Effective Degrees of Freedom of the Pearson's Correlation Coefficient under Serial Correlation

Highlights

• Autocorrelation biases the standard error of Pearson's correlation and breaks the variance-stabilising property of Fisher's transformation.
• Severity of resting state fMRI autocorrelation varies systematically with region of interest size, and is heterogeneous over subjects.
• Commonly used methods (see mis directory) to adjust correlation standard errors are themselves biased when true correlation is non-zero due to a confounding effect.
• We propose a “xDF” method to provide accurate estimates of the variance of Pearson’s correlation -- before or after Fisher’s transformation -- that considers auto-correlation of each time series as well as instantaneous and lagged cross-correlation.
• Accounting for the autocorrelation in resting-state functional connectivity considerably alters the graph theoretical description of human connectome.

Table of contents

• Introduction
• Configurations
  • Dependencies
  • Octave
• Examples
  • Using xDF
  • Constructing Functional Connectivity (FC) Maps
    • FDR-based Statistically Thresholded Functional Connectivity
    • CE-based proportionally Thresholded Functional Connectivity
    • Unthresholded Functional Connectivity
• Simulating time series of arbitrary correlation and autocorrelation structure
  • Correlated but White Time Series
  • Uncorrelated but Autocorrelated Time Series
  • Correlated and Autocorrelated Time Series

Introduction

Collection of scripts to implement the xDF method introduced in

Afyouni, Soroosh, Stephen M. Smith, and Thomas E. Nichols. "Effective Degrees of Freedom of the Pearson's Correlation Coefficient under Serial Correlation." bioRxiv (2018): 453795.

The xDF.m may be used to:

• Estimate the variance of Pearson's correlation coefficients
• Calculate the z-statistics maps of functional connectivity
• Estimate the p-values for such correlation coefficients

Configurations

For now, the xDF has only been implemented in MATLAB. Although we will be releasing the Python version in a near future. You need MATLAB statistical toolbox to run the script.

To clone the repository use:

git clone https://github.com/asoroosh/xDF.git

or alternatively download the zip file.

Dependencies

The xDF.m should work without requiring any external function. However it is comprised of several internal modules which are also available individually in mis/:

• xC_fft.m: approximates the cross correlations using Wiener–Khinchin theorem
• AC_fft.m: approximates the autocorrelations using Wiener–Khinchin theorem

Octave

xDF is also available for Octave via xDF_octave.m. Note that you require statistics package to run the script in Octave:

pkg install -forge io
pkg install -forge statistics

We have only tested the script on Octave 4.4.1.

Examples

Using xDF

Suppose X is a matrix of size IxT where I is number of regions and T is number of data-points,

1. Estimating the variance without any regularisation on the autocorrelation function:

[V,Stat]=xDF(X,T)
VarRho % is matrix of IxI where each element is variance of correlation coefficient between corresponding elements.
Stat.p % matrix of IxI of uncorrected p-values
Stat.z % matrix of IxI of z-scores

2. Estimating the variance with Tukey tapering method, on cut off M=sqrt(T) as suggested in Chatfield 2016.

[VarRho,Stat]=xDF(X,T,'taper','tukey',sqrt(T))

Similar regularisation with M=2*sqrt(T) as in Woolrich et al 2001:

[VarRho,Stat]=xDF(X,T,'taper','tukey',2*sqrt(T))

3. Estimating the variance with adaptive truncation method. Which we showed that generates the most accurate estimates.

[VarRho,Stat]=xDF(X,T,'truncate','adaptive','verbose')

4. Estimating the variance with shrinking as tapering method & without controlling the variance.

[VarRho,Stat]=xDF(X,T,'truncate','adaptive','TVOff')

For more options see the usage section of the function.

Constructing Functional Connectivity (FC) Maps

Here we show how you can use xDF to form functional connectivity of BOLD signals of I region of interest and T time-points. The examples cover both statistically and proportionally thresholded FCs as well as unthresholded FCs.

FDR-based Statistically Thresholded Functional Connectivity

[VarRho,Stat]=xDF(ts,T,'truncate','adaptive','TVOff')
FC = fdr_bh(Stat.p).*Stat.z; %FC of IxI

Function fdr_bh is an external function [+]. It can also be found in .../xDF/FCThresholding/StatThresholding/

CE-based proportionally Thresholded Functional Connectivity

[VarRho,Stat]=xDF(ts,T,'truncate','adaptive','TVOff')
densrng = 0.01:0.01:0.50;
[~,CE_den]=CostEff_bin(Stat.z,densrng)
FC = threshold_proportional(Stat.z,CE_den); %FC of IxI

Function threshold_proportional is an external function from Brain Connectivity Toolbox [+].

Unthresholded Functional Connectivity

[VarRho,Stat]=xDF(ts,T,'truncate','adaptive','TVOff')
FC = Stat.z; %FC of IxI

Simulating time series of arbitrary correlation and autocorrelation structure

If you are interested in reproducing results in the paper or sanity check the xDF. You can simulate N time series of desired correlation matrix of C and autocorrelation of A using function corrautocorr. We are showing example of three scenarios for pair time series of length T.

Correlated but White Time Series

>> ts = corrautocorr([0 0],0.9,eye(T),T);
>> corr(ts')

ans =

    1.0000    0.9014
    0.9014    1.0000

Uncorrelated but Autocorrelated Time Series

C_T1 = MakeMeCovMat([0.9:-.1:.1],T); %autocorrelation matrix of first time series (AR9=0.9:0.1)
C_T2 = MakeMeCovMat(0.4,T); %autocorrelation matrix of first time series (AR1 = 0.4)
A = cat(3,C_T1,C_T2); % TxTxI time series autocorrelation matrices
>> ts = corrautocorr([0 0],0,A,T);
>> rhohat = corr(ts')

rhohat =

    1.0000   -0.0350
   -0.0350    1.0000

AC_ts = AC_fft(ts,T); % estimate the autocorrelations

ac_x  = AC_ts(1,1:T-1);
ac_y  = AC_ts(2,1:T-1);

ac_x =

    1.0000    0.8962    0.7922    0.6924    0.6069    0.5357    0.4651    0.3770    0.2810    

ac_y =

    1.0000    0.4138    0.0143    0.0269    0.0765    0.0509    0.0116   -0.0213   -0.0587   

Correlated and Autocorrelated Time Series

C_T1 = MakeMeCovMat([0.9:-.1:.1],T); %autocorrelation matrix of first time series (AR9=0.9:0.1)
C_T2 = MakeMeCovMat(0.4,T); %autocorrelation matrix of second time series (AR1 = 0.4)
rho = 0.4;
A = cat(3,C_T1,C_T2); % TxTxI time series autocorrelation matrices
C = [1 rho; rho 1]; % IxI correlation matrix

ts = corrautocorr([0 0],C,A,T); %simulates the time series using Cholesky decomposition

AC_ts = AC_fft(ts,T); % estimate the autocorrelations

ac_x  = AC_ts(1,1:T-1);
ac_y  = AC_ts(2,1:T-1);

ac_x =

  1.0000    0.9021    0.8049    0.6960    0.5850    0.4955    0.4091    0.3137    0.2109

ac_y =

  1.0000    0.4885    0.0580   -0.0824   -0.1152   -0.1112   -0.0734   -0.0893   -0.0139

rhohat = corr(ts(1,:)',ts(2,:)')

rhohat =

  0.1239

Realise that simulating a pair of time series with different autocorrelation with a specified (log-zero) correlation is challenging. If the original pair of white time series (before "colouring") have correlation rho, they will not have correlation rho after colouring. This can be corrected using Eq. S9 in Section S3.1 as following

Sigma_X  = toeplitz(ac_x);
Sigma_Y  = toeplitz(ac_y);

Kx = chol(Sigma_X);
Ky = chol(Sigma_Y);
rhohat_corrected = rhohat./(trace(Kx*Ky')./T);

rhohat_corrected =

  0.3888

This disconnect between original and induced correlation is a simulation artifact. It remains the case that Pearson's sample correlation is approximately unbiased for the lag-zero correlation even if the time series are autocorrelated.

Obviously, these are just quick examples; more accurate estimates should be achieved via hundreds of iterations.