Tutorial 2, Part 1: Impact of uncentered measurements on functional connectivity

The standard correlation matrix assumes data is distributed according to a one-way multivariate model. This assumption is violated in spatio-temporal measurements that come from fMRI.

Content

1. Simulate null functional connectivity using centered one-way multivariate normal

2. Simulate null functional connectivity using row-uncentered two-way multivariate normal

3. Histogram of correlation coefficients without centering two-way data

Conclusion: Standard pairwise correlation intended for one-way measurements is a statistically biased estimator for uncentered two-way measurements

Simulate one-way multivariate data.

Default simulation uses p=50 brain regions and m=200 BOLD fMRI volumes or measurements with null correlation matrix sigma

In [3]:
% Simulate one-way multivariate data 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Dimensions or Features 
p = 50; 
% Measurements or Observations
m = 200; 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Multivariate, i.i.d normal data
rcen_mean = @(mu)(repmat(mu,[m 1])); 
rcen_mvnsim = @(mu,sigma)(rcen_mean(mu) + randn(m,p)*sqrtm(sigma)); 
title = @(tstring)(title(tstring,'fontsize',16)); 

mu = zeros(1,p); 
sigma = eye(p); 
X = rcen_mvnsim(mu,sigma); 
figure;
subplot(1,2,1); 
imagesc(rcen_mean(mu)); colorbar;
title('Row-centered, Mean')
subplot(1,2,2); 
Shat_centered = corr(X); 
imagesc(Shat_centered); colormap(summer);  colorbar; axis image equal;
title(sprintf('Sample Correlation. \n  Centered mean m=%d, p=%d',m,p)); set(gca,'fontsize',18)
figure;
hist_norm = 1;
hist(triu(Shat_centered,1),100,hist_norm);ylim([0 hist_norm*1.1]); xlim([-.5 .5]); 
title('Histogram of off-diagonal correlations')
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Simulate null functional connectivity using row uncentered two-way model

Default simuluation now adds a random offset for each measurement. This can be modified by setting rsignal to some non-zero value.

In [4]:
% Simulate row uncentered two-way multivariate data 
% Vary amount of the row offset by changing signal. Default signal = 1; 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
rsignal = 2; csignal = 0;  rand('seed',0); 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %Constant offset for each feature
% row_offset = ones(m,1)*.25*sqrt(signal);
%
% Random offset for each observation
row_offset = rsignal + randn(m,1)*.25*sqrt(rsignal);     
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Random offset for each feature. Default 0. 
mu = csignal + randn(1,p)*.25*sqrt(csignal);     
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

rncen_mean = @(mu)(repmat(mu,[m 1])+repmat(row_offset,[1 p])); 
rncen_mvnsim = @(mu,sigma)(rncen_mean(mu) + randn(m,p)*sqrtm(sigma));

Examine the uncentered data

In [8]:
Xnc = rncen_mvnsim(mu,sigma); 
disp('Center the features (columns or brain regions or voxels)')
Xnc = bsxfun(@minus,Xnc,mean(Xnc));
mu_c_hat = mean(Xnc);
mu_r_hat = mean(Xnc');
disp('.....')
disp('Mean signal corresponding to Regions.  Region 1, ...., Region 5')
mu_c_hat(1:5)
disp('Mean "brain wide" signal at each measurement due to measurement error. Observation 1, ...., Observation 5')
mu_r_hat(1:5)

Center the features (columns or brain regions or voxels)
.....
Mean signal corresponding to Regions.  Region 1, ...., Region 5
ans =

  -8.2601e-16  -4.7296e-16  -1.1458e-15  -1.0514e-15  -9.7478e-16

Mean "brain wide" signal at each measurement due to measurement error. Observation 1, ...., Observation 5
ans =

  -0.154242   0.075572   0.353733  -0.690877   0.017067

Histogram of correlation coefficients

Though all population correlation coefficients are zero 0. Sample correlation coefficients are statistically biased due to uncentered measurements. Excercise: Try increasing the number of measurements $m$.

In [10]:
Xnc = rncen_mvnsim(mu,sigma); 
figure;
subplot(1,2,1); 
imagesc(rncen_mean(mu)); colorbar; 
title('Row uncentered, Mean')
subplot(1,2,2); 
Shat_ncentered = corr(Xnc);  
imagesc(Shat_ncentered); colormap(summer);  colorbar; axis image equal;
title(sprintf('Sample Correlation. \n Uncentered mean m=%d, p=%d',m,p)); set(gca,'fontsize',18)
figure;
hist_norm = 1;
hist(triu(Shat_ncentered,1),100,hist_norm);ylim([0 hist_norm*1.1]); xlim([-.5 .5]); 
title('Histogram of off-diagonal correlations')
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