Autocorrelation functions

Eric Weeks - personal pages - research - time series analysis

My Adventures in Chaotic Time Series Analysis

weeks@physics.emory.edu

1. Meet the time series
2. Fourier Transforms
3. Mutual information to find delay coordinates
4. Plotting attractors
5. Attractors in 3-D
6. Autocorrelation functions (you are here)
7. Poincare sections
8. 1-D Maps
9. More later. Perhaps fractal dimensions.

A0. Links and related information

For an explanation of what these pages are all about, select topic 1 above.

Thanks to Josko Poljak for prompting me to check the autocorrelation functions.

What is autocorrelation? Define the standard deviation sigma_x of a time series x as:

sigma_x = sqrt( < ( x - < x > )^2 > )

That is, take the difference between each point in a time series and the mean of that time series. Square this quantity, find the average over all points, and take the square root; that's the standard deviation. If I could get latex2html to work I'd put in a cleaner formula. The angle brackets in the above formula connote taking the average.

The correlation r between two time series x and y is then defined as:

r = < ( x- < x > ) ( y- < y > ) > / (sigma_x sigma_y)

The autocorrelation r(tau) is given by the correlation of the series with itself; use x(t) and x(t+tau) as the two time series in the correlation formula.

This is a fairly sketchy explanation; you might be interested in the Numerical Recipes explanation. I wrote a program to do the autocorrelation calculation; click here for more information. My program is more inefficient than the Numerical Recipes routine, but is public domain.

Lorenz attractor:

This was generated with the commands: cat lorenz.gz | gawk '{print $2}' | autocor -t 0.01 -d500 | psdraw -l0.05 -S 10 10 -X - - -1 1 -Z 255 0 128 -A -y 0.5 -a "time" -b "correlation" -T "Autocorrelation - Lorenz" < autolor.ps

Rössler attractor:

This was made with:
cat rossler.dat | autocor -t 0.1 -d1000 | psdraw -l0.05 -S 10 10 -X - - -1 1 -Z 0 128 128 -A -y 0.5 -a "time" -b "correlation" -T "Autocorrelation - Rossler" < autoros.ps

Hénon attractor:

This was produced with the command: cat ../henondat.gz | autocor -d30 | psdraw -l0.05 -S 10 10 -X - - -1 1 -Z 128 25 0 -A -y 0.5 -a "time" -b "correlation" -T "Autocorrelation - Henon" < autohen.ps

Experimental data: periodic

Generated with the commands:

cat exptper.dat | autocor -t 0.1 -d 500 | psdraw -l0.05 -S 10 10 -X - - -1 1 -Z 0 0 255 -A -y 0.5 -a "time (s)" -b "correlation" -T "Autocorrelation - expt periodic" < autoper.ps

Experimental data: quasi-periodic-2

Generated with the commands:

cat exptqp2.dat | autocor -t 0.1 -d 1000 | psdraw -l0.05 -S 10 10 -X - - -1 1 -Z 0 0 255 -A -y 0.5 -a "time (s)" -b "correlation" -T "Autocorrelation - expt qp2" < autoqp2.ps

Experimental data: quasi-periodic-3

Generated with the commands:

cat exptqp3.dat | autocor -t 0.4 -d 250 | psdraw -l0.05 -S 10 10 -X - - -1 1 -Z 0 0 255 -A -y 0.5 -a "time (s)" -b "correlation" -T "Autocorrelation - expt qp3" < autoqp3.ps

Experimental data: chaotic (?)

Generated with the commands:

cat exptcha.dat | autocor -t 0.1 -d 1000 | psdraw -l0.05 -S 10 10 -X - - -1 1 -Z 0 0 255 -A -y 0.5 -a "time (s)" -b "correlation" -T "Autocorrelation - expt chaotic" < autocha.ps

The oscillation is because there is still some periodic component to the data, similar to the Rossler attractor.

So far the analysis has given me several different times. From the Fourier transform I have certain characteristic periods. From the mutual information calculation I have the first minimum. I also have the autocorrelation first zero-crossing.

System Period Mutual info
1st minimum Mutual info
1st maximum Autocorrelation
1st zero-crossing

Lorenz Attractor ? 0.16 0.22 2.09

Rössler Attractor 6.07 1.52 3.10 1.50

Hénon Map 2.2 "1" none "1"

Expt-periodic 6.9 0.9 2.1 1.7

Expt-q-periodic-2 13.0 2.5 6.8 2.9

Expt-q-periodic-3 6.1 2.0 3.2 2.0

Expt-chaotic 6.5 2.2 3.5 2.1

System	Period	Mutual info 1st minimum	Mutual info 1st maximum	Autocorrelation 1st zero-crossing

Lorenz Attractor	?	0.16	0.22	2.09
Rössler Attractor	6.07	1.52	3.10	1.50
Hénon Map	2.2	"1"	none	"1"
Expt-periodic	6.9	0.9	2.1	1.7
Expt-q-periodic-2	13.0	2.5	6.8	2.9
Expt-q-periodic-3	6.1	2.0	3.2	2.0
Expt-chaotic	6.5	2.2	3.5	2.1

Note that all times are in natural units (seconds for experimental data, iterations for Hénon map, the time step of the equations for the Rössler and Lorenz systems). Links in the table will take you to the appropriate location where the information was found.

Previous page: 3-D plots of strange attractors
Next page: Nothing, yet. Something, soon. Probably Poincare sections.

Links...

Graphs were made with a program I wrote, psdraw. The program makes PostScript output which I then converted to GIF. Rotations & 3-D boxes done with rotate, another program I wrote. Both psdraw and rotate are public domain.
If you have any questions or comments, send me email; email address below.

Current address:
Eric R. Weeks
weeks@physics.emory.edu
Department of Physics
Emory University
Atlanta, GA 30322-2430