advanced spectral methods for climatic time series - Atmospheric ...

40, 1 / REVIEWS OF GEOPHYSICS Ghil et al.: CLIMATIC TIME SERIES ANALYSIS ● 3-3
TABLE 1. (continued)
Symbol Definition Method Section
nt discretely sampled time 1
Uk(f) W
discrete Fourier transform of wk(t) sliding-window length of wavelet
MTM
SSA
3.4
2.4
W(a, b)
W(k) Wm(k) wk(t) Xˆ (t)
wavelet transform of {X(t)} using b-translated and a-dilated ((t b)/a)
lag window for S
X(f) with smoothing parameter
Bartlett window for S
X(f) with window length m
kth taper
observed time series
SSA
BT
BT
MTM
2.4
3.2
3.2
3.4
1
Xn X
X(nt) 1
(p)
pth-order differentiation of X(t) with respect to time t, d p X/dt p
1
{X(t)} univariate time series in t or t
X˜(t) M-dimensional augmented vector of X(t)
1
2.1
X0 XI(t) X˜(t), X˜(nt)
mean of {X(t)}
reconstructed X(t)
continuous- and discrete-time reconstructed signal
SSA
WLTs
MTM
2.3
2.4
3.4
{X(t)} multivariate time series in t or t
{X˜
l}
Yˆ
k(f)
multichannel augmented vector of {X˜(t)}
discrete Fourier transform of {X(t)w k(t)}
time-scale ratio
POP
SSA
MTM
SSA
4.1
4.2
3.4
2.4
(t) additive noise MTM 3.4
explained contribution to variance MTM 3.4
k, X R k
kth eigenvalue and eigenvalue matrix of CX projection of CR onto EX weight of kth taper
number of degrees of freedom of spectrum
SSA
SSA
MTM
MTM
2.2
2.3
3.4
3.4
(t) random forcing 1
k, EX kth eigenvector and eigenvector matrix of CX SSA 2.2
2
variance of random forcing 1
X(k), ˆ
X(k)
(x)
characteristic delay time
true and estimated lag k autocorrelation function of X(t)
mother wavelet of variable x
SSA
BT
WLTs
2.3
3.2
2.4
unexplained contribution to variance MTM 3.4
a Method and section number correspond to where the symbol appears first or is used differently from previous sections.
tifying certain features of the time series. These properties
can then help understand the system’s behavior and
predict its future.
[3] To illustrate the ideas and methods reviewed here,
we shall turn to one of the best known climatic time
series. This time series is made up of monthly values of
the Southern Oscillation Index (SOI). It will be introduced
in section 2.2 and is shown in Figure 2 there.
[4] At this point we merely note that physical processes
usually operate in continuous time. Most measurements,
though, are done and recorded in discrete
time. Thus the SOI time series, as well as most climatic
and other geophysical time series, are available in discrete
time.
1.1. Analysis in the Time Domain Versus the
Spectral Domain
[5] Two basic approaches to time series analysis are
associated with the time domain or the spectral domain.
We present them at first in the linear context in which
the physical sciences have operated for most of the last
two centuries. In this context the physical system can be
described by a linear ordinary differential equation
(ODE), or a system of such equations, subject to additive
random forcing.
[6] It goes well beyond the scope of this review paper
to introduce the concepts of random variables, stochastic
processes, and stochastic differential equations. We refer
the interested reader to Feller [1968, 1971] for the
former two concepts and to Arnold [1974] and Schuss
[1980] for the latter. Many of the standard books on
classical spectral methods that are cited in sections 3.1
and 3.2 also contain good elementary introductions to
stochastic processes in discrete and, sometimes, continuous
time.
[7] We concentrate here on time series in discrete
time and consider therefore first the simple case of a
scalar, linear ordinary difference equation with random
forcing,
M
Xt 1 aj Xt M j t. (1)
j1
Its constant coefficients a j determine the solutions X(t)
at discrete times t 0, 1, , n, . In (1) the random
forcing (t) is assumed to be white in time, i.e., uncor-