Next Big Thing: Mathematics helps find hidden patterns in complex data

I’m interested in many areas of science, but if I really want to understand something I have to drill down to the maths behind it to really get to the heart of the issue. To really figure out how something is working, it always comes down to mathematics.

Most of my training is in a subject called dynamical systems. This involves any system that is changing over time – that’s the dynamic aspect. The weather, the climate, the stock market – pretty much everything in everyday life is some kind of process that’s changing over time.

These dynamical systems have many different aspects – you can have statistical descriptions of these processes, or you can have geometric descriptions. In a mathematical model of weather, for example, you might have different variables such as air pressure, air temperature and humidity. You can then view the evolution of these three variables over time as a trajectory in three-dimensional space. And this generates a distortion of three-dimensional space over time, so you get this geometric view of the dynamical process. In low-dimensional settings, I picture this geometric view as like a piece of rubber, and that piece of rubber is getting stretched and compressed and distorted by the process.

Very early on I realised I liked this geometric way of thinking about how processes change in time, and I still work in this area. But I’ve also moved towards statistical descriptions. As we know, it’s hard to predict the weather more than a week out because of chaos theory. People recognise that you can’t make accurate long-term predictions, but what is more reliable are predictions about statistics – predictions about means or standard deviations or other finer statistical descriptions.

One difficulty with chaos theory is that the dynamics are non-linear. If they were linear, coming back to the rubber analogy, it would mean your piece of rubber is stretched by a linear factor of, say, two or three. You double or triple the length. But what is more common is that you don’t do this uniformly along the piece of rubber. Some parts of the rubber might be stretched a little bit, some might be compressed, some might be stretched a lot. There’s a non-uniformity or non-linearity to the stretching or the compression. And this makes analysis very hard.

To really figure out how something is working, it always comes down to mathematics.

A lot of my techniques are built around lifting these low-dimensional spaces into much higher dimensional spaces where you can again get a linear description. You pay more by introducing many more dimensions, but you simplify the dynamics by making it linear. From there, there are a whole bunch of new techniques that you can use – operator theory, spectral theory – that give you the kind of analysis you want much faster.

Some of my research focuses on extracting long-lived persistent features. Take, for example, the ocean. In the ocean, you have persistent fast currents, like the East Australian Current running from north to south down our east coast bringing warm water from the north down towards Sydney where it spins off a lot of eddies that might be a hundred kilometres across, like big rotating vortices of water. These eddies are responsible for carrying water with certain properties long distances. For example, the Indian Ocean is very warm and salty compared to the South Atlantic, which is much cooler and fresher. A lot of heat and salt is transported by these eddies going westward from the Indian to the South Atlantic. Here, the input information for my analysis would be ocean circulation currents, the velocities of water, all driving the dynamical system of the ocean. From this, you want to extract these long-lived objects like the eddies.

Part of my recent ARC Laureate is concerned with extensions of my work on extracting long-lived objects, such as eddies in the ocean and vortices in the atmosphere.

There’s a non-uniformity or non-linearity to the stretching or the compression.

In the ocean, you’re in two or three dimensions. My aim is to take imagery from the satellites whizzing around the Earth taking photos of sea surface and land surface temperature. Let’s say every couple of days you get a snapshot of sea surface temperature. If you think of an image in one of these snapshots as having, say, 10,000 pixels, what you have is in effect the evolution of 10,000 pixels, from day-to-day or week-to-week, and you want to pull out the persistent feature information from that series of images. But this is now high-dimensional, because you’re in 10,000 dimensions – you have 10,000 pixels. This is one direction I’m wanting to head in to extend one of these low-dimensional operations to higher dimensions.

By developing these new mathematical tools, we should be able to better discover hidden patterns in complex data, and improve our prediction of key climate drivers, which are connected to droughts, heatwaves, bushfires and major flooding.

I’m continually developing new methods. Back around 2010, there were several existing techniques, but I thought I could do better. I needed some examples, so I had to get out there and convince people. I had to contact some atmospheric science people and some oceanography people and say, “Look, do you have any questions of this type that I can try to answer with these new methods?” It was very much a case of me going out to them. And there’s still a bit of that, because I continually have to prove that my methods are working. But more and more, it’s the other way around – people come to me with problems and ask if I can help out.

As told to Graem Sims.