2D space and 1D time evolution of a random field

[tworitdash]

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

1. Define a 2D grid $[x, y]$ with $n \times n$ points
2. Define 1D time axis $[t]$ with $n_t$elements
3. Find the lagrangian distance between the points in space with the velocity in $x$ and $y$, i.e., $[v_x, v_y]$

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

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

4. 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.

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.

 

[Stephen Tashi]

1. Define a 2D grid $[x, y]$ with $n \times n$ points

3. Find the lagrangian distance between the points in space with the velocity in $x$ and $y$, i.e., $[v_x, v_y]$

If $(x_i, y_i)$ and $(x_j, y_j)$ are two points on 2-D grid then why would the distance between them be a non-constant function of time? Is the grid deforming with time?

 

[tworitdash]

If $(x_i, y_i)$ and $(x_j, y_j)$ are two points on 2-D grid then why would the distance between them be a non-constant function of time? Is the grid deforming with time?

The Lagrangian distance is a function of the grid positions but also the velocity and time. So, as time progresses, this distance between $(x_i, y_i)$ and $(x_j, y_j)$ changes. You can check the formula of D on step number 3. It is just a notion by which we find the distance of the new position of $(x_i, y_i)$ i.e., $(x_i - v_xt, y_i - v_yt)$ from the point $(x_j, y_j)$.

 

[Stephen Tashi]

You can check the formula of D on step number 3. It is just a notion by which we find the distance of the new position of $(x_i, y_i)$ i.e., $(x_i - v_xt, y_i - v_yt)$ from the point $(x_j, y_j)$.

According to that formula, with $x_i = x_j = 0, y_i = y_j = 0, v_x = 3, v_y = 0, t = 1$ we have $D(1,(0,0),(0,0)) = 3$. So is it correct to call $D$ a distance formula?

 

[tworitdash]

According to that formula, with $x_i = x_j = 0, y_i = y_j = 0, v_x = 3, v_y = 0, t = 1$ we have $D(1,(0,0),(0,0)) = 3$. So is it correct to call $D$ a distance formula?

It is Lagrangian distance. It takes into account the time also. So, at $t = 1$ and with a velocity in x as $v_x = 3$, the new grid position of $(x_i, y_i)$ is $(-3, 0)$ for example. The Lagrangian distance is calculated from this moment in time $t = 1$ of the position $(x_i, y_i)_{t = 1} = (-3, 0)$ to the position of $(x_j, y_j)_{t=0} = (0, 0)$ at time $t = 0$. So, it is a distance in space and time. That is why the distance is $3$ instead of 0. You are probably thinking of Eulerian distances where you check absolute distance between two points when time is fixed (snapshot). Here, one set of co-ordinates move in time with a defined velocity in time and you find the distance from the other set of co-ordinates which for example didn't move in time. Then, all sorts of permutations is done for all time lags and all spatial positions.

 

[Stephen Tashi]

Is there any other way I can solve this?

In general, a given covariance matrix goes not determine a unique stochastic process that has that covariance matrix. So I doubt that you can solve this problem if "solve" is taken to mean finding a unique solution.

If you are asking how to find an AR process that has the given covariance matrix, I'm not sure how to do it. I can only speculate. For example, the AR model with random variable $V$ given by $X(t) = V(t) + e^{-c t} X(t-1)$ has a time varying variance. So perhaps you could use a model of the form:
$X_{i,j}(t) = V_{i,j}(t) + \sum_{k=1}^n \sum_{m=1}^n e^{-c_{i,j,k,m} t} X_{k,m}(t-1)$