The story of this blogpost goes back to July 2017, Santa Fe. I was a student at SFI Complex Systems Summer School, where I had a series of very interesting lectures on chaotic systems and nonlinear time series analysis. During one of the lectures, we had plots of two different time series, one coming from a chaotic system without any noise, and one being purely noise. The question was, of course, which one is which. I tried to demonstrate the same for you just below. One of the time series is the downsampling of the Mackey-Glass equation which shows a chaotic behaviour, and the other is simply white noise. Hard to say just by looking at it. You can see the underlying dynamics of the Mackey-Glass equation when you move the cursor to the chaotic plot.
There are many studies concerned with this fundamental issue of distinguishing chaos from noise, which can be a real challenge. Both chaotic and random signals exhibit irregular temporal fluctuations, both have a fast-decaying auto-correlation function, and both are hard to forecast. But the nature of their dynamics are quite different : chaos is deterministic, noise is not [1]. That's the reason why the question of chaos versus noise is so attractive : low-dimensional nonlinear determinism may allow short-term predictability [2]. In a deterministic system, short-term futures of two almost-identical states should be similar, whereas in a pure noise process, that is improbable [1].
This idea of "future of my peers is my future" was first proposed by Lorenz in 1969 to predict the atmosphere [3]. He introduced the "Lorenz Method of Analogues", which searches the known state-space trajectory (in this case the state variables are temperature, pressure, distribution of wind, water vapor, etc.) for the nearest neighbor of a given state, and takes that neighbor’s forward path as the forecast. Essentially, it is a nearest-neighbor prediction.
There are many studies concerned with this fundamental issue of distinguishing chaos from noise, which can be a real challenge. Both chaotic and random signals exhibit irregular temporal fluctuations, both have a fast-decaying auto-correlation function, and both are hard to forecast. But the nature of their dynamics are quite different : chaos is deterministic, noise is not [1]. That's the reason why the question of chaos versus noise is so attractive : low-dimensional nonlinear determinism may allow short-term predictability [2]. In a deterministic system, short-term futures of two almost-identical states should be similar, whereas in a pure noise process, that is improbable [1].
This idea of "future of my peers is my future" was first proposed by Lorenz in 1969 to predict the atmosphere [3]. He introduced the "Lorenz Method of Analogues", which searches the known state-space trajectory (in this case the state variables are temperature, pressure, distribution of wind, water vapor, etc.) for the nearest neighbor of a given state, and takes that neighbor’s forward path as the forecast. Essentially, it is a nearest-neighbor prediction.