Calculating mean-squared displacement (msd) with MATLAB

Question

This is part of a matlab intro course for biologists. I have my data points (for a single particle!) in a matrix with 4 columns (time, x, y, z) and a few thousand rows. What

Answer 1

Note that there's a MATLAB class available to do MSD analysis: MSD analyzer, including extensive documentation and tutorials.

Basic usage:

ma = msdanalyzer(2, SPACE_UNITS, TIME_UNITS);
ma = ma.addAll(tracks);
ma = ma.computeMSD;

Answer 2

If you're calculating the displacement only with respect to the first position, then you don't actually average anything, since MSD(1) is the average distance your object travels within one time step. So your reasoning is spot on.

However, I would calculate MSD the other way around: Get all displacements at dt=1 (1-2,2-3,3-4,...) and average. This is MSD(1). Then, you get all displacements at dt=2 (1-3,2-4,...) and average. This is MSD(2). And so on.

A useful property of Matlab is that you can vectorize some computations, i.e. do the calculation on an entire array, rather than doing them element-by-element. So if you have an array a of 100-by-1 coordinates, the difference between each coordinate and the next one is b=a(2:100)-a(1:99), or more generally, b=a(2:end)-a(1:end-1), so b(1) is a(2)-a(1), b(2) is a(3)-a(2) etc.

To calculate MSD from your array data (where I assume that time is in equal steps!), you'd write

nData = size(data,1); %# number of data points
numberOfdeltaT = floor(nData/4); %# for MSD, dt should be up to 1/4 of number of data points

msd = zeros(numberOfDeltaT,3); %# We'll store [mean, std, n]

%# calculate msd for all deltaT's

for dt = 1:numberOfDeltaT
   deltaCoords = data(1+dt:end,2:4) - data(1:end-dt,2:4);
   squaredDisplacement = sum(deltaCoords.^2,2); %# dx^2+dy^2+dz^2

   msd(dt,1) = mean(squaredDisplacement); %# average
   msd(dt,2) = std(squaredDisplacement); %# std
   msd(dt,3) = length(squaredDisplacement); %# n
end