deterministic nature of the underlying dynamics of surface ... · 1504 r. c. sreelekshmi et al.:...

Ann. Geophys., 30, 1503–1514, 2012www.ann-geophys.net/30/1503/2012/doi:10.5194/angeo-30-1503-2012© Author(s) 2012. CC Attribution 3.0 License.

AnnalesGeophysicae

Deterministic nature of the underlying dynamics of surface windfluctuations

R. C. Sreelekshmi1, K. Asokan2, and K. Satheesh Kumar1

1Department of Futures Studies, University of Kerala, Thiruvananthapuram, Kerala, 695 034, India2Department of Mathematics, College of Engineering, Thiruvananthapuram, Kerala, 695 016, India

Correspondence to:K. Satheesh Kumar ([email protected])

Received: 14 March 2012 – Revised: 6 August 2012 – Accepted: 3 September 2012 – Published: 8 October 2012

Abstract. Modelling the fluctuations of the Earth’s surfacewind has a significant role in understanding the dynamics ofatmosphere besides its impact on various fields ranging fromagriculture to structural engineering. Most of the studies onthe modelling and prediction of wind speed and power re-ported in the literature are based on statistical methods or theprobabilistic distribution of the wind speed data. In this pa-per we investigate the suitability of a deterministic model torepresent the wind speed fluctuations by employing tools ofnonlinear dynamics. We have carried out a detailed nonlin-ear time series analysis of the daily mean wind speed datameasured at Thiruvananthapuram (8.483◦ N,76.950◦ E) from2000 to 2010. The results of the analysis strongly suggest thatthe underlying dynamics is deterministic, low-dimensionaland chaotic suggesting the possibility of accurate short-termprediction. As most of the chaotic systems are confined tolaboratories, this is another example of a naturally occurringtime series showing chaotic behaviour.

Keywords. Atmospheric composition and structure (Gen-eral or miscellaneous)

1 Introduction

Surface wind plays a crucial role in climate and weather sys-tem of the Earth. It has significant impact on agriculture, nav-igation, structural engineering calculations and reduction ofatmospheric pollution as well as the economy of the regionas an alternate energy source (Martın et al., 1999; Elliott,2004; Bantaa et al., 2011; Finzi et al., 1984). Recent surgeof interest in research related to wind power is due to its po-tential as an alternate source of energy because of the fastdepletion of natural resources of Earth. Presently, there is

extensive literature on various areas related to wind energyacquisition and utilisation such as wind speed modelling andprediction, wind power production and wind resource quan-tification (e.g.Finzi et al., 1984; Martın et al., 1999; Elliott,2004; Celik, 2004; von Bremen, 2007; Mabel and Fernan-dez, 2009; Kavasseri and Seetharaman, 2009; Bantaa et al.,2011).

Wind speed modelling and forecasting is an important as-pect of wind power generation – yet one of the most diffi-cult due to the myriads of factors affecting it – and over theyears many tools have been developed for this purpose. Agood number of such tools rely on statistical methods, eithermoving average models such as ARMA and ARIMA fittedto the time series of wind speed (Kamal and Jafri, 1997; Tor-resa et al., 2005; Cadenas and Rivera, 2007; Kavasseri andSeetharaman, 2009) or models based on probability distribu-tion of wind speed (Hennessey, 1977; Celik, 2004; Mathewet al., 2011). Models based on artificial neural networks havealso been developed by many authors for making short-termpredictions of wind speed and generated wind power (Mo-handes et al., 1998; Cadenas and Rivera, 2007; Bilgili et al.,2007; Monfared et al., 2009).

As a matter of fact, none of these forecasting methods,based on time series analysis or meteorological models, is ca-pable of significantly reducing the prediction error comparedto the elementary method of persistence (Sfetsos, 2002) andthis is usually attributed to the high fluctuations and variabil-ity in wind speed. Exploring the source of these random fluc-tuations in wind speed, especially whether it is stochastic ordeterministic, is therefore important both for understandingthe nature of the dynamics and for improving the tools usedfor prediction. A few attempts have been made in this direc-tion, although the focus was not on determining the nature of

Published by Copernicus Publications on behalf of the European Geosciences Union.

1504 R. C. Sreelekshmi et al.: Deterministic nature of surface wind fluctuations

the dynamics, rather on comparing stochastic versus deter-ministic models constructed from time series of scalar mea-surements of wind speed.Palmer et al.(1995) have analysedseveral time series of wind components and X-band Dopplerradar signals gathered over an area of ocean surface and havefound the presence of a low-dimensional dynamical attractorin the case of time series of the horizontal wind speed as wellas the vertically polarized radar reflectivity. They were alsoable to achieve better short-term predictions from the deter-ministic models than from statistical models. However, theanalysis carried out on daily mean wind speed (DMWS) databy Ragwitz et al.(2000) suggests that, on average, no reduc-tion of the prediction error can be achieved by using a nonlin-ear model instead of a linear stochastic model. However, theycould predict intermittent gusts with significantly higher ac-curacy. These studies are however limited by short times se-ries (Hirata et al., 2008). Martın et al.(1999) have analysedthe wind speed data by splitting them into deterministic andstochastic components. Their analysis shows that the deter-ministic component has 1-year, 24-h and 12-h periods. Thesecycles have also been observed by other authors in surfacewind studies (Brett and Tuller, 1991; Gavalda et al., 1992).The 1-year and 24-h periods are the natural Earth cycles. The12-h period for wind speed series is well-defined and cor-responds to the daytime and nighttime maxima due to thefull development of the land–sea breezes. These periodicitiespresent in the wind speed time series clearly show presenceof determinism in the data. However, it is not clear whetherthe apparentstochasticcomponent is strictly stochastic orarises out of chaotic underlying dynamics.

In general, the predictability and degree of determinismof atmospheric parameters depend on the time scale con-sidered (Palmer, 1993), and this applies in particular to thecase of wind speed predictions. The various models for windspeed predictions reviewed above are mostly in time scalesranging from hours to a few days. What follows from thisdiscussion is that wind is believed to have both determin-istic and stochastic components in the time scales consid-ered, but the manner in which these components interact isstill elusive and a matter of debate. The rotation of Earthand solar heat radiation are two major causes of the sur-face wind in addition to the local topography. The Earth’srevolution is clearly deterministic. However, many authorshave argued that the solar radiations are stochastic in na-ture and hence the underlying dynamics of the surface windshould be governed by both deterministic as well as stochas-tic factors. On the other hand, our previous analysis of thedata of total electron content (TEC), which is strongly influ-enced by the solar radiation, shows strong evidence of de-terministic low-dimensional character of the underlying dy-namics (George et al., 2002; Kumar et al., 2004). The sur-face wind speed is a similar atmospheric parameter to so-lar influence but its dynamics is further complicated by thelocal conditions such as topography. Hence, it is worth in-vestigating whether a stochastic or a deterministic model is

most suitable for the underlying dynamics of surface windfluctuations. In this work we carry out a detailed system-atic analysis of the time series of daily mean wind speed(DMWS) measured at Thiruvananthapuram, Kerala, India(8.483◦ N,76.950◦ E ; elevation: 64 m) using tools of nonlin-ear dynamics for the period from year 2000 to 2010. Notethat the length of the time series is about the length of a solarcycle. The data were obtained from National Climatic DataCentre (http://www.ncdc.noaa.gov). We demonstrate, usingthe DMWS-data, that the dynamics of wind speed is essen-tially deterministic with a low-dimensional chaotic charac-ter. The chaotic behaviour is what makes the long-term pre-dictions of wind speed erroneous, but it should be possibleto obtain better short-term predictions using the determinis-tic model than would otherwise be made with the statisticalmethods. It is reported that short-term predictions of one tosix hours ahead at intervals of 10 min are important in powerdispatching systems (Mabel and Fernandez, 2009).

We assume that there is also a stochastic component inthe data arising mainly from measurement and averaging er-rors. The averaging errors are a result of considering themeanwind speeds and not the actual wind speeds equidis-tant in time as it should be for a time series. The effects ofthese errors are assumed to contribute an additive noise tothe data which is independent of the true deterministic dy-namics of the system. Hence, the first step in our analysisis to remove the effect of this noise process using a suitablenoise reduction technique to reveal the true dynamics behindthe data. The denoised data still contain irregular persistentfluctuations, which upon analysis using tools of non-lineardynamics reveals many attributes of a chaotic system witha low-dimensional attractor. Since some of these attributesmay also be found in linear stochastic processes, we furthersubject the denoised data to a detailed surrogate analysis toconfirm that the underlying dynamics is indeed deterministicand could not be described by a linear Gaussian stochasticmodel. Most of these analyses were carried out using toolsimplemented in the TISEAN package (Hegger et al., 1999).

2 Time delay coordinates and attractor reconstruction

The time series of DMWS is plotted in Fig.1 which showsthat the wind speed exhibits persistent temporal fluctuations.The underlying mechanism giving rise to these irregular fluc-tuations could either be stochastic or a deterministic systemexhibiting chaotic behaviour. Prior to the advent of chaos the-ory through the pioneering work ofLorenz(1963), it was be-lieved that random-like fluctuations such as the one in Fig.1could only originate from a stochastic system and not froma deterministic system. However, chaos theory has demon-strated that deterministic systems can also lead to behaviourthat is quite complex and, like stochastic systems affectedby noise, unpredictable in the long term. Deterministic dy-namical systems, which evolve continuously over time, are

Ann. Geophys., 30, 1503–1514, 2012 www.ann-geophys.net/30/1503/2012/

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R. C. Sreelekshmi et al.: Deterministic nature of surface wind fluctuations 1505

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Fig. 1. Time series of the measured daily mean wind speed(DMWS) in knots.

described by a state vectorx(t) and an equation of motion:

x ≡dx

dt= f (x). (1)

Such systems are usually characterised by an attractor, whichis a bounded subset of the phase space reached asymptoti-cally by a set of trajectories over an open set of initial condi-tions as timet → ∞.

A striking feature of some dynamical systems is that thetrajectories on the attractor may exhibitsensitive dependenceon initial conditions. This means that trajectories startingfrom neighbouring initial conditions may separate from eachother at an exponential rate, evolving independently of eachother and in an apparently uncorrelated manner after a suf-ficiently long period of time, and yet remain confined to abounded subset of the phase space.Chaos is the boundedaperiodic behaviour in a deterministic system that showssensitive dependence on initial conditions(Alligood et al.,1997). A detailed illustration of the sensitivity to initial con-ditions of a chaotic system, particularly in the setting of at-mospheric prediction, has been presented byPalmer(1993).The term chaos is reminiscent of the intricate dynamics ex-perienced by the trajectories on the attractor; the exponentialdivergence stretches out the trajectories as it evolves in time,which is then folded back to remain confined to a finite re-gion of the phase space. The attractor is the result of thesesequences of stretching and folding repeated indefinitely.

Exponential divergence of trajectories on the attractormakes long-term predictions in a chaotic system difficultwhile the stretching and folding of trajectories cause mea-surements of quantities that depend on the state space to lookrandom. Together, these attributes often make a chaotic sys-tem indistinguishable from a truly stochastic system. In fact,many systems that were earlier dubbed as stochastic werelater shown to be chaotic (Alligood et al., 1997; Ott, 1993).

Most of the time the state vectorx(t) is not measured di-rectly but indirectly at discrete time intervalsτ using a scalarmeasurement functiony(t) = h(x(t)) leading to a time seriesyi = y(iτ ). The central idea in time series analysis is that thedynamics of the state vectorx(t) on the attractor can be re-captured from the time seriesy(t) using a technique calledattractor reconstruction, first suggested byPackard et al.(1980) and successfully used by many others. This techniqueis based on the fact that the dynamics of then-dimensionalstate vectorx(t) on the attractor is topologically identical tothat of them-dimensional delay vector:

y(t) = (y(t), y(t + τ), · · ·y(t + (m − 1)τ ), (2)

which was constructed from samples ofy(t) taken at regu-lar time intervalsτ (also called “delay”), under the mappingx(t) −→ y(t) which is an embedding under rather generalconditions. The embedding theorem ofTakens(1981) andits extensions (Sauer et al., 1991; Sauer and Yorke, 1993)furnishes the mathematical theory behind reconstruction andasserts that the embedding is valid for almost all values oftime delayτ and all smooth measurement functionsh aslong asm > 2D whereD is the box-counting dimension ofthe attractor. This means that the dynamical and geometricalcharacteristics of the original system, in particular the geo-metrical invariants such as the fractal dimension, Lyapunovexponents and entropies, are preserved in the reconstructedspace and can be computed from the flow defined byy(t)

(Kantz and Schreiber, 1997; Ott et al., 1994). The analysisof the DMWS-data, presented in the next section, relies onattractor reconstruction for computing many such character-istics.

3 Analysis of the denoised data

For the purpose of our analysis of the DMWS-data, weheuristically assume that the dynamics underlying wind canbe modelled by a deterministic system with state vectorx(t).The daily mean wind speed can be regarded as observationsof a measurement functiony(t) = h(x(t)) made at regularintervals. However, since we are considering themeanwindspeeds and not the actual wind speeds at regular daily inter-vals, regarding them as observations made at equal intervalsof time contributes an averaging error. We assume that theseaveraging errors along with other measurement errors can to-gether be modelled as an additive noise process with zeroaverage and delta correlation. It is therefore important to re-duce the effect of this noise before analysing the data. Forreducing noise we have applied the noise reduction methodof Schreiber(1993), which employs a locally constant ap-proximation of the dynamics to reduce noise.

Despite the apparent random oscillations, recurring annualvariations are evident in the times series plot (Fig.1) of theDMWS-data. To confirm this we have plotted the space-timeseparation plot of the data (Fig.2), which helps in identifying

www.ann-geophys.net/30/1503/2012/ Ann. Geophys., 30, 1503–1514, 2012


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Fig. 2. Space-time separation plot for the time series in Fig.1 forτ = 1 andm = 14.

temporal correlations inside the time series (Provenzale etal., 1992). Each point in the plot represents a pair of pointson the trajectory with their relative separation in time alongthe horizontal axis and separation in space along the verti-cal. Marked variations are observed in the graph at multiplesof around 365 days indicating annual variations. The mod-ulation effects due to these annual variations were reducedin subsequent analysis by applyingepoch analysison thedata, by deducting from each of the data points which are365 days apart their average value (Kumar et al., 2004). Theresulting time series showed prominent variations, in periodsof 28 days, arising from lunar influence. Hence, epoch anal-ysis was repeated for these 28-day variations as well. Theplot of the resultant denoised and detrended time series isshown in Fig.3 and its space-time separation plot in Fig.4which clearly show considerable reduction in the effect ofthe annual and lunar variations. The autocorrelation function(Eq.3, discussed later) of the observed time series is plottedin Fig. 5a and of the detrended time series in Fig.5b whichalso show that the temporal correlation due to the annual vari-ation and lunar influence has significantly been reduced bythe epoch analysis. As is clear from the Fig.3, the denoiseddetrended data still show persistent temporal fluctuations.

As a first step in the analysis of the denoised data, we de-termine the embedding parameters – the delayτ and the em-bedding dimensionm – for the proper reconstruction of theattractor using the method discussed in the previous section.The embedding theorems do not place any restriction what-soever on the delayτ and the embedding dimensionm, buttheir choice can nonetheless affect the inferences deducedfrom reconstruction significantly, especially when the datacome from experiment. Small delays, for example, result inhighly correlated vectorsy(t) leading to unduly larger val-ues for the correlation dimension, while large delays yieldvectors with fairly uncorrelated components resulting in datarandomly distributed in the embedding space (Kantz and

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Fig. 3. Time series of the detrended denoised daily mean windspeed shown in Fig.1.

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Fig. 4. Space-time separation plot for the detrended time series forτ = 1 andm = 14.

Schreiber, 1997). Proper choice of the time delay is, there-fore, important, and a first guess of a suitable delay may beobtained from the autocorrelation function of the sample datayi given by

ρ(τ) =

∑i(yi − y)(yi+τ − y)∑

i(yi − y)2(3)

wherey is the sample mean. The value ofτ , at which the au-tocorrelation attains its first zero or its first local minimum, isusually an optimal choice for the delay (Kantz and Schreiber,1997).

Another tool to determine an optimal delay, which takesinto account non-linear correlations also, is the method oftime-delayed mutual information suggested byFraser andSwinney(1986). In this method, a quantity called averagemutual information is computed for various delays as a mea-sure of the predictability ofy(t + τ) given y(τ). The mu-tual informationI (τ ) for a given delayτ is calculated by



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Fig. 5. (a) The autocorrelation function of the observed DMWSdata.(b) The autocorrelation function of the detrended time series.

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Fig. 6. Mutual information of the time series of the measuredDMWS data.

regarding the sequences(yi) and(yi+τ ) as values of randomvariablesX andY and using the formula

I (τ ) =

∑y∈Y

∑x∈X

p(x,y) log2

(p(x,y)

p(x)p(y)

)(4)

wherep(x,y) is the joint probability mass function ofX andY , andp(x) andp(y) are the marginals. The probabilitiesare calculated by constructing a histogram of the data points.A good choice for time delay is then the value ofτ at whichthe graph of mutual information exhibits a marked minimum.For the DMWS-data, the plots of autocorrelation (Fig.5b)and mutual information (Fig.6) suggest a value aroundτ = 1as an optimal choice for the delay. Our preliminary analysiswith valueτ = 2 also gave identical results for the choice ofembedding dimension.

As for the choice of the embedding dimensionm, it shouldbe large enough for the attractor to fully unfold in the embed-

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Fig. 7. The fraction of false nearest neighbours as a function ofthe embedding dimensionm for the detrended time series withτ =

1,ω = 25, showing that anym ≥ 13 can be considered optimal.

ding space but choosing too large of anm may cause the vari-ous algorithms to underperform (Kantz and Schreiber, 1997).A practical method for choosing the right embedding dimen-sion, proposed byKennel et al.(1992), is to find the fractionof false neighboursas a function of the embedding dimen-sion. False neighbours arise when the current dimension isnot large enough for the attractor to unfold its true geome-try, leading to crossing of trajectories due to projection ontoa smaller dimension. The method checks the neighbours inprogressively higher dimensions until it finds only a negli-gible number of false neighbours in passing from dimensionm to m + 1. The first time the fraction of false neighboursattains a minimum indicates a suitable value for the embed-ding dimension. For the present data, Fig.7 plots the frac-tion of false neighbours as a function of embedding dimen-sion and it can be seen that an optimal choice ofm must behigher than 13, since form ≥ 13 the fraction of false neigh-bours becomes negligibly small. We have chosenm = 14 forthe further analysis. It may be noted that, for most of thepractical purposes, the important embedding parameter is theproductmτ of the embedding dimension and the delay timebecausemτ is the time span represented by an embeddingvector. Only a precise knowledge ofm is required to exploitthe determinism of the underlying dynamics with minimalcomputational effort (Kantz and Schreiber, 1997). The de-lay representation of the denoised detrended time series withm = 14 andτ = 1 is shown in Fig.8. The definite structurein the Fig.8 indicates the deterministic nature of the data.

A quantitative measure of the structure and self-similarityof the attractor is provided by various dimension estimates,such as the box-counting dimension, the Hausdorff dimen-sion etc., all of which generalise the Euclidean definition ofdimension to take care of the self-similar structure of chaoticattractors at arbitrary fine scales. As it turns out, chaotic at-tractors commonly have non-integer dimensions. Among the



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Fig. 8. The delay representation of the detrended time series.

various dimension estimates, the easiest to compute from agiven time series, and which has also become the standardnow, is the correlation dimension introduced byGrassbergerand Procaccia(1983). The correlation dimensionD2 is de-fined in terms of the correlation integralC(ε), which is de-fined as the probability that a pair of points chosen randomlyon the attractor is separated by a distance less thanε. On theattractor, the correlation integral is empirically found to scalelike C(ε) ∝ εD2 asε → 0, so that the correlation dimensionmay be estimated as the slope of the curve of lnC(ε) versusln(ε) given by

D2 = limε→0

d lnC(ε)

d lnε. (5)

In practical computations involving a single time series andN data points ofm-dimensional delay vectorsyi , the corre-lation integralC(ε) is approximated by the correlation sumC(ε,m) given by (Kantz and Schreiber, 1997)

C(ε,m) =2

N(N − 1)

N∑i=1

N∑j=i+1

2(ε − ‖yi − yj‖), (6)

for sufficiently largeN , where2(a) = 1 if a > 0, 2(a) = 0if a ≤ 0. The scaling exponent in Eq. (5), when calculatedusing the correlation sumsC(ε,m), typically increases withm and saturates to a final value for sufficiently largem whichis then taken as an estimate forD2. In practice, one computesthe local slopes with the following equation:

D2(ε,m) =d lnC(ε,m)

d lnε(7)

and plots them as a function ofε for variousm; the valuecorresponding to a plateau in the curves is identified as anapproximation toD2. There are, however, some subtletiesto be taken care of in the computation of correlation di-mension. While only the spatial closeness of points shouldbe accounted for in Eq. (7), the actual computations may

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Fig. 9. The local slopesD2(ε,m) for the detrended time series form ranging from 14 to 16 withτ = 1,ω = 25 giving a plateau forsmall values ofε and giving an estimate ofD2 = 3.7967± 0.0116.The convergence of the plateau for higher dimensions is also evidentin Fig. 11a indicating evidence of low dimensionality.

be affected by the temporal closeness of points as well. Toguard against this, points that are closer in time by less thana Theiler windowω – which is approximately equal to theproduct of the time delay and the embedding dimension –are excluded while calculating the correlation sum (Theiler,1986). Hegger et al.(1999) have suggested that the value ofω should be chosen generously.

Figure9 plots the local slopesD2(ε,m) for the DMWS-data with the previous choice of delay and for embeddingdimensions ranging from 14 to 16 using 25 as value forTheiler window. The curves exhibit convergence for largerm, an indication of low dimensionality of the attractor, andsuggest a value ofD2 = 3.7967± 0.0116. This shows that,while the original system may be affected by a multitude offactors, the eventual behaviour can be characterised by a low-dimensional attractor.

As mentioned previously, chaotic systems are charac-terised by their sensitive dependence on initial conditions,meaning that trajectories that start from neighbouring ini-tial conditions may diverge exponentially over time. Letx0be any point on the basin of attraction, and consider an in-finitesimal sphere of perturbed initial conditions. This spheredistorts into an ellipsoid as the system evolves in time (Alli-good et al., 1997). Letεk(t), k = 1,2, · · · ,n denote the lengthof thek-th principal axis of the ellipsoid. In general we canwrite εk(t) = εk(0)eλk t where theλks may be positive, zero,or negative, and are called theLyapunov exponents. The Lya-punov exponents quantify the average rate of divergence orconvergence of nearby orbits, and the existence of a positiveLyapunov exponent is one of the most striking signatures ofchaos (Ott, 1993). In such a system the growth of the sepa-rationδ(t) between two neighbour trajectories will be even-tually dominated by the maximum Lyapunov exponentλ, so



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Fig. 10. The curve ofS(1n) for the embedding dimensionsm =

14,15,16 withτ = 1,ω = 25. The maximum Lyapunov exponentλ

of the detrended time series is the slope of the dashed line 0.0265±0.0008.

that‖δ(t)| = ‖δ(0)‖eλt , and hence

λ = limt→∞

1

tln

‖δ(t)‖

‖δ(0)‖. (8)

In practice one computesλ by plotting lnδ(t) versust , whichshould fall nearly on a straight line, the slope of which thengives an estimate ofλ. Lyapunov exponents are invariant un-der smooth transformations of the attractor; hence, they arepreserved under delay reconstruction and may be estimatedfrom a time series. There are many algorithms for estimat-ing the maximal Lyapunov exponent from time series, all ofwhich implement the above ideas to delay vectors in the em-bedding space. Most popular among them is the Kantz algo-rithm (Kantz, 1994; Kantz and Schreiber, 1997), which pro-ceeds by computing the sum:

S(1n) =

1

N

N∑n0=1

ln

(1

‖U(yn0)‖

∑yn∈U(yn0)

∥∥∥yn0+1n − yn+1n

∥∥∥)(9)

for a pointyn0of the time series in the embedded space and

over a neighbourhoodU(yn0) of yn0

with diameterε. If theplot of S(1n) against1n is linear over small1n and for areasonable range ofε, and all have identical slope for suf-ficiently large values of the embedding dimensionm, thenthat slope can be taken as an estimate of the maximum Lya-punov exponent (Kantz, 1994; Kantz and Schreiber, 1997).For our time series, Fig.10 shows curves ofS(1n) form = 14,15,16 which increase linearly with1n and then set-tle down. An estimate for the maximum Lyapunov exponentas obtained from the figure isλ = 0.0265±0.0008. The com-putations were repeated for various values of the embeddingdimension and the diameter of the neighbourhoodU(yn0

),all of which gave results identical to the above. The estimated

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Fig. 11. (a)The local slopesD2(ε,m) of original time series for theembedding dimensionsm = 1, . . . ,24 with τ = 1,ω = 25. (b) Thesame for the phase randomized time series.

positive value of the maximum Lyapunov exponent indicatesthat the underlying system is chaotic.

A colour noise time series can mimic many characteristicsof a chaotic time series. In order to make a distinction be-tween these two, we compared the DMWS time series withits phase randomized time series. The phase randomizationof a chaotic signal can destroy its profile, whereas a colournoise time series retains its profile (Pavlos et al., 1992). Thephase randomized time series of DMWS data was obtainedby representing it by Fourier series and then reconstructingthe time series after adding a random phase distribution. Wecalculated the local slopes of the logarithmD2(ε,m) of thecorrelation sum for both the original time series and the phaserandomized time series and plotted the values in Fig.11a andb respectively. These figures clearly show that phase random-ization destroys deterministic profile.

The estimated values of the correlation dimension and thefraction of false nearest neighbours obtained for various em-bedding dimensions show that the underlying dynamics ofthe fluctuations in the DMWS data is low-dimensional. Thepositive value of the maximum Lyapunov exponent indicatesthat the underlying system is chaotic. The comparison of theDMWS data with its phase randomized time series furtherconfirms the chaotic nature of the underlying system.

4 Comparison with surrogate data

The analysis of the DMWS-data in the previous section re-veals a number of features that are characteristic of time se-ries originating from non-linear deterministic systems, whichare chaotic. However, many of these features could also beexhibited by stochastic systems driven by a linear Gaussianprocess, which may possibly be distorted by some non-linearprocess. So to further validate the results of the previous



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1 2 3 4 5 6 7 8 9 10

Fra

ctio

n of

Fal

se N

eare

st N

eigh

bour

s

Embedding Dimension

(a)

MeanMean +- SD

Original

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

1 2 3 4 5 6 7 8 9 10

S

Embedding Dimension

(b) S2

Fig. 12. (a)The mean values of the fraction of false nearest neigh-bours of the surrogates with standard deviation.(b) Plot of the sig-nificance of differenceS versusm.

section, we must ascertain that the source of the complex be-haviour exhibited by the DMWS-data is not stochastic.

The method of surrogate data (Theiler et al., 1992) iswidely used as a tool for discriminating whether the sourceof random fluctuations in time series data is deterministic orstochastic. It is basically a statistical test to formally rejectthe hypothesis that the observed data convey a linear noiseprocess. The method proceeds by first formulating a null hy-pothesis, which is usually an assumption that the observeddata are random, and then generating an ensemble of timeseries of random numbers, called surrogate data, which areconsistent with the null hypothesis and are otherwise similarto the original data. In other words, these surrogate data arewhat independent, repeated observations of the process thatgenerated the original data would yield if that process wereconsistent with the null hypothesis. Then one compares thevalues of some discriminating statistic, such as correlationdimension, computed from the given data to the distributionof values obtained from the surrogates. If the values differsignificantly, then the null hypothesis may be rejected.

0

5

10

15

20

25

30

0.1 1

D2(

ε,m

)

ε

(a) MeanMean +- SD

Original

0

5

10

15

20

25

30

35

40

45

50

0.1 1

S

ε

(b) S2

Fig. 13. (a)The mean values of the local slopes of the surrogateswith standard deviation.(b) Plot of the significance of differenceSversusε. Here the normalised data sets are used, andm = 14,τ = 1,

andω = 25.

In what follows we apply the surrogate data analysis to testthe null hypothesis that the observed time series is a linearGaussian noise process. We used the algorithm ofSchreiberand Schmitz(1996) to generate a set of 40 surrogates con-sistent with the null hypothesis. Generated by the amplitude-adjusted Fourier transform method, the surrogates preservethe amplitude distribution, power spectrum and autocorrela-tion of the DMWS-data, so that they can be regarded as whatthe realisations of the process underlying the DMWS-datawould be like, if that process had the properties enjoinedby the null hypothesis. The null hypothesis is tested using,as discriminating statistic, both geometrical and dynamicalcharacteristics such as fraction of false nearest neighbours,the local slopes of the correlation sums and the curves ofS(1n) which are related to the maximal Lyapunov exponent.Each of the above characteristics are calculated for both theoriginal data and the surrogate data, and the null hypothesisis accepted or rejected depending on the value of the signif-icance of difference given by (Mitschke and Dammig, 1993;



-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

0 5 10 15 20 25 30 35 40

S(Δ

n)

Δn

(a)

MeanMean ± SD

Original

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35 40

S

Δn

(b) S2

Fig. 14. (a)The mean values ofS(1n) of the surrogates with stan-dard deviation.(b) Plot of the significance of differenceS versus1n.

Pavlos et al., 1999)

S =µ − µorig

σ(10)

whereµ andσ are the mean and standard deviation of thecharacteristic computed from the surrogates andµorig is themean of the characteristic on the original data. It is estimatedthat we may reject the null hypothesis with 95 % confidenceif S > 2, which means that the probability is 95 % or morethat the observed time series is not a realisation of a Gaussianstochastic process (Pavlos et al., 1999).

Figure12a plots the mean values of fraction of false near-est neighbours of all the surrogates and values one standarddeviation away from the mean, alongside the values of frac-tion of false nearest neighbours of the DMWS-data. We canobserve that the curves of fraction of false nearest neighboursversusm of all the surrogates deviate significantly from thecorresponding curve of the original data for a range of theembedding dimensions. As shown in Fig.12b, the signifi-cance of differenceS for the fraction of false nearest neigh-

1.040

1.050

1.060

1.070

1.080

1.090

1.100

1.110

1.120

0 5 10 15 20 25 30 35 40 45

Pre

dict

ion

erro

r

The original and surrogates

surrogatesoriginal

Fig. 15. The plot of the prediction errors form = 14,τ = 1 for thesurrogates (denoted by circles) and that of the original series (de-noted by filled square), showing the determinism in the time series.The significance of differenceS = 4.15.

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0 5 10 15 20 25 30 35 40 45

Tim

e re

vers

e as

ymm

etry

sta

tistic

The original and surrogates

surrogatesoriginal

Fig. 16. The plot of the time reversal asymmetry statistic for delayτ = 14 for the surrogates (denoted by circles) and that of the orig-inal series (denoted by filled square), showing the determinism inthe time series. The significance of differenceS = 2.95.

bours reaches up to 9, and hence the null hypothesis can besafely rejected.

Next we compared the local slopes of the correlation sums.Figure13a compares the local slopesD2(ε,14) of the corre-lation sums (Eq.7), of the DMWS-data with the mean valuesof the slopes of all the surrogates along with values one stan-dard deviation away from the mean. It is clear that the val-ues of the slopes of the surrogates deviate considerably fromthose of the original data, especially in the region of smallerε. As is clear from Fig.13b, the significance of difference islarge enough to reject the null hypothesis.

We further compared the original data with their surro-gates usingS(1n) of Eq. (9) as the test statistic. Figure14acompares the curves ofS(1n) of the surrogates with those



1.0

1.1

1.1

1.1

1.1

1.1

1.1

1.1

1.1

5 10 15 20 25 30

Pre

dict

ion

erro

r

Embedding dimension

Fig. 17. The plot of the prediction error versus embedding dimen-sionm.

of the original data plotted for delayτ = 1 Theiler win-dow ω = 25 and embedding dimensionm = 14. We observestrong differences between the values ofS(1n) correspond-ing to the original data and the surrogates. The significanceof differenceS, shown in Fig.14b, is larger than 2 for all1n ≤ 40. Here again, based on the values ofS, we can rejectthe null hypothesis.

The alignment of the neighbouring segments of trajecto-ries in a flow, which ultimately leads to a definite structurefor the attractor if the dynamics is deterministic, can be usedas a criterion to distinguish determinism from stochastic dy-namics. A straightforward way to quantify this is the non-linear prediction error which computes the deviations of thevalues predicted using past data from the actual values in thetrajectory. It is reported that the non-linear prediction erroris a consistently good tool for discriminating non-linearity(Schreiber and Schmitz, 1997). We calculated the predictionerrors by using a locally constant approximation to predictfuture values (Tong, 1983; Hegger et al., 1999), and the root-mean-square prediction errors of each of the 40 surrogatesand the original data were computed. The results are dis-played in Fig.15 which shows that the prediction errors aresignificantly lower for the original data than all the surrogateswith S = 4.15, and hence the null hypothesis can be rejected.The time reversal asymmetry statistic defined by

T rev=

〈(yn − yn−τ )3〉

〈(yn − yn−τ )2〉(11)

is frequently used as a measure of deviations from time re-versibility which is a characteristic of linear systems. Fig-ure16 showsT rev for all the surrogates and the original data,and it is seen that time reversal asymmetry of the originaldata is larger than that of the surrogates withS = 2.95; hence,we can reject the null hypothesis.

To summarise, based on the results of these series of statis-tical tests comparing the DMWS-data with their surrogates,

we can reject the null hypothesis with 95 % confidence leveland infer that the DMWS-data do not originate from a linearGaussian process. This further confirms that the results re-ported in the previous section are not an artefact of a stochas-tic system but of a system that is indeed deterministic witha low-dimensional chaotic attractor. Although deterministic,the chaotic nature of the data makes long-term predictionsprone to errors, but short-term predictions can be made withfairly good accuracy by carefully chosen methods adapted tothe data. The average prediction errors of DMWS-data basedon a locally constant approximation are shown in Fig.17 asfunction of embedding dimension. As is clear from the fig-ure, the prediction error becomes smaller and stabilised forembedding dimensionm ≥ 14 which, besides being a furtherjustification for our choice ofm = 14 in the previous analy-sis, furnishes another piece of evidence for the determinismin the data. However, the locally constant approximation isby no means the most suitable for all types of data, and aproper choice of prediction method requires a careful anal-ysis of the data against the various prediction schemes. Thiswill be addressed in a future work.

5 Conclusions

We have carried out a detailed analysis of the daily meanwind speed measured at Thiruvananthapuram from 2000 to2010 using tools of non-linear time series analysis. The pur-pose of the study was to examine whether the persistent irreg-ular temporal fluctuations exhibited by the data arose fromdeterministic or stochastic dynamics of the underlying sys-tem. The analysis reveals that the underlying dynamics ofDMWS-data is deterministic, low-dimensional and chaotic.The estimated values of correlation dimension and the frac-tion of false nearest neighbours as a function of embeddingdimension indicate the low dimensionality of the system,and the positive value of the maximum Lyapunov exponentshows that the system is chaotic. The reduction and stabiliza-tion of prediction errors with increase of embedding dimen-sion is further evidence for determinism. A detailed surrogatedata analysis, using a number of measures as discriminatingstatistic, shows that the characteristics shown by the data arenot of a stochastic system exhibiting chaos-like behaviour,and corroborates the deterministic character of the system.The analysis further shows that the chaotic profile does notarise from the pseudo-characteristics of a colour noise timeseries. While most of the chaotic systems reported in the lit-erature are confined to laboratories, this is a natural systemshowing chaotic behaviour.

Acknowledgements.The authors are grateful to Manoj Changat,Department of Futures Studies, Univ. of Kerala for his support andencouragement. They also wish to thank the anonymous referees fortheir constructive comments and useful suggestions to improve theoriginal manuscript.



Topical Editor P. Drobinski thanks two anonymous referees fortheir help in evaluating this paper.

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