se of a Statistical Forecast Criterion to Evaluate Alternative ... - CiteSeerX

se of a Statistical Forecast Criterion to Evaluate Alternative Empirical Spatial Oscillation Pattern Decolnposition Nlethods in Climatological Fields Charles Kooperberg and Finbarr O'Sullivan Technical Report No. 276 Department of Statistics University of vVashington Seattle, WA 98195. August 1, 1994

Abstract ;-,patlal

n"Hpr"R

of oscillation in clllTlatolog;lcal r,~rnrclR aid

atrno:spllerlc dynamics. The

ph,~n()n1,en()loJi!;lcal understand-

identification of

oscillation patterns has

been realized using a collection of techniques including empirical orthogonal functions (EOFs) canonical correlation analyses (CCAs) and first order Markov

via principal oscillation

patterns (POPs). Each of these methods represent the motion of the field under study as a linear combination of a number of spatial patterns forced by its own characteristic temporal oscillation

In practice it can be difficult to determine which of these alternative ap-

is most appropriate for a given data set. Also for a given method the number of

nf"()ar.l1es

spatial patterns required to adequately capture the temporal and spatial motion of the field is often unclear. In this paper we consider the use of a statistical forecast criterion for resolving these issues. The approach is illustrated by applications of EOF, CCA and POPs to some well studied data from the northern hemisphere 500 hPa height record, as well as to some simulated data. The one-step ahead statistical forecast errors are also used to compare the performance of various decomposition methods.

1.

Introduction empirical study of fluid motion in atmospheric sciences and oceanography presents a number mtpr.',:;1.1nfY

st,l,ti:stiicaJ problems.

the physical principles and

of motion de-

scr'lorn,:; evolution of such systems over shorter periods are quite well established (e.g. Smagorinsky "Wallace and Hubbs 1977, and Holton 1992), summary characterization of the low frequency heh":VlCIT of the system in terms of spatial and temporal variation is more difficult except for simple

problems such as the wave equation in a homogeneous medium. Thus interest in the

of empirical methods which are motion over

few

has been substantial

of de:scrlbln,:;

errlpi:ricl,l identification of spatial patterns of te(:hnJqliJeS including empirical orthogonal patterns

has been realized (EO

a collection

correlations

(POPs).

these

il,H,ilJVSeS

are well established,

a

set it is often unclear which technique and in

form is most appropriate.

a

is often a substantial degree of subjectivity involved in

application of these

methods. In this paper we propose a statistical methodology based on a statistical forecast error The approach involves constructing 'optimal' one-step ahead

rules on half the

and then computing the performance of those rules on the remaining half of the data. The approach is applied to a 47 year record (January 1946 through June 1993) of Northern Hemisphere extra-tropical geopotential (500 hPa) height based on daily operational analyses from NationaJ Meteorological Center (NMC). The full resolution NMC grid is a 1977 point octagonal superimposed on a polar stereographic projection of the Northern Hemisphere, extending to 20 degrees north (Wallace et at.

1992). The data are projected onto a 445-point half-resolution

grid. The series was made approximately stationary by removing the mean field and the first three harmonics ofthe annual cycle, separately for every grid point. The data were then time-averaged to produce five-day averages. In this paper we examine the complete series of 3467 five-day averages, as well as the separate 47 winter (November-March) series and the combined 47 summer (MaySeptember) series of total length 1410 each, though for reasons that will become clear in Section

:1, our analyses in Section 2 will only involve the, first half of these data. We In

by

a brief description of three methods to determine the spatial patterns.

3 we

the idea of the one-step ahead statistical forecast error, as well as its

implementation. In Section 4 we apply this methodology to the geopotential height field, as well as to some TnT'pr,u,t

2.

error

\Ve conclude III

situations.

some

on

ahead statistical

2.1.

Empirical Orthogonal Functions (EOFs)

n'llflC:Ip,al ODlnpOl1e][lts

411d,lY~1~

IS

used in

a

Its development for multivariate time series

applications (e.g. Mardia et

a

anarV'SlS owes much to the work of of l:SclllJ'I:W;(3r and Priestley and co-authors (Brillinger 1981, . In the atmospheric science literature principal components are more frequently known as empirical orthogonal functions (Barnett and Preisendorfer 1987, Hore11981, Jolliffe 1986, Lorenz 19.56). Generalizations of EOF for the examination of coupled in fields, include canonical correlation analysis and singular value decomposition analysis (Barnett and Preisendorfer 1987, Jjn~tn(3rt(m

et al. 1991, Wallace et aZ. 1991).

Empirical orthogonal function analysis is based on the spectral decomposition of the (marginal) covariance of the measured field, i.e. 2: 0

Yare x( t)). This covariance matrix is Hermitian and its

decomposition is

p

2: 0

L

U DU*

djUjUj.

j=1

symbol

* denotes the transpose of the complex conjugate.

The matrix U is orthonormal, with

columns Uj for j = 1,2 .. . p, and D is a diagonal matrix with as elements the eigenvalues of 2: 0 ,

:2: O. The columns of U represent orthogonal linear combinations of the process with maximal marginal variance. Projecting onto the first

J(

columns gives

K

x(t)

L

(t)

+ 17K(t),

(l

k=1

t)

=

Note that

and

while 17K( t) is a vector of length p. For small. nUl110.~r

of

J( ~V4"l

g W

0

0>

'i

~

0

u..

EOF CCA POPs BP-CCA BP-POPs

'"'" g

-- ---

'"

r-

5

0

10

15

Number of Factors

Winter

§

'" 0>

E w g

'.,g"

II u..

'"'" '"

r-

EOF CCA pOPs BP-CCA BP-POPs

---------

- - - - --:::-_~-

-::::..--

---:..::. '":-::..-

---

EOFs 0 0

.,.-

l() 0)

....0 ........

0

0)

\\

r\ \.

\ \. \ .......

,, '\

w

-( J)

aJ 0

l()

OJ

....