NEXCF – Cross-Correlation Estimates for Non-Equidistantly Sampled Time Series

(for MATLAB^®)

General Notes

NEXCF estimates auto- and cross-correlation functions for time series with arbitrary, non-equidistantly sampled time series using a Gaussian kernel-based technique.

Installation and Usage

To install, simply copy the file nexcf.m into your path. You will find an example by calling help nexcf.

C = NEXCF(TX, X, TY, Y) returns the Gaussian kernel cross-correlation function estimates between the time series X and Y, for which the sampling times are given in the column vectors TX and TY. The lags for which the correlation functions are estimated are given in the vector lag =-T:DT:T, where DT is usually the mean or median sampling interval. If this vector is not provided, it is created such that lag=-T:DT:T with T = range([max(TX(1),TY(1)),min(TX(end),TY(end))])/2 and DT = T/10. If it is not specified, the default kernel width is 0.25*DT.

C = NEXCF(TX, X, TY, Y, LAG, H) computes the correlation using a kernel width H, given in units of DT. H=0.25 amounts to a kernel width of 0.25*DT.

C = NEXCF(TX, X, TY, Y, LAG) returns the cross-correlation function at lags given in the vector LAG.

[C,LAGS] = NEXCF(...) returns the correlation function and the vector of time lags (LAGS).

Example

We take two time series from stochastic processes X and Y, {tx,x} and {ty,y}, which are defined as follows:

% Sampling times 
tx = (1:1001)'; ty = (1:1001)'; 

% Original signals 
x = randn(1001,1);
x(2:end) = 0.5*x(1:end-1)+randn(1000,1); 
y = randn(1001,1);
y(2:end) = 0.7*x(1:end-1)+randn(1000,1);

We have, however, in reality, observed only 50% of the points in y, yielding a time series {ty_miss,y_miss}:

rem = randi(length(ty), floor(length(ty)*0.5),1); 
ty_miss = ty; ty_miss(rem) = [];
y_miss = y; y_miss(rem) = [];

We can now calculate the ACF for the time series {ty_miss,y_miss} as follows:
First we create a lag vector,

lag = -50:50;

and we define the kernel width we want to employ:

H = 0.25; 

Now the ACF of {ty_miss;y_miss} is given by

Cy = nexcf(ty_miss, y_miss, ty_miss, y_miss, lag, H); 
plot(lag,Cy), xlabel('Lag'), ylabel('Auto-correlation C_{y}')

and the CCF, indicating in this example the lag and the magnitude of the coupling, is given by

Cxy = nexcf(tx, x, ty, y, lag, H);
plot(lag,Cxy), xlabel('Lag'), ylabel('Cross-correlation C_{xy}')

  

References

1. Rehfeld, K., Marwan, N., Heitzig, J., Kurths, J.: Comparison of correlation analysis techniques for irregularly sampled time series, Nonlin. Proc. Geophys., 18(3), 389-404, 2011.

Download

nexcf.m

Author

Kira Rehfeld