# How to express the “size” of measurement noise in a dynamical system

I have a discrete [non-linear] dynamical system $$x_{n+1} = f(x_{n})$$. There is measurement error, so my observables are a time series $$\left\{ \hat{x}_{n}\right\} _{n=1}^{N}$$ where $$\hat{x}_{n}=x_{n}+\eta_{n}$$ and $$\eta_{n}\sim\mathcal{N}\left(0,\sigma^2\right)$$ i.i.d.

It seems to me that $$\sigma$$ is not enough to convey the "size" of the noise, which should be somehow expressed in comparison to the size of the dynamics. Is there a canonical way of doing this? Perhaps as a fraction of the dynamics' size, but what would that be? The range of the time series? Its standard deviation? What if it were multiple separate time series originating from the same dynamical system (perhaps with different parameters) and which are subjected to the same type of measurement noise? should it be measured as a fraction of the size of phase space?

• @stafusa What part was unclear? I'd be happy to clarify – D.M. Apr 16 '20 at 14:32
• Hey D.M. First, is the noise a measurement error or, as stated later, something "applied" to the system, i.e., the system is noisy? But, coming to think of it, I think that in both cases it makes sense to measure de error as relative to the system size - or, formulating it in another way, with respect to the unity, if the phase space volume is normalized. – stafusa Apr 16 '20 at 14:39
• Fair point, Iv'e reformulated. I put it that way since in my case these are synthetic time series which I generate on a computer, so I do indeed "apply" random noise to the time series (add independent random noise to the value at each time step) after generating them. – D.M. Apr 16 '20 at 15:06
• Just to make sure that I get your question: Your are interested in time series, which is a topic in statistics, where you additionally have a measurement error. Is this correct? – Semoi Apr 16 '20 at 20:18
• @Semoi I'm interested in a very specific type of time series - ones which represent sequences of observations derived from a low dimensional dynamical system. In particular, a system whose dynamics I know (and therefore also know e.g. the size and shape of phase space and the various attractors) – D.M. Apr 18 '20 at 7:07

Any such "size" parameter trying to compare the noise to the dynamics of the system would have to be intrinsically tied to the behavior of f(x) and what you seek to convey with it.

As an extreme example, consider a dynamic system with a discontinuity. As the system gets very close to that discontinuity, it becomes critically important to know whether the system appears to have fallen off the edge because it actually did, or if that's just a measurement error.

On the other hand, if I have a non-linear system which I am trying to make as linear as possible, concepts like Total Harmonic Distortion become very applicable.

If I'm trying to achieve a goal state, I could measure how much this noise affects my final state. For example, if my goal is to land on the moon, I can propagate these errors forward and see how large of a "landing patch" I end up with (or whether I will land at all!)

It may also be useful to look at Kalman filtering as a way to describe these sources of noise with respect to the other uncertainties in your system.

Indeed, $$\sigma$$ gives the absolute size of the noise, whose effect on the system depends on its type and its parameters. In your case these are hidden in function $$f(x)$$, so it is hard (or even impossible) to say anything specific.

As a revealing example you could analyze the case of linear function $$f(x) = -\gamma x,$$ and check that the width of the limiting distribution for $$x$$ is proportional to $$\sigma^2/gamma$$ or something similar.