Taken’s Embedding Theorem for Non-Mathematicians
In 1981, Floris Takens published the paper “Detecting Strange Attractors in Turbulence” which has since been cited (according to Google) over 14,000 times. If you look at the original paper, it is very technical and hard to understand for people — like me — who do not have a great understanding of algebra, analysis, topology, or dynamical system theory. However, the implications of the theorem stretches far beyond these fields. This post aims to distill the theorem for better understanding while, hopefully, not losing much information. Like image compression algorithms, we will inevitably lose some information, but hopefully, the picture is still there.
Background of the Theorem
The domain of the theorem is primarily in the context of dynamical systems theory. In other words, when we talk about Taken’s theorem, it is generally in the context of differential equations. As a quick recap, differential equations describes the interactions between different variables and the rate of change of these variables. Differential equations are extremely versatile and can model a lot of natural systems such as:
Population of animals
The spread of infectious diseases
Carbon cycles, chemical reactions, national economies, forest fires… you get the point. They are very versatile.
Motivation of Theorem
Let’s stick with the example of being biologists very interested in the population of rabbits. The population of all the animals can be very complicated. In practice, there may be many such animals interacting all together. In a simple example, there may be just wolves and rabbits. But, in a realistic example, there may be wolves, rabbits, bears, hawks, coyotes, and more. However, we can’t know which animals are interacting with the rabbit population and maybe we can’t even track all of these other animals.
Can we know the underlying dynamics of our rabbits if we can only observe rabbits and not any of the other animals? Do we need to measure the bear, wolf, and/or hawk populations as well? If so, which ones?
Taken’s Theorem
Taken’s theorem surprisingly says that we only need to observe the rabbit population to be able to understand the underlying differential equations. Let’s take a look:
Excuse me?
Ok, let’s break this down a bit. First, the function y would be one of our observables of the dynamics, so that could be our measurement of the rabbit population. The function φ just moves the system to a different time. So, y(x) is our rabbit population on January 1st while, for example, y(φ(x)) is our rabbit population on January 2nd and y(φ²(x)) is our rabbit population on January 3rd.
So, that means the function Φ is a collection of rabbit population at different times. This data is an “embedding” which just means that it contains the same information as if we had access to all of the different populations the rabbits were interacting with!
In conclusion, Taken’s theorem says that having access to all of the different variables of a dynamical system is equivalent to having a number of one variable sampled at sufficiently many different time points. What are the implications of this? Well, it means that we can definitely use a deterministic time series to forecast into the future. We will explore the consequences of this in another post.
