Robust stochastic seasonal precipitation scenarios

INTERNATIONAL JOURNAL OF CLIMATOLOGYInt. J. Climatol. 26: 2077–2095 (2006)Published online 5 May 2006 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/joc.1351

ROBUST STOCHASTIC SEASONAL PRECIPITATION SCENARIOS

IOANNIS KIOUTSIOUKIS,* SPYRIDON RAPSOMANIKIS and REA LOUPADemocritus University of Xanthi, Environmental Engineering Department, Laboratory of Atmospheric Pollution and Pollution Control

Engineering, Xanthi, Greece

Received 10 May 2005Revised 11 January 2006Accepted 10 March 2006

ABSTRACT

In this paper, a stochastic statistical forecasting methodology is employed for long-term predictions of winter precipitationover Greece. Lagged climatic indices and North Atlantic (NA) sea-level pressure (SLP) field are explored as potentialpredictors of the teleconnection. Rather than employing traditional stationary models, two dynamic regression-modellingschemes are analysed and validated and their parameter variation is interpreted. Dynamic regression models, in contrastto static (constant parameter) regression models, have time variable parameters (TVPs) evaluated through recursiveoptimisation and are suitable for analysis of non-stationary phenomena like most atmospheric processes.

The analysis of the spectrum with non-stationary models points out that the most influential seasonal components ofthe winter precipitation anomalies have periods of 14 and 3.5 years, explain 40% of its variance, possess significantamplitude change and correlate significantly with the North Atlantic Oscillation Index Anomaly (NAOIA) and SouthernOscillation Index Anomaly, indicating their climatic origin. Furthermore, the forecasting skill of the dynamic models(R2 = 0.71), in addition to reproducing the peaks, was found superior even to the hindcasting skill of the stationarymodel (R2 = 0.55). Copyright ! 2006 Royal Meteorological Society.

KEY WORDS: non-stationary time series analysis; North Atlantic Oscillation; Southern Oscillation; winter precipitation anomaly; Greece

1. INTRODUCTION

Seasonal climate prediction seeks to forecast the likely state of the climate several months in advance. Itsscientific basis lies in the lower boundary conditions of the atmosphere (e.g. sea-surface temperatures (SST),land surface characteristics) that once perturbed, alter the likelihood of the occurrence of weather regimes onseasonal and long-term timescales (Goddard et al., 2001). Statistical and dynamical models have been utilizedto extract these potentially predictable low-frequency signals.

The dominant mode of seasonal climate variability on a global scale is the El Nino – Southern Oscillation(ENSO) phenomenon that arises from the strong ocean–atmosphere interactions internal to the tropical Pacificand overlying atmosphere. ENSO (Trenberth, 1997) seems to have a global influence beyond influencingtropical climate. The shifts in the location of the organised rainfall in the tropics and the associated latentheat release alters the heating patterns of the atmosphere that forces large-scale waves in the atmosphere thatin turn establish teleconnections. In practice, most remote climate impacts (or teleconnections) arise throughthe propagation of SST disturbances that are excited in the tropical Pacific.

In addition to teleconnections directly linked to SST changes, some arise from natural preferred modes ofthe atmosphere associated with the mean climate state and the land–sea distribution. The most prominent arethe Pacific-North American (PNA) and the North Atlantic Oscillation (NAO).

The NAO (Hurrell, 2001; Wanner et al., 2001; for a review) is one of the most dominant patterns ofwintertime atmospheric circulation variability. Although NAO is evidently a mode of variability internal

* Correspondence to: Ioannis Kioutsioukis, Democritus University of Xanthi, Environmental Engineering Department, Laboratory ofAtmospheric Pollution and Pollution Control Engineering, Kimmeria Campus, GR-67100, Xanthi, Greece; e-mail: [email protected]

Copyright ! 2006 Royal Meteorological Society

2078 I. KIOUTSIOUKIS, S. RAPSOMANIKIS AND R. LOUPA

to the atmosphere, there is an increasing evidence that the observed changes in the NAO may well bea response of the system to observed changes in SSTs and there are some indications that the warmingof tropical oceans is a key part of this. Relations between tropical SSTs and extra-tropical climate thathave been recently documented include links between tropical Pacific SST and the NAO (Hoerling et al.,2001) and the significant influence of El Nino on the climate of the North Atlantic (NA) region (Mathieuet al., 2004). Cassou and Terray (2001) presented evidence that the winter European atmospheric variabilityis connected to anomalous oceanic conditions in the North Atlantic and the Pacific. In addition, Suttonand Allen (1997) presented evolving NA SST anomaly patterns and suggested that their propagation canyield downstream predictability several years ahead. Consequently, since SST evolution is slow relative tocharacteristic atmospheric timescales (Hansen and Bezdek, 1996), NA presents considerable potential forpredictability of European climate more than a year in advance (Rodwell et al., 1999; Kushnir, 1999).

The Mediterranean is a small-scale coupled atmosphere ocean system with a short response time andwith high estimated human-induced climate change impacts (Hulme et al., 1999; Palutikof et al., 1996). Theunderstanding of the relevant processes and physical mechanisms and their link to large-scale climate willimprove the management of climate-related risks in the region. Over the last years, several papers haveaddressed the relationship mechanisms between climatic variability in the Mediterranean and fluctuations inthe global circulation. ENSO episodes and the NAO are among the key factors that influence the periodicityof droughts in the Mediterranean region (e.g. Haylock and Goodness, 2004; Eshel and Farrell, 2000; Hurrell,1995; Fernandez et al., 2003). Both time series (ENSO and NAO) do not show a relatively simple cyclicalbehaviour; additionally, the well-established association between the NAO circulation mode and the surfaceclimate of Europe has recently been shown to be non-stationary (Rodo et al., 1997) and asymmetric (Cassouet al., 2004).

The period from October to March (hereafter winter) is the epoch with the highest precipitation amountsin the Mediterranean as a whole. Despite the large spatio-temporal variability of precipitation, a significantfraction of its variation can be explained by large-scale circulation changes at different heights as exploredin recent studies. Eshel et al. (2000) studied the eastern Mediterranean (EM) winter rainfall variability interms of subsidence anomalies associated with NA sea level pressure (SLP) anomalies. The linear statisticalmodel trained from this teleconnection closely matched the observed precipitation anomalies (PRECA) inhindcast mode but its forecast skill was moderate. Dunkeloh and Jacobeit (2003) analysed the Mediterraneanrainfall variability for winter as well as for the other seasons in terms of NA – European area geopotentialheights (HGT) at 500 and 1000 hPa. The first set of canonical correlation patterns were explaining 30% of theprecipitation variance and were depicting the Mediterranean Oscillation (MO) (Conte et al., 1989; Palutikofet al., 1996), which in addition correlates significantly with the Northern hemisphere modes of the NAO.Xoplaki et al. (2004) studied winter precipitation variability in the Mediterranean using surface (SLP, SST)as well as lower and upper troposphere variables (HGT: 850, 700, 500 and 300 hPa) in a canonical correlationanalysis (CCA). They found that the SST does not improve the overall performance while a linear combinationof the other five predictors explains 30% of the Mediterranean winter precipitation variability. The normalisedtime coefficient of the first canonical correlation pattern of precipitation correlates with the negative NAOand is responsible for the decadal and long-term variation in precipitation. Moreover, the inclusion of upperlevel predictors did not improve the winter predictability in the southeastern Mediterranean. Consequently,higher predictability is identified for the west and the northern part of the Mediterranean and lower for thesouth and the east.

At a smaller scale, the amount and distribution of precipitation in Greece is highly irregular in boththe spatial and temporal dimensions. Maheras et al. (2004) examined the rainfall variability over Greece inrelation to the 500 hPa circulation types (14 in total) established on the high correlation between seasonalprecipitation amounts (predictant) and the frequency of the cyclonic circulation types at 500 hPa (predictors).The multiple linear regression (MLR) models per station achieved an adjusted R2 in the range 0.2–0.8 forwinter (December–February) and much lower for the other seasons. The authors conclude that the unexplainedvariance is due to phenomena that could not be explained only by changes in the circulation-type frequency.

Dynamic regression models, in contrast to static (constant parameter) regression models, have time variableparameters (TVPs) evaluated through recursive optimisation (Young et al. (2004); Young (1984); Young

Copyright ! 2006 Royal Meteorological Society Int. J. Climatol. 26: 2077–2095 (2006)DOI: 10.1002/joc

ROBUST STOCHASTIC SEASONAL PRECIPITATION SCENARIOS 2079

(1999) and the references therein). The approach has a Bayesian origin with the evolution of the stochastic(unobserved) state space (SS) variables assumed to be described by a generalised random walk (GRW) process.The hyperparameters of the GRW process are estimated from the data unless they are known a priori. Theskill of the dynamic models has been validated in several different applications, ranging from environmentalto economic systems (Young; 1998).

In summary, there is enough physical evidence that ENSO influences the NA region and in turn NAaffect the European and Mediterranean climate. Those teleconnections are based on well-documented physicalschemes (Cullen and deMenocal, 2000; Lamb and Peppler, 1987). This work attempts to further understandingof seasonal precipitation variability over Greece from the perspective of the changes in large-scale dynamics.Dynamic regression models are employed to cope with the non-stationarity of the predictor–predictantrelationship. In particular, the main objectives of the analysis are:

(1) To analyse the winter PRECA over Greece for 1951–2002 and explore its principal modes of oscillationusing non-stationary time series analysis.

(2) To quantify the influence of NAO/ENSO in the winter PRECA for Greece.(3) To develop robust winter precipitation predictions, at least one year in advance, for Greece (an area with

lower predictability than the northwest part and the whole Mediterranean on average) through the use ofdynamic models and explore their effectiveness.

2. DATA ANALYSIS

2.1. Predictant

Monthly precipitation data for Greece were derived from 26 stations of the National MeteorologicalService network (Figure 1) covering the period from October 1950 to March 2002. The stations are quitehomogenously distributed over the domain. The generation of the annual time series of winter (PRECA canbe viewed as a three-step procedure. First, we construct for each station monthly PRECA (point estimate) bysubtracting the climatological mean (for the particular station and month over the period 1951–2002) from therespective monthly value. Then, winter PRECA (point estimate) are calculated for each station by taking theaverage of the six-monthly PRECA (from October to March). Finally, the winter PRECA for the whole studyarea (area estimate) is constructed by averaging winter rainfall anomalies using the cell declustering technique(Isaaks and Srivastava, 1989). According to the approach, the entire area (domain 20–27 °E, 35–41°N) isdivided into rectangular regions (cells) and each sample (meteorological station) receives a weight inverselyproportional to the number of samples that fall within the same cell. The spatial distribution of the mean winterprecipitation field is shown in Figure 2(a). An uneven distribution is evident with the rainiest areas located inthe western part of Greece and the driest in the central part of the domain. The detrended winter PRECA (i.e.the dependent variable in the analysis) time series (Figure 2(b)) correctly identified the winter 1962–1963as the wettest of the whole period and the winter 1989–1990 as the driest (Maheras and Anagnostopoulou,2003). The spatial distribution of the PRECA for the extreme years (Figure 2(c,d)) shows that the westernpart of Greece is affected mostly by extreme weather events (drought and excess rainfall).

2.2. Predictors

2.2.1. Fields. Two sources of monthly NA sea-level pressure anomaly (SLPA) data are used:

(1) Atlas of Surface Marine Data (DaSilva et al., 1994) spanning from January 1948 to December 1994 witha spatial resolution of 1° over the area 90.5 °W–0.5 °W and 0.5 °N–89.5 °N, comprising 2627 (sea) gridpoints.

(2) Climate Data Assimilation System (NCEP/NCAR Reanalysis) spanning from January 1948 to the presentwith a spatial resolution of 2.5° (Kalnay et al., 1996) over the area 87.5 °W–2.5 °W and 2.5 °N–87.5 °N,comprising 382 (sea) grid points.



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Figure 1. Distribution of the meteorological stations over Greece

The steps to generate the input time series for the large-scale 2D fields (Eshel et al., 2000) are brieflyreported here:

(1) Noise filtering . A 3-month average of the 2D monthly SLPA dataset is created. Thus, for example, theDecember map of each year is replaced with the mean value of November–December–January.

(2) Correlation pattern. The prognostic skill of the NA-SLPA over the detrended PRECA is investigated forlead times ranging from 1 to 18 (lagged teleconnection). Correlation maps are constructed betweenthe PRECA time series and the detrended NA-SLPA time series at each grid point and for allconsidered lead times. Only locally significant correlations (p < 0.05) are reserved with the rest setto zero.

(3) Field significance. The significance of each correlation map (18 in total) as a whole is examined throughMonte Carlo analysis (Livezy and Chen, 1983). We fit an AutoRegressive Moving Average (ARMA)model to the PRECA time series and generate 1000 different realisations of it using its statistical properties(spectral properties and variance). The areal fraction of the domain with locally significant correlationsis calculated for each realisation (and lead time) and the cumulative probability density function (cdf) isconstructed for each lead. The 95th percentile of the cdf is taken as the field significance threshold valuefor that lead.

(4) Time series projection. The detrended monthly (3-month average) NA-SLPA map of each year is projectedon the correlation map Cl of the leads (l) that passed the significance tests yielding the annual laggedSLPA time series (the bar indicates the 3-month average SLPA and the superscript T the transpose of thematrix):

SLPAl(t) = SLPAlTCl (1)

We assume here that the 3D matrices have been reshaped into their 2D equivalents, i.e. the (X,Y,T) matrixwith dimension NX*NY*NT has been reshaped to (S,T) with dimension (NX*NY)*NT.



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Figure 2. (a) Geographical distribution of winter (October–March) precipitation amounts (mm). (b) Winter rainfall anomaly timeseries (mm/month). (c) Geographical distribution of winter (October–March) PRECA (mm/month) for the wettest year (1962–1963).

(d) Geographical distribution of winter (October–March) PRECA (mm/month) for the driest year (1989–19990)

2.2.2. Indices. In addition to the 2D fields, the following monthly climatic indices were exploited:

• North Atlantic Oscillation Index Anomaly (NAOIA) (Climate Prediction Centre: http://www.cpc.ncep.noaa.gov/data/teledoc/telecontents.html).



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• Southern Oscillation Index Anomaly (SOIA) (Climate Prediction Centre).• Tropical North Atlantic Index (TNAI) (anomaly of the average of the monthly SST from 5.5 °N to 23.5 °N

and 15 °W to 57.5 °W) (Climate Diagnostic Center, http://www.cdc.noaa.gov/index.html).

The selection of the input time series from the climatic indices is explored through correlation analysis forlead times up to 10 months.



3. MODELLING SCHEMES

The PRECA forecasts are established on inputs such as projected NA-SLPA and climatic indices at variouslead times. The input–output mapping is calculated using dynamic regression models with time variableparameter such as dynamic linear regression (DLR), dynamic harmonic regression (DHR) and dynamicauto-regressive exogenous (DARX) variables. Traditional MLR models are also calculated for comparativepurposes.

The approach to TVP estimation in this paper is established on the basis of a multiple input–single outputunobserved components (UC) model of the general form:

yt = Tt + St + f (ut ) + et (2)

The subscript t symbolises the discrete-time of the observed values t = 1, 2, . . . , N (sample size). Tt is thetrend element, St is the periodic (seasonal/cyclical) component, u denotes the vector of the auxiliary variablesused for forecasting and finally et is the discrete-time white noise (!N(0, ! 2)) element.

Equation (2) describes the observation equation of the SS model. The stochastic evolution of the statevector is assumed to be expressed by a GRW process:

xti = Fixt"1

i + Gi"ti I = 1, 2, . . . , k (analogous to the total number of TVPs) (3)

with "ti the zero mean white noise vector (independent from the observation noise et ) with covariance Qi and

Fi =!# $0 %

", Gi =

!& 00 '

"(4)

The optimisation of the GRW model hyperparameters (elements of Fi , Gi and the noise variance ratio(NVR) defined as Qi/!

2) is achieved in a recursive context accomplished by several approaches (maximumlikelihood (ML) optimisation, minimisation of the sum of squares of the recursive multiple-step-aheadprediction error, frequency domain optimisation) (Young, 1999). Fixed interval smoothing (FIS) estimates(derived from Kalman’s work on optimal SS filter theory in the time domain; Kalman, 1960; Young andPedregal, 1999) of the TVPs are then calculated recursively under the assumption that the stochastic SS statevector parameters vary as a GRW process (the term ‘fixed interval’ refers to the interval covered by the totalsample size N ).

Applications of the full UC model are generally limited due to problems in the simultaneous estimate ofall the components. Indeed, special cases of the general UC model have been proven valuable practically(Young, 1999):

(1) The dynamic linear regression (DLR) model is

yt = Tt +m#

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btiu

ti + et (5)

where bti are the regression coefficients and ut

i the ‘regressors’ (auxiliary input variables). Note that when theregression coefficients are constant we obtain the normal regression model.(2) The dynamic harmonic regression (DHR) model is

yt = Tt +Rs#

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ait cos((i t) + bit sin((i t) + et (6)

where the summation represents the variable amplitude periodic (seasonal or cyclical) term.



(3) The dynamic auto-regressive exogenous variables (DARX) model is

yt = Tt +n#

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"aityt"i +#

bitxt"&"1 + et (7)

where & is a pure time delay. In DARX, in addition to auxiliary variables, past values of the output variableare also used.

For DLR and DARX, the hyperparameters (NVR, elements of Fi and Gi) are optimised via ML based onprediction error decomposition. The approach, although established on a strong theoretical basis, has somepractical disadvantages such as the intense dependence on the length of the series, the shape of the likelihoodsurface (e.g. multiple minima), etc. (Young, 1999). To counter these limitations, DHR model utilizes a specialform of optimisation in the frequency domain; specifically, optimisation is based on fitting the DHR modelpseudo-spectrum to the logarithm of the AutoRegressive (AR) spectrum. The approach, although it is notas general as ML, leads to a considerably better-defined optimum in the objective function-hyperparameterspace with consequent advantages to DHR modelling (convergence time and the number of parameters thatcan be optimised simultaneously).

The final step in the analysis using linear dynamic modelling techniques would be the physical explanationof the nature of the parameter variation in terms of other variables. The analysis can then be further extendedwith the use of more complex dynamic models arriving at fully non-linear stochastic SS models.

4. RESULTS

4.1. Hindcast with DHR – analysis of the PRECA time series

The identification of the frequency values in the DHR model is accomplished with reference to the ARspectrum. The Akaike information criterion (AIC) (Akaike, 1974) pointed out the best model order as AR(20)followed by AR(14). The AR(14) spectrum is selected in the analysis because it represents the spectralproperties while being the most parsimonious. Figure 3(a) demonstrates the AR(14) spectrum; the peaksappear at periods close to 14, 7, 3.5 and 2 years indicating seasonal/cyclical oscillations with a fundamentalperiod of 14 years with most significant periods at 14 and 3.5 years.

Periodogram analysis of the North Atlantic Oscillation Index (NAOI) demonstrate a 7.5-year cyclicalcomponent (Werner and Schonwiese, 2002) for winter. Sutton and Allen (1997) identified the existence ofdecadal fluctuations (regular period of 12–14 years) in NA SST. Furthermore, typical ENSO cycles arebetween 3 and 4 years (Schneider and Schonwiese, 1989). Hence, PRECA cycles at first seem to be relatedto typical NAO and ENSO cycles. The analysis of whether this coincidence is random or not is investigatedin the next paragraphs.

We therefore built a DHR model with the five harmonics estimated from the data and a trend component.The 11 TVPs (Tt , ait , bit , i = 1, 2, 3, 4, 5) are modelled as random walk (RW) processes (the NVR val-ues are as follows NVR(Tt ) = 0.2692, NVR(a1t ) = NVR(b1t ) = 0.0677, NVR(a2t ) = NVR(b2t ) = 0.00004,NVR(a3t ) = NVR(b3t ) = 0.0498, NVR(a4t ) = NVR(b4t ) = 0.00003 and NVR(a5t ) = NVR(b5t ) = 0.0023).The estimates of the TVPs for the particular interval (1951–2002) are then obtained in a recursive manner(Equations (6) – in its SS form – and 3) using a two-step (prediction-correction) version of Kalman filteringfollowed by an optimal smoothing procedure (FIS). The hindcasted PRECA is shown in Figure 3(b) andthe seasonal components in Figure 3(c). The hindcast skill of the DHR model in terms of the coefficient ofdetermination is R2 = 0.94.

The negative precipitation trend is a fact for the second half of the twentieth century in Greece (e.g.Amanatidis et al., 1993; Giorgi, 2002). We estimate this to 0.57 mm/winter/year as identified from the non-linear trend component. Three harmonics have essential contribution to the total seasonal component. Thefundamental harmonic (14 years) possess an amplitude increase and explains 22% of the PRECA variance.In addition, it correlates negatively with NAOIA (r = "0.41, p < 0.02). The 3.5 years harmonic exerts an



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Figure 3. (a) The AR(14) spectrum of the PRECA time series, (b) Comparison of DHR model output with measured data, (c) the TVPs

amplitude oscillation, explains 17% of the PRECA variance and correlates with SOIA (r = "0.52, p < 0.04).Finally, we observe that the local minima of the fundamental harmonic coincide with the plateau (at zeroamplitude) of the 2 years harmonic, which explains 8% of the PRECA variance, demonstrating an interactionbetween the two oscillations. The origin of this mode is unknown (correlates weakly with all indices, includingthe Mediterranean Oscillation Index) and probably expresses the superposition-interaction of the existingforcings.



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Consequently, the principal cycles of the PRECA time series appear at periods of 14 and 3.5 years, explaintogether 40% of its variance and in addition they correlate significantly with NAOIA and SOIA, which showsimilar principal cycles.

4.2. Hindcast with MLR and DLR

4.2.1. Stationary models. Before analysing the detrended input–output time series for the period1951–2002, we compare the results obtained using the two different SLPA (input) datasets for the over-lapping period (1951–1992). The whole analysis is performed, according to Section 2.2.1, and MLR modelsare built based on the highest skill SLPA predictors. The two different input datasets revealed almost identicalresults as shown in Table I. These results also agree with the findings of Eshel et al. (2000), whose PRECAtime series was averaged over the whole EM.

The analysis is now performed to the full dataset (1951–2002). The leads that passed the field significancetest (we focus on long-term forecast and hence we do not consider leads prior to 8 months) as well as thet-test of the lead’s hindcast skill in univariate mode are then combined in a backward stepwise MLR analysis.The best models in terms of R2, mean square error (MSE) and Malow’s Cp (Weisberg, 1985) are presentedin Table II. The best bivariate model captures only 33% of the PRECA variability (the result was 49% forthe 42-year dataset), whereas the overall best model has three predictors and achieves a Pearson correlationcoefficient of 0.63. The reason for this notable change in the model skill, when data from the last decade areadded, is shown in Figure 4. Weaker correlation fields accompanied with smaller fraction of the domain withsignificant correlations produce regressor time series with less significance. From the physical point of view,



Table I. Hindcast skill (Pearson correlation between observed and modelledPRECA) of MLR models trained with lagged (by 9, 15, 16 and 18 months)SLPA predictors from different resolution SLPA datasets (I:DaSilva and II:CDAS) over 1951–1992. All correlations are t-test significant at p < 10"5

Predictors(lagged SLPA)

Pearson correlation(SLPA from DaSilva)

Pearson correlation(SLPA from CDAS)

(9, 16) 0.70 0.70(15, 18) 0.71 0.68

Table II. Hindcast skill of MLR models trained with different combi-nations of lagged (by 10, 12, 15 and 18 months) SLPA predictors fromCDAS SLPA dataset (1951–2002). All correlations are t-test significant

at p < 10"4. MSE is the mean square error

Predictors R2 MSE Mallow’s Cp

(12, 18) 0.33 119.3 7.0(10, 18) 0.30 124.8 9.5(15, 18) 0.29 126.8 10.4(10, 12, 18) 0.35 115.7 7.4(12, 15, 18) 0.40 106.8 3.5(10, 15, 18) 0.35 115.3 7.2(10, 12, 15, 18) 0.41 105.8 5

those differences reflect the probabilistic nature of the teleconnection relationship arising from fluctuations inatmospheric circulation that are inherently unpredictable in some cases. From the mathematical point of view,there exist no hindcast univariate correlation significant at the 0.001 level (there were two at the 1951–1992dataset: leads 9 and 16); the ratio of the percentage of the domain with significant points, calculated as1951–2002 over 1951–1992 is 0.54 for the 9 month lead and 0.65 for the 16 month lead.

We now explore the relationship of the PRECA with the climatic indices (Table III). Significant correlationsare identified for lead times of 8–9 months. In other words, the winter PRECA over Greece is influenced byanomalies in the climatic indices (calculated based on SLP and SST anomalies) of the previous spring. Thisresult matches the earlier finding based on NA-SLPA. Significance level is p < 0.001 for the highlightedNAOIA and SOIA and p < 0.01 for the highlighted TNAI.

Thus, we will train an MLR model utilising the lagged by 8–9 months climatic indices (NAOIA MAY,SOIA APR and TNAI MAY). The MLR model achieved R2 = 0.42 (MSE = 104.1) slightly better than theone based on the SLPA (12, 15, 18). Using all six inputs for training, the upper bound of R2 in MLR modereaches 0.55 (MSE = 80.3).

4.2.2. Non-stationary models. The spatial and temporal non-stationarity of the correlation field influencedconsiderably the hindcast skill of the MLR models trained with SLPA predictors. The same result wasobtained for the MLR trained with climatic indices as predictors. The DHR model (Figure 3b), establishedon a probabilistic theoretical basis, tackled with the probabilistic nature of the teleconnections and achieved ahindcast score of R2 = 0.94. We now explore the behaviour of another dynamic model trained with auxiliaryvariables, that is, the DLR model.

The DLR model is built using as regressors the set – NAOIA MAY, SOIA APR, TNAI MAY, SLPA12,SLPA15 and SLPA18, i.e. the winter PRECA is analysed in terms of SLPA variations of the previ-ous winter (12 m), the previous early autumn (15 m) and the previous early summer (18 m) as wellas with variations in the climatic indices of the previous spring. The six TVPs (bit , i = 1, 2, 3, 4, 5



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Figure 4. Correlation patterns between PRECA and SLPA from CDAS-1 dataset over 1951–1992 (left) and 1951–2002 (right) for leadtimes 9, 10, 12, 15, 16 and 18 months. This figure is available in colour online at www.interscience.wiley.com/ijoc



Table III. Pearson correlation between PRECA andclimatic indices. Correlations are significant at

p < 0.01

NAOIA SOIA TNAI

J – – –F – – –M – "0.38 0.33A – !0.48 0.38M !0.47 – 0.41J – – 0.36J – – –A – – –S – –O – –N – – 0.41D – – 0.37

Values in bold are significant at p < 0.001

and 6) are modelled as both random walk and integrated random walk (IRW) processes (The NVR’sobtained after ML unconstrained optimisation for RW (IRW) are given as follows NVR(NAOIA) = 7.29 #10"3(1.46 # 10"4), NVR(SOIA) = 7.88 # 10"9(3.36 # 10"13), NVR(TNAI) = 4.00 # 10"7(7.87 # 10"8),NVR(SLPA12) = 1.43 # 10"7(3.12 # 10"9), NVR(SLPA15) = 3.76 # 10"6(4.43 # 10"27), NVR(SLPA18)= 1.48 # 10"6(1.57 # 10"26)). FIS estimates of the TVPs are shown in Figure 5. The coefficient of modeldetermination is R2 = 0.71 for DLR and R2 = 0.55 for the constant parameter MLR shown dotted; DLRmodel explain better with the large deviations from the mean. The statistical significance of the estimatedTVPs is examined through the properties of the normalised recursive residuals. If the assumptions regardingthe dynamic regression problem are satisfied, those quantities should be a zero mean, serially uncorrelatedsequence of random variables with changing variance (Young, 1984). The residuals were found statisticallyindependent of the input functions as validated from the lag-correlation of the autocorrelation function andthe cross-correlation function.

The best models built on two, three, four, five and six regressors are also given in Table IV. The SLPAauxiliary variables have higher variability than the climatic variables. This variation is considered in thedynamic models resulting in lower MSE, even when the correlation coefficient gives indistinguishable results.

If we constrain the NVRs to higher values (NVR = 0.01) than the ones estimated after optimisation, weobtain a model that ‘fits’ the PRECA time series (R2 = 0.9888). The negative effect is less smooth variations(Figure 5(b)) of the regression coefficients (especially for the coefficients of SLPA). Those significantparameter variations then suggest the use of higher order non-linear models. However, such an analysisis prohibitive due to the limited length of the time series. Alternatively, if possible, one should explain thevariation of the TVPs in terms of known variables (to improve the forecast skill of the model).

We have demonstrated that the analysis with dynamic models is superior in comparison with the constantparameter models. However, their forecast skill must be evaluated before any inferences are made.

4.3. Forecast

The ability of the models (MLR, DLR and DHR) to forecast one year in advance the PRECA is nowexamined. The 52-year time series due to its autocorrelation has approximately 24 degrees of freedom; hencea forecast for one degree of freedom should be applied to 2.2 years. We apply the omission approach andsequentially withdraw 3 years of data (50 cases) and repeat the whole procedure presented in Section 2 forthe censored dataset. Then we test the skill of both the constant parameter models and the dynamic models tocorrectly forecast over the gap in the input data. The observed time series is compared with the mean forecast



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Figure 5. (a) Comparison of DLR model output (RW: full line, IRW: full line with dots) with measured data (circles) for unconstrainedoptimisation. The output of the constant parameter model is shown as dotted. (b) The corresponding TVPs, for unconstrained optimisation

(left) and for TVPs constrained to equal values (right) (dashed-dotted line: RW, full line: IRW, dotted line: MLR coefficient)



Table IV. Hindcast skill of MLR and DLR (unconstrained optimisation)models trained with lagged SLPA (by 12, 15 and 18 months) predic-tors and climate indices (1951–2002). Notation: Climate Group (C1:NAOIA MAY, C2: SOIA APR and C3: TNAI MAY), SLPA Group (S1:SLPA12, S2: SLPA15 and S3: SLPA18). Correlations are t-test significant

at p < 10"4. CC is the Pearson Correlation

Predictors MLR DLR

CC MSE CC MSE

C(1,2,3) 0.65 104.1 0.66 100.1S(1,2,3) 0.63 106.8 0.76 77.4C(1,2) 0.61 112.8 0.63 108.6C(1,2)S(2) 0.67 97.1 0.74 80.1C(1,2)S(1,2) 0.71 87.9 0.77 71.9C(1,2)S(1,2,3) 0.73 82.6 0.84 54.1C(1,2,3)S(1,2,3) 0.74 80.3 0.84 53.1

Values in bold are significant at p < 10"5). CC is the Pearson correlation and MSEthe mean square error.

Table V. Forecast skill of MLR and DLR (unconstrained optimisation). Notation:see Table IV. Correlations are t-test significant at p < 10"3

Predictors MLR DLR

CC MSE CC MSE

C(1,2,3) 0.57 122.1 0.60 115.5S(1,2,3) 0.32($) 171.2 0.47($$) 142.0C(1,2)S(2) 0.53 130.7 0.61 111.7C(1,2)S(1,2) 0.56 123.4 0.64 106.5C(1,2,3)S(1,2,3) 0.54 130.1 0.64 107.3

Value with $ is significant at p < 0.03; value with $$ is significant at p < 0.01).

time series, calculated as the mean of the forecasts for each year (there exist one forecast for 1951 and 2002,two forecasts for 1952 and 2001 and three forecasts for all other years). Table V demonstrates the results forDLR/MLR models (unconstrained optimisation).

Forecasting using only climate indices is more robust than forecasting based on SLPAs. In forecast mode,the projected SLPA time series is recomputed each time after the sequential withholding, repeating all thesteps given in Section 2 (noise filtering, correlation pattern, field significance and time series projection).The recalculation of the correlation field for each censored dataset assigns high variability to the SLPA timeseries (i.e. the SLPA time series is very sensitive to the correlation field) resulting in higher forecast skill tothe model utilising only climate indices and lower to the model built only on SLPAs (also recall that highersignificance correlations with PRECA were observed for climatic predictors than for SLPA predictors in thefull dataset), although in hindcast mode their skill was equal (however, the forecast results are also alikewhen random withholding is applied). For comparative reasons, the MLR forecast built only on SLPAs forthe period 1951–1992 achieved r = 0.51. Mixed inputs improve the skill only in DLR mode, while in MLRmode the addition of SLPA regressors does not always give better forecasts.

Figure 6 demonstrates results for DLR/MLR using all six regressors. DLR models explain more of thePRECA variability in comparison with the stationary models, although they require additional information(forecasting of the regression coefficients variation for the withheld sample). The scatterplots of the forecast



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error (MSE) in terms of the standardised regression coefficients (Figure 6(b)) shows clearly the principalimportance of the NAOI input. The nominal values (hindcasting mode) are shown as dotted. Significantimprovement of the results by means of constrained optimisation, as seen before, is highly improbable dueto the higher variability of the regression coefficients.

The analysis in now repeated for the DHR model. Figure 7 illustrates the DHR forecasts after sequential3-year withdrawal and their associated standard errors, with the NVRs modelled as RW (top) and IRW



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(bottom). The Pearson correlation between forecasts and observations is 0.77 and 0.84, for RW and IRW,respectively, while the respective MSE is 114.2 and 79.7. IRW statistics are better, however, those forecastsare associated with higher standard errors.

Forecasting is less complicated when using DHR than DLR because the regressors in the former are well-known periodic functions (sine and cosine) that can be easily identified inside or outside the sample. TheDHR results are also quite unaltered even for larger segments of missing data. The high robustness of theDHR model allows us to force the model to forecast beyond the end of the series. An 8 years forecast (years



2003–2010) is also presented in Figure 7. Here we assume than an 8-year period is reasonably short forretaining the same dynamics in the system (e.g. the human-induced pressure is non-radically different). Wecould expect, given the uncertainty expressed by the standard errors, the winter during 2004–2005 will bedrier in comparison with the climatological mean.

5. CONCLUSIONS

The relationship between winter PRECA over Greece and disturbances linked to SLP and SST is investigatedwith the aim to develop robust seasonal precipitation predictions. Teleconnections between NA and EM aswell as between tropical and extra-tropical climate form the scientific basis for the predictability.

Initially, the data are validated against published results for the EM over the period 1951–1992.Subsequently, fluctuations in the EM – NA teleconnection were identified and quantified, which limitedradically the applicability of traditional MLR models. The effectiveness of two (DLR and DHR) sophisticateddynamic regression models was then examined as the most natural choice to manage non-stationary processes.Significant parameter variation was found in the case of DLR that was growing with increasing input lag.However, the development of non-linear state dependent regression models was prohibited from the timeseries length. Even with such an inherent limitation, DLR achieved more than 10% higher R2 than traditionalregression models in forecast mode, with the latter failing systematically in predicting high variations fromthe mean.

On the other hand, the analysis of the PRECA time series through the DHR model pointed out somefundamental properties of the spectrum like its harmonic terms and their associated seasonal components. Itwas found that the amplitude of the most influential seasonal components with periods of 14 and 3.5 yearsis significantly varying over time and specifically increasing in the former and oscillating in the latter;additionally, the respective terms correlate significantly with NAOIA and SOIA, indicating their climaticorigin. Finally, the forecasting capabilities of DHR were examined. Model predictions appeared very robustwhen missing data were included in the analysis reaching a forecast skill (r = 0.84) higher even than thehindcast skill of the traditional regression models.

The dynamic model with TVPs reflecting the inherent variability of the time series produced the bestresults. This is explained partly by the periodicity of the regression coefficients in the latter as well as to thehigher total number of TVPs. Given the uncertainty arising from the internal variability of the atmosphereand the variations in the strength and spatial distribution of SST anomalies, the dynamic scheme providedrobust forecasts accompanied with uncertainty estimates. The forecast, although in aggregated form, couldbe always used in the direction of reduced risk/cost or increased production by, e.g. farmers. Its influence iscurrently investigated in the skill of a stochastic daily downscaling scheme for the same region.

ACKNOWLEDGEMENTS

The authors wish to thank Prof. P. Young and C. Taylor for providing an extended license of the CAPTAINtoolbox (Young et al., 2004) and Associate Prof. D. Melas for the fruitful discussions. The authors also wishto thank the two anonymous reviewers for their significant comments on the earlier version of this manuscript.

REFERENCES

Akaike H. 1974. A new look at the statistical model identification. IEEE Transactions on Automatic Control 19: 716–723.Amanatidis G, Paliatsos A, Repapis C, Barttzis J. 1993. Decreasing precipitation trend in the Marathon area, Greece. International

Journal of Climatology 13: 191–201.Cassou C, Terray L. 2001. Dual Influence of Atlantic and Pacific SST anomalies on the North Atlantic/Europe winter climate.

Geophysical Research Letters 28: 3195–3198.Cassou C, Terray L, Hurrel J, Deser C. 2004. North Atlantic winter climate regimes: Spatial asymmetry, stationarity with time, and

oceanic forcing. Journal of Climate 17: 1055–1068.Conte M, Giuffrida A, Tedesco S. 1989. The Mediterranean Oscillation. Impact on Precipitation and Hydrology in Italy. Conference on

Climate, Water. Publications of the Academy of Finland: Helsinki; 121–137.



Cullen HM, deMenocal PB. 2000. North Atlantic influence on tigris-euphrates streamflow. International Journal of Climatology 20:853–863.

DaSilva A, Young A, Leritus S. 1994. Atlas of Surface Marine Data. Algorithms and Procedures. NOAA Atlas NESDIS 6, U.S , Vol. 1.Department of Commerce: Washington, DC.

Dunkeloh A, Jacobeit J. 2003. Circulation dynamics of Mediterranean precipitation variability 1948–98. International Journal ofClimatology 23: 1843–1866.

Eshel G, Farrell B. 2000. Mechanisms of eastern Mediterranean rainfall variability. Journal of the Atmospheric Sciences 57: 3219–3232.Eshel G, Cane M, Farrell B. 2000. Forecasting eastern Mediterranean Droughts. Monthly Weather Review 128: 3618–3630.Fernandez J, Saenz J, Zorita E. 2003. Analysis of wintertime atmospheric moisture transport and its variability over Southern Europe

in the NCEP-Reanalyses. Climate Research 23: 195–215.Giorgi F. 2002. Variability and trends of sub-continental scale surface climate in the twentieth century. Part I: observations. Climate

Dynamics 18: 675–691.Goddard L, Mason SJ, Zebiak SE, Ropelewski CF, Basher R, Cane MA. 2001. Current approaches to seasonal-to-interannual climate

predictions. International Journal of Climatology 21: 1111–1152.Hansen DV, Bezdek HE. 1996. On the nature of decadal anomalies in North Atlantic sea surface temperature. Journal of Geophysical

Research 101: 8749–8758.Haylock M, Goodness C. 2004. Interannual variability of European extreme winter rainfall and links with mean large-scale circulation.

International Journal of Climatology 24: 759–776.Hoerling MP, Hurrell JW, Xu T. 2001. Tropical origins for recent North Atlantic climate change. Science 292: 90–92.Hulme M, Barrow E, Arnell N, Harrison P, Johns T, Downing T. 1999. Relative impacts of human-induced climate change and natural

variability. Nature 397: 688–691.Hurrell JW. 1995. Decadal trends in the North Atlantic oscillation: Regional temperatures and precipitation. Science 269: 676–679.Hurrell JW. 2001. The North Atlantic Oscillation. Science 291: 603–605.Isaaks EH, Srivastava RH. 1989. Applied Geostatistics. Oxford University Press: Oxford.Kalman R. 1960. A new approach to linear filtering and prediction problems. ASME Transactions Journal Basic Engineering 83D:

95–108.Kalnay E, Kanamitsu M, Kistler R, Collins W, Deaven D, Gandin L, Iredell M, Saha S, White G, Woollen J, Zhu Y, Chelliah M,

Ebisuzaki W, Higgins W, Janowiak J, Mo KC, Ropelewski C, Wang J, Leetmaa A, Reynolds R, Roy J, Dennis J. 1996. TheNMC/NCAR 40-year reanalysis project. Bulletin of the American Meteorological Society 77: 437–471.

Kushnir Y. 1999. Europe’s winter prospects. Nature 398: 289–291.Lamb PJ, Peppler RA. 1987. North Atlantic oscillation: Concept and application. Bulletin of the American Meteorological Society 68:

1218–1225.Livezy R, Chen W. 1983. Statistical field significance and its determination by Monte-Carlo techniques. Monthly Weather Review 111:

46–59.Maheras P, Anagnostopoulou C. 2003. Circulation types and their influence on the interannual variability and precipitation changes in

Greece. In Mediterranean Climate, Variability and Trends. Springer-Verlag: Berlin, 215–239.Maheras P, Tolika K, Anagnostopoulou C, Vafiadis M, Patrikas I, Flocas H. 2004. On the relationship between circulation types and

changes in rainfall variability in Greece. International Journal of Climatology 24: 1695–1712.Mathieu PP, Sutton RT, Dong B, Collins M. 2004. Predictability of winter climate over the North Atlantic European region during

ENSO events. Journal of Climate 17: 1953–1974.Palutikof JP, Conte M, Casimiro Mendes J, Goodess CM, Espirito Santo F. 1996. Climate and climate change. In Mediterranean

Desertification and Land Use, Brandt CJ, Thornes JB (eds). John Wiley and Sons: London.Rodo X, Baert E, Comin F. 1997. Variations in seasonal rainfall in southern Europe during the present century: relationships with the

North Atlantic oscillation and the El Nino-southern oscillation. Climate Dynamics 13: 275–284.Rodwell MJ, Rowell DP, Follan CK. 1999. Oceanic forcing of the wintertime North Atlantic oscillation and European climate. Nature

398: 320–323.Schneider U, Schonwiese C. 1989. Some statistical characteristics of El Nino/southern oscillation and North Atlantic oscillation indices.

Atmosfera 2: 167–180.Sutton RT, Allen MR. 1997. Decadal predictability of North Atlantic sea surface temperature and climate. Nature 388: 563–567.Trenberth K. 1997. The definition of El Nino. Bulletin of the American Meteorological Society 78: 2771–2777.Wanner H, Bronnimann S, Casty C, Gyalistras D, Luterbacher J, Schmutz C, Stephenson D, Xoplaki E. 2001. North Atlantic

oscillation–concepts and studies. Surveys in Geophysics 22: 321–382.Weisberg S. 1985. Applied Linear Regression. Wiley & Sons: New York.Werner A, Schonwiese C. 2002. A statistical analysis of the North Atlantic oscillation and its impact on European temperature. The

Global Atmosphere and Ocean System 8: 293–306.Xoplaki E, Gonzalez-Rouco J, Luterbacher J, Wanner H. 2004. Wet season Mediterranean precipitation variability: influence of large-

scale dynamics and trends. Climate Dynamics 23: 63–78.Young P. 1984. Recursive Estimation and Time Series Analysis. Springer-Verlag: Berlin.Young P. 1998. Data-based mechanistic modelling of environmental, ecological, economic and engineering systems. Environmental

Modelling & Software. 13: 105–122.Young P. 1999. Nonstationary time series analysis and forecasting. Progress in Environmental Science 1: 3–48.Young P, Pedregal D. 1999. Recursive and en-bloc approaches to signal extraction. Journal of Applied Statistics 26: 103–128.Young PC, Taylor CJ, Tych W, Pedregal DJ, McKenna PG. The Captain Toolbox. Centre for Research on Environmental Systems and

Statistics. Lancaster University: 2004. (www.es.lancs.ac.uk/cres/captain).


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