Extreme event statistics of daily rainfall: Dynamical systems approach
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Extreme event statistics of daily rainfall: Dynamical systems approach
G. Cigdem Yalcin
Department of Physics, Istanbul University, 34134, Vezneciler, Istanbul, Turkey
Pau Rabassa and Christian Beck
Queen Mary University of London, School of Mathematical Sciences, Mile End Road, London E1 4NS, UK
We analyse the probability densities of daily rainfall amounts at a variety of locations on theEarth. The observed distributions of the amount of rainfall fit well to a q -exponential distributionwith exponent q close to q ≈ .
3. We discuss possible reasons for the emergence of this powerlaw. On the contrary, the waiting time distribution between rainy days is observed to follow anear-exponential distribution. A careful investigation shows that a q -exponential with q ≈ . I. INTRODUCTION
The statistical analysis of precipitation data is an in-teresting problem of major environmental importance [1–4]. Of particular interest are extreme events of rainfall,which can lead to floodings if a given threshold is ex-ceeded. From a mathematical and statistical point ofview, it is natural to apply extreme value statistics tomeasured rainfall data, but it is not so clear which classof the known extreme value statistics, if any, is applica-ble, and how the results differ from one spatial locationto another. Another interesting quantity to look at isthe waiting time between rainy days. Extreme events ofwaiting times in this context correspond to droughts. Soan interesting question is what type of drought statisticsis implied by the observed distribution of waiting timesbetween rainfall periods if this is extrapolated to verylong waiting times.In this paper we present a systematic investigation ofthe probability distributions of the amount of daily rain-fall at variety of different locations on the Earth, and ofwaiting time distributions on a scale of days and hours.Our main result is that the observed distributions ofthe amount of rainfall are well-fitted by q -exponentials,rather than exponentials, which suggests that techniquesborrowed from nonextensive statistical mechanics [5] andsuperstatistics [6] could be potentially useful to betterunderstand the rainfall statistics. An entropic exponentof q ≈ . q -exponentials asymptotically decay with a powerlaw, we discuss the corresponding extreme value statis-tics that is highly relevant in the context of floodingsproduced by extreme rainfall events. We also investigatethe waiting time distribution between rainy events, whichis much better described by an exponential, although anentropic exponent close to 1 such as q ≈ .
05 seems togive the best fits. We discuss possible dynamical reasons for the occurrence of q -exponentials in this context.One possible reason could be superstatistical fluctua-tions of a variance parameter or a rate parameter. Letus explain this a bit further. Superstatistical techniqueshave been discussed in many papers [6–19] and they rep-resent a powerful method to model and/or analyse com-plex systems with two (or more) clearly separated timescales in the dynamics. The basic idea is to consider forthe theoretical modelling a superposition of many sys-tems of statistical mechanics in local equilibrium, eachwith its own local variance parameter β , and finally per-form an average over the fluctuating β . The probabilitydensity of β is denoted by f ( β ). Most generally, the pa-rameter β can be any system parameter that exhibitslarge-scale fluctuations, such as energy dissipation in aturbulent flow, or volatility in financial markets. Anotherpossibility is to regard β as the rate parameter of a localPoisson process, as done, for example, in [20]. Ultimatelyall expectation values relevant for the complex system un-der consideration are averaged over the distribution f ( β ).Many applications have been described in the past, in-cluding modelling the statistics of classical turbulent flow[7, 21–23], quantum turbulence [24], space-time granular-ity [25], stock price changes [13], wind velocity fluctua-tions [26], sea level fluctuations [27], infection pathwaysof a virus [28], migration trajectories of tumour cells [29],and much more [10, 20, 30–33]. Superstatistical systems,when integrated over the fluctuating parameter, are ef-fectively described by more general versions of statisticalmechanics, where formally the Boltzmann-Gibbs entropyis replaced by more general entropy measures [15, 17].The concept can also be generalized to general dynami-cal systems with slowly varying system parameters, see[34] for some recent rigorous results in this direction.Our main goal in this paper is to better understand theextreme event statistics of rainfall at various example lo-cations on the Earth. We will start with a careful analysisof experimentally recorded time series of the amount ofrainfall measured at a given location, whose probabilitydensity is highly relevant to model the corresponding ex-treme event statistics [35–39]. Ultimately of course allthis rainfall dynamics can be formally regarded as be-ing produced by a highly nonlinear and high-dimensionaldeterministic dynamical system in a chaotic state, pro-ducing the occasional rainfall event, hence it is useful tokeep in mind the basic results of extreme event statis-tics for weakly correlated events as generated by mix-ing dynamical systems. Recently there has been muchactivity on the rigorous application of extreme valuestheory to deterministic dynamical systems [40–48] andalso to stochastically perturbed ones [49–51]. A remark-able feature of the dynamical system approach is thatthere exist some correlations between events, and hencethe extreme value theory used to tackle it must accountfor this correlation going beyond a theory that is justbased on sequences of events that are statistically inde-pendent. In the superstatistics approach, correlations arealso present, due to the fact that parameter changes takeplace on long time scales, but the relaxation time of thesystem is short as compared to the time scale of theseparameter changes, so that local equilibrium is quicklyreached.What is worked out in this paper is a comparison withexperimentally measured rainfall data, to decide whichextreme event statistics should be most plausibly appliedto various questions related to amount of rainfall andwaiting times between rainfall events. Extreme value the-ory quite generally tells us (under suitable asymptotic in-dependence assumptions) that there are only three possi-ble limit distributions, namely, the Gumbel, Fr´echet andWeibull distribution. But are these assumptions of near-independence satisfied for rainfall data, and if yes, whichof the above three classes are relevant? This is the subjectof this paper. We will also discuss simple deterministicdynamical system models that generate superstatisticalprocesses in this context.The paper is organized as follows. In Section II wepresent histograms of rainfall statistics, extracted fromexperimentally measured time series of rainfall at variouslocations on the Earth. What is seen is that the prob-ability density of amount of rainfall is very well fittedby q -exponentials. We discuss the generalized statisticalmechanics foundations of this based on nonextensive sta-tistical mechanics with entropic index q , with q ≈ . q -exponentials, but this time with q much closer to 1. Asimple superstatistical model for this is discussed in sec-tion IV, a Poisson process that has a rate parameter thatfluctuates in a superstatistical way. We review standardextreme event statistics in section V and then, in sec-tion VI, based on the measured experimental results of rainfall statistics, we develop the corresponding extremevalue statistics. In section VII we analyse the ambigui-ties that arise for the extreme event statistics of waitingtimes, depending on whether we assume the waiting timedistribution is either an exact exponential or a slightlydeformed q -exponential as produced by superstatisticalfluctuations. Finally, in section VIII we describe a dy-namical systems approach to superstatistics. II. DAILY RAINFALL AMOUNTDISTRIBUTIONS AT VARIOUS LOCATIONS
We performed a systematic investigation of time se-ries of rainfall data for 8 different example locations onthe Earth (Figs. 1-8). The data are from various pub-licly available web sites. When doing a histogram of theamount of daily rainfall observed, a surprising featurearises. All distributions are power law rather than expo-nential. They are well fitted by so-called q -exponentials,functions of the form e q ( x ) = (1 + (1 − q ) x ) / (1 − q ) (1)which are well-motivated by generalized versions of sta-tistical mechanics relevant for systems with long-rangeinteractions, temperature fluctuations and multifractalphase space structure [5, 6]. Of course the ordinary ex-ponential is recovered for q →
1. Whereas the data ofmost locations are well fitted by q ≈ .
3, Central Englandand Vancouver have somewhat lower values of q closer to1.13.One may speculate what the reason for this power lawis. Nevertheless, the formalism of nonextensive statis-tical mechanics [5] is designed to describe complex sys-tems with spatial or temporal long-range interactions,and q -exponentials occur in this formalism as generalizedcanonical distributions that maximize q -entropy S q = 1 q − X i (1 − p qi ) , (2)where the p i are the probabilities of the microstates i .Ordinary statistical mechanics is recovered in the limit q →
1, where the q -entropy S q reduces to the Shannonentropy S = − X i p i log p i . (3)The generalized canonical distributions maximize the q -entropy subject to suitable constraints. In our case theconstraint is given by the average amount of daily rain-fall at a given location. The way rainfall is producedis indeed influenced by highly complex weather systemsand condensation processes in clouds, so one may spec-ulate that more general versions of statistical mechanicscould be relevant as an effective description. Also for hy-drodynamic turbulent systems [7, 21] and pattern form-ing systems [52] these generalized statistical mechanics Amount of rainfall
Central England - (1931 - 2014)
FIG. 1: Observed probability distribution of amount of dailyrainfall (minus average value) in Central England (red datapoints). The blue curve is a fit with a q -exponential e q ( − x ).The parameter q is fitted for the various locations. For Cen-tral England, q = 1 . Amount of rainfall
Darwin - (1975- 2014)
FIG. 2: As Fig. 1, but for Darwin ( q = 1 . methods have previously been shown to yield a good ef-fective description. The amount of rain falling on a givenday is a complicated spatio-temporal stochastic processwith intrinsic correlations, as rainy weather often has atendency to persist for a while, both spatially and tem-porally. The actual value of q for the observed rainfallstatistics reflects characteristic effective properties in theclimate and temporal precipitation pattern at the givenlocation. For temperature distributions at the same lo-cations as in Fig.1-8, see [53].. III. WAITING TIME DISTRIBUTIONSBETWEEN RAINY DAYS
Another interesting observable that we extracted fromthe data is the waiting time distribution between rainyepisodes. We did this both for a time scale of days and
Amount of rainfall
Dubai - (1968 - 2014)
FIG. 3: As Fig. 1, but for Dubai ( q = 1 . Amount of rainfall
Eureka - (1951 - 2014)
FIG. 4: As Fig. 1, but for Eureka ( q = 1 . a time scale of hours. A given day is marked as rainyif it rains for some time during that day. The waitingtime is then the number of days one has to wait untilit rains again, this is a random variable with a givendistribution which we can extract from the data. Resultsfor the waiting time distributions are shown in Fig. 9-14. Amount of rainfall
Hongkong - (1997 - 2014)
FIG. 5: As Fig. 1, but for Hongkong ( q = 1 . Amount of rainfall
Ottawa - (1939 - 2014)
FIG. 6: As Fig. 1, but for Ottawa ( q = 1 . Amount of rainfall
Sydney - (1910 - 2014)
FIG. 7: As Fig. 1, but for Sydney ( q = 1 . What one observes here is that the distribution is nearlyexponential. That means the Poisson process of nearlyindependent point events of rainy days is a reasonablygood model.At closer inspection, however, one sees that again aslightly deformed q -exponential, this time with q ≈ . Amount of rainfall
Vancouver - (1937 - 2014)
FIG. 8: As Fig. 1, but for Vancouver ( q = 1 . t-
Central England - waiting times
FIG. 9: Waiting time distribution between rainy days for Cen-tral England (red data points). The blue curve is a fit with a q -exponential. For waiting time distributions, the parameter q is much closer to 1 ( q = 1 .
05 for Central England). t-
P(t)
Darwin - waiting times
FIG. 10: As Fig. 9, but for Darwin ( q = 1 . out in the next section, one may explain this with a super-statistical Poisson process, i.e. a Poisson process whoserate parameter – on a long time scale– exhibits fluctua-tions that are χ -distributed, with a rather large numberof degrees of freedom n . IV. SUPERSTATISTICAL POISSON PROCESS
We start with a very simple model for the return timeof rainfall events (or extreme rainfall events) on any giventime scale. This is to assume that the events follows aPoisson process. For a Poisson process the waiting timesare exponentially distributed, p ( t | β ) = β exp( − βt ) . Here, t is the time from one event (peak over threshold)to the next one, and β is a positive parameter, the rateof the Poisson process. The symbol p ( t | β ) denotes the t-
Eureka - waiting times
FIG. 11: As Fig. 9, but for Eureka ( q = 1 . t-
Ottawa - waiting times
FIG. 12: As Fig. 9, but for Ottawa ( q = 1 . conditional probability density to observe a return time t provided the parameter β has certain given value.The key idea of the superstatistics approach [6, 20] canbe applied to this simple model, thus constructing a su-perstatistical Poisson process. In this case the parameter β is regarded as a fluctuating random variable as well, butthese fluctuations take place on a very large time scale.For example, for our rainfall statistics the time scale on t-
Sydney - waiting times
FIG. 13: As Fig. 9, but for Sydney ( q = 1 . t-
Vancouver - waiting times
FIG. 14: As Fig. 9, but for Vancouver ( q = 1 . which β fluctuates may correspond to weeks (differentweather conditions) whereas our data base records rain-fall events on an hourly basis.If β is distributed with probability density f ( β ), andfluctuates on a large time scale, then one obtains themarginal distribution of the return time statistics as p ( t ) = Z ∞ f ( β ) p ( t | β ) = Z ∞ f ( β ) β exp( − βt ) . (4)This marginal distribution is actually what is recordedwhen we sample histograms of the observational data.By inferring directly on a simple model for the distribu-tion f ( β ), a more complex model for the return times canbe derived without much technical complexity. For exam-ple, consider that there are n different Gaussian randomvariables X i , i = 1 , . . . , n , that influence the dynamics ofthe intensity parameter β as a random variable. We maythus assume as a very simple model that β = P ni =1 X i with E ( X i ) = 0 and E ( X i ) = 0. Then the probabilitydensity of β is given by a χ -distribution: f ( β ) = 1Γ( n/ (cid:18) n β (cid:19) n/ β n/ − exp (cid:18) nβ β (cid:19) , where n is the number of degrees of freedom and β is ashape parameter that has the physical meaning of beingthe average of β formed with the distribution f ( β ).The integral (4) is easily evaluated and one obtains the q -exponential distribution: p ( t ) ∼ (1 + b ( q − t ) / (1 − q ) , where q = 1 + 2 / ( n + 2) and b = 2 β / (2 − q ).To sum up, this model generates q -exponential distri-butions by a simple mechanism, fluctuations of a rateparameter β . Typical q -values obtained in our fits are q = 1 . q = 1 .
05 for waiting timebetween rainfall events.
V. EXTREME VALUE THEORY FORSTATIONARY PROCESSES
Classic extreme value theory is concerned with theprobability distribution of unlikely events. Given a sta-tionary stochastic process X , X , . . . , consider the ran-dom variable M n defined as the maximum over the first n -observations: M n = max( X , . . . , X n ) . (5)In many cases the limit of the random variable M n maydegenerate when n → ∞ . Analogously to central limitlaws for partial sums, the degeneracy of the limit can beavoided by considering a rescaled sequence a n ( M n − b n )for suitable normalising values a n ≥ b n ∈ R . In-deed, extreme value theory studies the existence of nor-malising values such that P ( a n ( M n − b n ) ≤ x ) → G ( x ) . (6)as n → ∞ , with G ( x ) a non-degenerate probability dis-tribution.Two cornerstones in Extreme Value Theory are theFisher-Tippet Theorem [54] and the Gnedenko Theorem[55]. The former asserts that if the limiting distribu-tion G exist, then it must be either one of three pos-sible types, whereas the latter theorem gives necessaryand sufficient conditions for the convergence of each ofthe types. A third cornerstone in Extreme Value The-ory are the Leadbetter conditions [35, 56]. These are akind of weak asymptotic independence conditions, underwhich the two previous theorems generalize to station-ary stochastic series satisfying them. Let us review theseresults in somewhat more detail.In the case where the process X i is independent iden-tically distributed (i.i.d.) the Fisher-Tippett Theoremstates that if X , X , . . . is i.i.d. and there exist sequences a n ≥ b n ∈ R such that the limit distribution G isnon-degenerate, then it belongs to one of the followingtypes: Type I : G ( x ) = exp ( − e − x ) for x ∈ R . This distribu-tion is known as the Gumbel extreme value distri-bution (e.v.d.).
Type II : G ( x ) = exp ( − x − α ), for x > G ( x ) = 0,otherwise; where α > Fr´echet e.v.d.
Type III: G ( x ) = exp ( − ( − x ) α ), for x ≤ G ( x ) = 1,otherwise; where α > Weibull e.v.d.A further extension of this result is the Gnedenko The-orem, which provides a characterization of the conver-gence in each of these cases. Let X , X , . . . be an i.i.d.stochastic process and let F be its cumulative distribu-tion function. Consider x M = sup { x | F ( x ) < } . The following conditions are necessary and sufficient for theconvergence to each type of e.v.d.: Type I:
There exists some strictly positive function h ( t )such that lim t → x − M − F ( t + xh ( t ))1 − F ( t ) = e − x for all real x ; Type II: x M = + ∞ and lim t →∞ − F ( tx )1 − F ( t ) = x − α , with α >
0, for each x > Type III: x M < ∞ and lim t → − F ( x M − tx )1 − F ( x M − t ) = x α , with α >
0, for each x > F ( x ). VI. EXTREME EVENT STATISTICS FOREXPONENTIAL AND Q-EXPONENTIALDISTRIBUTIONS
The rainfall data were well described by q -exponentials, but waiting time distributions were ob-served to be close to ordinary exponentials, with q onlydeviating by a small amount from 1. Let us now discussthe differences in extreme value statistics that arise fromtheses different distributions.In the case where X is distributed as an ordinary ex-ponential function with parameter λ , we have1 − F ( t ) = (cid:26) exp( − λt ) if t >
01 if t < . (7)It is not difficult to check that the exponential distri-bution belongs to the Gumbel domain of attraction. Inother words, the extreme events associated to the expo-nential distribution will be Gumbel distributed.Recall that the q -exponential function is defined asexp q ( t ) := [1 + (1 − q ) t ] / (1 − q ) , with 1 ≤ q <
2. A random variable X is q -exponentialdistributed (with parameter λ ) if its density function isequal to (2 − q ) λ exp q ( − λx ). In such a case, its hazardfunction is1 − F ( t ) = (cid:26) exp q ′ ( − λt/q ′ ) if t >
01 if t < . (8)where q ′ = 1 / (2 − q ).Using the Gnedenko theorem if follows that the q -exponential distribution belongs to the Fr´echet domainof attraction. In this case the shape parameter of theFr´echet distribution α is equal to − qq − . VII. MODEL UNCERTAINTY
Extremely large waiting times for rainfall events corre-spond to droughts. Clearly, it is interesting to extrapo-late our observed waiting time distributions to very large P k q = 1q = 1.05q = 1.3 FIG. 15: Horizontal axis: Multiple k of the mean, verticalaxis: probability P for an event at that level. time scales. However, in Section III we saw that in mostcases it is difficult to discern if the waiting time distri-bution is that of a Poisson process, distributed as anexponential, or if it is a q -exponential with q close toone. This can make a huge difference for extreme valuestatistics. The aim of this section is to assess the impactof choosing one or the other model.Consider k a constant and X a random variable mod-elled either by an exponential or a q -exponential. To nor-malize the problem, we can scale our analysis in terms ofthe mean. In other words, we look at the probability of X being bigger than a multiple, say k -times, the mean of X .If X is distributed like an exponential with parameter λ , its mean is equal to 1 /λ and its hazard function isgiven by (7). Then it is easy to check that P ( X > kE ( X )) = exp ( − k ) . On the other hand, if X is distributed like a q -exponential with parameter λ , its mean is equal to1 /λ (3 − q ) (provided 1 ≤ q < /
2) and its hazard func-tion is given by (8). In this case we have P ( X > kE ( X )) = exp q ′ (cid:18) − − q − q k (cid:19) . Recall that the exponential distribution can be under-stood as the limit of the q -exponential as q goes to 1.This is also true for the probability above, which con-verges to exp ( − k ) as q goes to 1. In Fig 15 we plottedthe probability of an event of level k for different valuesof q . For instance the probability of having an observa-tion bigger than 5 times the mean is 0 . q = 1,0 . q = 1 .
05 and 0 .
026 when q = 1 .
3. When welook at the probability of an observation bigger than 10the mean, it is 0 . . . q = 1 or the very similarvalue q = 1 .
05 for the observed waiting time distribution. This illustrates the general uncertainty in model buildingfor extreme rainfall and drought events [57].
VIII. DYNAMICAL SYSTEMS APPROACH
Ultimately, the weather and rainfall events at a givenlocation can be regarded as being produced by a veryhigh-dimensional deterministic dynamical system ex-hibiting chaotic properties. It is therefore useful to ex-tend the superstatistics concept to general dynamical sys-tems, following similar lines of arguments as in the recentpaper [34].The basic idea here is that one has a given dynamics(which, for simplicity, we take to be a discrete mapping f a : Ω → Ω on some phase space Ω) which depends onsome control parameter a . If a is changing on a large timescale, much larger than the local relaxation time deter-mined by the Perron-Frobenius operator of the mapping,then this dynamical system with slowly changing controlparameter will ultimately generate a superposition of in-variant densities for the given parameter a . Similarly,if we can calculate return times to certain particular re-gions of the phase space for a given parameter a , then inthe long term the return time distribution will have tobe formed by taking an average over the slowly varyingparameter a . Clearly, the connection to the previous sec-tions is that a rainfall event corresponds to the trajectoryof the dynamical system being in a particular subregionof the phase space Ω, and the control parameter a corre-sponds to the parameter β used in the previous sections.Let us consider families of maps f a depending on a con-trol parameter a . These can be a priori arbitrary mapsin arbitrary dimensions, but it is useful to restrict theanalysis to mixing maps and assume that an absolutelycontinuous invariant density ρ a ( x ) exists for each valueof the control parameter a . The local dynamics is x n +1 = f a ( x n ) . (9)We allow for a time dependence of a and study the long-term behavior of iterates given by x n = f a n ◦ f a n − ◦ . . . f a ( x ) . (10)Clearly, the problem now requires the specification of thesequence of control parameters a , . . . , a n as well, at leastin a statistical sense. One possibility is a periodic orbitof control parameters of length L . Another possibilityis to regard the a j as random variables and to specifythe properties of the corresponding stochastic process inparameter space. This then leads to a distribution ofparameters a .In general, rapidly fluctuating parameters a j will leadto a very complicated dynamics. However, there is a sig-nificant simplification if the parameters a j change slowly.This is the analogue of the slowly varying temperatureparameters in the superstatistical treatment of nonequi-librium statistical mechanics [6]. The basic assumptionof superstatistics is that an environmental control pa-rameter a changes only very slowly, much slower thanthe local relaxation time of the dynamics. For maps thismeans that significant changes of a occur only over alarge number T of iterations. For practical purposes onecan model this superstatistical case as follows: One keeps a constant for T iterations ( T >> T iterations to a new value a , after T iterations oneswitches to the next values a , and so on.One of the simplest examples is a period-2 orbit in theparameter space. That is, we have an alternating se-quence a , a that repeats itself, with switching betweenthe two possible values taking place after T iterations.This case was given particular attention in [34], and rig-orous results were derived for special types of maps wherethe invariant density ρ a as a function of the parameter a is under full control, so-called Blaschke products.Here we discuss two important examples, which are ofimportance in the context of the current paper, namelyhow to generate (in a suitable limit) a superstatisticalLangevin process, as well as a superstatistical Poissonprocess, using strongly mixing maps. Example 1 Superstatistical Langevin-like pro-cess
We take for f a a map of linear Langevin type[58, 59]. This means f a is a 2-dimensional map givenby a skew product of the form x n +1 = g ( x n ) (11) y n +1 = e − aτ y n + τ / ( x n − ¯ g ) (12)Here ¯ g denote the average of iterates of g . It has beenshown in [58] that for τ → t = nτ finite this deter-ministic chaotic map generates a dynamics equivalent toa linear Langevin equation [60], provided the map g hasthe so-called ϕ -mixing property [61], and regarding theinitial values x ∈ [0 ,
1] as a smoothly distributed ran-dom variable. Consequently, in this limit the variable y n converges to the Ornstein-Uhlenbeck process [58, 59] andits stationary density is given by ρ β ( y ) = r β π e − βy (13)The variance parameter β of this Gaussian depends onthe map g and the damping constant a . If the parameter a changes on a very large time scale, much larger than thelocal relaxation time to equilibrium, one expects for thelong-term distribution of iterates a mixture of Gaussiandistributions with different variances β − . For example,a period 2 orbit of parameter changes yields a mixture oftwo Gaussians p ( y ) = 12 r β π e − β y + r β π e − β y ! . (14)Generally, for more complicated parameter changes onthe long time scale T , the long-term distribution of iter-ates y n will be given by a mixture of Gaussians with a suitable weight function h ( β ) for τ → p ( y ) ∼ Z dβ h ( β ) e − βy (15)This is just the usual form of superstatistics used instatistical mechanics, based on a mixture of Gaussianswith fluctuating variance with a given weight function[6]. Thus for this example of skew products the super-statistics of the map f a reproduces the concept of super-statistics in nonequilibrium statistical mechanics, basedon the Langevin equation. In fact, the map f a can beregarded as a possible microscopic dynamics underlyingthe Langevin equation. The random forces pushing theparticle left and right are in this case generated by de-terministic chaotic map g governing the dynamics of thevariable x n . Generally it is possible to consider any ϕ -mixing map here [58]. Based on functional limit theo-rems, one can prove equivalence with the Langevin equa-tion in the limit τ → g . Numerical evidence has been presentedthat in this case often q -Gaussians with q > Example 2 Superstatistical Poisson-like process
Take f a : Ω → Ω to be a strongly mixing map, forexample the binary shift map f a ( x ) = 2 x mod 1 on theinterval Ω = [0 , ǫ , say I ǫ = [1 − ǫ,
1] and a generictrajectory of the binary shift map for a generic initialvalue x = P ∞ i =1 σ i − i , where σ i ∈ { , } are the dig-its of the binary expansion of the initial value x . Wedefine a ‘rainfall’ event to happen if this trajectory en-ters I ǫ (of course, for true rainfall events the dynamicalsystem is much more complicated and lives on a muchhigher-dimensional phase space). It is obvious that theabove sequence of events follows Poisson-like statistics,as the iterates of the binary shift map are strongly mix-ing, which means asymptotic statistical independence fora large number of iterations. Indeed between successivevisits of the very small interval I ǫ , there is a large numberof iterations and hence near-independence. Hence the bi-nary shift map generates a very good approximation ofthe Poisson process for small enough ǫ , and the waitingtime distribution between events is exponential.We may of course also look at a more complicated sys-tem f a , where we iterate a strongly mixing map f a whichdepends on a parameter a , and where the invariant den-sity ρ a of the map f a depends on the control parameter ina nontrivial way. Examples are Blaschke products, stud-ied in detail in [34]. If the parameter a varies on a largetime scale, so does the probability p ǫ = R I ǫ ρ a ( x ) dx ofiterates to enter the region I ǫ , and hence the rate param-eter of the above Poisson-like process will also vary. Theresult is a superstatistical Poisson-like process, generatedby a family of deterministic chaotic mappings f a . In thisway we can build up a formal mathematical framework todynamically generate superstatistical Poisson processes. IX. CONCLUSION
We started this paper with experimental observations:The probability densities of daily rainfall amounts at avariety of locations on the Earth are not Gaussian or ex-ponentially distributed, but follow an asymptotic powerlaw, the q -exponential distribution. The correspondingentropic exponent q is close to q ≈ .
3. The waitingtime distribution between rainy episodes is observed tobe close to an exponential distribution, but again a care-ful analysis shows that a q -exponential is a better fit,this time with q close to 1.05. We discussed the corre-sponding extreme value distributions, leading to Gumbeland Fr´echet distributions. We made contact with a veryimportant concept that is borrowed from nonequilibriumstatistical mechanics, the superstatistics approach, and pointed out how to generalize this concept to stronglymixing mappings that can generate Langevin-like andPoisson-like processes for which the corresponding vari-ance or rate parameter fluctuates in a superstatisticalway. Of course rainfall is ultimately described by a veryhigh dimensional and complicated spatially extended dy-namical system, but simple model systems as discussedin this paper may help. Acknowledgement
G.C. Yalcin was supported by the Scientific ResearchProjects Coordination Unit of Istanbul University withproject number 49338. She gratefully acknowledges thehospitality of Queen Mary University of London, Schoolof Mathematical Sciences, where this work was carriedout. The research of P. Rabassa and C. Beck is supportedby EPSRC grant number EP/K013513/1. [1] Silva, V.B.S; Kousky, V.E.; Higgins R.W. Daily Precipi-tation Statistics for South America: An Intercomparisonbetween NCEP Reanalyses and Observations.
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Chaos: An In-terdisciplinary Journal of Nonlinear Science
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Statistics & Probability Letters
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Trans-actions of the American Mathematical Society
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