Describe the dynamics of a fluctuating vector I have a time series for the $\mathbf{v}(t) = (x,y,z)$ components of a vector quantity. It is fluctuating in time, and has a non-trivial autocorrelation function which I want to somehow elucidate. The system is anisotropic and asymmetric in the x-y plane, so in the end I can obtain three distinct autocorrelation functions:
$$
\begin{align}
    A_{xx}(t) &= \langle x(t)x(0) \rangle\\
    A_{yy}(t) &= \langle y(t)y(0) \rangle\\
    A_{zz}(t) &= \langle z(t)z(0) \rangle
\end{align}
$$
as well as two cross-correlations
$$
\begin{align}
    A_{xy}(t) &= \langle x(t)y(0) \rangle\\
    A_{yx}(t) &= \langle y(t)x(0) \rangle
\end{align}
$$
That is quite a bit of data and I am still unable to grasp how my vector moves, so I am looking for ideas on how to synthesize/analyse the data. Would it make sense to construct a time-dependent tensor
$$
A(t) = \begin{pmatrix}
\langle x(t)x(0)\rangle & \langle x(t)y(0)\rangle & 0\\
\langle y(t)x(0)\rangle & \langle y(t)y(0)\rangle & 0\\
0 & 0 & \langle z(t)z(0)\rangle \\
\end{pmatrix}
$$
and then diagonalize it using standard means, which would give 3 autocorrelations of the principal axes, plus some time dependent tilt angle $\theta(t)$? 
I can also obtain the cross-product correlation
$$
\langle \mathbf{v}(t)\times \mathbf{v}(0) \rangle = \langle x(t)y(0) - y(t)x(0)\rangle \mathbf{\hat{z}}
$$
which looks like a physically sensible quantity but I am still having a hard time understanding the motion. By the way, the average $\langle x\rangle = \langle y\rangle = \langle z\rangle = 0$.
Do you have any ideas on how to analyze/visualize such data to obtain more insight about the motion?
 A: In statistics Box et al. developed the so called time series analysis. You could look into that. Here a simple intro.


*

*I would definitely try to understand the behavior by looking at the time and frequency structure of the signal. The Fourier transform allows you to identify periodicities. 

*Substrat your model from the signal to identify further structure.

A: I hesitate to offer practical advice on the numerical analysis, but if the auto-correlation functions may be found, then you may be able to determine if the dynamics is already well known.
For example, white noise is characterised by the correlation function,
$$\langle v_i(t_1)v_j(t_2)\rangle \sim \delta_{ij}\delta(t_2-t_1)$$
which gives rise to a variance in the position linearly dependent on $t$. So, if you can determine something close to a closed form expression from the data, you may be able to identify a known type of noise/process. 
Furthermore, thanks to the Wiener–Khinchin theorem, the correlation is related to the Fourier transform of the power spectral density. A closed form expression is not required for this, as you can simply perform a discrete numerical Fourier transform. It may be that the spectral density is more useful in identifying the dynamics, if they are known at all. 
A: I may have come up with a solution. Let us focus on the x-y plane only, where the data is cross-correlated. Let us pick some random starting point $(x(0), y(0))$ and try to predict the the future position by
$$
x(t) = g_{11}(t)x(0) + g_{12}(t)y(0) + W_x(t)\\
y(t) = g_{21}(t)x(0) + g_{22}(t)y(0) + W_y(t)
$$
where $(W_x, W_y)$ is stochastic noise and $g_{ij}(t)$ are some deterministic correlation functions that I propose to obtain in the following manner:
$$
\langle x(t) x(0) \rangle = g_{11}(t)\langle x(0)^2 \rangle + g_{12}(t) \langle y(0)x(0)\rangle\\
\langle x(t) y(0) \rangle = g_{11}(t)\langle x(0)y(0) \rangle + g_{12}(t) \langle y(0)^2\rangle\\
$$
We can solve this to obtain
$$
\begin{pmatrix}
g_{11}(t) \\ g_{12}(t)
\end{pmatrix} = 
\frac{1}{\langle x^2 \rangle \langle y^2 \rangle - \langle xy \rangle^2}
\begin{pmatrix}
\langle y^2\rangle  & -\langle xy \rangle \\
-\langle xy\rangle  & \langle x^2 \rangle \\
\end{pmatrix} \begin{pmatrix}
\langle x(t)x(0) \rangle \\ \langle x(t)y(0) \rangle
\end{pmatrix}
$$
and similarly for $g_{21}, g_{22}$. Then I select some points on a circle and plot Eq. 1 (without the noise), to show how the vector relaxes on average:


