Inferring long memory processes in the climate network via ordinal pattern analysis

3.6 MB·English

by  Marcelo Barreiro

Inferring long memory processes in the climate network via ordinal pattern analysis
Marcelo Barreiro and Arturo C. Marti
Instituto de F´ısica, Facultad de Ciencias, Universidad de la Repu´blica, Igua´ 4225, Montevideo, Uruguay
Cristina Masoller
Departament de Fisica i Enginyeria Nuclear, Universitat Politecnica
de Catalunya, Colom 11, E-08222 Terrassa, Barcelona, Spain
(Dated: January 28, 2011)
We use ordinal patterns and symbolic analysis to construct global climate networks and uncover
long and short term memory processes. The data analyzed is the monthly averaged surface air
1
1 temperature (SAT field) and the results suggest that the time variability of the SAT field is deter-
0 mined by patterns of oscillatory behavior that repeat from time to time, with a periodicity related
2 tointraseasonal oscillations and to El Nin˜o on seasonal-to-interannual time scales.

n PACSnumbers: 05.40.-a,05.40.Ca,05.45.Tp,02.50.-r
a Keywords: Climateanalysis,complexnetworks,ordinalpatterns, symbolictimeseries
J

7
2 We analyze climatological data from a complex I. INTRODUCTION
networks perspective, using techniques of non-

] linear time-series symbolic analysis. Specifically,
h Complexnetworkshavebeenintensivelystudiedinthe
p we employ ordinal patterns and binary represen- lastyearsbecausetheyrepresentmanyrealsystemssuch
- tations to analyze monthly-averaged surface air
astheInternet,ecological,socialandmetabolicnetworks,
o
temperature (SAT)anomalies. Bycomputingthe
a genes, cells and the brain [1]. Global climate modeling
mutual information of the time-series in regular
. is also a hot topic nowadays because of its huge eco-
s grid points covering the Earth’s surface and then
c nomic and social impact for future generations. Giving
i performing global thresholding, we construct cli- the complexity of the inter-relations between the differ-
s
mate networks which uncover short-term mem-
y ent elements that constitute our environment, it is im-
ory processes, as well as long ones (5-6 years).
h portant to analyze climatological data from a complex
p Ourresultssuggestthatthetimevariabilityofthe
network perspective. However, despite the intensive ef-
[ SAT anomalies is determined by patterns of os-
fort in research done in these two interdisciplinary and
cillatory behavior that repeat from time to time,
3 fascinating fields, just very few studies have combined
with a periodicity related to intraseasonal varia-
v both [2–8]. These studies have shown that network the-
4 tions and to El Nin˜o on seasonal to interannual ory can yield light into interesting, previously unknown
6 time scales. The present work is located at the
features of our climate.
5 triple intersection of three highly active inter-
1 disciplinary research fields in nonlinear science: Tsonis and Swanson [4] and Yamasaki, Gozolchiani
. and Halvin [3] have shown that the climate network is
0 symbolic methods for nonlinear time series anal-
significantlyaffectedbyElNin˜o,asduringElNin˜oyears
1 ysis, network theory, and nonlinear processes in
manylinks ofthe networkarebroken. TsonisandSwan-
0 the earth climate. While a lot of effort is be-
1 ing done in order to improve our understanding son [4] constructed cross-correlation-based networks of
: the SAT field for El Nin˜o and for La Nin˜a years and in-
v of natural complex systems, with many different
vestigated their structure. They found that the El Nin˜o
i methods for mapping time series to network rep-
X networkpossessessignificantlyfewerlinksandlowerclus-
resentations being investigated and employed in
r tering coefficientand characteristicpathlength than the
complex systems such as the human brain, our
a
La Nin˜a network. They conjectured and verified that,
work is the first one aimed at characterizing the
because El Nin˜o network is less communicative and less
globalclimatenetworkintermsofoscillatorypat-
stable than La Nin˜a one, during El Nin˜o years temper-
terns that tend to repeat from time to time, with
ature predictability is lower compared to La Nin˜a years.
various time scales. By mapping these processes
Using a different approach, Yamasaki, Gozolchiani and
into a global network, using ordinal patterns and
Halvin [3] arrived at a similar conclusion. They devel-
binary representations, we find that the struc-
oped a method which allows to follow time variations
ture of the network changes drastically at differ-
of the network structure by observations of fluctuations
ent time scales.
inthe correlationsbetweennodes. Themethodallowsto
distinguishbetweenthetwoqualitativelydifferentgroups
of network links, blinking links that appear and disap-
pearinashorttime,androbustlinksthatrepresentlong
lastingrelationsbetweentemperaturefluctuationsintwo
regions. Assumingthatbrokenlinksareduetostructural
2
changesinthenetwork,bytrackingthesechangesinsev- statistics. TheSAT dataavailable(describedinthenext
eralzonesastrongresponsetoElNin˜owasreveled,even section) limited us to construct ordinalpatterns of max-
in geographical regions where the mean temperature is imum length 5, which allows to consider time-scales up
not affected by El Nin˜o. to 5 years or 5 months. To overcome this limitation we
Dongeset al. [6]comparedthestructuralpropertiesof employed “binary representations”, by which the time-
networks constructed by using, as a measure of dynami- seriesofSAT anomaliesweretransformedintosequences
calsimilarity betweenregions,linear and nonlinearmea- of 0s and 1s. These binary representations allowed to
sures: the linear Pearson correlation coefficient and the consider processes with longer time-scales, up to 6 years
nonlinearmutualinformation. Theyanalyzedtwosetsof or 6 months. We will show in what follows that ordinal
data: the SAT anomalies obtained from large-scale cli- patterns andbinaryrepresentationsaretools that, when
mate simulations by the coupled atmosphere-ocean gen- employedwithinacomplexnetworkperspective,arevery
eralcirculationmodelsandtheSATanomaliesreanalysis powerfulfortheanalysisofclimatologicaldata. Byrevel-
data sets. A high degree of similarity using the two ap- inglongtermandshorttermmemoryprocessestheypro-
proaches (linear and nonlinear similarity measures) was vide additional information to that obtained from con-
found on the local and on the mesoscopic topological ventional time-series analysis and thus they help to a
scales;however,importantdifferenceswereuncoveredon better understanding of our complex climate.
the global scale, particularly in the betweenness central- Thisarticleis organizedasfollows: SectionII presents
ity field. In [7] Donges et al. employed the mutual in- the description of the data analyzed and a summary of
formation to reveal wave-like structures of high-energy the methodology employed. Section III presents the re-
flow, that could be traced back to global surface ocean sultsobtainedwithordinalpatternsandbinaryrepresen-
currents. Their results point to the major role of the tations, anda comparisonwith the methodologies previ-
oceanicsurfacecirculationincouplingandstabilizingthe ously employed by other authors (i.e., the linear cross-
global temperature field in the long-term mean. correlation[4]andthenonlinearhistogram-basedmutual
When computing the mutual information, in order to information [6]). Section IV contains a discussion of the
detect patterns and correlations in the variability of two results and the conclusions.
nodes, a critical issue is defining probability distribu-
tion functions (PDFs) that fully take into account the
temporal orderin which the SAT anomaliesoccur in the
time-series. Histogram-based PDFs do not take into ac-
II. DATA AND METHODS
count this temporal order, and thus, are not optimal for
capturing subtle correlated oscillatory patterns. Alter-
natively, one can use time-delay embedding techniques We present the analysis of the monthly averaged sur-
to represent the time series as a trajectory in a high- faceairtemperature(SAT field,reanalysisdatafromthe
dimensionalspace;however,theinformationprovidedby National Center for Environmental Prediction/National
themutualinformationisstronglydependentontheem- Center for Atmospheric Research, NCEP/NCAR [12]).
bedding technique, the time-delay, and the phase space As in [2, 4, 6,7], anomalyvalues areconsidered(i.e., the
partition [9]. actual temperature value minus the monthly average).
An alternative methodology, originally proposed by The data covers a regular grid over the earth’s sur-
Bandt and Pompe (BP) [10] allows to define probabil- face with latitudinal and longitudinal resolution of 2.50.
itydistributionfunctionsthatfullytakeintoaccountthe These N = 10226 grid points are considered the nodes
time ordering of the SAT anomalies. The BP method (or vertices) of a network (or graph), and the exis-
is based on comparing values in the time-series to con- tence of a link (or edge) between any two nodes de-
struct ”ordinal patterns”. By computing the PDF of pends on the “weight” of the link that measures the
the possible ordinalpatterns,variousinformation-theory degree of statistical similarity between the climate dy-
quantifiers,such as the permutationentropy,the mutual namics in those two nodes. The data covers the period
information,complexitymeasures,etc. canbecomputed. January1949-December2006,and therefore in eachgrid
The BP method has been successfully employed to ana- point i (i = 1...N) we have M = 696 data points,
lyze time-series generated from physical, biological and {xi(t),t = 1...M}. W = {wij,i,j = 1...N} is the
social systems (see, e.g., [11] and references therein). matrix that contains the weights that characterize the
links between any two nodes. Since we don’t attempt to
When employing the BP methodology the precise val-
uncoverdirectionality inthe couplingsamong the nodes,
ues of the SAT anomalies are neglected (as the method
we will consider a symmetric measure of statistical simi-
is basedoncomparingrelative values in the time-series);
larity that results in symmetric weights.
however, as we will show, with the BP method one can
identify patterns of oscillatory behavior that tend to re- In[2,4]theseweightswerequantifiedwiththeabsolute
peat from time to time, with various time scales. A value of the linear cross-correlation coefficient; in [6, 7],
drawback of the BP method is that, in order to cap- with the mutual information, a nonlinear measure that
ture long memory processes, long time series are needed is a function ofthe probability density functions (PDFs)
to compute the PDFs of the ordinal patterns with good that characterizethe time series in the two nodes, p (m)
i
3
and p (n), as well as of the joint probability, p (m,n), Weusethefollowingprocedure: first,wecheckthatwe
j ij
only take into account significant network connections.
pij(m,n) To do this we compute the weight matrix W from ran-
Wij =Mij =Xpij(m,n)log . (1)
pi(m)pj(n) domly shuffled time series in each node. The random
m,n
elementsofthis 10226×10226matrixhaveaverynarrow
The mutual information, which can also be written as PDF which, in principle, allows the use of the maximum
matrix element, w , as a significant limit. Then, we
max
Mij =Si+Sj −Sij, (2) compute W with the original time series and consider
that there is a significant link between the nodes i and
where S = − p logp , S = − p logp and S =
i P i i j P j j ij j if w > w , otherwise, we set w = 0. While there
− p logp , indicates the amount of information of ij max ij
P ij ij areseveralmethodstoeliminatenon-significantlinksand
{x (t)}, we obtain by knowing {x (t)}, and vice versa.
i j the evaluation of statistical significance is still an open
M measures the degree of statistical interdependence
ij problem(see,e.g.,thediscussionin[6]),thisprocedureis
of the time series; if they are independent, p (m,n) =
ij computationally cost-efficient and we will show in what
p (m)p (n) and M =0.
i j ij follows that allows to uncover meaningful climate net-
To uncover correlated“patterns” of oscillatory behav-
works. The drawback is that it is a rather strong test
ior in the SAT anomalies, we employ the methodologies
that eliminates weak but significant links, and as a re-
referred to as ordinal patterns and symbolic analysis,
sults the networks tend to be very spare. The final step
which are based on comparing consecutive values in the
is to chose a threshold τ to select the strongest links.
time series, to compute the PDFs in Eq.(1). We begin
As in [6], we chose τ such that the resulting networks
by presenting the ordinal pattern methodology [10].
have a pre-determined number of links. In the follow-
First, in each grid point i, the time series {x (t)} is
i ing we present results for networks that have 1% of the
divided into M−D overlappingvectorsof dimension D.
total possible links (which will be referred to as “low-
Then, each element of a vector is replaced by a number
threshold” networks) and networks containing 0.1% of
from0toD−1,inaccordancewithitsrelativemagnitude
the total possible links (referred to as “high-threshold”
in the orderedsequence (0 correspondingto the smallest
networks). For easy comparison and to visualize the ef-
andD−1tothelargestvalueineachvector). Forexam-
fect of thresholding, we also present the networks con-
ple, with D =3 the vector (v0,v1,v2)=(6.8, 11.5, 1.1), taining all the significant links, which will be referred to
gives the ordinal pattern 201 because v2 < v0 < v1. In as the “zero-threshold”networks.
thisway,eachvectorhasassociatedan“ordinalpattern”
The networks are represented graphically as two-
(OP) composed by D symbols, and the symbol sequence
dimensional maps by plotting the area-weighted connec-
comes from a comparison of neighboring values. Last,
tivity[2,4,6,7], whichis thefractionofthe totalareaof
one computes the PDF of the D! possible ordinal pat-
the earth to which each node i is connected,
terns. For example, with D = 3 the 3! = 6 different
patterns are (012,021,102,120,201 and210),andthus, NA cos(λ )
thePDFiscalculatedwith6bins. Tohaveagoodstatis- AWC = Pj ij i , (3)
tics one must have M −D >>D! (i.e., # of OPs in the i PNj cos(λj)
time series >> # of possible OPs).
Because in each time series we have M = 696 data where λi is the latitude of node i and Aij = 1 if nodes
points,tocomputethePDFswithgoodstatisticswelimit i and j are connected (i.e., if wij ≥ τ) and 0 otherwise.
to consider only D = 4 and D = 5. Ordinal patterns of The cosine terms correct for the fact that in a surface
D ≤ 3 do not provide good resolution for computing spherical network defined on a regular planar grid, the
the mutual information, Eq. 1, because the PDFs are nodes correspond to regions of different area.
calculated with very few bins (for D = 6, there are only
6 ordinal patterns, and thus, only 6 bins).
III. RESULTS
With climatological data meaningful ordinal patterns
can be formed either by comparing consecutive years or
consecutive months. Specifically, if we use D = 3, when A. Ordinal Patterns Analysis
comparing consecutive years, the OPs in node i are de-
fined by (x (t),x (t+12),x (t+24)), t=1,...,M −24; The networks obtained when the ordinal patterns are
i i i
whencomparingconsecutivemonths,theyaredefinedby definedbycomparingSATanomaliesinconsecutiveyears
(x (t),x (t+1),x (t+2)), t=1,...,M −2. and in consecutive months are displayed in Figs. 1-4. In
i i i
To decide whether there is a link between two nodes, eachpanelthevaluesofthethreshold,τ,andoftheedge-
we perform global thresholding [13], i.e., we define a density,
threshold τ (which is the same for all pairs of nodes)
and assume that there is a link between i and j if the ρ= PNi,jAij , (4)
weight of the link is above the threshold, i.e., wij ≥τ. N(N −1)
Clearly, a careful selection of the threshold is crucial
for uncovering the backbone of the network [6]. are indicated.
4
τ = 0 ρ = 0.027 τ = 0.227 ρ = 0.01 τ = 0.504 ρ = 0.001
0.218 0.08 0.0115
60N 60N 60N
0.1744 0.064 0.0092
30N 30N 30N
0.1308 0.048 0.0069
0 0 0
0.0872 0.032 0.0046
30S 30S 30S
0.0436 0.016 0.0023
60S 60S 60S
0 0 0
0 90E 180E 90W 0 90E 180E 90W 0 90E 180E 90W
FIG.1. Zero-threshold(left),low-threshold(center)andhigh-threshold(right)networksconstructedbycomputingthemutual
informationfromordinalpatternsoflengthD=4definedbycomparingSATanomaliesinconsecutiveyears. The2Dplotsare
color-coded suchthatthewhite(red)regionsindicatethegeographical areaswithzero(largest) areaweighted connectivity. In
each panel thevalues of thethreshold, τ, and of the edge-density,ρ,are indicated.
τ = 0 ρ = 0.006 τ = 0.674 ρ = 0.001
0.0565 0.009
60N 60N
0.0452 0.0072
30N 30N
0.0339 0.0054
0 0
0.0226 0.0036
30S 30S
0.0113 0.0018
60S 60S
0 0
0 90E 180E 90W 0 90E 180E 90W
FIG. 2. Zero-threshold (left) and high-threshold (right) networks constructed by computing the mutual information from
ordinal patterns of length D = 5 defined by comparing SAT anomalies in consecutive years. The 2D plots are color-coded
such that the white (red) regions indicate the geographical areas with zero (largest) area weighted connectivity. The weaker
links lose a bit of memory (compare thezero-threshold networks with D=4 and D=5) while the strong links do not, as the
high-threshold networks are the nearly same for D=4 and D=5.
For consecutive years the networks with D = 4, Fig. waves. Changes in the precipitation associated with El
1, and D = 5, Fig. 2 are very similar showing highest Nin˜o also induce stationary Rossby waves in the north-
connectivity for the tropical region. ern and southern extratropics that generate long range
connections called atmospheric teleconnection patterns.
For zero-threshold (left panels in Figs. 1 and 2) the
ExamplesofthesestructuresarethePacific-NorthAmer-
tropical Pacific shows the largest connectivity, particu-
ican pattern that affects the northern Pacific and North
larly in the central and eastern side of the basin; the
America, and the Pacific-South American pattern that
tropical Atlantic and Indian oceans follow. In the ex-
propagates in the southern Pacific toward South Amer-
tratropics there are patches of high connectivity off the
ica. TheseanomalousstructuresconnectthetropicalPa-
western coast of Canada in the Northern Hemisphere
cificwithremotelocationsandaffectthelocalclimateby
(N.H.) and in the south Pacific in the Southern Hemi-
changing,for example, the advectionof heator moisture
sphere (S.H.). This connectivity structure is more pro-
into a region.
nounced for D = 5, although some of the weak links
lose memory. These characteristics hint to El Nin˜o as a For non-zero-thresholds (center and right panels in
fundamental player in setting up these connections [14]. Figs. 1 and 2) only the strongest links remain and the
The El Nin˜o phenomenon occurs on interannual time networks clearly show again an El Nin˜o-like structure in
scalesandconsistsinananomalouswarmingofthe east- thetropicalPacific. SecondarymaximaintheIndianand
ern equatorial Pacific. This warming in turn warms up Atlanticoceansarepresentinthelow-thresholdnetworks
thelocalatmosphereandinfluencesothertropicalregions butaresignificantlyweakenedinthehigh-thresholdones.
through the excitation of equatorial Kelvin and Rossby The continental regions have overall very low connec-
5
τ = 0 ρ = 0.018 τ = 0.186 ρ = 0.01 τ = 0.476 ρ = 0.001
0.041 0.0215 0.0015
60N 60N 60N
0.0328 0.0172 0.0012
30N 30N 30N
0.0246 0.0129 0.0009
0 0 0
0.0164 0.0086 0.0006
30S 30S 30S
0.0082 0.0043 0.0003
60S 60S 60S
0 0 0
0 90E 180E 90W 0 90E 180E 90W 0 90E 180E 90W
FIG.3. AsFig. 1butD=4ordinalpatternsdefinedbycomparingSATanomaliesinconsecutivemonths. The2Dplotsofthe
areaweightedconnectivityarecolor-coded suchthatthewhite(red)regionsindicatethegeographical areaswithzero(largest)
area weighted connectivity.
τ = 0 ρ = 0.004 τ = 0.656 ρ = 0.001
0.007 0.0015
60N 60N
0.0056 0.0012
30N 30N
0.0042 0.0009
0 0
0.0028 0.0006
30S 30S
0.0014 0.0003
60S 60S
0 0
0 90E 180E 90W 0 90E 180E 90W
FIG. 4. Zero-threshold (left) and high-threshold (right) networks constructed by computing the mutual information from
ordinal patterns of length D = 5 defined by comparing SAT anomalies in consecutive months. The 2D plots of the area
weightedconnectivityarecolor-coded suchthatthewhite(red)regionsindicatethegeographical areaswithzero(largest) area
weighted connectivity.
tivity which translates in the low predictability of sur- muminthetropicalband. Thisnetworkstructureisalso
face temperature anomalies on interannual time scales. seenwhenusingordinalpatternsformedby5consecutive
Within these continental regions, the largest connectiv- months, Fig. 4.
ity is seenoverAsia andNorthAmerica, the latter max-
imum being perhaps due to the Pacific North American
pattern induced by El Nin˜o [15].
B. Binary representations
The networks obtained when the ordinal patterns are
definedbycomparingtemperatureanomaliesinconsecu- To capture longer memory processes one should use
tivemonths,Figs. 3,4,present,forzero-andlow-thresh- larger D values; however, for D = 6 there are 6! = 720
old,similarfeaturesasforconsecutiveyears,althoughthe possible ordinal patterns, and since we have time series
networks are more homogeneous. There is a maximum with less than 700 data points, there is not enough data
in the equatorial Pacific, a secondary maximum in the to calculate ordinal patterns PDFs with good statistics.
IndianoceanandextratropicalmaximaoverAsia,North As discussed in the introduction, a solution to over-
America and southern subtropics. On the other hand, comethisproblemisemploying“binaryrepresentations”,
thehighthresholdnetworkshowsthatthestrongestlinks by which the time series {x (t)} is transformed into a
i
are located in the extra tropics. We speculate that this sequence {v (t)} of 0s and 1s, using the following rule:
i
could be a result of the modulation of the temperature v (t) = 0 if x (t) ≥ 0 and v (t) = 1 otherwise (since
i i i
variancebytheseasonalcycle,whichisstrongestoverthe the x values are temperature anomalies, we are taking
i
northern hemisphere continental masses and has a mini- into account whether the SAT field is above or below its
6
τ = 0 ρ = 0.041 τ = 0.218 ρ = 0.01 τ = 0.512 ρ = 0.001
0.3075 0.0805 0.009
60N 60N 60N
0.246 0.0644 0.0072
30N 30N 30N
0.1845 0.0483 0.0054
0 0 0
0.123 0.0322 0.0036
30S 30S 30S
0.0615 0.0161 0.0018
60S 60S 60S
0 0 0
0 90E 180E 90W 0 90E 180E 90W 0 90E 180E 90W
FIG.5. AsFig. 1butemployingbinaryrepresentation. Zero-threshold(left),low-threshold(center)andhigh-threshold(right)
networksconstructedbycomputingthemutualinformationfrompatternsoflengthD=4definedbycomparingSATanomalies
in consecutiveyears. The 2D plots are color-coded such that thewhite (red) regions indicate thegeographical areas with zero
(largest) area weighted connectivity. In each panel thevalues of thethreshold, τ, and of the edge-density,ρ,are indicated.
τ = 0 ρ = 0.021 τ = 0.331 ρ = 0.01 τ = 0.566 ρ = 0.001
0.2005 0.0885 0.01
60N 60N 60N
0.1604 0.0708 0.008
30N 30N 30N
0.1203 0.0531 0.006
0 0 0
0.0802 0.0354 0.004
30S 30S 30S
0.0401 0.0177 0.002
60S 60S 60S
0 0 0
0 90E 180E 90W 0 90E 180E 90W 0 90E 180E 90W
FIG. 6. As Fig. 5 but with D=5.
monthly averaged value). We can then define “binary ordinal patterns, Figs. 3, 4: for short memory processes
patterns” of dimension D (e.g., for D = 3 the possible (or for low threshold) the maximum connectivity is in
patters are 000, 001, 010, 100, 011, 110, 101 and 111) the tropics, while for longer time scales (or for higher
and compute their PDF. The number of different pat- threshold)the extratropicsshowlargestnumber oflinks.
terns is 2D, andthus, we cancalculate PDFs of patterns As discussed before, this could be the result of the mod-
ofD =6(26 =64)withgoodenoughstatistics. Patterns ulation of the temperature variance by the seasonal cy-
withD ≤3donotprovidegoodresolutionforcomputing cle, which would be the main process that connects grid
themutualinformation,Eq. 1,becausethePDFsarecal- points in very low density networks (gridpoints that are
culated with very few bins. Therefore, in the following, connected by very strong links). Our results could also
we consider D =4, 5, and 6. hintattheroleoflandsurfaceconditionslikesnoworsoil
humidity inincreasingthe persistence ofsurfacetemper-
Figures 5-10 present the results when the binary pat-
atureanomaliesoverthenorthernhemispherecontinents.
terns are defined by consecutive years and months.
Overall, these results agree well with the fact that tem-
Forconsecutiveyears,Figs. 5-7,thenetworksobtained
peratureteleconnectionsfromthetropicalPacifictendto
when using binary representations are very similar to
last no much longer than a season in the different parts
those found with ordinal patterns. The tropical regions
of the world.
are quite uniformly well connected (although a Pacific
maximum is clear) while the extratropics show localized
regions of high connectivity likely due to atmospheric
As a way to test the interpretation of the above pre-
teleconnections forced from the tropics, particularly for
sented results, in terms of the symbolic methodology of
low density networks.
time-seriesanalysiscapturingtwodifferenttime-scalesof
The networks obtained for consecutive months, Figs. the Earth’s climate, seasonal and interannual, we con-
8-10, show that as the threshold or as D increases there structed binary patterns of fixed dimension, D =6, that
are overall similar changes in structure as those seen for cover three different time intervals:
7
τ = 0 ρ = 0.008 τ = 0.645 ρ = 0.001
0.088 0.0095
60N 60N
0.0704 0.0076
30N 30N
0.0528 0.0057
0 0
0.0352 0.0038
30S 30S
0.0176 0.0019
60S 60S
0 0
0 90E 180E 90W 0 90E 180E 90W
FIG. 7. AsFig. 5 but with D=6. Comparing with Figs. 5 and 6 onecan see that there is a good agreement with theresults
obtained previously with ordinal patterns: theweaker links tend to lose memory,while thestrongest links donot.
τ = 0 ρ = 0.022 τ = 0.186 ρ = 0.01 τ = 0.481 ρ = 0.001
0.114 0.0475 0.0015
60N 60N 60N
0.0912 0.038 0.0012
30N 30N 30N
0.0684 0.0285 0.0009
0 0 0
0.0456 0.019 0.0006
30S 30S 30S
0.0228 0.0095 0.0003
60S 60S 60S
0 0 0
0 90E 180E 90W 0 90E 180E 90W 0 90E 180E 90W
FIG. 8. AsFig.5 but thenetworks constructed with binary representation comparing anomalies in D=4 consecutivemonths.
The 2D plots of the area weighted connectivity are color-coded such that the white (red) regions indicate the geographical
areas with zero (largest) area weighted connectivity.
i) covering one-year, the patterns are composed as extra tropical connections dominate, while on interan-
nual scales,the “El Nin˜o” is the key player in setting up
[xi(t),xi(t+2),xi(t+4),...,xi(t+10)]; teleconnections worldwide.
As it was previously discussed, the significance of the
ii) covering two-years,the patterns are composed as network links was tested in comparison with links com-
putedfromsurrogatetime-seriesineachnode. Sincesur-
[x (t),x (t+4),x (t+8),...,x (t+20)];
i i i i rogatedatadoesnotpreservetheautocorrelationproper-
tiesoftheoriginaltime-series,tofurthertestthevalidity
ii) covering three-years, the patterns are composed as
of the previously presented results, we did the following
test: wecomputedthemutualinformationusingtheorig-
[x (t),x (t+6),x (t+12),...,x (t+30)].
i i i i inal time series in node i and the time-inverted series in
node j. The results show no significant spatial structure
TheresultsarepresentedinFig.11,wereonecanseehow
in the area weighted connectivity plots (not shown).
the network changes. For a time interval of one year the
extra tropics have the largest number of links and there
are very few in the tropical region. On the other hand,
for a time interval of three years, the extra-tropics keep C. Comparison with other measures
about the same number of connections while in tropi-
cal Pacific“El Nin˜o” stands out (note the different color It is interesting to compare the results obtained us-
scales in the panels in Fig.11). In summary, confirming ing ordinal patterns and binary representations with
theresultspreviouslyfoundwithordinalpatternsandbi- those obtained using conventional techniques of time-
nary representationscomposed by consecutive years and series analysis, as the linear cross-correlation coefficient
by consecutive months, on intra-seasonaltime scales the (as in Ref. [4]) and the mutual information, computing
8
τ = 0 ρ = 0.012 τ = 0.533 ρ = 0.001
0.0245 0.0015
60N 60N
0.0196 0.0012
30N 30N
0.0147 0.0009
0 0
0.0098 0.0006
30S 30S
0.0049 0.0003
60S 60S
0 0
0 90E 180E 90W 0 90E 180E 90W
FIG. 9. As Fig. 8 but with D=5.
τ = 0 ρ = 0.005 τ = 0.608 ρ = 0.001
0.0095 0.0015
60N 60N
0.0076 0.0012
30N 30N
0.0057 0.0009
0 0
0.0038 0.0006
30S 30S
0.0019 0.0003
60S 60S
0 0
0 90E 180E 90W 0 90E 180E 90W
FIG. 10. AsFig. 8 but with D=6.
the PDFs from standardhistograms of amplitude values mutualinformationuncoversmainlytheinterannualnet-
(as in Refs. [6, 7]). Figure 12 displays the zero, low work. These methodologies fail to separate the two dis-
and high threshold networks when the weights are cal- tinct time-scales (intra-seasonal and inter-annual) that
culated with the absolute value of the cross-correlation are clearly seen when using symbolic analysis and the
coefficient coefficient and Fig. 13, when they are calcu- time series are transformed in sequences of patterns by
latedwithmutualinformation,withthePDFscalculated comparing consecutive years or consecutive months.
from histograms of temperature anomaly values. In this
case the PDFs were computed employing 32 bins and in
each time-series the values of the SAT anomalies were IV. CONCLUSIONS
re-normalized such that each time-series has zero mean
andstandarddeviationequaltoone(asinRef. [6,7],for
Concluding, we have shown that ordinal patterns and
easier comparison).
symbolic analysis applied to anomalies of the surface air
The2Dplotsoftheareaweightedconnectivityaresim- temperature are powerful tools for the analysis of the
ilar to those previously reported in Ref. [6] and also, large-scaletopology of the climate network. The success
to those seen in Figs. 1, 2, 5, 6, where the ordinal of these methods is based on an appropriate partition
patterns and the binary representations are formed by of the phase space that results in ordinal patterns and
comparing consecutive years. “El Nin˜o” is the main fea- binaryrepresentationshavingPDFsthatcharacterizethe
ture uncovered. There are also regions with relatively diversity of patterns present in the climate dynamics.
high number of links in the northern hemisphere conti- A main advantage of the methodology proposed here
nents and southern subtropics, but the high connectiv- is that by varying the dimension of the pattern and the
ity in the extratropics seen previously in Figs. 3, 4, 8, year-month comparison, one can uncover memory pro-
9, and 10 is not observed. In other words, employing cesses with different time scales. We found that both,
the cross correlation coefficient or the histogram-based monthly and yearly patterns reveal long memory pro-
9
τ = 0.62 ρ = 0.001 τ = 0.625 ρ = 0.001 τ = 0.635 ρ = 0.001
0.0015 0.0015 0.0045
60N 60N 60N
0.0012 0.0012 0.0036
30N 30N 30N
0.0009 0.0009 0.0027
0 0 0
0.0006 0.0006 0.0018
30S 30S 30S
0.0003 0.0003 0.0009
60S 60S 60S
0 0 0
0 90E 180E 90W 0 90E 180E 90W 0 90E 180E 90W
FIG. 11. High-threshold networks constructed with binary representations, with patterns of D = 6 covering time-intervals of
oneyear(left), twoyears(center)andthreeyears(right). The2Dplotsofthearea weightedconnectivityarecolor-coded such
thatthewhite(red)regionsindicatethegeographical areaswithzero(largest) areaweighted connectivity. Seetextfordetails.
τ = 0 ρ = 0.095 τ = 0.643 ρ = 0.01 τ = 0.932 ρ = 0.001
0.433 0.066 0.006
60N 60N 60N
0.3464 0.0528 0.0048
30N 30N 30N
0.2598 0.0396 0.0036
0 0 0
0.1732 0.0264 0.0024
30S 30S 30S
0.0866 0.0132 0.0012
60S 60S 60S
0 0 0
0 90E 180E 90W 0 90E 180E 90W 0 90E 180E 90W
FIG.12. Zero-thresholdnetwork(left)andnon-zero-thresholdnetworks(centerandright)constructedbyestimatingtheweights
withtheabsolutevalueofthecross-correlation coefficient. The2Dplotsoftheareaweightedconnectivityarecolor-codedsuch
that the white (red) regions indicate thegeographical areas with zero (largest) area weighted connectivity.
cesses, and that depending on the time scale considered knowledges support from the European Community’s
the climate network can change completely. Seventh Framework Programme(FP7/2007-2013)under
The fact that ordinal patterns and symbolic analysis Grant Agreement N 212492 (CLARIS LPB. A Europe-
givemeaningfulinformationindicatesthatthetimevari- South America Network for Climate Change Assess-
ability of the anomaly SAT field is strongly determined ment and Impact Studies in La Plata Basin). C.M.
by patterns of oscillatory behavior that tend to repeat acknowledges support from the ICREA Academia pro-
from time to time. gramme, the Ministerio de Ciencia e Innovacio´n, Spain,
Overall we found that on seasonal time-scales the ex- projectFIS2009-13360andtheAgenciadeGestiod’Ajuts
tratropical regions, mainly over Asia and North Amer- Universitaris i de Recerca (AGAUR), Generalitat de
ica, present the strongest links while in interannualtime Catalunya, through project 2009 SGR 1168.
scales, the tropical Pacific clearly dominates.
ACKNOWLEDGMENTS
The authors thank the two anonymous referees for
their very useful comments and suggestions. M.B. ac-
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τ = 0 ρ = 0.039 τ = 0.167 ρ = 0.01 τ = 0.456 ρ = 0.001
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