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

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• tworitdash
In summary: 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.
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.

Last edited:
tworitdash said:
1. Define a 2D grid $[x, y]$ with $n \times n$ points

tworitdash said:
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
Stephen Tashi said:
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)##.

tworitdash said:
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?

Stephen Tashi said:
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.

tworitdash said:
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) ##

tworitdash

## 1. What is a random field?

A random field is a mathematical concept used to describe a collection of random variables that are defined over a certain space. It can be thought of as a function that assigns a random value to each point in a given space.

## 2. How is 2D space and 1D time evolution represented in a random field?

In a random field, 2D space and 1D time evolution are represented by the two dimensions of the space and the variable that changes over time. This allows for the analysis and prediction of how the random field will change over both space and time.

## 3. What is the significance of studying the 2D space and 1D time evolution of a random field?

Studying the 2D space and 1D time evolution of a random field allows for a better understanding of how random variables behave in a given space over time. This can have practical applications in fields such as physics, engineering, and finance.

## 4. How is the evolution of a random field modeled?

The evolution of a random field is typically modeled using statistical methods, such as stochastic processes or differential equations. These models take into account the randomness of the field and the effects of various factors on its evolution.

## 5. What are some real-world examples of random fields?

Random fields can be found in various natural phenomena, such as the distribution of temperature in a room, the fluctuations of stock prices, or the movement of particles in a fluid. They are also used in computer graphics to generate realistic textures and patterns.

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