Analysis of spatiotemporal signals: A method based on perturbation theory
A. Hutt,
1,
* C. Uhl,
1
and R. Friedrich
2
1
Max-Planck-Institute of Cognitive Neuroscience, Stephanstrasse 1a, 04103 Leipzig, Germany
2
Institute for Theoretical Physics, University of Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
Received 18 December 1998
We present a method of analyzing spatiotemporal signals with respect to its underlying dynamics. The
algorithm aims at the determination of spatial modes and a criterion for the number of interacting modes.
Simultaneously, a way of filtering of nonorthogonal noise is shown. The method is discussed by examples of
simulated stable fixpoints and the Lorenz attractor. S1063-651X9901908-X
PACS numbers: 05.45.-a, 02.50.Sk
I. INTRODUCTION
In various scientific fields the analysis of spatiotemporal
patterns emerging from complex systems plays an important
role. An investigation of measured multidimensional data al-
lows us to learn more about the internal dynamics of the
system. It represents the basis for microscopic modeling of
interactions in investigated systems e.g., 1. Some typical
fields of application are chemical reactions 2, meteorology
e.g., 3 and hydrodynamics 4 or biological systems as
analyzing electroencephalography EEG or magnetoen-
cephalography MEG data 5–7.
Depending on the intended use, different kinds of data
processing techniques can be applied. An often used method
for linear data analysis is known as principal component
analysis PCA8 or Karhunen-Loe
`
ve expansion. Spatial
modes are calculated based on maximizing signal projections
on these modes. It leads to orthogonal spatial and temporal
modes and gives a measure for the contribution of each
mode to the signal. Modes with a signal contribution above a
certain threshold are considered as relevant, those below the
threshold as irrelevant. However, this method fails to sepa-
rate signal from noise, if signal and noise are not orthogonal
on each other, and if noise parts contribute more than parts
of the relevant signal to the data. Furthermore an estimation
of the number of interacting modes depends on the choice of
the threshold. Underlying dynamic structures are neglected
by this linear data technique.
A nonlinear approach aiming at extracting interacting
modes and the underlying dynamics has been presented, e.g.,
in 9,10. However, the numerical effort of these nonlinear
approaches is considerably high, especially with an increas-
ing dimensionality of the underlying dynamical system. An
estimation of the number of interacting modes is also still an
open question.
In this paper we will present a nonlinear technique based
on linear perturbation theory, which focuses on internal
deterministic dynamic patterns and extracts signal dynamics
from noisy data sets. It improves PCA suspending the con-
dition of orthogonality and allows an objective estimation of
interacting spatial modes. Due to the linear equations to be
solved, the method leads to a fast and robust algorithm.
The perturbational approach is based on a ground state of
the PCA modes, which represents the exact solution of mini-
mizing a cost function leading to a complete orthogonal ba-
sis. We introduce a perturbation by an additional term in the
cost function for a determination of signal dynamics. Using a
mathematical methodology similar to Hartree and Fock
11,12, we obtain dynamically coupled spatial modes. A
criterion for the estimation of the number of interacting
modes can be derived. We obtain the relevant signal sub-
space independent of an orthogonality relation between sig-
nal and noise, due to our special choice of a biorthogonal
basis.
II. METHOD
A. Principal component analysis „PCA…
A N-dimensional spatiotemporal signal can be described
by a vector q( t ) of dimension N. In order to determine sig-
nificant parts of the signal, one can decompose the signal
into spatial and temporal modes v
i
and x
i
( t ) by PCA. The
properties are determined by a cost function
V =
i =1
N
q t - q–v
i
v
i
2
q
2
+
i , j =1
N
ij
v
i
• v
j
-
ij
,
V
v
k
=0, 2.1
where ¯ denotes time average and
ij
are Lagrange mul-
tipliers to fulfill the orthogonality constraint.
Standard cost functions of PCA lead to degenerated solu-
tion spaces. To obtain the known equations of PCA directly,
here one sums up the single errors to the signal and fixes the
amplitudes as projections on the modes. This breaks the in-
variance with respect to linear transformation.
It leads to an eigenvalue problem
Cv
k
=
k
v
k
, 2.2
with C= q( t ) q( t ) / q
2
, orthogonal spatial modes v
k
and
amplitudes x
k
( t ) =q( t ) • v
k
, where denotes the dyadic
product. They obey Eqs. 2.3,
v
k
• v
l
=
kl
, x
k
t x
l
t =
k
kl
. 2.3 *Electronic address: hutt@cns.mpg.de
PHYSICAL REVIEW E AUGUST 1999 VOLUME 60, NUMBER 2
PRE 60 1063-651X/99/602/13509/$15.00 1350 © 1999 The American Physical Society