Identification of slowly reacting variables in a dynamic system using the Wasserstein metric

Speaker: Prof. Dr. Sjoerd Verduyn Lunel
Department:
Applied Analysis
Subject:
Identification of slowly reacting variables in a dynamic system using the Wasserstein metric
Location:
Utrecht University
Date: 2016-0
3-08
Author: Romano van Genderen

Of the seminars given during the Dutch National Mathematics Symposium I visited, I chose to share the seminar given by professor Verduyn Lunel with you. I chose this particular professor because I have already once visited one of his lectures on real single-variable analysis, and this topic because it has the most practical applications.

He first started slowly with the basics. He explained that a time series is a series of measurements of the same thing, done once every time interval t. A time series can be generated through measurement of a physical phenomenon or by a simpler model function. As an example, which was also recently mentioned in Computational Science, he used the Newton-Raphson method. This formula defines a variable x_{n+1} based on its previous value x_{n}. This also happens in another well-known formula, namely Hénon’s equation.

Next, he explained about attractors. When you have a time series, it has an attractor. This is a point or set of points A, where if the time series gets in the vicinity of A,in a set of points called V, it will never leave V. In the case of the Newton-Ralphson method, it is the root of the function, in the case of Hénon’s equation, a complicated 2-dimensional shape. Every time series has an attractor.

But the question he asked, can you predict the attractor when you have a time series. Kennel et al (1992) showed that this is possible by grouping specific terms in the time series into vectors and adding a lag between the terms. For example, the time series (1, 2, 3, 4, 5, 6) can be grouped like (1, 4), (2, 5), (3, 6). These are vectors in R^2, but sometimes other dimensions or lags are required.

So now you have a set of vectors {v_1, … v_n} ∈ R^n. Next, you should put the distance between these vectors inside a matrix. So now you have the matrix:

ea89f30c06c85680bb3b42f4f44add32.png

You can plot the columns of these matrices in a vector space. This leads to a point cloud. The fact that you have transformed a time series into a point cloud allows the most innovative mathematical object in this seminar to be used, the Wasserstein metric. This metric is a specific way to assign “distance” or “difference” between two point clouds. The way to visualise the Wasserstein distance is to think of all points in a point cloud as small heaps of sand. In that case, the Wasserstein distance is the least amount of work to be done on the heaps in the configuration of the first point cloud to push them into the second configuration. If these point clouds are very similar, you need just a little work to push the heaps of sand. But if they differ a lot, a lot of work is needed. So a low Wasserstein distance means a low difference between the two point clouds. The parameters in the model equation change the Wasserstein distance. And because the Wasserstein distance is related to the attractor as shown before, the attractor shifts position when the parameters change.

flow_eqw_opt.jpg

Fig 1. Image explaining the principle behind the Wasserstein metric, the gray heaps u are moved to the darker heaps w. Scott Cohen, Stanford University.

Now for all these things to come together. I showed that you can change a time series into a point cloud. You can calculate an objective distance or a measure of difference between two point clouds using the Wasserstein metric. So you can objectively say how different two time series are. This has many practical uses. But because prof. Verduyn Lunel was already a bit short in time, he only mentioned two.

The first is if you use a specific point in an MRI as your time series. In this case, you can objectively see the difference between two points in time. Instead of letting a doctor guess if something is severe or not, risking potential bias or human mistakes, you now have a simple number stating how much difference there is between a sick and a healthy patient. This also helps set a simple number for when intervention is needed.

The second use, which was a case study prof Verduyn Lunel participated in at the Academic Medical Centre in Amsterdam, was to distinguish between asthma and Chronic Obstructive Pulmonary Disease. Using time samples of patients breathing into what he called “Digital Noses”, he could distinguish between the patients having asthma and COPD, even noticing that one patient was suffering from both, a fact that could not have been observed earlier.

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