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

atrno:spllerlc dynamics. The 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

nf"()ar.l1es is most appropriate for a given data set. Also for a given method the number of

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 has been substantial

interest in the of empirical methods which are of de:scrlbln,:;

motion over few

errlpi:ricl,l identification of spatial patterns of has been realized a collection

te(:hnJqliJeS including empirical orthogonal (EO correlations il,H,ilJVSeS

patterns (POPs). these 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 (May­

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

In 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 \Ve conclude some on ahead statistical

TnT'pr,u,t error III situations.

2.

2.1. Empirical Orthogonal Functions (EOFs)

n'llflC:Ip,al ODlnpOl1e][lts 411d,lY~1~ IS a used in

a applications (e.g. Mardia et

anarV'SlS owes much to the work of of l:SclllJ'I:W;(3r

Its development for multivariate time series

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 isp

2:0 UDU* L 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) Lk=1

(t) + 17K(t), (l

t) = and Note that Zk( t) is a scalar (possibly complex

while 17K(t) is a vector of length p. For J( sufficiently large, the total marginal variance

small. this we a representation of the process in terms a sum

nUl110.~r of J( ~V4"l<M palGtelms statistically

two for combined series, the first 24 summer series first 24 winter series. From

L·.~... ~.~ 2 we note that the first EOF in the winter and the summer are quite different. In particula.r,

first two EOFs in the winter are very to first two EOFs for the combined series.

pattE~rn are recognized as mixtures of North Atlantic Oscillation (NAO) and Pacific

mJ:'rl£'"n (PNA) pattern. These patterns are well in the atmospheric sciences

(Barnston and Livezey 1987, Dickson and Namias 1976, Wallace and Gutzler 1981). The fact

both EOFs are combinations of the NAO and PNA pattern suggests that from an interprative

point of view it may be useful to rotate the EOFs (Barnston and Livezey 1987, Richman 1988).

Vile also computed the EOFs based on the second half of the data (1970-1994). The results

(not shown) were compared to the patterns in Figure 2. The first EOF for the second half of the

combined series the summer series and the winter series, looked very much like the corresponding

patterns ba.sed on the first half of the series. The second EOF, however, did not resemble the

corresponding patterns for the first half. They seemed to be a combination of the second to fourth

EO F for the first half.

IFigures 1 and 2 about here I

2.2. Canonical Correlations Analysis (CCA)

Anotller multivariate statistical method that ha.s been widely used for the identification of spatial

n"1Lt"'rn,, is canonical correlation analysis. 'When analyzing two fields x and y canonical correlation

a b so that the marginal correlation between a*x(t) and b*y(t) is

In the atmospheric sciences, this has found a useful technique for analyzing two

c011pJed fields Barnett Preisendorfer Brethertorr et at. 1991). When only one

tecltlm''lue can the carLOnlCal correl''l.tlCll1 p,atterrlS bl~tween

LS

orthonormal and D is a diagonal matrix with as elements the singular values of 1:~1/21:1

Bretherton et ai. (1991) for more details.

U and V are not identicaL In particular, U are spatial patterns at

time t that are correlated with spatial patterns V at time t + 1. However we would expect the

dominant patterns at time t, U to look like the dominant patterns at time t + 1, V. For the 500mb

height field this is indeed the case.

The patterns that are obtained from (2) are often not very satisfactory. This is because the

estimate of 1:0 is often quite noisy, in fact, it is often almost singular. Thus when inverted, the

smallest, and least interesting, eigenvalues and eigenvectors of 1:0 dominate the decomposition.

To circumvent this problem, it is customary to first project the data on the first k empirical

orthogonal functions of x(t), before applying the canonical correlations algorithm. This is known

as the Barnett-Preisendorfer (BP) approach to CCA (Barnett and Preisendorfer 1987). Bretherton

et ai. (1991) suggest to take k such that the first k EOFs explain between about 70 and 80% of

the variance in the field.

Either for regular CCA or for BP-CCA, we can decompose the field in a spatial-temporal way,

identical to (1), on basis of U or on basis of V. For the 5-day lag that we studied, typically U and

V are very similar. In particular, in Figure 3 we show the first two patterns at time t, U1 and U2 ,

for the first half of the complete series, the first 24 summer series and the first 24 winter series

using BP-CCA. In fact, the first two patterns at time t+ 1, VI and V2 were extremely similar to U1

and U2 in all three cases. For the combined series we projected the data on the first 23 empirical

orthogonal functions, which explain 80.1% of variation. For the summer series we projected

on the first 26 EOFs, which explain 80.0% of the variance, and for the winter series we projected

on first 19 EOFs, which explain 79.6% of the variance.

As can be seen more than

res:uI1~s were dltterent. though they seemed to convey the same information.

2.3. Principal Oscillation Patterns (POPs)

This technique was introduced by Hasselmann (1988) and by Von Storch et ai. (1988) and has

become widely used in the atmospheric science literature. Its extension to complex fields is rec:entiy

explored by Biirger (1993). The method is based on an AR(l) representation for the process

x(t) = Bx(t - 1) + t(t)

where t(t) is assumed to be a mean zero uncorrelated (in time) process. The coefficient matrix B

is expanded in terms of its eigenvectors, Uj, and corresponding eigenvalues Aj, both of which may

be complex. The eigenvalues are ordered by modulus, IAll 2': IA21 2': ... 2': IApl.

Typically, the coefficient matrix B is estimated by iJ = 2:1 2:01. In particular, the decomposition

of iJ is nowA -1

B U = 2:1 2:0 U = UD. (3)

It is interesting to note the similarity between the CCA decomposition (2) and the POPs decom­

position (3). Typically, some of the eigenvectors of iJ are real, while some pairs may be complex.

Von Storch et al. (1988) interpret the real eigenvectors as standing oscillatory patterns while they

interpret the complex eigenvectors as propagating patterns. If a pair of eigenvectors is complex

can be chosen freely. As in Von Storch et ai. (1988), we will choose this phase such

that the norm of the real component of the eigenvectors is maximized.

"'!J'1.vl.Xl p,atterrlS derived from EOF and eie:en'vec:tors of iJ are

VV""llJLC' to de(:onap()se in terms the D::l.lttp·rns squares

J( malGnX consist,ing

interpretations of the spatial pattE~rns.

U - 1) is the residual. By construction, if the moduli of the

values of B P'j I) become small for j, then for J( the

of is negligible. Thus the coherent time variation of is captured in terms of the complex

"'11<1,41<U patterns Uk being forced by complex AR(l) the POPs represen-

tation above in which the Zk'S are defined by least squares regression differs from the prescription

by Von Storch et. al. (1988) which use of the spectral decomposition of B. We use

regression approach to improve the correlation with the original field. This improves the utility

the PO Ps representation for the purposes of statistical forecasting.

The inverse of the covariance matrix in (3) makes it evident that POPs may have the same

defect as CCA, caused by the smallest eigenvalues of the sample covariance matrix 2::0 • Therefore,

the common approach to POPs is to project the data on the first k empirical orthogonal functions

before computing the principal oscillation patterns. Because of the similarity with the BP approach

to eCA, we will call this the Barnett-Preisendorfer approa.ch to POPs (BP-POPs). In Figure 4 we

show the first to principal oscillation patterns for the first half of the complete series, the summer

series and the winter series. Except for the second pattern during the summer, for which we show

real part of the pattern, all of these patterns turned out to be real.

As can be seen from Figure 4, the principal oscillation patterns suffer from the same drawback

as the CeA patterns: they are considerably more noisy than the EOF patterns. Furthermore, the

POPs patterns explain even less variance than the eeA patterns.

in EOF, eeA and POPs are quite different.

F """fi-"",..e are <1eTme<1 \\fJtrl011lt ",,,«~,"r1 to the structure of forcing terms's.

3. The one-step ahead statistical forecast error

3.1. Definition

Which method to use? Suppose that one is analyzing a climatological field, should one use EOF,

POPs, BP-CCA or BP-POPs to compute the spatial patterns that determine a spatial­

temporal decomposition? Another question is how many patterns one should consider? Although

one would typically only interpret a 'few patterns, for a larger climatological modeling effort one

would want to include any patterns that have more signal than noise. Indeed, the number of

:SPi:\,\,Ji:Ll oscillation patterns, K, acts as a regularization parameter. If K is too small there will be

substantial bias in the model; on the other hand if K is too large the model will tend to overfit the

data. As one approach in answering these two questions we consider the one-step ahead statistical

forecast error.

Let x(t) be a climatological field, and let U be a p X P matrix, containing the spatial patterns

aB columns in decreasing order of importance. Let WK be the p X K matrix consisting of the first

K columns of U. Let zK(t) be the least squares projection of x(t) on WK obtained using (4). Note

that (4) reduces to zK(t) = Wj{x(t) when U is orthonormal, as is the case for the EOF and CCA

decompositions.

Suppose that we know the temporal oscillation patterns up to time t, and we wish to predict

complete field at time t + 1, x(t + 1). A measure of how good of a summary of the field zK

WK are would be the difference between the prediction x( t + 1) of x(t + 1) on the basis of

information contained in the projection ofx(t) onto the patterns WK, i.e. zK(s) for s:S; t. To

formalize let FE(K, U) be the expected value of the squared norm of the prediction error,

+ 1) - x(t + 1)11 2/Ellx(t + 12. The of F E(K, U) is, that it is the

Tr ,,,r-Tl rUT of variance that cannot be explained if ahead based upon

ml0rll1atlon U"JH1,i:LUleU m on <

of a decomposition U we are going to use, we cn()m;e that va.lue of J( that minimizes

U).

HpfoT'P we can apply the idea of minimizing the OW,,-SlGen

to develop methods to predict x(t + 1) and to estimate the forecast error FE itself. These

are the topics of the next two sections.

3.2. Forecast Rules

If it is assumed that the process x( t) is stationary and Gaussian, the best predictor of x( t +1) with

respect to the forecast error is linear in zK. That is,

x(t + 1) =L <PIZK(t -I),1>0

(5)

where the <PI are p X K matrices of coefficients. We call 1 the lag. For a given set of lags we can

estimate the <PI using least squares.

In practice we have to limit the number of lags 1 for which <PI is estimated, since if we would

estimate <PI for too many the estimates would become noisy and the forecast error would be too

large. At the other extreme if we only estimate <Po, we may ignore information useful in the

prediction of x( t + 1) if zK (t - 1) actually contains information about x( t + 1). Thus the role of

set of lags is comparable to the role of K.

To find the best set of lags we use the technique backwards deletion. That is, we start out

set potential for which we estimate compute the forecast

error as described in the next section. Using standard least squares techniques we then decide

and continue 'by

this way weo.+1

set ofuseful. We drop that term from

are dropped, in which case we

is

Se(lU(~rl{:e of estimates lorec;'l,st error, are

to aoo U ,,[H Y construct it. In the atmospheric sciences we are in the fortunate circumstance that

amount of data is frequently very large. Therefore, it is possible to divide the data into two

, compute the patterns and prediction models based on one half of the data, and estimate

SL,:l<U::5Ll'C,U forecast error based upon the other half of the data.

first half of the series x(t) is called the training series, the second half is called the test

series. (The terminology training set and test set come from the field of classification, c. f. Breiman

et. af. (1984).) We compute the patternsU based on the training series. To compute the forecast

error associated with the first I( patterns of the decomposition, we compute zK, using (4), both

for the training series and the test series. For a given set of lags the coefficients <PI are estimated

using by least squares, using only the training data. Then the predicted values x are computed for

the test series, using the estimates of <PI obtained from the training set. It should now be noted

that x(t + 1) itself is not used in the computation of x(t + 1). Furthermore, the patterns U and

the coefficients <PI are all computed based on the training series, while x(t + 1) itself is part of the

test series.

For the first I( patterns of this particular decomposition U and a fixed set of lags the forecast

error is now computed by

FE(I( U)= Ltllx(t)-x(t)112, Lt Ilx(t)112 '

the sum is over the test set.

4. Examples

4.1. Applications to Geopotential Height Field Data

field dataprevious section to the geopotential

statistical lor'ec;ast error as

for i( = 2 or J( 3 it often was {O,l}. tor J( > 3 we found that set that

forecast error only included 0, that is, best predictor t + 1 uses

zI< (t) and no earlier information. It should be noted here that our observations were five

averages.

As can be seen from these plots, the difference between four of these methods is small while CCA

performs much worse in all cases. Close examination reveals that BP-CCA and EOF frequently

have the smallest forecast errors. It is somewhat surprising that EOF performs as well as BP-CCA.

As discussed in Section 2, BP-CCA takes the time dependence of x(t) into account, while EOF

does not do this explicitly. As can be seen from Figure 5, EOF is still as good as BP-CCA, with

respect to forecasting. Somewhat surprisingly, POPs and BP-POPs perform worse than BP-CCA

and EOF with respect to the forecast error. This may be caused by the fact that the field, though

highly dependent, is not very well modeled by a first order autoregressive process.

In Table 1 we summarize 100% minus the minimum forecast error for the various methods for

, 1 to 4 patterns. From the table we read that EOF performs slightly better than BP-CCA for the

combined series and the summer, while these two method perform equivalent for the winter series,

In Figure 6 we show the one-step ahead statistical forecast error for EO F, for a longer range

J(. (Since we projected on the first about 25 EOFs, we clearly cannot show the BP-CCA and

BP-POPs curves beyond about J( As can be seen from this graph, of J(

the error is for the cOlnbm€~d

for winter series vVe can

number of patterns J( to minimize the one-step ahead statistical forecast error is substan-

in either case. In particular, the 54 patterns for the combined series explain 95.3% of

variance in x(t), the 33 patterns for the summer series explain 93.2% of the variance in

the first 38 patterns for winter series explain 88.3% of the variance in x(t). Even the

minilmllm statistical forecast error is substantial. However, these forecast errors are substantially

than those obtained when taking x( t) = x( t - 1), which are above 100% for all three series.

4.2. Simulation

In the example for the geopotential height data, we noticed that while EOF ignores the time

dependence when determining the spatial patterns, it still ends up being one of the methods with

smallest forecast error. There is no theoretical reason why EOF should work that well. On

the contrary, it is easily possible to construct a dataset in which the patterns that explain most

of the variation in x(t), and are thus selected by EOF, are not at all the patterns that are easy

to predict. In Figure 7 we show the autocorrelation functions of the temporal oscillation patterns

of ten orthogonal patterns (not shown themselves), which we used to generate a process x(t) with

p = 10. In particular, we weighted these ten patterns such that the amount of variance explained

by each pattern in a simulated dataset of length 1000 is as indicated in the figure.

An EO F analysis on the first 500 observations, identified the ten patterns in the sequence 1 to

10. However, when we computed the forecast error on the second 500 observations, we found, not

that patterns do not predict at all (see Figure 8). Combining the percentage

variance explained and the autocorrelation patterns shown in 7, it seems more sensible

sequence 5, to pr,edlct:abllllty

curve lalJiel€,Q

on our simulated example (Kooperberg and O'Sullivan

obte(:tI1/e as the one advocated by Deque (

5. Discussion

a comparable

In the atmospheric sciences there has been considerable interest in the development of empirical

methods which are capable of decomposing spatial and temporal variation into a small number

of fixed patterns. Techniques that are used include empirical orthogonal functions, canonical cor­

relation analysis and principal oscillation patterns. All of these methods their advantages.

Patterns that are obtained using empirical orthogonal functions explain a large fraction of the vari­

ance of the field, however, the spatial patterns need not reflect anything to do with the evolution

of the field, which is often of prime interest for modeling purposes. On the other hand, canonical

correlations analysis and principal oscillation patterns focus on components with a strong tem­

poral auto-correlation pattern,but there is no constraint that the patterns are necessarily highly

correlated with the original field.

In this paper we discuss a criterion that can be a tool when comparing different methods.

The one-step ahead forecast error measures how well one is able to predict the next instance of

the field, based on the summary patterns up to now. Clearly a good prediction requires both

high correlation between the patterns and the field and a strong temporal auto-correlations of the

patterns. The one-step ahead forecast error can also be used to get aJ;l idea how many patterns of

a certain decomposition best summarize a field.

In practice the decision which decomposition to use and how many modes of the decomposition

should be retained are likely to made based on many different considerations. We belief that

st<:.tu,tl,:::al tor'eC<1St error may be one

Acknovvledgelnents

~'''t<Hlvd h.clon,er,bel'!!: was supported in part by NSF grant DMS-94.03371. Finbarr O'Sullivan was

SU1Dp.[}rt,ed in part by NIH grant CA-57903. authors want to thank Professor John M. W;:i,Ua,ce.

and James Renwick of University of Washington, Department of Atmospheric

Snpr"r-p<: for encouragement and many helpful discussions.

References

P., and R. W. Preisendorfer, 1987: Origins and levels of monthly and seasonal forecast

skill for United States surface air temperature determined by canonical correlations analysis.

MOll. Ww. Rev., 115, 1825-18,50.

Barnston, A. G. and R. E. Livezey, R. E., 1987: Classification, seasonality, and persistence of

low-frequency atmospheric circulation patterns. Mon. Wea. Rev, 115, 1083-1126.

Breiman, L., J. H. Friedman, R. A. Olshen and C. J. Stone, 1984: Classification and Regression

Trees. Wadsworth, 358pp.

Bretherton, C. S., C. Smith, and J. M. Wallace, 1992: An intercomparison of methods for finding

coupled patterns in climate data. J. Climate, 5,541-,560.

Brillinger, D. R., 1981: Time Series: Data Analysis and Theory. Holden Day, -540pp.

G., 1993: Complex principal oscillation pattern analysis. J. Climate, 6, 1972-1986.

LJt'qUt M., 1988: lO-day of Northern Hemisphere winter 500-mb height by the

operational 40A, 26-36.

circulation and climate.l\lallllas. 1976: North American influences on

Wea. 104,

and J.

North Atlantic

p

Kooperberg, and F. O'Sullivan, 1994: Predictive Oscillation Patterns: a synthesis of methods

for spatial-temporal decomposition of random fields. Technical Report No. 277, Department

oj Statistics, University oj Washington, Seattle.

Lorenz, . N., 1956: Empirical orthogonal functions and statistical weather prediction. MIT,

Department of Meteorology, Sci. Rep. 1,49pp.

Mardia, K. V., J. T. Kent, and J. M. Bibby, 1979: Multivariate Analysis. Academic Press, 521pp.

Priestley, M. B., 1987: Spectral Analysis and Time Series. Academic Press, 890pp.

Richman, M.B., 1988: Procrustes target analysis: a multivariate tool for identification of climate

fluctuations. J. Geophys. Res., 93 D9, 10,989-11,003.

Smagorinsky, J., 1969: Problems and promises of deterministic extended range forecasting. Bul.

Amer. Meteor. Soc., 50, 286-310.

Von Storch, H., T. Bruns, L Fischer-Bruns, and K. Hasselmann, 1988: Principal oscillation pattern

analysis of the 30- to 60-day oscillation in general circulation model equatorial troposphere.

J. Geophys. Res., 93 D9, 11,022-11,036.

Wallace, J. M. and P. V. Hobbs, 1977: Atmospheric Sciences: an Introductory Survey. Academic

Press, 467 pp.

Wallace, J. M. and D. S. Gutzler, 1981: Teleconnections in geopotential height field during the

Northern Hemisphere winter. Mon. Wea. Rev., 109, 784-812.

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sea surface temperature and 500-mb height anomalies. J. Climate, 5, 561-576.

Figure Captions

1. first two empirical orthogonal functions and their corresponding temporal oscillation

"",1'+OTn" for the combined series. On the right side of the figure we show for a part of Zk(t), as

as the first 30 autocorrelations and an estimate of the log-spectral density of Zk( t). The fields

are dimensionless. The thick contour represents the zero-level, dashed contours are negative, solid

contours positive.

Figure 2. The first two empirical orthogonal functions for the combined series, the summer series

winter series.

Figure 3. The first two canonical correlation patterns in the U domain for the combined series,

summer series and the winter series.

Figure 4. The first two principal oscillation patterns for the combined series, the summer series

and the winter series.

Figure 5. The one-step ahead forecast error (in %) for various methods as a function ofthe number

of patterns.

Figure 6. The one-step ahead forecast error (in %) for empirical orthogonal functions for the

combined series, the summer series and the winter series.

7. Autocorrelation functions for the empirical orthogonal functions a simulated dataset.

a turlctilon

8. The one-step ahead forecast error (in

number of patterns.

for the simulated dataset, for various methods

Tables and Figures

Combined Series Summer Series Winter Series

J( 1 2 3 4 1 2 3 4 1 2 3 4

EOF 3.8 6.2 9.0 11.1 1.9 3.9 5,4 7.8 5,4 8.2 11.1 13.7

CCA 0.1 2.2 .36 4.7 0.5 2.1 2.6 3.6 1.3 2.6 3.2 5.2

BP-CCA 2.3 4.2 8.0 10,4 1.6 2,4 4.2 6.0 5.5 7.8 10.5 14.4

POPs 1.6 3.7 6.6 7.4 1.4 2.2 3.7 4.9 3.9 6.3 9.6 10.7

BP-POPs 1.5 3.7 6.5 8.4 1.5 2.3 2.9 4.4 2.1 5.1 9.0 11.1

30

• I I

2520

S ectrum1510

Autocorrelations

I I

5o

I II

EOF I - combined - 8.9% variance

0.0 0.1 0.2 0.3 0.4 0.5

Autocorrelations

30

• I I25

1968

20

1967

15

Spectrum

Part of z t

1968

10

1965

o

EOF 2 - combined -7.3% variance

0.0 0.1 0.2 0.3 0.4 0.5

two arrlnlrlr::l.i

a

EOF 1 combined 8.9% variance

EOF 1 - sunmler - 6.9% variance

EOF 1 - winter - 11.6% variance

EOF 2 - combined 7.3% variance

EOF 2 - summer - 6.2% variance

EaF 2 - winter - 8A% variance

CCA 1 - combined variance CCA 2 - combined 3.2% variance

CCA 1 summer - 3.7% variance

CCA 1 - winter - 8.3% variance

CCA 2 - summer - 3.5% variance

CCA 2 - "'inter - 7.9% variance

POPs 1 combined - 2.9% variance POPs 2 - combined 4. variance

POPs 1 - summer - 4.3% variance

POPs 1 - winter - 4.4% variance

POPs 2 - summer - 2.5% variance

POPs 2 winter - 5.2% variance

Combined Series

---------- ---::::::--...-- - ~-:::-::_::-:o_,..... .J

o 5 10 15

Number of Fac10fS

Summer

~

'"0>

g 0W 0>

'i~ '"0 '"u..

g

'"r-0

EOFCCAPOPsBP-CCABP-POPs

5 10

-- ---

15

Number of Factors

Winter

---

--------- -- - - --:::-_~--::::..-----:..::. '":-::..-

§

'"0>

E gw'"., EOFg

'"II '" CCAu..pOPsBP-CCABP-POPs

'"r-

CombinedSummerWinter

00.,.-

l()0)

.... 00 0)........w--(J) l()aJ OJ0<D....0u... 0

OJ

l()('-..

0('-..

EOFs

\\r\ \.\ \.\ ......., \ .., \"

'\ ...•...................... . .~~~~--_ ..-......

\\

'-...'-...

" -'------..,..,...--------

...--/_/-------

o 20 40 60 80 100

Number of Factors

Figure 6. The one-step ahead forecast error (in %) for empirical orthogonal functions for the

combined series, the summer series and the winter series.

EOF or 1 - 25.12 % variance

'"d

" ""d d

c , , , , , . •• t I ., t t iiI t. • t • t C

d d

5 10 15 20 25 3Q 0 5 10 15 20 25 30

lag lag

EOF or 3 - 13.62 %, variance EOF nr 4 - 10.77 % variance

'" '"~

d c d.2

.! ~e

"~

5 5 "~

d 0 d$!

"" "c . I ..• i i I I . •• t I I I " I . I .•• I • 0 I t I • I I j . Id . id •• f •• I' • t t • t •

0 5 10 15 20 25 30 0 5 10 15 20 25 3Q

lag lag

EOF or 5 - 9.88 % variance EOF or 6 - 7.34 % variance'<

I '"c l.()

.111111

111, 11,111111

' I1 II

c d,2 d

I! III.2

10 101i 1il:: 0 i5 "':

! 0 0 0

~" l.() "9 0 I I I I I .• I I. t •• f •• . I . i fit Id . " . I

0 5 10 15 20 25 3Q 0 5 10 15 20 25 3Q

lag lag

EOF or 7 - 5.66 % variance EOF nr 8·3.72 % variance'<

'".~

0

i <D

i ""0

g 0'$

"'< • I . I I , I I I " I I " I I , •. I . I . '"0

. , . , 0

5 10 15 20 25 30 0 5 10 15 20 25 30

lag

q

'"d

""d

oo

....e........w.....(/)

aso(!)....eu...

o(J)

l()ro

o 2

EOF~~~~~~ ...... -. eeA-- --- POPS--- deterministic

4 6 8 10

Number of Factors

Figure 8. The one-step ahead forecast error (in %) for the simulated dataset, for various methods

as a function of the number of patterns.