260
Chapter 14: Patterns in Time: SSA and MSSA
of freedom than spatial ones, allowing the development of interesting spectral
properties, whereas in EEOFs, the state vector contains only a few'lags and a
large number of spatial points. In both cases, one seeks the dominant spacetime patterns of the analysed signals, taking into account both their spatial
and temporal correlations.
By contrast with EOF analysis, time or space-time patterns provide dynamical information about the underlying physical system, that is, about the
evolution of the state vector. SSA was originally devised by Broomhead and
King (1986) as a method of dynamical reconstructionj Without any knowledge of the evolution equations of a dynamical system, one can reconstruct
its complete dynamics with the sole knowledge of a large set of observations
from only one variable, under certain hypothesis. In Section 14.2, we recall
the main motivations for the development of SSA within the framework of
dynamical systems and we define the delay-coordinate space which is the
space where the application of EOF expansion is called SSA. SSA and MSSA
turn out to be powerful statistical tools. Their characteristics and properties
are developed in Section 14.3. Two applications of MSSA are presented in
Section 14.4.
14.2 Reconstruction and Approximation of
Attractors
14.2.1 The Embedding Problem
The climate's time evolution is quantified by a set of partial differential equations in a manner similar to other physical systems. Schematically speaking,
these equations can be written in the form of a continuous dynamical system:
8X t = F(X)
8t
t
(14.1)
where X is the vector describing the state of the climate at a given instant.
Such a system has the fundamental deterministic property that, given the
state vector Xt at time t, only one state vector Xt+, is possible at a future
time t+8. When computers are to be used to approximate solutions of (14.1),
both time and space discretisation are applied, so that the system (14.1) is
transformed into a discrete dynamical system,
(14.2)
where ß is the integration time step. Spatial discretisation enforces Xt to
be a vector of finite dimension m. In the case of the climatic system, the
function G is a nonlinear function that contains forcing and dissipation. In
that case, the dynamical system (14.2) is generally chaotic, that is, has an
evolution that does not tend, as t goes to infinity, to be stationary or periodic,
Chapter 14: Patterns in Time: SSA and MSSA
of freedom than spatial ones, allowing the development of interesting spectral
properties, whereas in EEOFs, the state vector contains only a few'lags and a
large number of spatial points. In both cases, one seeks the dominant spacetime patterns of the analysed signals, taking into account both their spatial
and temporal correlations.
By contrast with EOF analysis, time or space-time patterns provide dynamical information about the underlying physical system, that is, about the
evolution of the state vector. SSA was originally devised by Broomhead and
King (1986) as a method of dynamical reconstructionj Without any knowledge of the evolution equations of a dynamical system, one can reconstruct
its complete dynamics with the sole knowledge of a large set of observations
from only one variable, under certain hypothesis. In Section 14.2, we recall
the main motivations for the development of SSA within the framework of
dynamical systems and we define the delay-coordinate space which is the
space where the application of EOF expansion is called SSA. SSA and MSSA
turn out to be powerful statistical tools. Their characteristics and properties
are developed in Section 14.3. Two applications of MSSA are presented in
Section 14.4.
14.2 Reconstruction and Approximation of
Attractors
14.2.1 The Embedding Problem
The climate's time evolution is quantified by a set of partial differential equations in a manner similar to other physical systems. Schematically speaking,
these equations can be written in the form of a continuous dynamical system:
8X t = F(X)
8t
t
(14.1)
where X is the vector describing the state of the climate at a given instant.
Such a system has the fundamental deterministic property that, given the
state vector Xt at time t, only one state vector Xt+, is possible at a future
time t+8. When computers are to be used to approximate solutions of (14.1),
both time and space discretisation are applied, so that the system (14.1) is
transformed into a discrete dynamical system,
(14.2)
where ß is the integration time step. Spatial discretisation enforces Xt to
be a vector of finite dimension m. In the case of the climatic system, the
function G is a nonlinear function that contains forcing and dissipation. In
that case, the dynamical system (14.2) is generally chaotic, that is, has an
evolution that does not tend, as t goes to infinity, to be stationary or periodic,