Eric Weeks
 personal pages  research
 time series analysis
My Adventures in Chaotic Time Series Analysis 
weeks@physics.emory.edu 
A0. Links and related information
For an explanation of what these pages are all about, select topic 1 above.
This page: top  lorenz  rossler  henon  expt: periodic  qperiodic2  qperiodic3  chaotic  bottom
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.
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
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
This page:
top 
lorenz 
rossler 
henon 
expt: periodic 
qperiodic2 
qperiodic3 
chaotic 
bottom
Hénon:
time series

power spectrum

mutual information

attractor

attractor 3D

autocorrelation

poincare

1D maps
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
This page:
top 
lorenz 
rossler 
henon 
expt: periodic 
qperiodic2 
qperiodic3 
chaotic 
bottom
Experimental/periodic:
time series

power spectrum

mutual information

attractor

attractor 3D

autocorrelation

poincare

1D maps
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
This page:
top 
lorenz 
rossler 
henon 
expt: periodic 
qperiodic2 
qperiodic3 
chaotic 
bottom
Experimental/quasiperiodic2:
time series

power spectrum

mutual information

attractor

attractor 3D

autocorrelation

poincare

1D maps
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
This page:
top 
lorenz 
rossler 
henon 
expt: periodic 
qperiodic2 
qperiodic3 
chaotic 
bottom
Experimental/quasiperiodic3:
time series

power spectrum

mutual information

attractor

attractor 3D

autocorrelation

poincare

1D maps
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
This page:
top 
lorenz 
rossler 
henon 
expt: periodic 
qperiodic2 
qperiodic3 
chaotic 
bottom
Experimental/chaotic:
time series

power spectrum

mutual information

attractor

attractor 3D

autocorrelation

poincare

1D maps
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 zerocrossing.
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: 3D plots of strange attractors
Next page: Nothing, yet. Something, soon. Probably Poincare
sections.
This page: top  lorenz  rossler  henon  expt: periodic  qperiodic2  qperiodic3  chaotic  bottom