Assessing the direction of climate interactions by means of complex networks andinformation theoretic tools

J. I. Deza,1, ∗ M. Barreiro,2 and C. Masoller1

1Departament de Fısica i Enginyeria Nuclear, Universitat Politecnica de Catalunya,Colom 11, E-08222, Terrassa, Barcelona, Spain.

2Instituto de Fısica, Facultad de Ciencias, Universidad de la Republica, Igua 4225, Montevideo, Uruguay.(Dated: October 6, 2014)

An estimate of the net flow sense of climate interactions in different geographical regions is madeby constructing a directed climate network from a regular latitude-longitude grid of nodes, usinga directionality index (DI) based on conditional mutual information. Two datasets of surface airtemperature anomalies—one monthly-averaged and another daily-averaged—are analyzed and com-pared. The links of the obtained networks are interpreted in terms of known atmospheric tropicaland extra-tropical variability patterns. Specific and relevant geographical regions are selected, thenet direction of propagation of the atmospheric patterns is analyzed and the direction of the inferredlinks is validated by recovering some well-known climate variability structures. These patterns arefound to be acting at various time-scales, such as atmospheric waves in the extra-tropics or longerrange events in the tropics. This analysis demonstrates the capability of the DI to infer the netdirection of climate interactions and may have a sound impact in improving the present under-standing of climate phenomena and climate predictability. As an interdisciplinary paper coveringinformation theory and network analysis together with climate studies, this work also stands outas an application of different theoretical techniques to the solution of a real-world problem of highcurrent interest.

PACS numbers: 92.70.Gt, 89.70.-a, 87.18.SnKeywords: Climate Networks, Directionality, Information Theory

Information-theoretic tools are usedto construct directed climate networksfrom time-series analysis of observedclimatological data. Specifically, sur-face air temperature (SAT) anomaliesare considered. Two data sets—onemonthly-averaged and another daily-averaged— are used. Directed linksamong the network nodes are definedvia an analysis of the net direction ofinformation transfer. A predictabilitymeasure—based on conditional mutualinformation—quantifying the amountof information in a time-series x(t), con-tained in τ time units in the past of an-other time series, y(t). The resultingdirected network is then studied anda full agreement with state-of-the-artknowledge in climate phenomena hasbeen found, validating this methodol-ogy for inferring the net directionalityof climate interactions, directly fromthe data. No weather assumptions ormodels are made, except for the appro-priate setting of the parameter τ whichis sensible to the shorter or longer auto-correlation of the time series.

∗ juan.ignacio.deza@upc.edu.

I. INTRODUCTION

Network theory is a well-known framework for describ-ing complex systems composed of many interacting com-ponents [1–4]. Many systems can be straightforwardlyrepresented in terms of a well-defined set of nodes cou-pled among them via links that have a clear physicalinterpretation. This is the case, for example, of airportnetworks [5], of social interactions [6] or the Internet [7],just to name some. In other systems it is not clear howto define the relevant nodes, and/or there is no obviousinteraction that can be used to define links. An exampleof this situation is the Earth climate system, in which thelattice of grid points from measurements or models, is de-fined to be the network set of nodes, and it depends onthe resolution of the dataset analyzed; and many clima-tological fields—such as surface air temperature (SAT)or the geopotential height (HGT) at a certain pressurelevel—can be used to define links via an analysis of sig-nificant correlations [8–10].

These so called climate networks (CN), representingthe statistical similarity structure of a spatio-temporalresolved climatological variable, strongly dependent onthe definition of nodes and links [11]. Another drawbackis that the regular spatial sampling results in a small-world topology [12], and thus, a careful interpretation ofthe inferred network is required. Nevertheless, CNs havebeen successfully employed to analyze climate featuresincluding the global connectivity [13–15], the identifica-tion of community structures [16, 17] and the study ofthe possible collapse of the meridional overturning circu-

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lation (MOC) [18] on the north Atlantic, between others.

A relevant drawback of this correlation analysis,which uses symmetric similarity measures (such as cross-correlation or mutual information), is that it yields non-directed networks where the presence of a link revealsinter-dependency but the direction associated (if any) ofthe underlying interaction is not established. For im-proving the understanding of climate phenomena and itspredictability, it is of foremost importance not only to beable to infer the presence of a link between two nodes,but also, to infer the direction of this interaction.

A path to overcome this limitation is by constructingweighted climate networks, where the weight of each linkis composed of two numbers: the correlation strengthbetween the two nodes and the time delay which max-imizes this correlation strength. The sign of the timedelay gives information about the direction of the link.Using this approach, the high sensitivity of the networklinks to El-Nino events was demonstrated, even in geo-graphical regions far from the Pacific ocean [19, 20].

An alternative approach for assessing the directional-ity of climate interactions involves the use of GrangerCausality. In [21] it was argued that the inclusion ofseveral variability patterns like NAO (North AtlanticOscillation), PDO (Pacific Decadal Oscillation), ENSO(El Nino–Southern Oscillation), and NPI (North PacificIndex)—which occur naturally and explain an importantpart of the global atmospheric variability—into a nonlin-ear network-like prediction method, largely improves thepredictability of global temperature over seasonal timescales; even if the time scales of the patterns used is typ-ically in the order of decades. This study suggests acausal directional influence between these major oceanicand atmospheric modes and global temperature variabil-ity, not only over their own time-scales but also over muchshorter ones.

Granger causality has also been used to test interde-pendence between ENSO and the Indian monsoon [22].A non-symmetric bidirectional and even alternating char-acter of coupling was found that extends previous knowl-edge about the presence of negative correlation and in-tervals of phase synchronicity between the processes.

A third approach for directionality detection is basedon information-theoretic measures [23]. For example,in [24] information transfer from larger to smaller timescales was detected in daily-averaged surface air tempera-ture (SAT) time series as the causal influence of the phaseof slow oscillatory phenomena with periods of about 6-11years in the amplitudes of the variability, characterizedby smaller timescales, from a few months to 4-5 years.

The reliability of directed climate networks detected bybivariate nonlinear methods based on information theorycompared to those generated by linear Granger causalityanalysis, was studied in [25]. Several algorithms for esti-mating transfer entropy with a wide range of parameterchoices were considered. As transfer entropy is a specialcase of the conditional mutual information, it reduces toGranger causality for linear Gaussian processes, and it

usually requires longer time-series for accurate estima-tion. It was shown that all the causality methods consid-ered provided reproducible estimates of climate directedinteractions, with the linear method outperforming thenonlinear ones in terms of reliability.

In this paper directed climate networks are constructedby using a predictability measure which, to the best ofour knowledge, has not yet been applied to climatolog-ical data. This bivariate analysis quantifies the amountof information contained in a time-series, x(t), aboutthe past of another time series, y(t), τ time units be-fore. More specifically, we use the directionality index,which quantifies the net direction of information flowbetween two time series, and has been successfully em-ployed for the analysis of numerical data generated fromcoupled oscillators, and empirical data from cardiores-piratory recordings and electroencephalographic record-ings, to name just a few [26–28].

The objective of this work is to demonstrate that thepresent methodology for inferring the net direction of in-formation transfer indeed unveils climatologically rele-vant phenomena. Clear patterns of variability are un-covered, both in the tropics and in the extra-tropical re-gions, and their direction of propagation is shown to be ingood agreement with the current understanding of mainclimate phenomena. The method is shown to be partic-ularly useful for the analysis of daily-averaged data, aswell-defined atmospheric patterns are uncovered, whichare not seen with monthly-averaged data.

The paper is organized as follows. Section II presentsthe data analyzed and the method used for constructingdirected climate networks. The statistical significancetest is also discussed. Section III presents the results andcomparison between directed and non-directed climatenetworks is made, where relevant patterns of global atmo-spheric variability and interpreted in terms of well-knownclimatological phenomena. Finally Sec. IV presents theconclusions.

II. DATA AND METHODS

A. Data

We consider two datasets, both corresponding to SATanomalies from the reanalysis of the National Centerfor Environmental Prediction/National Center for Atmo-spheric Research, (NCEP/NCAR)[29]. The data covers aregular grid over the Earth’s surface with latitudinal andlongitudinal resolution of 2.5◦, resulting in N = 10226grid points or network nodes.

The first data set corresponds to monthly-averagedSAT data. Since the data cover the period from Jan-uary 1949 to December 2013, in each node we have atime series of 780 data points. The SAT anomalies arecalculated as the actual temperature values minus themonthly average, and they are normalized by the stan-dard deviation. Each time-series is linearly detrended.

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The second data set consists of daily-averaged SATfrom the same source, with the same spatial resolution,and covering the same period of time (the length of eachtime series is 23725 data points). In this case, for cal-culating the anomalies, the daily average has been sub-tracted from the actual temperature value, and the leapdays have been discarded. The time series are also de-trended and normalized by the standard deviation.

B. Directionality measure

We consider the SAT anomalies time series in twonodes, X(t) and Y (t), which are characterized by proba-bility distribution functions (PDFs) pX , pY , and by theirjoint PDF, pXY . To assess the directionality of the linkbetween these two nodes we use the directionality index(DI) as defined in [26, 30]:

DIXY (τ) =IXY (τ)− IY X(τ)

IXY (τ) + IY X(τ), (1)

where IXY (τ), IY X(τ) is the conditional mutual infor-mation defined as:

IXY (τ) = I(X;Y |Xτ )

= H(X|Xτ ) +H(Y |Xτ )−H(X,Y |Xτ ); (2)

IY X(τ) = I(Y ;X|Yτ )

= H(Y |Yτ ) +H(X|Yτ )−H(Y,X|Yτ ), (3)

with Xτ = X(t − τ), Yτ = Y (t − τ) and H(X|Y ) beingthe conditional entropy [26, 30].

In the more general case, I(X;Y |Z) indicates theamount of information shared between X(t) and Y (t),given the effect of Z(t) over Y (t). This way, to as-sess the information transfer from X(t) to Y (t), Z(t)is replaced by the past of time-series X(t). This wayI(X;Y |Xτ ) quantifies the amount of information sharedbetween X(t) and Y (t), given the influence of X(t − τ)over Y (t). Analogously, to assess the information trans-fer from Y to X, Z(t) is replaced by the past of Y (t).This special case of conditional mutual information isalso referred as transfer entropy.

The directionality index, DIXY , then quantifies thenet information flow. From the definition of DIXY ,Eq.(1), it is clear that DIXY = −DIY X . Also, −1 ≤DIXY ≤ 1: DIXY = 1 if and only if IXY 6= 0, IY X = 0(i.e., the information flow is X → Y and there is noback coupling Y → X) and DIXY = −1 if and only ifIXY = 0, IY X 6= 0 (i.e., the information flow is Y → Xand there is no back coupling X → Y ).

Naturally, τ > 0 is a parameter that has to be tunedappropriately to the time-scales involved in the series. Ifτ is too small DIXY (τ) will capture short time scale di-rectionality, and may fail if the time series behave toosimilarly on those time scales as they do if they are sub-jected to the same external forcing. On the other hand,if τ is too large, larger than the decorrelation time of the

time-series, the effect of the past X over Y (and of Y overX) will be negligible and DIXY (τ) will be a small andin principle random value. In the next section the cri-terium employed for accessing the statistical significanceof the DI values will be discussed, and in section III,a throughout study of the effect of varying the parame-ter τ will be performed. It is worth noticing that in thetwo data sets considered, τ = 1 has a different meaning:in the monthly-averaged SAT time-series, the minimumvalue of τ that we can consider is one month, while inthe daily-averaged data, the minimum value of τ is oneday.

To compute the PDFs associated to each time series,and the joint PDFs we use 10-bin histograms of values.An alternative approach is to use a symbolic transforma-tion known as ordinal analysis [26, 31–33]. Some prelimi-nary studies shows that using ordinal analysis offers as anadditional advantage the possibility of finding the direc-tionality of the links at different time-scales; this studyis still in progress.

It it important to note a drawback of using the direc-tionality index for network construction: it does not dis-tinguish indirect from direct information transfer. Giventhree time-series, X, Y and Z, positive and significantDI values IXY > 0 and IXZ > 0 will also imply a sig-nificant value of IY Z—either positive or negative—evenif there is no “direct” information in Y about the fu-ture or the past of Z, as the information will have been“indirectly” contained in X.

C. Statistical significance analysis

To address the significance of DI values, surrogatedata is generated using the bootstrap surrogate (BS) al-gorithm [34].

For each possible link between any pair of nodes (i.e.,for each pair of time-series), an ensemble of 100 BS sur-rogates is calculated and the DI value computed fromthe original time-series is compared with that computedfrom the surrogates. A DI value is considered significantif it is larger (or smaller) than µ±3σ, with µ and σ beingthe mean and standard deviation of the DI distributioncomputed from the surrogate ensemble. The distributionof surrogate DI values is usually bell-shaped motivatingthe use of classical statistics. The more time and memoryconsuming 99nth percentile test was also used, yieldingsimilar results.

In addition, mutual information (MI) values are gen-erated, and a second significance test is then applied,consisting on accepting only the DI links that have a sig-nificant value on the MI network. The criterium used forconsidering a MI value significant is the same as abovewith the distinction that MI values are defined positive.This second significance test is necessary as the aim isinferring the effective direction only for links determinedto be significant, quantified via the MI measure as in[33].

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FIG. 1. (Color Online). This figure shows the procedure of constructing directionality maps from the raw DI calculations.In panels (a) and (d) the unfiltered DI maps are shown for the central Pacific and the Indian oceans respectively. In panels(b) and (e) the statistically significant MI is calculated using 100 bootstrap surrogates. These results are combined in (c) and(f) where only the links accepted by the MI are considered and then filtered using the same bootstrap statistical significancetechnique.

FIG. 2. (Color Online). Plot of the DI and MI values of allthe links of two nodes, one in the Pacific Ocean and the otherin the Indian Ocean (the nodes are the same as in Fig. 1).In blue incoming links are indicated and outgoing links arein red. The black dots indicate the disregarded links (eitherbecause MI is not significant, or because MI is significant butDI is not). Significant links are plotted in red (outgoing links,DI > threshold) and in blue (incoming links, DI < threshold).

A graphical explanation of the full procedure is shownin figure 1 where for two nodes (one over the Pacific andone over the Indian ocean) the unfiltered DI maps areshown in panels (a,d), the significant MI values are dis-played in panels (b,e) and finally the significant DI valueshaving passes both tests, in panels (c,f). The DI mapsshow in red positive values of DI, which mean outgoinglinks, while the incoming links are shown in blue.

In order to visualize the results of the criterion em-

ployed to assess the significance of (MI, DI) values, Fig.2 displays, as examples, the DI and MI values of all thelinks of the two nodes, one located in the central Pacificand the other in the Indian ocean (the nodes are thesame as in Fig. 1). Significant links are those which haveMI and |DI| values above the corresponding thresholds.We remark that two thresholds, one for MI and one forDI, are defined for each link, from 100 surrogates. Theblack dots indicate the disregarded links (either becauseMI is not significant, or because MI is significant butDI is not). Significant links are plotted in red (outgoinglinks, DI > threshold) and in blue (incoming links, DI <threshold).

Figure 2 indicates that high MI values do not implyhigh DI values. In Fig. 2(b) one can notice that most ofthe blue dots are located in a narrow range of MI valueswhile they are distributed in terms of DI values. Aninspection of panel 1(f) shows that the blue links come tothe node in the Indian Ocean from a well-defined regionin the central Pacific Ocean. On the other hand, onecan notice in Fig. 2(b) that the few red dots are moredistributed in the MI, DI plane; and in Fig. 1(f) we notethat red outgoing links connect the node in the IndianOcean to various regions on Earth.

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FIG. 3. (Color Online). Effect of τ on tropical areas usingmonthly-averaged data. In this case a point in central pacific(the same point used in [33] is considered). The values of τchosen are: (a) 1 month, (b) 3 months, (c) 6 months, (d) 12months. Notice the decorrelation of the time series for largeτ . Incoming links are in blue while outgoing are in red.

FIG. 4. (Color Online). Effect of τ = 1 month over the extra-tropics. Two points, one in each hemisphere were considered.Significant values of DI for (a) southern South America, de laPlata basin. (b) Labrador Sea, are shown. As the decorrela-tion time on the extra-tropics is very fast, no good results arefound even for τ = 1. Figures (c) and (d) show significant MIover the same points in order to show this is not a problemof statistics. Incoming links are in blue while outgoing are inred.

III. RESULTS

Firstly results of the analysis of the monthly-averageddata set will be presented, which allow comparison tothe undirected networks reported in [33], afterwards re-sults of the analysis of the daily-averaged data set willbe shown, uncovering additional patterns of atmospheric

variability, not observed with monthly-averaged data be-cause of its coarse time resolution.

A. Analysis of monthly-averaged SAT anomalies

As stated in the methodology the correct choice of thevalue of τ is necessary for obtaining consistent results. Asτ can be only an integer, using monthly averaged data,its minimum value will be of one month. In the trop-ical areas the influence of the ocean on the surface airtemperature is a dominant characteristic. Moreover, be-cause of the large heat capacity of water and the ocean’sdynamics, the sea surface temperature (SST) anomaliesvary in the scale of months. Calculating DI for a point inthe central pacific (NINO3 area) for different values of τyields the results shown in Fig. 3. The point consideredis the same as in in Fig. 1 (a-c) and moreover, the panel1(c) is the same as panel 3(a).

For τ = 1 the figure shows the central Pacific influ-enced by (in blue) the eastern Pacific and influencing(in red), presumably through atmospheric teleconnec-tions, the global tropics and the extratropical Pacificocean. However, as τ grows the number of significantconnections decreases, suggesting that the time-scale ofdecorrelation of the surface air temperature is less than 6months. This is consistent with the persistence time scaleof 3 to 6 months of observed sea surface temperature.

The extratropical atmosphere shows larger internalvariability than the tropics and the impact of the ex-tratropical SST on the atmosphere is much more limitedthan in the tropics. Thus, the variability of extratropi-cal SAT is dominated by synoptic atmospheric dynamicsand has time scales of a few days. Longer persistencetime scales might appear in the extratropics if the re-gion is influenced by tropical SST. This motivates theuse of a small value of τ when considering extra-tropicalvariability.

In Fig. 4 The DI and MI maps for two points in theextra-tropics are shown. Panels 4 (a,c) show links relatedto a point in southeastern South America, while Panels4 (b,d) show links related to a point in the Labrador sea,whose characteristics are linked to the North AtlanticOscillation (NAO)[35]. The top panels show DI for τ = 1while the bottom panels show MI.

Consistent with the previous description, the extra-tropical SAT show only some incoming links from thetropical region for τ = 1 month. The point over theLabrador Sea seems to show also some outgoing links tothe northeast, although there is no clear structure. Abetter depiction of links is found using daily data (seebelow).

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FIG. 5. (Color Online). Comparison of monthly-averaged and daily-averaged SAT datasets for the tropical area. The pointsconsidered are those in fig. 1. On panels (a) and (c) monthly data for τ = 1 month is shown—Figs. 1 (c) and (f) are respectivelyrepeated. On panels (b) and (d) daily data for τ = 30 is presented. Results are consistent and the resolution using daily datais better in both cases. Incoming links are in blue while outgoing are in red.

B. Analysis of daily-averaged SAT anomalies

1. Comparison between datasets

In order to obtain more temporal resolution daily datahas been used. Fig. 5 shows a comparison betweenDI formonthly data –panels 5(a) and 5(c)—and for daily data—panels 5(b) and 5(d). Corresponding to the same pointsconsidered in figure 1. In order to adequately comparethe datasets, τ was chosen to be equivalent in both mapschoosing τ = 1 month and τ = 30 days for the monthlyand daily data respectively.

The maps using monthly and daily mean data showsimilar features and no inconsistencies are found. Fur-thermore, the one constructed using daily data capturesmuch better the local and remote dependences and di-rectionality of the links. Areas with significant links arebetter defined and some regions that are known to be in-fluenced by equatorial Pacific SST, like the tropical northAtlantic [36], clearly appear using daily data, but onlyvery roughly using monthly data. Thus, the increase intemporal resolution improves the representation of thelinks related to tropical regions.

2. Directionality on the extra-tropics

The improvement in characterizing directionality us-ing daily data is even larger in the extratropics. As wementioned above, the extratropical SAT is strongly de-pendent on synoptic scale perturbations (a time scale ofa few days). Thus, the use of daily data should allow touncover these relationships and investigate the directionof the links as the lag increases. To do so we consider thepoint in southeastern South America shown in Fig. 4 andconstruct the directionality network for several values ofτ ranging from a day to one month ( Fig. 6). For synop-tic time scales of a few days the methodology uncoversthe existence of a wave train connected to southeasternSouth America propagating with a southwest-northeastdirection. Moreover, there is a clear separation line be-tween regions with incoming and outgoing links. Thisconfiguration is characteristic of the progression of a frontthrough the reference point and does not imply that theSAT over the reference point influences the region to thenortheast but it only happens to be in the path of theperturbation. As the lag time increases, the extratropicalwave train associated with synoptic time scales fades andonly the points in the tropics remain, consistent with an

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influence of the equatorial Pacific on the region on longertime scales, perhaps related to ENSO.

A similar behavior is seen taking as reference pointthe SAT over the Labrador Sea (Fig. 7). For small val-ues of τ the progression of a front is clearly detectedusing this procedure: given the mean westerly winds atthese latitudes, the front moves from west to east andis clearly marked as the boundary between the incomingand outgoing links. It is also seen for τ = 3 suggestingthat in about three days the front reaches the Mediter-ranean region affecting temperatures there. Again, asτ increases mainly the tropical links remain. However,even for τ = 30 there is a well defined region of outgo-ing links that remain over the Labrador Sea, suggestingthat the SAT in the region may have relatively long timescales of variability, perhaps related to the North AtlanticOscillation.

3. Directionality on tropical Pacific Ocean

On figure 8 the DI for τ = 30 days has been plottedfor points covering all the equatorial Pacific. They beginat 95.0◦ W –panel 8(a) – and end at 125.0◦ E –panel8(d). Clearly, the influence the Pacific ocean exerts is al-most global, over tropics and extra-tropics, in agreementwith previous studies (e.g. in [37]). The DI allows toshow that —even if there are feedbacks and the Pacific isaffected by extratropical perturbations and other oceanbasins (e.g. the tropical Atlantic)— the influence is effec-tively from the Pacific to the rest of the world. Moreover,the maps show that the largest influence is done by theequatorial Pacific close to the dateline. This is clear inthe extratropical atmosphere, as well as in the tropicalnorth Atlantic. On the other hand, the connection tothe Indian ocean and south Atlantic is not so sensitiveof the point considered over the equatorial Pacific. Asthe reference point moves further west from the date-line the influence decreases substantially, only remainingweak connections to the tropical north Atlantic and In-dian oceans. The methodology can thus be applied tofind the best region to construct an index that describesthe Pacific influence over the area where climate anoma-lies are studied.

Notice that all maps show a blue tongue of incominglinks to the east of the point considered; it it is seen firstin panel 8(a) and extends westward until covering thewhole Pacific ocean in 8(d). This feature is related tothe existence of the equatorial cold tongue and the factthat easterly trades blow over the equator thus advectingair from the east to the west of the point.

IV. CONCLUSIONS

The climatological relevance of directed climate net-works constructed via information-theoretic tools hasbeen shown. The presence of significant links was in-ferred using mutual information while the direction ofthose links was inferred using the directionality index.Using monthly-averaged SAT data the results from ourprevious work [33] have been recovered and the analysisextended to infer the net direction of propagation of theinformation. The inference method was tested againstthe value of the parameter τ , that represents the timerequired for the information to travel from one node toanother. It was found that by adequately tuning τ thenetwork connectivity varied revealing the various time-scales of atmospheric processes; too short values of τfailed to capture several long-range links, while for toolarge values of τ (above the length of correlations in thedata) the connectivity of the network decreased drasti-cally.

In addition, when considering daily-averaged SATdata, the analysis revealed variability patterns consis-tent with known features of the global climate dynam-ics. In the extra-tropics the long time average synopticweather was correctly inferred: as specific examples twogeographical regions in different hemispheres—one nodein de la Plata basin and another in the Labrador Sea—were considered, and the link direction revealing wavetrains propagating from west to east, in both hemisphereswas shown.

As a future work, it would be interesting to apply thismethodology to analyze how the climate network changesin the different seasons. Another interesting issue is therole of permutation entropy. Computing the direction-ality index using the ordinal patterns approach [32, 33],which would allow constructing networks revealing atmo-spheric processes with short time-scales (of days-months)or with long time-scales (few years). These studies are inprogress and will be reported elsewhere.

ACKNOWLEDGMENTS

The research leading to these results has received fund-ing from the European Community’s Seventh Frame-work Programme FP7/2007-2011 under grant agreementn◦ 289447 (ITN LINC). C.M. also acknowledges finan-cial support of grant FIS2012-37655-C02-01 of the Span-ish Ministerio de Ciencia e Innovacion, and the ICREAACADEMIA award.

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FIG. 6. (Color Online). Effect of τ using daily data in the extra-tropics. Southern hemisphere, the point is the same as infig. 4(a). The values of τ are: (a) 1 day, (b) 3 days, (c) 7 days, and (d) 30 days. Smaller values of τ can capture wave trainspropagating from west to east, while for values of τ of over a week the decorrelation is higher and only a longer time scaleinfluence from the Pacific ocean persists. Incoming links are in blue while outgoing are in red.

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FIG. 7. (Color Online). As in fig 6 but for a point the northern extra-tropics, on the Labrador Sea—the same as in fig.4(b). The values of τ are: (a) 1 day, (b) 3 days, (c) 7 days, and (d) 30 days. The Labrador sea area is related to a source ofatmospheric variability of the north Atlantic ocean that affects Europe. As in the last figure, smaller values of τ can capturewave trains propagating from west to east, over Europe, while values of τ of over a week loose this effect and only a longertime scale influence from the Pacific ocean persists. Incoming links are in blue while outgoing are in red.

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FIG. 8. (Color Online). The zonal change of directionality over the equatorial pacific is shown. In all cases τ = 30 days. From(a) near the south American coast, to (d) in the western Pacific ocean. As seen in the maps, most of the points over centraland eastern Pacific have an important effect over a large part of the world; especially over the tropical areas and the rest ofthe Pacific ocean—notice that in (a)-(c) the teleconnections remain basically the same, although the intensity varies. Incominglinks are in blue while outgoing are in red.