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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.
Wallace, J. C. Smith, and C. S. Bretherton, 1992: Singular value decomposition of wintertime
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.