Discretize in two dimension

I have data points or data co-ordinates in 2D. Let's say random trajectory for 10000 time steps

x= randn(10000,1);

y= randn(10000,1);

I want to discretize the trajectory in 2D phase space X and Y of equal grids sizes. And then try to calculate the probability fluxes between each grid i.e transition between boxes with time. The transition can be in and out in x or y direction in 2D space.

Any help is appreciated.

[John D'Errico]

Far too vague to have an answer. I think MAYBE all you need to do us to use round, to move your x and y coordinates to the grid points. If you want a better answer than that, then you need to possibly give an example. Or be far more clear aout your goal.

Answer 1

[nick]

Hi Alakesh,

I understand that you want to compute the probability flux between each transition in a 2D space for a random trajectory that is discretized with equal grid size in the X and Y directions. To discretize a 2D trajectory into a grid and calculate the probability fluxes between the grid cells, you can use the 'histcounts2' function to create a 2D histogram of the data points to discretize the space.

Then, you can calculate the transitions between grid cells over time by looping through transitions between bins. Here's a sample MATLAB code that demonstrates the same:

x= randn(10000,1);
y= randn(10000,1);
numBinsX = 10;
numBinsY = 10;

% Define edges for the bins
xEdges = linspace(min(x), max(x), numBinsX+1);
yEdges = linspace(min(y), max(y), numBinsY+1);

[N, ~, ~, binX, binY] = histcounts2(x, y, xEdges, yEdges);

% Initialize transition matrix
transitions = zeros(numBinsX, numBinsY, numBinsX, numBinsY);

% Calculate transitions
for t = 1:length(x)-1
    currentBinX = binX(t);
    currentBinY = binY(t);
    nextBinX = binX(t+1);
    nextBinY = binY(t+1);
    
    % Ensure valid bins before counting transition
    if currentBinX > 0 && currentBinY > 0 && nextBinX > 0 && nextBinY > 0
        transitions(currentBinX, currentBinY, nextBinX, nextBinY) = ...
            transitions(currentBinX, currentBinY, nextBinX, nextBinY) + 1;
    end
end

% Initialize probability flux matrix
probabilityFlux = zeros(numBinsX, numBinsY, numBinsX, numBinsY);

% Calculate probability flux
for i = 1:numBinsX
    for j = 1:numBinsY
        totalTransitions = sum(sum(transitions(i, j, :, :)));
        if totalTransitions > 0
            probabilityFlux(i, j, :, :) = transitions(i, j, :, :) / totalTransitions;
        end
    end
end

You can refer to the following documentation to learn more about 'histcounts2' function ;

https://www.mathworks.com/help/releases/R2024a/matlab/ref/histcounts2.html

Hope this helps.