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I have some real measurements from a process and I happened to look at the mutual distribution of (x(t), xdot(t)). I found that they seem to follow 2d normal distribution around (mu, 0). See image, the y-axis is xdot.

x-xdot-histogram

Is there some well known process which exhibit this property? What kind of dynamic equations on x(t) cause this?

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You're getting some quite interesting Answers, but I suppose you will get something more interesting if you describe the type and timings of the measurements you're making and the overall setup of the experiment. You say in your response to dmckee that it's a complex dynamic system, but you have measured and reported here only one variable at a discrete set of times, $x(t_i)$. The complete time-series might be analyzable in ways that would give more information than your $x(t_i),\dot x(t_i)=\frac{x(t_{i+1})-x(t_i)}{t_{i+1}-t_i}$ plot. If it's a complex system, are there other measurable DoFs? –  Peter Morgan Mar 4 '12 at 13:33
    
In particular, a Fourier analysis of the time-series data might give a better, more easily quantified impression of how close your data is to a simple harmonic oscillator and of what types of noise there are in the data. I'm curious also whether the timings $t_i$ of your measurement of $x_i=x(t_i)$ are accurately measured and/or are accurately periodic. –  Peter Morgan Mar 4 '12 at 13:43
    
This is a biological measurements from cells, and x is the voltage which fluctuate quite randomly, due to external (non-measured) events. No observable oscillations there! –  Uri Cohen Mar 4 '12 at 17:30
    
Nonetheless, looking at the Fourier transform of your data may be illuminating. For the sake of my curiosity, how many data points do you have, over what time-scales? –  Peter Morgan Mar 4 '12 at 18:00
    
With the histogram I presented I have 120 seconds sampled using 20K samples/sec, but I have much more data. –  Uri Cohen Mar 4 '12 at 19:26
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5 Answers

up vote 3 down vote accepted

It a widely known and experimentally useful fact in nuclear and particle physics that the position and momentum distributions of bound systems are related to one another by a Fourier transform.

Is the system you are inspecting bound?

The tails in the data that Nathaniel notes suggest that it is not fully bound, which means the Fourier relationship between the two distributions will be only approximate.

To the extend that one of the distributions is Gaussian, you would expect the other to be Gaussian as well, as this is a property of the Gaussian under a Fourier transform.

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The case here is of a complex dynamic system which is indeed bounded, so your comment is very interesting. Can you suggest some intuition why do such relation occur (relation by Fourier transform)? –  Uri Cohen Mar 3 '12 at 21:18
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An harmonic oscillator.

When evolving with time, its joint distribution in (p,x) is given by the Boltzman distribution: $e^{-H(p,x)}$, but the energy along a trajectory is constant. Nevertheless if write explicitly the hamiltonian you will find that

$e^{-H} = e^{-p^2/2 - x^2/2}$

and although the energy is constant the individual distributions of $x$ and $p=\dot x$ are gaussians.

Including masses, etc... you can get different width for the gaussians.

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It doesn't look that much like a normal distribution to me - particularly on the x axis, the right-hand tail looks heavier than the left, whereas the left one is much longer.

But, generally speaking, normal distributions tend to arise when lots of small, independently distributed random numbers (of any distribution) are added together. (The theorem that shows this is called the "central limit theorem".) So if your process has something that keeps more or less randomly perturbing x and xdot then all those perturbations will be added together, and this is the sort of result you should expect to see.

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statistical independence of the small random numbers that are being added is crucial for the proof of the central limit theorem. It does look as if the distribution is not quite a normal distribution, however visual data rarely gives enough detail to give quantitative measures of how asymmetric a system dynamics is. –  Peter Morgan Mar 3 '12 at 14:00
    
@PeterMorgan good point - I've edited the answer to make clear that the small random numbers must be independently distributed. –  Nathaniel Mar 3 '12 at 18:11
    
Normal distribution appear in many diverse cases due to CLT, as you said, but the most interesting bit here is that (x, xdot) is joinly distributed as gaussian. What kind of process randomly pertubate both x, xdot? –  Uri Cohen Mar 3 '12 at 21:16
    
@Peter Morgan: They don't have to be independent, just have correlations that make them asymptotically independent--- nearby ones can be strongly correlated. –  Ron Maimon Mar 4 '12 at 2:10
    
@UriCohen Well, I suppose that given what Ron just said, if a process randomly perturbs the velocity then this will add up to asymptotically independent perturbations to the position, and you'll get something like this. –  Nathaniel Mar 4 '12 at 13:03
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If x(t) is a random process it is quite unlikely that the derivative xdot(t) exists. So your description looks somewhat problematic.

It seems that you have a Wiener process (= random walk, Brownian motion). See http://en.wikipedia.org/wiki/Wiener_process Here the changes in x are Gaussian and uncorrelated with x itself. Then x itself also follows a Gaussian distribution (but xdot exists only as a distribution).

More generally, you might have an Ornstein–Uhlenbeck process. See http://en.wikipedia.org/wiki/Ornstein–Uhlenbeck_process

The small deviations from normality might mean that your process is actually slightly nonlinear.

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Ok, this is experimental measurements so obviously I don't have access to xdot, just to (x(t+dt)-x(t))/dt for dt of my choise (sampling frequency). I believe the variance of x is fixed as t increases, which is inconsistent with a Wiener process. I'll recheck this. –  Uri Cohen Mar 3 '12 at 21:24
    
In any case, you need to model this by an appropriate stochastic differential equation. Or, if your sampling is equidistant, a discrete time sti=ochastic model might be better; it is certainly tractable with much less baggage. In the discrete case, you can try to fit an AR or ARMA process, or a linear state space model, and if this is not sufficient, consider adding nonlinear terms. For AR, see gps.caltech.edu/~tapio/arfit –  Arnold Neumaier Mar 3 '12 at 21:34
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An equation for $x(t)$ of the form

$$\ddot{x} = -\kappa (x-\mu) - \dot{x} + \sqrt{2T} \eta(t)$$

where $\eta(t)$ is a zero-mean, unit-variance, Gaussian white noise, i.e.,

$$\langle \eta(t) \rangle = 0; \qquad \langle \eta(t) \eta(t') \rangle = \delta(t-t')$$

will generate normal distributions for both $x$ and $\dot{x}$ with

$$\langle x(t) \rangle = \mu; \qquad \langle x^2(t) \rangle - \mu^2 = \frac{T}{\kappa}$$

and

$$\langle \dot{x}(t) \rangle = 0; \qquad \langle \dot{x}^2(t) \rangle = T$$.

From fitting the data you can extract $\mu$, $\kappa$ and $T$.

This is nothing but the Langevin equation for a particle in a harmonic trap.

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