Previously we described what a random walk is and demonstrated some simple code to perform this walk in one dimensional space. Today we will provide some simple code for how to perform such a walk in two-dimensional space. In the following post, we’ll look at the general case, and then we’ll get into some simulations. Here’s a sample of how our output will look like:

2D random walk in MATLAB

Just like before, we’ll first begin by initializing our function “random_walk”, do some simple argument error checking, and utilize MATLAB’s built-in “rand” and “round” functions.

% Random Walk in MATLAB
function position = random_walk(numSteps, numDimensions, plotResults)

% default values
if nargin < 3
    plotResults = false;
end
if nargin < 2
    numDimensions = 2;
end
% Use 40 as default number of steps
if nargin < 1
    numSteps = 40;
end

% setup random walk values
randValues = round(rand(numSteps, numDimensions));

While in the one-dimensional case, we just had to worry about movement of -1 or 1 along the single axis, now we have four possibilities, specifically [1,0], [-1,0], [0,1], and [0,-1], in the (x,y) directions, respectively. Here’s how we will handle having these four possibilities. We’ll search through our array “randValues” and replace any (0,0) or (1,1) entries with (-1,0) and (0,-1). We can use the built-in MATLAB function repmat to do this for all instance of (0,0) or (1,1) in the array.

% One dimensional random walk
if numDimensions == 1
    randValues(randValues == 0) = -1;
% Two dimensional random walk
elseif numDimensions == 2
    % lets do the following conversions in 2D
    % [0, 0] move backwards in x, i.e. [-1, 0]
    negX = ismember(randValues,[0 0], 'rows');
    randValues(negX,:) = repmat([-1 0], sum(negX), 1);
    % [1, 0] move forwards in x
    % [0, 1] move forwards in y
    % [1, 1] move backwards in y, i.e [0, -1]
    negY = ismember(randValues,[1 1], 'rows');
    randValues(negY,:) = repmat([0 -1], sum(negY), 1);
end

Just like before, we now increment our position with each step:

% start at origin and initialize the position
position = zeros(numSteps+1, numDimensions);
for k = 2:numSteps+1
    position(k,:) = position(k-1,:) + randValues(k-1,:);
end

and now the example we show at the beginning of this post is as follows:

position = random_walk(40,2,true)

position =

     0     0
     0    -1
    -1    -1
    -1     0
    -1     1
    -2     1
    -2     2
    -2     1
    -2     0
    -2    -1
    -2     0
    -3     0
    -4     0
    -3     0
    -4     0
    -4     1
    -5     1
    -4     1
    -3     1
    -2     1
    -3     1
    -2     1
    -2     2
    -1     2
    -2     2
    -2     1
    -2     2
    -3     2
    -2     2
    -1     2
    -2     2
    -2     1
    -1     1
    -1     2
    -1     3
    -2     3
    -3     3
    -2     3
    -1     3
     0     3
     0     2

Stay tuned for more, and let us know if you have any comments!