November 5, 2013

I’m in the process of starting up a blogging series, to write about Data Analysis in Haskell. This is a sort of introduction/manifesto…

### What’s “Data Analysis”?

To me, data analysis means turning data (bits and bytes) into knowledge (some higher level of conceptual information). It can be exploratory (i.e. we don’t really know what we’re doing when we start, we just discover along the way) or something more organised. It usually involves a few common tasks: data intake and pre-processing, some “main analyses” and some post-processing and visualisation. The input data can range from a few measurements in the laboratory to enormous remote sensing data sets. The main analyses can range from simple data aggregation to complicated inverse modelling or even weirder stuff.

I’ve done a lot of this sort of work in the past, mostly in climate science or environmental remote sensing, and almost exclusively using Fortran, C++ or R. I’ve been using Haskell a lot over the past year and a bit though, and I’ve been thinking more and more that it’s a perfect language for this sort of work.

Most of the serious programming work I’ve done in this area has been kind of unsatisfactory. For developing new kinds of data analyses, you spend some time sitting down with pen and paper to work out the details of your algorithms, then you code them up (in C++ say). The code you write often bears little relationship to the equations you write down on paper, and you spend a lot of time bogged down in the minutiae of iterating over arrays, avoiding off-by-one errors, managing memory, and so on. It’s hard to make a direct link between your mathematics and your code, which leads to mistakes.

Haskell provides an opportunity to make this situation quite a bit nicer, I think. Some experiments I did a while ago with Lyapunov exponent calculations for dynamical systems convinced me that Haskell is a great vehicle for mixed algebraic/numerical calculations – you can easily work with symbolic representations of the mathematical structures you care about, then simplify and convert these to efficient numerical calculations. You’re unlikely to get performance that competes with Fortran that’s been lovingly hand-coded and optimised over decades of effort, but you’ll be able to move much more quickly while getting some help with the algebra and ending up reasonably efficient numerical code that you can be confident works.

### What I’m going to do

To put this somewhat hubristic series of pronouncements to the test, I’m going to try doing some “serious” data analysis examples in Haskell. I’m mostly interested in scientific applications, so that’s where I’m going to concentrate. The applications I have in mind include some atmospheric science (the field I did my PhD in), some non-linear time series analysis, some remote sensing work, some medical imaging, and a bunch of other fun stuff. Here’s an incomplete list of what I have on my “probable” list so far:

• Ultra-low frequency variability in the northern hemisphere atmosphere;
• Critical scaling in tropical rainfall rates;
• Inverse methods for medical X-ray CT imaging;
• Nonlinear time series methods for noise reduction and signal extraction (extracting foetal heart rate data from ECG signals);
• Land cover classification from MODIS/ASTER data – applications to hydrology of North African saline lakes;
• Landform classification from laser altimetry data (using data from the Mars Orbital Laser Altimeter).

What all of these applications have in common is that they require data intake from large and complicated data sets, often in obscure file formats, they require significant domain knowledge to make any real progress, they tend to require analyses that are mathematically complex and are in some way “non-standard”, and they require complex (or if not complex, at least good!) data visualisation.

The goal is to as much of this stuff in Haskell in a way that gives some pedagogical insight into what’s going on but is still efficient enough to do the job. (For some of these problems, that’s a tall order, but we’ll see how we go.)

### Warm-up

Now, I have a couple of slightly easier examples to start with (one that I’ve even done more or less end-to-end in C++ and R before now), but I wanted to start with something venerable to demonstrate some interesting Haskell things before I really get into the meat of data analysis examples.

In the world of data analysis algorithms, you don’t get much more venerable than the fast Fourier transform. The market for new FFT software has been quite effectively killed by FFTW (“the fastest Fourier transform in the West”), but that doesn’t mean there aren’t things to be learnt. So, the first few articles in this series are going to be about an attempt to implement a mixed-radix FFT in Haskell with empirical compile-time optimisation of the data decomposition ordering. This sort of empirical optimisation is one of the keys to FFTW’s success, and is also used by the ATLAS linear algebra libraries.

### What we’ll learn

There are a number of different things we’ll be able to look at with this:

1. A technique I call “toy computer algebra”. The heart of the FFT is a clever matrix decomposition that turns an $O(n^2)$ algorithm into an $O(n \log n)$ algorithm. This decomposition isn’t obvious, and for any reasonable size of matrix, working through the decomposition by hand is extremely tedious. You could just fire up Mathematica or Maple, but I prefer a different approach that’s really easy in Haskell, and that can give you a lot more insight into what’s going on.

2. For numerical code, profiling and benchmarking is often really important. Even once you’ve found a nice $O(n \log n)$ algorithm, real-world performance can be disappointing (damn those constant factors!). Once you have code that works, there’s no substitute for measuring performance. Haskell has some great facilities for benchmarking (primarily Bryan O’Sullivan’s criterion package) and we’ll do lots of performance measurement this way.

3. Once you can measure performance, you can use those measurements do drive optimisation. The FFT algorithm we’re going to look at has a great deal of flexibility in the order that calculations are performed, flexibility that can have a big impact on performance, depending on input data size, your machine’s processor, and cache characteristics. The enormous success of FFTW indicates pretty clearly that empirical optimisation is the way to go – you could perhaps come up with optimal execution plans for some data sizes for some known cache architectures, but you’d have to redo that analysis every time you moved to a new machine.

4. Haskell has a powerful metaprogramming facility called Template Haskell. I’m going to try to use this to move the benchmarking and FFT executaion plan optimisation to compile time. (This might be tricky…)

Anyway, next time, I’ll give a quick introduction to the discrete Fourier transform to give some grounding before we talk about the FFT.