its use should be guided by the problem under 13.6.1 General. The Singular Systems Anal-

consideration. ysis (SSA; see Vautard, Yiou, and Ghil [381]

Jolliffe [198] points out that rotation should be or Vautard [380]) and the Multichannel Singular

used routinely for subsets of EOFs that have equal, Spectrum Analysis (MSSA, see Plaut and Vau-

or near-equal, eigenvalues. The corresponding tard [317]) are time series analysis techniques

EOFs are not well de¬ned because of their used to identify recurrent patterns in univariate

degeneracy (cf. [13.1.8]), and thus the patterns time series (SSA) and multivariate time series

contained by the degenerate EOFs may be (MSSA). Mathematically, SSA and MSSA are

arbitrarily rotated within the space that they span. variants of conventional EOF analysis, but the

The sensitivity of the rotation to the normalization application of the mathematics is markedly dif-

of the EOFs becomes less relevant since all ferent. Vautard [380] reviews recent applications

eigenvalues are similar. of SSA and MSSA. Allen and colleagues [8, 9,

13.6: Singular Systems Analysis 313

10] have investigated various aspects of these are empirically determined averages (recall Sec-

tion 10.5) of length m. That is, ±k (t) is a ¬ltered27

methods.

version of the original time series Xt , with ¬lter

13.6.2 Singular Systems Analysis. Univariate weights that are given by the kth eigenvector.

time series Xt are considered in SSA. An m- When Xt is dominated by high-frequency varia-

dimensional vector time series Yt is derived from tions, the dominant eigenvectors will be high-pass

¬lters, and when most of the variance of Xt

Xt by setting:

is concentrated at low frequencies the dominant

Yt = (Xt , Xt+1 , . . . , Xt+m’1 ) .

T

(13.54) eigenvectors will act as low-pass ¬lters. The eigen-

A Singular Systems Analysis is an EOF analysis of vectors will generally not form symmetric ¬lters.

Thus we need to be aware that operation (13.57)

Yt .

The vector space occupied by Yt is called the causes a frequency-dependent phase shift.

As with ordinary EOF analysis, SSA distributes

delay-coordinate space.

The (zero lag) covariance matrix of Y, ΣY Y = the total variance of Yt to the m eigenvalues »i .

Cov Yt , Yt , is a T¨ plitz matrix.26 Element ( j, k) The total variance of Yt is equal to m times the

o

of Σ , say σ , is the covariance between the variance of Xt . Thus

YY jk

m

jth element of Yt (Xt+ j’1 ) and its kth element

»i = m Var(Xt ). (13.58)

Xt+k’1 . Thus

i=1

σ jk = γx x (| j ’ k|), The vector-matrix version of (13.57) is

±(t) = P Yt

where γx x (·) is the auto-correlation function of Xt .

where ±(t) and P are de¬ned in the usual

All off-diagonal elements of ΣY Y are identi¬ed

by |i ’ j| = „ and have the same value γx x („ ). way. Thus the auto-correlation function of the

multivariate coef¬cient process ±(t) is related to

Thus, matrix ΣY Y is band-structured and contains

all auto-covariances of Xt up to lag m ’ 1. The the auto-correlation function of Xt by

covariance and correlation matrices of Yt differ by

Σ±± („ ) = PΣY Y („ )P T

2

only a constant factor (1/σ X ). They therefore have

where ΣY Y („ ) is the matrix whose (i, j)th entry is

the same eigenvectors. The eigenvalues of the two

given by

matrices differ by the same constant factor.

The eigenvectors e i of ΣY Y , sometimes called [ΣY Y („ )]i, j = γx x („ + j ’ i). (13.59)

time EOFs, are interpreted as a sequence in time.

Note that

Each eigenvector e i is a normalized sequence of

Σ±± (0) = = diag(»1 , . . . , »m ).

m time-ordered numbers,

T

e i = e0 , . . . , em’1 ,

i i

(13.55) 13.6.3 Reconstruction in the Time Domain.

Also, as with ordinary EOFs,

that may be understood as a ˜typical™ sequence m

of events. The orthogonality of the eigenvectors Y = ±i (t)e i . (13.60)

t

Tk

= δ jk ,

in the delay-coordinate space, e j e i=1

is equivalent to the temporal orthogonality of Thus, using equation (13.60) to expand

j j

any two typical sequences (e0 , . . . , em’1 ) and Yt , Yt’1 , . . . , Yt’m+1 , we ¬nd that Xt

(e0 , . . . , em’1 ):

k k has m equivalent time expansions in the m

˜SSA-signals™:

m’1

m

j

ei eik = δ jk . (13.56)

Xt = ±i (t)e1

i

i=0

i=1

The EOF coef¬cients m

= ±i (t ’ 1)e2

i

m’1

±k (t) = yt , e k = Xt+i eik i=1

(13.57)

.

.

i=0

. (13.61)

26 The elements on each diagonal of a T¨ plitz matrix are m

o

= ±i (t ’ m + 1)em .

i

equal. That is, if A is an m — m matrix and if there are

constants c(’(n’1)) , . . . , c(n’1) such that Ai, j = c j’i , then i=1

A is T¨ plitz. Graybill [148] describes some of their properties

o

27 See Section 17.5.

(see Section 8.15).

13: Empirical Orthogonal Functions

314

Each of these expansions distributes the variance

1.0

1 1

of the SSA-signals differently. In fact, using the 1 1

1 1

orthogonality of the EOF coef¬cients, it is easily

0.5

2

shown that 2

3 3

m 2 4 4 4

2

Var(Xt ) = »i eki (13.62)

0.0

3 3

i=1

2

4 4 4

3 3

k.28

for all If we consider the normalized