# Correlation or no correlation?

By Marcus Wilson 25/03/2011 5

Here’s an example of how easy it is to see things that don’t exist. It’s from a real piece of research (mine). As  way of background, I’ve been doing some work with computer models of neurons in the cortex (NB this isn’t artificial neural networks, which were all the rage in the 1980/90s). Broadly speaking, I’ve been looking at the cross-over between two different models – (a) a very detailed model of neurons including explicity modelling of ionic currents across membranes that lead to action potentials (a neuron ‘firing’), and (b) a more statistical approach in which we only consider firing rates, rather than modelling every firing event.

What I’ve been trying to do is to take output from the simpler firing rate model (b) and reconstruct a pattern of firing events that is consistent with it – i.e. reverse engineer the detail. Then see how this compares with model (a), when the detail was in there to start with.

What I was hoping to see from the computer output was sequences of firing events that were strongly correlated in space. I’ll show you a picture of what I got (which is just one small piece of a much bigger simulation). Time is along the x-axis (horizontal), position (i.e. different neurons) along the y-axis (vertical), and colour gives the neuron voltage. A neuron firing is therefore shown as a small, bright, vertical bar.

Now, at first glance this looks promising. There’s a chunk of neurons in the middle that appear to be firing pretty much together. And at the bottom, neurons appear to be ‘pairing’, so a neuron fires with its neighbour. So, having plotted this, I was quite happy.

But then came the let-down. I actually analyzed the full output (much longer than the 0.5 seconds shown here, and more neurons too) in a systematic, mathematical manner, independent of my gut feeling.  The result? Any correlations I’m seeing are purely imagined (or, if they exist, are very small indeed). I am ‘seeing things’ in the picture. (NB Yes, there certainly are correlations in time – each neuron fires at a fairly constant rate, but there aren’t any correlations between neighbouring neurons, which is what I wanted to see.)

It’s often the case that we can see what we want to see in data, when it’s just presented to us in raw form. We’re great at ‘seeing’ patterns that just don’t exist. Analyze it systematically, and often these disappear. Frustrating in this case – I was hoping that they’d be real, but that’s not the result I get.

### 5 Responses to “Correlation or no correlation?”

• Falafulu Fisi says:

Marcus said…
Any correlations Iâ€™m seeing are purely imagined

I’m sure you’ve already using or have heard of RMT (Random Matrix Theory) originated in particle physics (from decades ago by Wigner), which today it has found wide applications in varieties of domains. RMT is apparently good in resolving correlation in the data where traditional statistical methods may fail to do. The following blog post is an excellent summary.

“Random matrix theory: a law that explains everything”
http://blogs.monografias.com/sistema-limbico-neurociencias/2010/04/08/random-matrix-theory-a-law-that-explains-everything/

Here is a good scholarly brief description of it.

“Random Matrix Theory”
http://www.scholarpedia.org/article/Random_matrix_theory

My interest in RMT is its application to economics (see #1 below) in portfolio risk management & model development. I checked out its other use and I was very surprised at how wide it’s applicability is.

#1) “RANDOM MATRIX THEORY AND FINANCIAL CORRELATIONS”
http://web.mit.edu/18.325/www/rmt_finance3.pdf

#2) “Application of Random Matrix Theory to Biological Networks”
http://arxiv.org/abs/q-bio/0503035

#3) “Random Matrix Theory Analysis on the Spatial Correlation of Section (road) Speed”
http://cedb.asce.org/cgi/WWWdisplay.cgi?175013

#4) “Random Matrix Models of String Theory”
http://theory.tifr.res.in/~mukhi/Physics/mukhi-anu.pdf

#5) “Random matrix theory and wireless communications”
http://portal.acm.org/citation.cfm?id=1166374&dl=

#6) “Constructing gene co-expression networks and predicting functions of unknown genes by random matrix theory”
http://www.biomedcentral.com/1471-2105/8/299

and many more…

• Marcus Wilson says:

Yes, I did a little bit with RMT when I did my PhD. My supervisor waxed lyrical about it; I, on the other hand, struggled to see what was going on. In this case I’m doing fairly simple correlation functions between rows. I’m not sure whether it’s worth pursuing the RMT just at the moment because clearly I don’t get what I hoped to get.

• This looks all very interesting, but is it possible to explain it to someone whose last science class was back in 1991? (i.e. me)

I really would like to know what the heck you’re talking about:)

• possum says:

@evitwit

I think what RMT is saying is something like this:
When you have a large amount of data, it is possible to find relationships between things that look as if they are important, but which are just flukes. Maybe two sets of data move up and down in very much the same way. But you are not sure if you have found something important or not. So you run the same mathematical analysis on a bunch of random numbers. If you can get similar relationships between random numbers as you can with your data, then what you have found in your data is probably just a fluke.

It is a way of trying to separate signal from noise and as such is useful for any discipline where you are trrying to find mathematical relationships between variables.

If I have this wrong, please post a better explanation, because I think I might be able to make use of this in my work.

• ah – the experiment was all about looking at relationships and correlations in data – i thought it was trying to explain something else and it wasn’t happening because he’d found there was no correlation in the data (and i was wondering what that something was – cool:)