Parallel programming with opencl and python

In the next few posts I’ll cover my experiences with learning how to program efficient parallel programs on gpus using opencl. Because the machine I got was a mac pro with the top of the line gpus (7 teraflops) I needed to use opencl, which is a bit complex and confusing at first glance. It also requires a lot of boilerplate code which makes it really hard to just jump in and start experimenting. I ultimately decided to use pyopencl, which allows us to do the boring boilerplate stuff in just a few lines of python and focus on the actual parallel programs (the kernels).

First, a few pointers on what I read. A great introduction to the abstract concepts of parallel programming is the udacity course Introduction to parallel programming. They use C and CUDA to illustrate the concepts, which means you can’t directly apply what you see there on a computer with a non nvidia gpu. To learn the opencl api itself, I used the book OpenCL in Action: How to Accelerate Graphics and Computation. As for pyopencl, the documentation is a great place to start. You can also find all the python code I used in github. Continue reading “Parallel programming with opencl and python”

How to fix scipy’s interpolating spline default behavior

Scipy’s UnivariateSpline class is a super useful way to smooth time series, especially if you need an estimate of the derivative. It is an implementation of an interpolating spline, which I’ve previously covered in this blog post. Its big problem is that the default parameters suck. Depending on the absolute value of your data, the spline produced by leaving the parameters at their default values can be overfit, underfit or just fine. Below I visually reproduce the problem for two time series from an experiment with very different numerical values.

Two time series with different numerical values and their derivatives below
Two time series with different numerical values and their derivatives below. The first is overfit, the second underfit.

My usual solution was just to manually adjust the $s$ parameter until the result looked good. But this time I have hundreds of time series, so I have to do it right this time. And doing it right requires actually understanding what’s going on. In the documentation, $s$ is described as follows:

Positive smoothing factor used to choose the number of knots. Number of knots will be increased until the smoothing condition is satisfied:

sum((w[i]*(y[i]-s(x[i])))**2,axis=0) <= s

If None (default), s=len(w) which should be a good value if 1/w[i] is an estimate of the standard deviation of y[i]. If 0, spline will interpolate through all data points.

So the default value of $s$ should be fine if $w^{-1}$ were an estimate of the standard deviation of $y$. However, the default value for $w$ is 1/len(y) which is clearly not a decent estimate. The solution then is to calculate a rough estimate of the standard deviation of $y$ and pass the inverse of that as $w$. My solution to that is to use a gaussian kernel to smooth the data and then calculate a smoothed variance as well. Code below:

 

Same timeseries with the variance estimate fix
Same timeseries with the variance estimate fix

Now, you may be thinking I only moved the parameter dependence around: before I had to fine tune $s$ but now there is a free parameter sigma. The difference is that a) the gaussian filter results are much more robust with respect to the choice of sigma;  b) we only need to provide an estimate of the standard deviation, so it’s fine if the result coming out is not perfect; and c) it does not depend on the absolute value of the data. In fact, for the above dataset I left sigma at its default value of 3 for all timeseries and all of them came out perfect. So I’d consider this problem solved.

I understand why the scipy developers wouldn’t use a method similar to mine to estimate $w$ as default, after all it may not work for all types of data. On the other hand, I think the documentation as it stands is confusing. The user would expect that parameters which have a default value would work without fine tuning, instead what happens here is that if you leave $w$ as the default you must change $s$ and vice versa.

Quick introduction to gaussian mixture models with python

Usually we like to model probability distributions with gaussian distributions. Not only are they the maximum entropy distributions if we only know the mean and variance of a dataset, the central limit theorem guarantees that random variables which are the result of summing many different random variables will be gaussian distributed too. But what to do when we have multimodal distributions like this one?

A gaussian mixture with 2 dimensional features
A gaussian mixture with 2 dimensional features

A gaussian distribution would not represent this very well. So what’s the next best thing? Add another gaussian! A gaussian mixture model is defined by a sum of gaussians $$P(x)=\sum_i w_i \, \mathcal{G}(\mu_i, \Sigma_i)$$ with means $\mu$ and covariance matrices $\Sigma$. Continue reading “Quick introduction to gaussian mixture models with python”