Matlab: Aligning data by using cross covariance

Question

I want to get the offset in samples between two datasets in Matlab (getting them synced in time), a quite common issue. Therefore I use the cross correlation function xcorr or t

Answer 1

Your data has a minor peak around 5 and a major peak around 101.

If I knew something about my data then I could might window around an acceptable range of offsets as shown below.

Code for initial exploration:

figure; clc;
subplot(2,1,1) 
plot(1:numel(b), b);
hold on
plot(1:numel(c), c, 'r');
legend('b','c')

subplot(2,1,2)
plot(crossCorr,'.b-')
hold on
plot(peakIndex,crossCorr(peakIndex),'or')
legend('crossCorr','peak')

Initial Image:

If you zoom into the first peak you can see that it is not only high around 5, but it is polynomial "enough" to allow sub-element offsets. That is convenient.

Image showing :

Here is what the curve-fitting tool gives as the analytic for a cubic:

Linear model Poly3:
     f(x) = p1*x^3 + p2*x^2 + p3*x + p4
Coefficients (with 95% confidence bounds):
       p1 =  8.515e-013  (8.214e-013, 8.816e-013)
       p2 = -3.319e-011  (-3.369e-011, -3.269e-011)
       p3 =  2.253e-010  (2.229e-010, 2.277e-010)
       p4 = -4.226e-012  (-7.47e-012, -9.82e-013)

Goodness of fit:
  SSE: 2.799e-024
  R-square: 1
  Adjusted R-square: 1
  RMSE: 6.831e-013

You can note that the SSE fits to roundoff. If you compute the root (near n=4) you use the following matlab code:

% Coefficients
       p1 =  8.515e-013
       p2 = -3.319e-011
       p3 =  2.253e-010
       p4 = -4.226e-012
% Linear model Poly3:
syms('x')
f = p1*x^3 + p2*x^2 + p3*x + p4

xz1=fzero(@(y) subs(diff(f),'x',y), 4)

and you get the analytic root at 4.01420240431444.

EDIT: Hmmm. How about fitting a gaussian mixture model to the convolution? You sweep through a good range of component count, you do between 10 and 30 repeats, and you find which component count has the best/lowest BIC. So you fit a gmdistribution to the lower subplot of the first figure, then test the covariance at the means of the components in decreasing order.

I would try the offset at the means, and just look at sum squared error. I would then pick the offset that has the lowest error.

Procedure:

• compute cross correlation
• fit cross correlation to Gaussian Mixture model
  • sweep a reasonable range of components (start with 1-10)
  • use a reasonable number of repeats (10 to 30 depending on run-to-run variation)
  • compute Bayes Information Criterion (BIC) for each level, pick the lowest because it indicates a reasonable balance of error and parameter count
• each component is going to have a mean, evaluate that mean as a candidate offset and compute sum-squared error (sse) when you offset like that.
• pick the offset of the component that gives best SSE

Let me know how well that works.