

Yes, a few figures and tables still bump into each other. I’m loath to do very fine-tuning because things will shuffle around a bit with minor changes to the words.

I’m thinking of running some online sessions around each chapter. Watch this space for information.

Comments on the COVID-19 model source code from Imperial

Derek Jones from The Shape of Code

At the end of March a paper modelling the impact of various scenarios on the spread of COVID-19 infections, by the MRC Centre for Global Infectious Disease Analysis at Imperial College, appears to have influenced the policy of the powers that be. This group recently started publishing their modelling code on Github (good for them).

Most of my professional life has been spent analyzing other peoples’ code, for one reason or another (mostly Fortran, then Pascal, and then C). I had heard that the Imperial software was written in C, but the released code is written in R (as of six hours ago there is the start of a Python version). Ok, I can work with R, but my comments will be general, since I don’t have lots of in depth experience reading R code.

The code comes from a research context, and is evolving, i.e., some amount of messiness is to be expected.

There is not a lot of code to talk about (248 lines setting things up, 111 lines for a Stan model, 371 lines of plotting code, and 85 lines of utility code). The analysis is performed by creating a model using the Stan statistical inference language (in which the high level structure of the problem is specified, compiled to a lower level form and then run; the Stan language is very similar to R). These days lots of problems are coded using a relatively small number of lines that call fancy libraries to do the heavy lifting. It is becoming rare to have to write tens of thousands of lines of code to solve a problem.

I have two points to make about the code, all designed to reduce the likelihood of mistakes being made by the person working on the source. These points mainly apply to the Stan code, because that is where the important stuff happens, but are equally applicable to all code.

  • Numeric literals are embedded in the code, values include: 2.4, 1.0, 0.5, 0.03, 1e-5, and 1e-9. These values obviously mean something to the person who wrote the code, and they can probably be interpreted by experts in the spread of virus infections. But why are they scattered about the code, rather than appearing together (as a sequence of assignments to variables with meaningful names)? Having all the constants in one place makes it easier to spot when a mistake has been made, e.g., one value has been changed without a corresponding change in another value; it also makes it easier for people new to the code to figure out what is going on,
  • when commenting out code, make it very obvious, e.g., have /********************** on its own line, and *****************************/ on its own line. Using just /* and */ makes it easy to miss that code has been commented out.

Why have they started a Python implementation? Perhaps somebody on the team is more comfortable working with Python (when deadlines loom, it is always best to go with what you know).

Having both an R and Python version is good, in that coding mistakes are likely to show up as inconsistencies in the results produced. It’s always good to have the output of two independently written programs to compare (apart from the fact it may cost twice as much).

The README mentions performance issues. I imagine that most of the execution time is spent in the Stan code, so R vs. Python is not a performance issue.

Any reader with expertise tuning Stan models for performance might like to check out the code. I’m sure the Imperial folk would be happy to hear about worthwhile speed-ups.

Source code chapter of ‘evidence-based software engineering’ reworked

Derek Jones from The Shape of Code

The Source code chapter of my evidence-based software engineering book has been reworked (draft pdf).

When writing the first version of this chapter, I was not certain whether source code was a topic warranting a chapter to itself, in an evidence-based software engineering book. Now I am certain. Source code is the primary product delivery, for a software system, and it is takes up much of the available cognitive effort.

What are the desirable characteristics that source code should have, to minimise production costs per unit of functionality? This is what an evidence-based chapter on source code is all about.

The release of this chapter completes my second pass over the material. Readers will notice the text still contains ... and ?‘s. The third pass will either delete these, or say something interesting (I suspect mostly the former, because of lack of data).

Talking of data, February has been a bumper month for data (apologies if you responded to my email asking for data, and it has not appeared in this release; a higher than average number of people have replied with data).

The plan is to spend a few months getting a beta release ready. Have the beta release run over the summer, with the book in the shops for Christmas.

I’m looking at getting a few hundred printed, for those wanting paper.

The only publisher that did not mind me making the pdf freely available was MIT Press. Unfortunately one of the reviewers was foaming at the mouth about the things I had to say about software engineering researcher (it did not help that I had written a blog post containing a less than glowing commentary on academic researchers, the week of the review {mid-2017}); the second reviewer was mildly against, and the third recommended it.

If any readers knows the editors at MIT Press, do suggest they have another look at the book. I would rather a real publisher make paper available.

Next, getting the ‘statistics for software engineers’ second half of the book ready for a beta release.

Reliability chapter of ‘evidence-based software engineering’ updated

Derek Jones from The Shape of Code

The Reliability chapter of my evidence-based software engineering book has been updated (draft pdf).

Unlike the earlier chapters, there were no major changes to the initial version from over 18-months ago; we just don’t know much about software reliability, and there is not much public data.

There are lots of papers published claiming to be about software reliability, but they are mostly smoke-and-mirror shows derived from work down one of several popular rabbit holes:

The growth in research on Fuzzing is the only good news (especially with the availability of practical introductory material).

There is one source of fault experience data that looks like it might be very useful, but it’s hard to get hold of; NASA has kept detailed about what happened using space missions. I have had several people promise to send me data, but none has arrived yet :-(.

Updating the reliability chapter did not take too much time, so I updated earlier chapters with data that has arrived since they were last released.

As always, if you know of any interesting software engineering data, please tell me.

Next, the Source code chapter.

The Renzo Pomodoro dataset

Derek Jones from The Shape of Code

Estimating how long it will take to complete a task is hard work, and the most common motivation for this work comes from external factors, e.g., the boss, or a potential client asks for an estimate to do a job.

People also make estimates for their own use, e.g., when planning work for the day. Various processes and techniques have been created to help structure the estimation process; for developers there is the Personal Software Process, and specifically for time estimation (but not developer specific), there is the Pomodoro Technique.

I met Renzo Borgatti at the first talk I gave on the SiP dataset (Renzo is the organizer of the Papers We Love meetup). After the talk, Renzo told me about his use of the Pomodoro Technique, and how he had 10-years worth of task estimates; wow, I was very interested. What happened next, and a work-in-progress analysis (plus data and R scripts) of the data can be found in the Renzo Pomodoro dataset repo.

The analysis progressed in fits and starts; like me Renzo is working on a book, and is very busy. The work-in-progress pdf is reasonably consistent.

I had never seen a dataset of estimates made for personal use, and had not read about the analysis of such data. When estimates are made for consumption by others, the motives involved in making the estimate can have a big impact on the values chosen, e.g., underestimating to win a bid, or overestimating to impress the boss by completing a task under budget. Is a personal estimate motive free? The following plot led me to ask Renzo if he was superstitious (in not liking odd numbers).

Number of tasks having a given number of estimate and actual Pomodoro values.

The plot shows the number of tasks for which there are a given number of estimates and actuals (measured in Pomodoros, i.e., units of 25 minutes). Most tasks are estimated to require one Pomodoro, and actually require this amount of effort.

Renzo educated me about the details of the Pomodoro technique, e.g., there is a 15-30 minute break after every four Pomodoros. Did this mean that estimates of three Pomodoros were less common because the need for a break was causing Renzo to subconsciously select an estimate of two or four Pomodoro? I am not brave enough to venture an opinion about what is going on in Renzo’s head.

Each estimated task has an associated tag name (sometimes two), which classifies the work involved, e.g., @planning. In the task information these tags have the form @word; I refer to them as at-words. The following plot is very interesting; it shows the date of use of each at-word, over time (ordered by first use of the at-word).

at-words usage, by date.

The first and third black lines are fitted regression models of the form 1-e^{-K*days}, where: K is a constant and days is the number of days since the start of the interval fitted. The second (middle) black line is a fitted straight line.

The slow down in the growth of new at-words suggests (at least to me) a period of time working in the same application domain (which involves a fixed number of distinct activities, that are ‘discovered’ by Renzo over time). More discussion with Renzo is needed to see if we can tie this down to what he was working on at the time.

I have looked for various other patterns and associations, involving at-words, but have not found any (but I did learn some new sequence analysis techniques, and associated R packages).

The data is now out there. What patterns and associations can you find?

Renzo tells me that there is a community of people using the Pomodoro technique. I’m hoping that others users of this technique, involved in software development, have recorded their tasks over a long period (I don’t think I could keep it up for longer than a week).

Perhaps there are PSP followers out there with data…

I offer to do a free analysis of software engineering data, provided I can make data public (in anonymized form). Do get in touch.