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I also asked this on math stack-exchange, but it is also relevant for the signal processing community.

I want to develop a 2D random field and its change with time with constant velocity. My process:

  1. Define a 2D grid (not the fields yet) $[x, y]$ with $n \times n$ points
  2. Define 1D time axis $[t]$ with $n_t$ elements
  3. Find the Lagrangian distance $D$ between the points in space and with time [ with advection velocities in $x$ and $y$, i.e., $[v_x, v_y]$ ]. For now $v_x$ and $v_y$ are constants.

$$ D[t, (x_i, y_i), (x_j, y_j)] = \sqrt{(x_i - v_xt - x_j)^2 + (y_i - v_yt - y_j)^2} $$

$D$ has size $[n_t \times n^2 \times n^2]$

  1. Use this distance matrix and the time axis to formulate a covariance cube of size $[n_t \times n^2 \times n^2]$

$$ \rho(D, t; \theta) = \frac{e^{-(D/b_s)^{c_s}} e^{-(t/b_t)^{c_t}}}{1 - \theta e^{-(D/b_s)^{c_s}} e^{-(t/b_t)^{c_t}}} $$

Where $[b_s, c_s]$ are spatial correlation parameters and $[b_t, c_t]$ are time correlation parameters. A non-zero $\theta$ makes this spatio-temporal correlation non-separable.

Output Expected: A 2D Random field and its variation with respect to time that moves with a velocity of $[v_x, v_y]$. I haven't used a 2D random field yet. I defined the grid only and based on Lagrangian distances of every pixel in space and time I defined a covariance matrix (Also includes the velocity information). I want to reproduce the 2D random fields out of this covariance structure.

So, I have the correlation of all spatial points with all spatial points in space $[n^2 \times n^2]$ and also the change in the spatial correlation with time $[n_t \times n^2 \times n^2]$

I have no clue how to proceed next. I read ideas in papers to make it a Multivariate autoregressive process (MAR) and find coefficients based on this covariance matrix.

The first confusion is if this AR model is for my random field itself with time and covariance is just a tool to solve that AR process or the AR model is for the 2D covariance matrices with time.

The second confusion is that if it is an AR process of the 2D random field itself with time, then, how this covariance matrix will help me solve for the random field? Because the covariance matrix is a $[n_t \times n^2 \times n^2]$ matrix.

Is there any other way I can solve this? I have the correlation structure already so in my head I somehow believe it should be simple, but when I think more I am confused.

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