Detecting functional relationships between simultaneous time series
C. L. Goodridge, L. M. Pecora, T. L. Carroll, and F. J. Rachford
Code 6345, U.S. Naval Research Laboratory, Washington, D.C. 20375
Received 12 September 2000; revised manuscript received March 8 2001; published 23 July 2001
We describe a method to characterize the predictability and functionality between two simultaneously gen-
erated time series. This nonlinear method requires minimal assumptions and can be applied to data measured
either from coupled systems or from different positions on a spatially extended system. This analysis generates
a function statistic,
c
0, that quantifies the level of predictability between two time series. We illustrate the
utility of this procedure by presenting results from a computer simulation and two experimental systems.
DOI: 10.1103/PhysRevE.64.026221 PACS numbers: 05.45.Tp, 05.10.-a
I. INTRODUCTION
A common challenge encountered by experimentalists in
nonlinear dynamics is how to relate pairs of time series, such
as those measured from two points on a spatially extended
system or from two coupled systems. Many nonlinear sys-
tems exhibit spatial as well as temporal dynamics and an
understanding of the spatial behavior is often vital to under-
standing the overall dynamics 1. Examples of such spa-
tiotemporal systems are ‘‘auto-oscillations’’ of magnetostatic
spin wave modes in ferrimagnetic films 2, the response of a
magnetostrictive ribbon to ac magnetic fields, and fluid mo-
tion in Taylor-Coquette flow 3 or Rayleigh-Bernard con-
vection 4. One way to characterize the spatial dynamics is
to simultaneously monitor a property of the system at two
different positions and determine the relationship between
the resulting data. The relationship between simultaneous
time series may also describe properties of the coupling be-
tween two coupled systems. A wide variety of linear tech-
niques are available to investigate the functionality between
concurrent time series, but these techniques often fail to pro-
vide any useful information if the relationship is nonlinear.
In this paper we will describe a general nonlinear tech-
nique that investigates the functionality between pairs of
time series with minimal assumptions about the nature of
either the data or the dynamics. This generality allows this
technique to be applied to a wide range of experimental sys-
tems and to account for more general functionality than
strictly linear. The result of this analysis is a function statis-
tic,
c
0, that quantifies the predictability and functionality
between the two time series and can be compared to results
from linear techniques such as the cross correlation. This
technique may be useful to experimentalists with time series
data as well as provide another tool for general time series
analysts.
The procedure builds on techniques designed to investi-
gate functionality between time series 5, especially those of
several of the authors 6,7. These earlier procedures calcu-
late statistics that quantify certain properties of functions re-
lating time series such as continuity or differentiability. This
analysis provides a way to calculate a function statistic that
is a measure of the predictability between the time series.
Roughly speaking, this statistic quantifies how well can we
predict the behavior of one time series if we know the be-
havior of the other time series. This technique can be ex-
tended to investigate the nature of the functional relationship
between the two time series by testing for nonlinearity in the
function. One important aspect of this technique is that it
uses the data to establish a limiting length scale—a limit of
relevance 8—rather than intuition or knowledge about the
system. This analysis is also general enough to be applicable
to both experimental and computational results.
II. PROCEDURE
Given two simultaneous time series h
i
, g
i
9, we con-
struct vectors x
i
and y
i
and attractors X and Y, such that
x
i
= h
i
, h
i +
,..., h
i +
* d -1
X
and
y
i
= g
i
, g
i +
,..., g
i +
* d -1
Y,
by time delay embedding. The parameters of the embedding,
the time delay and the embedding dimension d, are deter-
mined using the minimum of the autocorrelation function 1
and the false nearest neighbor algorithm of Abarbanel 10,
respectively 11. However, any combination of d and that
adequately captures the dynamics of the system should yield
useful results. Next we assume that a function F exists such
that y
i
=F ( x
i
). Function F is assumed to be continuous but
no other conditions are imposed. Since the determination of
F may not be trivial, an intermediate goal is to investigate
properties of the function. We will calculate a function sta-
tistic that allows us to describe whether function F actually
exists, how accurately we can make predictions between
time series, and if the function is nonlinear. To derive this
statistic, we assume that nearby points on X map to nearby
points on Y see Fig. 1, provided F exists. This behavior is
equivalent to the two time series being related by a continu-
ous function. Our function statistic,
c
0, is a measure of the
local predictability between the two time series. A high value
indicates strong predictability between the time series.
Here is an outline of the procedure to calculate
c
0. In all
of the following, we assume that we have measured the data
in such a way that h
i
and g
i
are sampled simultaneously and
we define x
i
and y
i
as corresponding points if the indices of
the first coordinates are equal. We systematically investigate
PHYSICAL REVIEW E, VOLUME 64, 026221
1063-651X/2001/642/02622110/$20.00 ©2001 The American Physical Society 64 026221-1