Symbolic Recurrence Plots a New Quantitative

8/11/2019 Symbolic Recurrence Plots a New Quantitative

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Eur. Phys. J. Special Topics 164 , 85104 (2008)c EDP Sciences, Springer-Verlag 2008

DOI: 10.1140/epjst/e2008-00836-2THE EUROPEANPHYSICAL JOURNALSPECIAL TOPICS

Symbolic recurrence plots: A new quantitativeframework for performance analysisof manufacturing networksR. Donner 1, a , U. Hinrichs 2 , and B. Scholz-Reiter 2

1 Institute for Transport and Economics, Dresden University of Technology, Andreas-Schubert-Str. 23,01062 Dresden, Germany

2 BIBA, University of Bremen, Hochschulring 20, 28359 Bremen, Germany

Abstract. During the last years, the concept of recurrence plots has received con-siderable interest as a tool for analysing nonlinear and non-stationary time series.However, in the case of discrete-valued observables or variations on very differenttime scales, problems may occur in direct interpretations of the results of recur-rence quantication analysis (RQA). As a potential solution, we suggest combiningthis approach with ideas from symbolic time series analysis, which allows an arbi-trary static or dynamic coarse-graining of the dynamics that goes beyond recentrecurrence plot based methods. As an illustrative application, we discuss how theresulting symbolic recurrence plots may be used for a quantitative investigation of the dynamics of discrete-valued inventory levels of cooperating rms in a manu-facturing network. Based on discrete-event simulations, measures from traditionalRQA are used to evaluate the performance of the individual rms under different

production strategies as well as order policies. The results of our investigations arean important step towards an anticipative knowledge about the performance of manufacturing systems under different conditions, which is of major importancefor the planning and control of both production and logistics.

1 Introduction

The understanding of complex systems does not only require efforts in modelling physical mech-anisms, but also analysing data of meaningful observables characterising this system. Followingthis idea, time series analysis has emerged as a branch of statistics since the late 19th century.However, concepts originated in traditional statistics typically yield measures of linear charac-teristics of the data, like their correlation functions or spectral (Fourier) decomposition. Withthe development of the theory of nonlinear dynamical systems, it has become evident that suchlinear methods are often insufficient for characterising the dynamics of nonlinear systems. As aresult, numerous novel complex systems based approaches have been developed, which combineto a widely applicable toolbox of nonlinear methods of time series analysis [14].

Although existing (linear as well as nonlinear) methods of time series analysis have foundwide applications, there are natural limitations to most of the underlying concepts. In partic-ular, many methods have been developed for dealing with continously distributed data only.Exceptions from this are mainly situated in elds where symbolic analysis is applied as a stan-dard tool [5,6]. Additional problems with applying some of the existing methods may occurif the variability of a system covers very different time scales (not necessarily in a self-similar

a e-mail: [email protected]


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86 The European Physical Journal Special Topics

or multifractal way). Also in such cases, a symbolic discretisation of the time series allowsa ltering of the interesting aspects of the dynamics. However, methods of symbolic timeseries analysis sometimes suffer from a slow convergence and their high computationaldemands.

An alternative nonlinear concept that has attracted increasing interest during the recentyears are recurrence plots and their quantitative description (see www.recurrence-plot.tk )[7,8]. Recurrence plots have several important advantages: First, unlike other standard meth-ods of chaotic data analysis, they are applicable to non-stationary data and may even be usedfor dening proper stationarity tests [9,10]. Second, the resulting recurrence quanticationanalysis (RQA) yields a variety of nonlinear measures [8,1114]. Moreover, although the com-putational efforts may be even higher than in the case of symbolic time series analysis, the con-vergence of the associated numerical algorithms may still be characterised by a reasonable rate.Finally, during the recent years, a large number of generalisations of recurrence plots have beenintroduced for specic purposes, for example, the concept of spatial recurrence plots [15,16].

In general, RQA typically characterises the short-term dynamics of a system. If one wishes tostudy specic long-term features, it may be necessary to apply a coarse-graining or other specicltering to the data. In such cases, there is a natural relationship between RQA and the analysisof symbolic time series. Recent methodological developments like order patterns recurrence plots[1720] and ordinal recurrence plots [18] have explicitly made use of this relationship. In thiswork, we present a general framework for combining ideas from recurrence quantication andsymbolic time series analysis, which includes the aforementioned methods, but goes beyond andthus offers related concepts that are adapted to specic questions of analysis.

As a particular application that has up to now hardly been treated by nonlinear methodsof time series analysis, we consider the behaviour of manufacturing networks under variousdifferent production parameters and strategies. The dynamics of such networks is captured byobservables like inventory levels (stocks) or order volumes, which are characterised by integer-valued distributions. Traditional methods of symbolic time series analysis may be well appliedto such data, however, their capability of giving meaningful information about the dynamics of a logistic system is intrinsically limited by the specic features of the resulting time series likethe distribution of their discrete values and a variability involving rather different time scales.As this calls for further methodological developments, we will discuss how our new concept of symbolic recurrence plots may be used for gaining additional information.

This paper is organised as follows: In section 2, we review the basic concepts of symbolictime series analysis and recurrence quantication analysis and their intrinsic strengths andpotential weaknesses. In addition, the new approach of symbolic recurrence plots is introduced.In section 3, we introduce an event-discrete model for a manufacturing network and discusshow symbolic time series analysis, recurrence quantication analysis and symbolic recurrenceplots may be used to gain quantitative information about the variability of logistic observables.Finally, some conclusions are presented, which have however to be further validated by system-atically considering different types of model systems and comparing the results of standard andsymbolic recurrence quantication analysis with those of other nonlinear methods.

2 Symbolic time series analysis and recurrence plots

2.1 Symbolic time series analysis

During the last years, symbolic time series analysis has found many applications in differentelds of research [5,6]. The key idea behind this concept is an appropriate encoding of the timeseries into a sequence of discrete symbols, which are then subjected to a specic statisticalanalysis. In the framework of deterministic discrete-time dynamical systems, one may ndgenerating partitions for which the assignment of symbol sequences to trajectories is unique upto a set of measure zero [2123]. Of course, the use of such generating partitions is desirablefor estimating dynamical invariants from time series. However, in applications, it turns outthat the estimation of a generating partition in the presence of noise is challenging [2426].


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Recurrence Plots 87

For choosing a simple and practical symbolic encoding, two major strategies are most commonlyused instead: A static encoding groups the data with respect to a set of pre-dened threshold values

( 1 , . . . , I ) and assigns symbols according to the respective group an observation belongsto, i.e.,

X (t) =i if i X (t) < i +1 (i < I )0 if X (t) < 1I if X (t) I

A dynamic encoding operates in a similar (threshold-based) way on a difference-ltered timeseries of order p, where the difference lter is recursively dened in the standard way as

1X (t) = X (t + 1) X (t) p X (t) = p 1X (t + 1) p 1X (t) ( p > 1).

For stationary systems, it is most common to use one threshold at 1 = 0, yielding a binaryencoding of the considered time series.

In addition to purely static or dynamic encoding, mixed forms are also possible.

There is no general rule for optimally selecting a particular transformation to a symbolicsequence (in particular, regarding the type of transformation and the number of symbols) [27].In contrast, such a selection is often determined by practical considerations like the requiredcomputational efforts and the time scales of dynamics to be resolved.

The idea of symbolic time series analysis can be understood as a coarse-graining of thedynamics of the system, which preserves the essential dynamical information. Such a coarse-graining may be useful in a variety of situations, for example, in the presence of a considerablyhigh noise level, mutual dynamical states, or a high number of discrete values of the consideredobservable with certain mutual similarities, yielding numerous applications in different eldsof research [5]. From the structural point of view, time series of observables with only fewdiscrete values have the same essential properties as such symbolic sequences, such that it is anatural approach to adapt the corresponding mathematical analysis techniques.

Methods of symbolic time series analysis may be used for estimating various dynamicinvariants like correlation functions [28]

C XY ( ) =aA

P XY aa ( ) (1)

(with A being the set of possible symbols (the so-called alphabet ) and P XY ab ( ) := P (X (t i ) =a, Y (t i + ) = b)), mutual information [29]

I XY ( ) =a,b A

P XY ab ( ) log2P XY ab ( )P Xa P Y b

, (2)

(with P Xa = P (X = a)), permutation entropy [30], or transfer entropy [31]. However, some

mainly technical problems have to be noted: First, the resulting estimates may be stronglyinuenced by the distribution of the symbols, which may crucially differ if the signicantparameters of the analysed complex system are only slightly changed. We will illustrate thisfact later in this paper. Second, a nonlinear dynamical system may evolve on very differenttime-scales. If this is the case, after a static (threshold-based) encoding long blocks with onlyone symbol emerge. Furthermore, Cellucci et al. [32] have shown that the appropriate denitionof the corresponding thresholds has important consequences for the resulting estimates of themutual information. In particular, results obtained from groups with equal probabilities havebeen found to be more reliable than such from groups of equal size. Note that this nding hasspecial implications for time series with discrete values or, more general, a large number of ties.In both cases, it may be hardly possible to dene symbols that occur equally often in the nalsequence, which causes a systematic bias of the results.


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As a further potential disadvantage that limits the practical applicability of symbolic timeseries analysis, the computational efforts required for an estimation of entropies and relatedquantities increase strongly with the number of possible symbols. This problem may be illus-trated by the estimation of the source entropy of the underlying system, which is traditionally

approximated by the limit of the conditional block entropies hk = H k +1 H k for k where

H k = N ( l )

i =1

p(k )i log2 p( k )i (3)

are the block entropies of a binary encoding, i.e., the Shannon entropies of symbolic sequencesof length k, with p( k )i being the probabilities of occurrence of all possible subsequences of thislength within the time series [5,33]. As the number of individual symbols n = I + 1 increases,there are N (k) = nk different subsequences of length k. Hence, for sufficiently large k necessaryto give a reliable estimate, the number of symbols to be considered has a crucial inuence onthe computational efforts required. Moreover, for having a reliable statistics, the original timeseries has to be long enough to cover every possible sequence of length k at least several times.Hence, increasing n also increases the demands with respect to the amount of available data.

Apart from the computational demands, the appropriate choice of the number and locationof thresholds (and, in case of dynamic encoding, also of the order of difference ltering) is a cru-cial point in symbolic time series analysis. Consequently, there have been efforts for developingalternative concepts of nonlinear time series analysis that do not share the mentioned practicaldisadvantages. In the following, we will discuss the potential power of recurrence quanticationanalysis (RQA) as a potential candidate.

2.2 Recurrence Quantication Analysis (RQA)

Whereas symbolic time series analysis is computationally very demanding and may cruciallydepend on the specic parameters of the method, other concepts originated from complex sys-tems sciences are more robust in giving estimates of dynamical invariants from time series.

Among these, the application of recurrence quantication analysis has recently attracted con-siderable interest [8]. In contrast to symbolic analysis with static encoding, this approach is alsowell applicable in the case of laminar phases, as the distribution of such phases is one speciccharacteristic of the dynamics [14].

Recurrence quantication analysis consists of a variety of nonlinear statistics that can bedened based on the recurrence matrix [7]

RXij = X (m ) (t i ) X ( m ) (t j ) (4)

of the time series X (which is here subjected to a suitable m-dimensional embedding). Here() is the Heaviside function and an appropriate predened threshold. The recurrence matrixreveals the full topological structure of this series, which means that under general conditions,the rank-order of all values in a univariate time series can be fully reconstructed from this matrix

[34]. Extensions of this idea to higher dimensions are available [35], however, the problem of discrete-valued data with only a limited range has not yet been systematically addressed in thiscontext.

Following this idea, a large number of dynamic invariants can be estimated by a sophisti-cated statistical evaluation of the recurrence matrix that does not necessarily require embedding[3638]. Apart from traditional nonlinear measures like correlation dimension, entropy, ormutual information, this recurrence quantication analysis has the advantage that it providesa variety of additional measures which may be used to quantify different aspects of the (nonlin-ear) dynamics of an observed system. These measures can be mainly classied into statistics of diagonal lines in the recurrence plot (indicating the presence of deterministic dynamics, as forsome time interval, X (t i + t ) X (t i )) and those based on vertical lines (which characteriselaminar dynamics with X (t j ) X (t i ) for t j = t i + 1 , . . . , t i + t ).


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In comparison with symbolic time series analysis, recurrence quantication analysis mayhave a considerable higher computational efficiency in the estimation of comparable quantities.However, whereas in the case of a symbolic sequence of length k and an encoding with nsymbols, there are nk different sequences that have to be considered, in the quantitative analysis

of recurrence plots, only those sequences are of interest which consist exclusively of zeros orones, respectively. Hence, although the RQA requires the consideration of a N N matrixwith at most N (N 1)/ 2 non-trivial entries instead of a sequence of length N , in the limit of large k (which are required to estimate the entropy of the observed system), the analysis of therecurrence plots may be signicantly faster than an estimation based on a symbolic encoding.

In a recent study, Letellier [39] has introduced a new approach for estimating the Shannonentropy of a system from its recurrence plot. He could show that the resulting estimate ismore robust with respect to noise contaminations than the entropy estimate based on symbolicencoding. In particular, this result is important as there is a close relationship between Shannonentropy and largest positiv Lyapunov exponent of the system, i.e., the Shannon entropy is asimple and meaningful complexity measure.

2.3 Symbolic recurrence plots

As a preliminary conclusion from the above considerations, recurrence quantication analysishas some potential advantages with respect to symbolic time series analysis, which makes itsapplication to practical problems from various elds of research a promising and meanwhilequite established alternative. However, in the case of observables that vary on very differenttime scales, a preprocessing by coarse-graining or other appropriate ltering of the data may behelpful for making the corresponding results statistically more reliable. If such a coarse-grainingis applied, there are obvious similarities to the case of discrete-valued time series, which aretypically analysed by concepts from the analysis of symbolic sequences. Summarising theseconsiderations, it seems promising to combine both concepts in one general framework. Inparticular, a sophisticated choice of symbolic encoding before computing recurrence plotsmay combine the benets of a symbolic coarse-graining (which reduces the data to theirrelevant information if the detailed values themselves are not relevant) with the topological

completeness of recurrence plots.

The idea of a combination of a symbolic encoding and recurrence plots is not new: The consideration of recurrences instead of individual time series values can already be

regarded as a specic kind of symbolic discretisation. In the case of order patterns recurrence plots [1720], symbols (which correspond here to

different order patterns) are dened by the order relationships between k subsequent pointsin a time series. In this case, a recurrence requires that a given sequence is completelyrecovered in another part of the time series. Formally, this corresponds to a dynamic sym-bolic encoding using a difference lter of order 1, which is followed by an aggregation of k 1 symbols to one sequence and assigning a new symbol to any possible set of values of this sequence.

Recently, the concept of ordinal recurrence plots [8,18] has been proposed as a hybridbetween standard and order patterns recurrence plots. In this case, a recurrence takes placeif two states X i and X i + that are separated by a time shift have similar values andthe same order relationships. Hence, ordinal recurrence plots are also closely related to asymbolic encoding. However, there are no xed thresholds for the dynamic variable itself asin the case of a static encoding, but just order relationships between in total four differentvalues of the time series (cf. [8]).

The above mentioned links between symbolic encoding and recurrence plots partly suffer fromthe fact that the occurrence of ties and resulting long laminar phases has not yet been taken intoaccount explicitly. However, such structures can be frequently observed in discrete-valued timeseries. Moreover, whereas order patterns and ordinal recurrence plots represent only specialcases, a combination of an arbitrary symbolic encoding with recurrence quantication analysis


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yields a more general class of complex systems based methods of time series analysis, whichwe will refer to as symbolic recurrence plots in the following. As the two most generic types of symbolic discretisation, this approach includes the consideration of symbol sequences after arst-order dynamic encoding (i.e., the use of order relationships between subsequent values of

the considered observable) corresponding to order patterns recurrence plots as well as that of a static encoding resulting in a coarse-graining of the possible range of values. Note, however,that in practical application, the use of symbols according to a mixed encoding may be of interest, for example, if only the qualitative value of an observable (e.g., low, medium, andhigh, which corresponds to a static encoding) and its current change (dynamic encoding) arerelevant for a classication of the dynamics.

In general, a symbolic recurrence plot can be formally dened via its associated recurrencematrix

Rij = ( d((t i ), (t j ))) (5)

where d((t i ), (t j )) is a measure of the proximity of pairs of symbols which has to beappropriately dened. For example, in the case of order patterns and ordinal recurrence plots,a recurrence has been dened by an exact repetition of a particular symbol. If one appliesthe same strategy to recurrence plots originated from a mixed encoding, the symbolic recur-

rence plots can be considered as the joint recurrence plots [40] of the two symbolic sequencesobtained from purely static and dynamic encodings, respectively. Note, however, that depend-ing on the concrete application, the use of a weaker proximity measure may be a considerableoption. Beside the formal permission of neighbouring symbols, in case of a mixed encoding,this leads to the use of general Boolean relations between the considered pairs of static anddynamic symbols. Whereas the strict denition of recurrence as described above would corre-spond to a conjection (AND relation) between the events equal static symbols and equaldynamic symbols, one might also allow for disjunctions (OR relations) or other logical rela-tions. For example, a disjunction might be useful if there are many symbols due to a largenumber of thresholds or a dynamic encoding using a difference lter of higher order. In thiscase, there might be a practical interest in conditions where either the values of the consideredobservables themselves or their changes over some time steps are similar.

3 Performance analysis of manufacturing networks

3.1 Problem setting

In todays economies, successful companies must be able to exibly and fastly adapt to agradually changing market demand as well as a varying availability and price of electricity,commodities and other resources required for the production process. In order to minimise thenancial risks associated with these practical challenges, a concentration on the individual corecompetences (i.e., an outsourcing of secondary working elds) has been a frequent reaction tothe gradual diversication of goods and services. As a result of these tendencies, economic net-works have successively developed, within which the mutual interactions between the involvedcompanies lead to a very complex dynamics [4144]. This kind of behaviour may however

enhance the occurrence of undesired effects like production breakdowns due to a lack of mate-rial, which can result from unpredictable oscillations of inventory levels [4549].As a consequence of the above considerations, the practical control of present-day manu-

facturing networks is a challenging task, since their structural and dynamic complexity makesthem reacting very sensitively to small disturbances. Conventional approaches to planning andcontrol of production systems are usually not able to completely avoid such nonlinear dynamiceffects. In order to understand the corresponding dynamical mechanisms in some detail, duringthe recent years, efforts have been made to understand, predict, and improve the behaviour of production networks using concepts from the theory of nonlinear dynamical systems [43,5052].However, available results about the dynamics of complex networks [53] are often hardlyapplicable to manufacturing networks since these specic systems are characterised by a setof relevant state variables with rather specic interactions. On the one hand, without any


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Recurrence Plots 91

)a )b

Fig. 1. Schematic illustration of the material ows in different network topologies for an example of three collaborating rms (indicated by circles): (a) linear supply chain, (b) symmetrically interactingmanufacturers. The direction of information ows (orders) is always opposite to the correspondingmaterial ow which is schematically shown by arrows.

cooperative behaviour, individual rms would act like selsh agents who aim on optimis-ing their individual performance on the cost of the community. However, cooperation meansintroducing additional dynamic feedbacks to the system. On the other hand, logistic observ-ables like inventory levels are inuenced by possibly changing external demand and supplyconditions and may therefore show variations on very different time scales. Hence, althoughthe topological complexity of manufacturing networks may be rather low compared to otherreal-world networks, their dynamic complexity can be considerably high.

In order to optimise the performance of networks of production and logistics, it is necessaryto develop a deep understanding of the essential mechanisms which determine their dynamics.As information about the detailed behaviour of real-world manufacturing networks is usuallysubjected to condelity restrictions and therefore only hardly accessible, it is suitable to rstlyrepresent the essential processes of manufacturing and logistics in an appropriate mathemat-ical model, which can then be used to simulate different realisations of the system [43]. Inorder to analyse the general features of manufacturing networks, there is a variety of different

approaches, which can be roughly distinguished into macroscopic (continuous material ow)and microscopic models (discrete material ow) on the one hand, and time-continuous anddiscrete-event simulations on the other hand. The study of the corresponding models allowsthe identication of economic potentials and a successive testing of various optimisation andcontrol strategies.

In a series of recent papers, discrete-event simulations have been used for investigatingthe dynamics of small-scale manufacturing systems with two and four manufacturers [5458].In particular, the impact of stochastic perturbations as well as different dynamic instabilitiesof these systems has been qualitatively characterised [58]. In the following, we will considerthe outcome of such discrete-event simulations from a methodological point of view regardingthe applicability of the different concepts of nonlinear time series analysis addressed in thispaper. After presenting the mathematical details of our model, we will systematically discussthe potential benets and problems of traditional symbolic time series analysis and standardas well as symbolic recurrence quantication analysis.

3.2 Description of the model

A given number of manufacturers can form networks of production and logistics with a lotof different topologies (see Fig. 1). The two probably most distinguishable cases are linearchains, where there is no backow of material, and symmetrically interacting manufacturers,where each company produces commodities that are delivered to all other cooperating rmsin the network. Whereas a linear chain is characterised by a unidirectional nearest-neighbourcoupling, the latter setting corresponds to an all-to-all bidirectionally coupled network thatyields the maximum number of feedback loops for a given number of nodes. A comparison of


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the behaviour of these two topologies would allow to systematically study the importance of feedback-induced instabilities for the dynamics of production networks in some detail. However,a systematic investigation of the effects of topology is beyond the scope of the presented work.As feedback loops of material and/or information ows are one of the most important sources

of (potential) instabilities in production systems [48], we will focus on the case of symmetricinteractions in the following.In this work, we use a discrete-event model for studying the dynamics of manufacturing

networks, as this approach yields a probably more natural description of the commodity owsthan continuous-ow models. To restrict the number of relevant parameters to a minimum, weconsider in the following only small-scale networks which consist of a low number of manufac-turers, which serve as the nodes of a manufacturing network. Such small-scale networks cantypically be found in the case of groups of factories that belong to the same company or pro-vide highly specied services or products. In addition to the network participants, every node isunidirectionally connected with one external supplier and one external costumer which repre-sent the external market. To avoid mixtures of different commodities and production processes,every manufacturer is assumed to have as many sort-pure buffers for individual commoditiesas it has suppliers, and a number of production lines that equals the number of costumersinside and outside the network. Consequently, every supplier equips exactly one buffer of therespective factory, whereas every production line delivers their nal goods only to one distinctcostumer. This setting allows a redenition of the nodes of the manufacturing network on amicroscopic scale, where every node now represents exactly one production line. Macroscopi-cally, these nodes form groups corresponding to the individual factories, which can be math-ematically formulated in terms of the resulting matrices that dene the topology of themanufacturing network (see below).

Every node in the network is able to control the production process by applying differentpossible strategies for ordering commodities as well as producing its goods. These strategiesare the same for all manufacturers and xed in every simulated scenario. With respect to theinitiation of a production process, one may roughly distinguish push and pull strategies:

In the case of a push strategy, the production process starts immediately if productioncapacity is available (i.e., the considered production line i is not blocked by a currently

processed job) and the commodities N j required for producing the associated good i arepresent in sufficient quantities N ij . Mathematically, one may describe this fact by an indi-cator variable Qi (t) which is one if a new production process for good i starts at time t,and zero elsewhere. The value of this production indicator is then determined by its pastvalues over a time interval p,i that would have been necessary for nishing one productionprocess, and the current values of the inventories N j (t) as

Q i (t) =n c

j =1 , N ij =0

(N j (t) N ij ) 1 s


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In the case of the symmetrically coupled network with three manufacturers (Fig. 1(b)), thecorresponding blocks contain 3 3 elements. It has to be noted that in the case of a pushstrategy, there are no orders to the respective suppliers required, i.e., there is no backowof information about the inventory levels and the state of the production opposite to the

ow of commodities.For making the dynamics described by the above equations self-consistent, one has to addan equation for the changes of the inventory levels of individual commodities. This can bedone in the following way:

N j (t) = N j (t 1) +n l

i =1

Aij Q i (t p,i t,i ) n l

i =1

N ij Q i (t), (7)

where nl is the number of production lines in the network, including those representingthe external supply of commodities. In the latter equation, the rst sum represents therespective supply of commodities, which is in our model realised exclusively by one distinctsupplier for every commodity j , which are instantaneously shipped after their productionhas nished. This setting gives rise to an additional transportation delay t,i associated withthe responsible production line i (and, hence, also with the associated commodity j ). Theadjacency matrix A contains the delivery relationships for all production lines as well asall external suppliers (additional rows) and costumers (additional columns). In the lattercases, we set p,i 0, i.e., we consider an unlimited availability of commodities at theglobal market. The second sum contains terms describing the successive use of material bythe individual production lines of the network, which is determined by the lot size matrixN . For controlling the access of different production lines to individual inventories, somepriorisation rule has to be dened. In the case of the push strategy, we have chosen a scenariowithin which the production line supplying the external market accesses the buffers rst,followed by the other lines in numerical order.

In the case of a pull strategy, the production of a certain good is started by an order of the respective costumer, provided that the required commodities are present. Otherwise,orders are send to the responsible suppliers. For including these orders in the frameworkdiscussed above, one may rst introduce an additional multiplicative factor (Oi (t) 1) inthe equation dening Q i (t). Here, Oi (t) is the number of unprocessed orders of a product iat time t . For speciying the dynamics of the latter observable, one may set

Oi (t) = Oi (t 1) Q i (t) +n c

j =1

Aijn l

k =1 , N kj =0

f 1(N j (t), N kj ) f 2(N j (t), N kj ). (8)

Here, f 1() is an indicator function determining whether an order has to be realised, andf 2() determines the associated order volume. In the simplest possible case, f 1() may becomeone in every time step if the respective inventory level falls below the value necessary forproducing one more product. The order volume is then determined by the difference betweenthe required amount of material and the actual stock level. In general, both functions aredened according to a specic order policy (see below).

The production process is immediately initialised if there is a demand (unprocessed order)and the required commodities are available in sufficient quantities. Orders from the externalmarket are realised with a xed rate in regular time intervals of typically 8 or 24 timesteps (where every time step will be identied with one hour in the real world in thefollowing). Unlike in the case of the push strategy, the scenario has been dened here suchthat the individual production lines access the buffers in increasing numerical orders inevery simulation step, disregarding whether they produce for the external market or someof the collaborators. This fact may however cause an accumulation of unprocessed orders atcertain production lines of the network if the material ows are not sufficiently balanced.The moment of control for the presence of unprocessed orders can in principle be chosenwithout restrictions. In our simulations, however, there is a xed order of control steps,starting with a check of unprocessed orders, followed by a control of the available inventory


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(and an eventual ordering of the corresponding commodity from the responsible supplier),and concluded (if possible) by the initialisation of a production process. All these processesare repeated in every simulation step in a quasi-parallel way for all production lines of thenetwork.

Due to the presence of orders initiating the individual production processes, in the case of apull strategy, the resulting dynamics of logistic observables like N j and O j is strongly affectedby the corresponding feedback of material and information ows and may therefore show rathercomplicated variability patterns. Among the corresponding phenomena, one may observe tran-sitions from periodic over quasi-periodic to chaotic dynamics, which take place in the case of missing logistic synchronisation, i.e., if lot sizes or order volumes are not sufficiently balancedamong the network [58].

As some special cases of a pull control, manufacturers are additionally able to dene thetimes and volumes of their orders in different ways. In our simulations, we have studied threedifferent order policies: In the presence of a provision or an order point policy , there are variableorder periods with xed order volumes as an additional model parameter. These orders areinitialised if the inventory levels have become either zero or less than a prescribed thresholdvalue, respectively, which is checked with a period of six simulation steps (corresponding to

intervals of six hours). In addition, the periodic order policy has been considered, which ischaracterised by xed order intervals (here: 8 hours), but variable order volumes that arecomputed as follows:

Oj = T av,j D j + z T av,j 2D j + D j 2 , (9)where T av,j Aij ( p,i + t,i ) is the average time until ordered material is available to thecostumer, D j and D j are mean value and standard deviation of the demand, and z is anadditional safety factor which can be specied. In our simulations, we have set z = 1 .65 andT av,j = 2 days (= 48 simulation steps) for all commodities. The average demands and their

standard deviations are computed by summing up over all order volumes of the respectivecommodity. For all policies, if an order for certain commodities is necessary at a given time, thecorresponding volume is determined by the total demand considering the unprocessed costumerorders in all production lines using these commodities.

In order to implement the discrete-event model described above in a user-comfortable way,we have used the commercial simulation software eM-Plant (see [57,58] for more information).This simulation environment allows a detailed mapping of quantities with practical relevancefor production processes, such as processing and transportation times, working-on-intervals, lotsizes, and order volumes. Although different logistic observables can be deduced from the simu-lations, including the time series of unprocessed orders Oj (t) as well as the average productionrates Q i = lim T 1T

T t =1 Q i (t), we will restrict our attention in the following exclusively to

the inventory levels N j (t).

3.3 Symbolic time series analysis

Although there have already been considerable efforts in quantitatively analysing the differenttypes of instabilities in small-scale manufacturing networks, the reliability of the correspondingresults is still debatable as long as traditional linear methods of time series analysis are con-sidered: Beside the fact that in event-discrete models, the relevant variables do not necessarilychange at equally spaced time points, microscopic models in general yield inventory levels thathave only integer values, whereas most concepts in time series analysis have been developed forcontinuously dened variables.

In [56,57], a rst-order dynamic encoding of the inventory level time series has been applied,which denes a ternary sequence according to the sign of the local increments, i.e.,

(t) :=1, N (t + 1) > N (t)0, N (t + 1) = N (t)

1, N (t + 1) < N (t). (10)


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In this case, the reliability of the estimated values of the different dynamic invariants such asthe source entropy or mutual information suffers from the fact that the symbol for the eventno change occurs signicantly more often than those for changes towards higher or lowervalues (with a typical abundance of 7095% in the case of the symmetrically coupled four-

node network under different order policies). Note that the explicit consideration of equality isdistinctively different from the denition of order patterns in [1720].In order to study the inuence of the choice of transformation for symbolic encoding, we

have additionally used sequences obtained from a static encoding with equal group sizes (i.e.,the range of possible values of the considered observable is divided into three intervals of thesame size) as well as one with equal group probabilities (i.e., the range of values is dividedby thresholds which have been chosen such that every interval contains the same number of observations 1). It should be noted that the results of symbolic time series analysis may in generaldepend on the sampling rate of observations. In particular, very high sampling frequencies couldresult in more and longer sequences of equal values in the original data and/or the symbolictime series. However, in the present study, the sampling rate of one hour has been chosen tocorrespond to the typical production and transportation time scales in our model describedabove, which are in turn not much smaller than the intervals of ordering which determine anessential part of the dynamics of the manufacturing network.

In the following, we will illustrate the qualitative behaviour of measures from symbolictime series analysis by considering a network of four symmetrically interacting manufacturers.In this case, the adjacency matrix A = ( Aij ) (which describes the presence of material owsfrom production line i to inventory j , including the external suppliers and costumers) reads asfollows:

A =

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 00 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

. (11)

Here, rows correspond to production lines, columns to inventories, and blocks to individualfactories, where the last four rows (columns) represent the external suppliers (costumers).

In Fig. 2, the resulting dynamics of this system is shown for one specic setting with threedifferent order policies. Processing and transportation times have been xed to one hour (cor-responding to one time step in the simulation). All non-vanishing lot sizes N ij have been setto 1, all order volumes f 2(N j , N ij ) = 5. The external market demand has been assumed asconstant. In the case of the order point policy, the critical inventory levels have been set to 8(lines producing for the external market) and 3 (lines producing for another node), respectively.

On the one hand, although the behaviour of the inventory levels differs strongly betweenprovision and order point policy, symbolic time series analysis with a dynamic encoding yields

1 Note that this is usually not exactly possible in the case of discrete-valued observations.


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Fig. 2. Left panels: time series of inventory levels N j (t ) for one production line (line 2 of factory 1)supplying another node (factory 2) in the symmetrically coupled four-manufacturer network. From topto bottom, the results are shown for order point, provision, and periodic order policy. Right panels:corresponding values of the conditional block entropies h k resulting from three different kinds of ternary

encodings ( : dynamic encoding according to the sign of the local increments, , o: static encodingwith two thresholds separating the range of values into groups of equal size or frequency, respectively).The limit of the conditional block entropies for l is an estimator for the source (Shannon) entropyof the system [33]. The specic model parameters are listed in the text.

almost the same estimates for the entropy of the underlying system, which can be approximatedby the limit of the conditional block entropies, i.e., the information gain when the length of thesymbolic sequences is extended by one [33]. This means that for the considered data, entropymay lack specicity and/or sensitivity for giving meaningful quantitative information aboutthe underlying system. However, the periodic order policy (with variable order volumes) ischaracterised by a much larger dynamic complexity, which is recovered by larger values of the


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entropies. On the other hand, one has to note that the entropy values estimated using a dynamicencoding are hardly consistent with those of a static one with the same number of symbols. Thisobservation suggests that the values estimated by this approach are strongly affected by theparticular distribution of the symbols. In particular, the drop of the conditional block entropies

at a block length of k = 6 (for static encoding) or k = 5 (for dynamic encoding), respectively,(which corresponds to the basic period with which orders can usually take place in the case of order point and provision policy [56,57]) does not occur in all cases.

3.4 Recurrence quantication analysis

As we have shown above, in the case of symbolic time series analysis, the resulting values of the Shannon entropy (but probably also those of other dynamic characteristics like symboliccorrelation functions or mutual information) may crucially depend on the choice of the symbolicencoding. Recently [57], we have therefore suggested to use measures from recurrence quanti-cation analysis (RQA) [8,1114] as an appropriate alternative. The application of this concepthas been originally motivated by the fact that estimates of dynamic invariants (in particular,the Shannon entropy and measures derived from it) based on recurrence plots may be morestable than such based on a symbolic encoding [39].

In section 2.2, we have already argued that RQA has indeed some important benets incomparison with symbolic time series analysis. However, there are also possible weaknessesof this approach. For example, the values of the individual characterstics may depend on thedistribution of the considered observable, which is supported by other studies [32,59]. In the caseof a manufacturing network, this distribution may strongly change if the production strategyis modied. In the following, we will therefore address the question of how the distribution of the inventory levels as a discrete-valued time series may inuence the results of RQA.

As a particular example, let us again study the short-term dynamics of the symmetricallycoupled four-manufacturer network from section 3.3, restricting our attention to the last 1000points of the time series shown in Fig. 2. Note that with the same length of the time series,symbolic analysis would hardly yield statistically reliable results due to the minimum require-ment on the size of the data set (taking 3 6 . . . 37 different symbolic sequences into account for an

approximation of the source entropy). In the case of RQA, there are no such strong demands.Note, however, that due to instationarities in the rst parts of every simulation run, the resultsof symbolic analysis (using the whole record) and RQA (using only the last part) may showsome differences.

Considering the inuence of the distribution of the data in some more detail, we subjectall time series of the respective stocks to monotonous continuous transformations of the typeN = N , where is some positive, not necessarily integer number. In the case of symbolicanalysis as described in section 2.1, such a transformation would only affect the results if astatic encoding is used. In particular, if the distribution of inventory levels (which is alwaysbounded to non-negative values) has a positive skewness, this skewness is enhanced for > 1.In the following, we will examine how the aforementioned transformation affects the results of recurrence quantication analysis.

As it follows from visual inspection of Fig. 2, the consideration of a non-zero critical

inventory level (i.e., the replacement of the provision by an order point policy) leads to amore complex dynamics of the manufacturing system which evolves on very different timescales. This general observation has also important consequences for the outcome of recurrencequantication analysis. In Fig. 3, some of the corresponding results are shown. In the caseof a provision policy, all RQA measures (not only those displayed in the gure) are almostindependent of the recurrence threshold . Moreover, the power-law transformation of the datadoes not lead to any signicant changes. The reason for this is that because the transformeddata are afterwards normalised to minimum and maximum values of zero and one, respectively,the resulting distribution does not substantially differ from that of the original data.

In contrast to the provision policy, the results for an order point policy are clearly inuencedby the choice of the recurrence threshold and the power-law index , which illustrates theimportance of the distribution of the data in this case. Among the different RQA measures, three


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Fig. 3. Results of recurrence quantication analysis for the last 1000 points of the original time seriesX from Fig. 2 and the power-law transformed data X = X for different values of : Recurrence rateRR (upper panels) and Shannon entropy of the diagonal lines H d (lower panels) for the order point(left) and provision policy (right). The different symbols correspond to = 1 (+), 2 (o), 3 ( ), 4 ( ),and 5 ( ).

different types of behaviour may be distinguished: A rst group, which includes the determin-ism and laminarity (i.e., the percentage of recurrence points which form continuous diagonaland vertical structures in the recurrence matrix), shows a fast convergence towards 1 (indi-cating complete regularity) as increases, with an enhanced convergence for larger values of . In a second group, which consists of the recurrence rate RR (i.e., the relative abundance of recurrences, see Fig. 3 for the corresponding values) and the mean and maximum lengths

of diagonal and vertical structures, the RQA measures increase with . Moreover, for theorder point policy, there is also a remarkable increase with increasing , which is an effectof the larger size of the considered neighborhood and the increasing skewness of the data.Finally, the Shannon entropy H d 2 of the length distribution of recurrent structures alongdiagonals in the recurrence matrix shows different values for small and large . In the caseof a provision policy, the corresponding transition is very sharp and again almost independentof . In contrast, for the order point policy, there is a range of moderate threshold values where the corresponding estimate becomes almost independent of the power-law index , i.e.,of the distribution of the data (cf. Fig. 3). This nding is of major interest, as it suggests thatthis measure may be used for a quantitative comparison of the dynamics of a manufacturingsystem under different situations.

With respect to the specic features of the inventory levels, at least two strategies appearpossible for making the traditional symbolic time series analysis and recurrence quantication

analysis better applicable: As a potential solution to minimising the inuence of the data distribution on RQA, onemay apply a continuous transformation of the data which leads to normally distributedvalues. A corresponding amplitude adjustment is also used in other concepts of nonlineartime series analysis, for example, in many surrogate data algorithms for statistically testingagainst a linear-stochastic behaviour in observed time series [60]. However, one has to notethat formally, a corresponding transformation is only applicable to continuously distributed

2 Note that the Shannon entropy H d of recurrent structures ( R ij = 1) on diagonals is not comparableto the source entropy of the system, as the latter one increases with increasing complexity of the system.As Letellier [39] has pointed out, there is a much more closer analogy between the source entropy andthe Shannon entropy of non-recurrent diagonal structures ( R ij = 0) in the recurrence plot.


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data, such that there may be problems in the case of discrete-valued observables like theinventories considered here.

The presence of numerous ties as well as laminar phases in our time series suggests anaggregation of the data, which may help to remove dynamic redundancies. For instance,

one possible approach would be to increase the sampling interval, for example, from60 minutes to 6 hours, corresponding to the minimum order intervals of provision and orderpoint policies used in our simulation. Note, however, that such a coarse-graining in time isonly useful in the case of xed time scales. If the processing and transportation times areaffected by stochastic factors, a higher temporal resolution of the dynamics allows a betterinsight into the dynamics of the system. In addition, one has to ask oneself up to whichlevel an aggregation of the data is useful. In our considered example, we have found that theestimated values of some of the RQA measures (such as the mean lengths and entropies of diagonal and vertical structures) may strongly vary with time and depend on the width of the considered time window, which is quite likely an effect of the different time scales of thedynamics involved here. Hence, a high level of aggregation would lose important informationabout the variable dynamics of the system on short and intermediate time scales.

If one wants to avoid the aforementioned technical problems, the use of symbolic recurrence

plots may offer a possible alternative.

3.5 Symbolic recurrence plots

In the following, we will illustrate the performance of symbolic recurrence plots when analysinginventory level time series. As a particular possibility, we will use a mixed encoding. For thispurpose, we rst assign two different symbolic sequences to the observed time series: {d (t i )}icorresponds to a dynamic encoding according to Eq. (10), and {s (t i )}i is dened by separatingthe range of observed values into three groups of equal probability, where the symbols 1(0, 1) stand for low (intermediate, high) inventory levels. In a second step, both sequences arecombined into one by setting

(t i ) = d (t i ) + 3 ( s (t i ) + 1) + 2 . (12)

We apply the strongest possible proximity measure, which denes a recurrence only if thepairs of symbols ( d (), s ()) are equal for two times ti and t j . The combination of static anddynamic encoding is here motivated by the fact that the exclusive use of a static encoding isproblematic due to the distribution of the data, while the general level of the use of a givenbuffer is of practical relevance and should therefore be considered in addition to the short-termuctuations captured by the dynamic encoding.

In Fig. 4, the resulting standard and symbolic recurrence plots are shown for the examplesalready discussed in sections 3.3 and 3.4. For order point and provision policy, one observes somesmall-scale rectangular structures in the standard recurrence plots, which correspond to laminarphases during which the inventory level does not change. Due to the coarse-graining of thedynamics in terms symbolic encoding, these structures merge in the symbolic recurrence plots.However, on this coarse-grained scale, the signicant large-scale patterns are recovered. For the

periodic order policy, the long-term variations of the inventories are much less pronounced (seeFig. 2), which is recovered in the resulting recurrence plot that shows much shorter laminarintervals, but also interrupted diagonal structures as a signature of deterministic behaviour.

In general, one has to point out that the performance of the symbolic recurrence plotsdepends on the particular choice of encoding. This fact can be well described by consider-ing the differences between order point and provision policy in our example: using exclusivelya dynamic encoding (i.e., computing an order patterns recurrence plot) would result in analmost completely lled recurrence matrix due to the frequent occurrence of the symbol 0.In contrast to this, the application of a purely static encoding leads to the same large-scalepatterns in the recurrence plot, which hence mainly correspond to the static classicationof the inventory levels. According to these results, one has to point out that the appropri-ate denition of symbols is a crucial point for the application of symbolic recurrence plots.


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Fig. 4. Standard (upper panels) and symbolic recurrence plots (lower panels, with a mixed encodingas described in the text) for the last 2000 points of the time series shown in Fig. 2, for order point,provision, and periodic order policy (from left to right).

In particular, our ndings concerning the different types of encoding cannot be generalised toan arbitrary dynamics of discrete-valued observables. In the presented case, the use of mixedencoding allows to qualitatively distinguish the dynamics under order point and provision policy(which is hardly possible using standard recurrence plots): with an order point policy, the typ-ical time scale for strong changes in the inventory levels is clearly larger than with a provisionpolicy. Note, however, that in both cases, the resolved maximum time scale of 2,000 simulationsteps is still rather small compared to the total length of the records. On the very long timescales shown in Fig. 2, the behaviour seems to be quite opposite.

3.6 Quantitative analysis of symbolic recurrences

As in the case of standard recurrence plots, the quantitative analysis of the symbolic recurrenceplots may be used to estimate a variety of different nonlinear quantitative characteristics.Depending on the applied encoding, these measures can be mainly determined by small- orlarge-scale patterns in the recurrence matrix. In particular, static encoding yields a coarse-graining of the dynamics and thus focusses on larger time scales, whereas in the case of adynamic encoding, the short-term dynamics is considered while variations on larger scales areneglected. In order to illustrate these general considerations in some detail, RQA measures havebeen estimated for original as well as symbolic time series of inventory levels from our model.The corresponding results can be found in Tab. 1.

For provision as well as order point policies, determinism and laminarity always indicatea complete regularity of the system, while there are signicantly lower values in the case of an order point policy. The latter observation may be explained by the complicated mechanismof determining the times of volumes of orders, which involves the aggregation of a possiblychaotic process and may therefore show similarities to a presence of stochastic components inthe dynamics.

More detailed information about the dynamics of the system can be inferred from a classof RQA measures that quantify the distribution of the sizes of continuous structures, includingaverage and maximum lengths as well as entropies of diagonal and vertical lines. For the last


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Recurrence Plots 101

Table 1. Results of recurrence quantication analysis using the last 2,000 points of the original timeseries and their symbolic versions using dynamic (see Eq. (10)), static (with three equally probablyclasses), and mixed encoding (see Eq. (12)). Apart from the recurrence rate RR , the determinismDET and laminarity LAM (i.e., the percentage of recurrent points in diagonal and vertical structures,respectively), the mean and maximum lengths of diagonal and vertical recurrent structures ( l d andl v ) as well as the Shannon entropy of recurrent points in diagonals H d are shown. For the parametersof the individual simulations, see Fig. 2.

RR DET LAM l d lmax

d H d l v l

maxv

Order Point Standard 0.0774 1.00 1.00 26.66 150 3.57 39.20 138Policy Dynamic 0.9429 1.00 1.00 17.39 533 2.90 32.41 137

Static 0.5593 1.00 1.00 193.98 1259 4.38 287.75 1260Mixed 0.5320 1.00 1.00 20.55 533 3.06 37.97 137

Provision Standard 0 .1874 1.00 1.00 50.99 351 4.34 72.74 276Policy Dynamic 0.9708 1.00 1.00 33.72 383 3.71 63.67 275

Static 0.4407 1.00 1.00 131.67 570 5.27 202.27 420Mixed 0.4218 1.00 1.00 33.83 351 3.70 61.47 275

Periodic Order Standard 0.0790 0.96 1.00 4.92 218 1.68 7.44 40

Policy Dynamic 0.7572 0.82 0.88 9.07 217 2.89 17.16 39Static 0.3416 1.00 1.00 14.52 220 3.41 26.15 154Mixed 0.2588 0.82 0.87 8.72 217 2.87 16.85 39

2000 points of all time series, corresponding lower values suggest that the dynamics in the caseof an order point policy might be more complex than under a provision policy (note again thatin contrast to the standard denition of entropies, higher Shannon entropies of diagonal struc-tures correspond to a less complex dynamics [39]). However, it follows from Tab. 2 that whenconsidering the entire time series of 40,000 points 3 , the values of the different RQA measuresfor both policies can hardly be distinguished from each other any more. This observation under-lines that analyses based on recurrence plots are mainly suited to characterise the short-termdynamics of the data rather than their long-term variability. Apart from provision and orderpoint policy, the periodic order policy is again found to result in the most complex behaviour,supporting the information gained from visual inspection of Fig. 2, entropy estimation fromsymbolic time series analysis, and traditional RQA.

The results listed in Tables 1 and 2 allow to evaluate the performance of different encodingstrategies in some detail. On the one hand, a purely static encoding resolves only large-scalestructures, which leads to very long diagonal as well as vertical structures. There is hardlyany information about the small-scale variability of the data. On the other hand, a dynamicencoding is able to characterise the short-term dynamics of the inventory levels. However, thefrequent occurrence of short laminar phases in the data (corresponding to the symbol 0)leads to a very high recurrence rate, which may cause problems in the meaningful estimationof other RQA measures. If one combines both strategies, however, one may obtain a propercharacterisation of the short-term dynamics of the data (which can be seen by the values of thedifferent RQA measures which do not signicantly differ from those obtained with a dynamic

encoding) without an extremely large amount of recurrences.Comparing the different order policies with each other, one may see that standard andsymbolic recurrence plots show a similar behaviour of the resulting RQA measures. For provisionand order point policy where there are also pronounced variations on longer time scales, bothtypes yields results that are also quantitatively comparable with each other, while there aredifferences in the case of the order point policy which leads to a dynamics without such large-scale patterns. In the latter case, the coarse-graining of the dynamics due to the symbolicencoding results in a coarse-graining of the structures in the recurrence plots and, hence, to anincrease of their average as well as maximum extensions. Following these observations, standard

3 The RQA has been performed without embedding of the data, using the Euclidean norm, = 1,and diagonal and vertical lines of minimum length 2.


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Table 2. Results of recurrence quantication analysis using the complete original time series (40,000data points for provision and order point policy, 10,000 for periodic order policy) and their symbolicversions using dynamic (see Eq. (10)), static (with three equally probably classes), and mixed encoding(see Eq. (12)). Apart from the recurrence rate RR , the determinism DET and laminarity LAM (i.e.,the percentage of recurrent points in diagonal and vertical structures, respectively) as well as the meanand maximum lengths and Shannon entropies of diagonal and vertical recurrent structures ( l d and lv )are shown. For the parameters of the individual simulations, see Fig. 2.

RR DET LAM l d lmax

d H d l v l

maxv H v

Order Standard 0.0315 1.00 1.00 22.64 720 3.15 43.18 300 2.39Point Dynamic 0.9519 1.00 1.00 20.78 1553 2.95 39.65 299 2.15Policy Static 0.3338 1.00 1.00 198.92 5064 5.08 395.46 4866 2.94

Mixed 0.3177 1.00 1.00 20.78 719 2.95 39.67 299 2.16Provision Standard 0.0141 1.00 1.00 24.25 606 3.12 46.23 300 2.31Policy Dynamic 0.9541 1.00 1.00 21.78 9941 2.87 41.64 299 1.94

Static 0.3336 1.00 1.00 391.08 9623 5.13 779.77 9624 3.72Mixed 0.3183 1.00 1.00 21.88 8184 2.88 41.83 299 1.94

Periodic Standard 0.0708 0.96 1.00 4.94 888 1.66 7.63 73 1.34

Order Dynamic 0.7560 0.82 0.88 9.24 887 2.89 17.67 72 2.20Policy Static 0.3437 1.00 1.00 15.85 888 3.49 29.60 208 3.16Mixed 0.2596 0.82 0.87 8.87 887 2.88 17.54 72 2.26

and symbolic recurrence plots contain comparable quantitative information about the variabilityof a time series. However, as symbolic encoding may be considered as a lter for interestingdynamic features in multi-scale problems, the consideration of symbolic RQA potentially opensadditional elds of applications for the concept of recurrence plots.

As for symbolic time series analysis and traditional RQA, one has to discuss the problem of convergence of the measures estimated from symbolic recurrence plots. In the case of stationarydata, one may argue that for all three types of methods, the quantitative characteristics reachstationary values as the amount of available data becomes sufficiently large. Comparing stan-

dard and symbolic RQA, it is reasonable to assume that the convergence may be faster in thesymbolic case due to the coarse-graining of the range of possible values. This implies that therate of convergence is closely related to the size of the alphabet, which is also a typical featurein traditional symbolic time series analysis.

4 Conclusions

As it has already been known from recent studies, recurrence plot based methods give morereliable estimates of dynamic invariants such as entropies than approaches based on a symbolicencoding [39]. In addition, recurrence plots may have a higher computational efficiency, however,this advantage becomes gradually less important as the length of the considered time series N increases. The results of our presented study particularly underline that the quantication of

recurrence plots is mainly suited for characterising the short-term dynamics of time series,whereas the detection of long-term variations requires either considerable computational effortsor an aggregation of the data before the analysis.

By combining the concept of recurrence plots with an appropriate symbolic encoding of thedata, the dynamic information can be reduced to its practically relevant part, which allowsa specic estimation of dynamically relevant quantities on different time scales. In particular,we have introduced the new concept of symbolic recurrence plots which may be useful for avariety of applications, including the treatment of discrete-valued and aggregated time series.The examples presented in this study illustrate the possible potentials of the new approach,however, a systematic comparison between standard and symbolic recurrence quanticationanalysis has been beyond the scope of the presented work. A more detailed comparison of thedifferent estimates of dynamic invariants (such as entropies) obtained from symbolic time series


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Recurrence Plots 103

analysis as well as standard and symbolic recurrence quantication analysis will be a subjectof future studies. In a similar way, we outline a straightforward extension of our considerationsto the case of bi- and multivariate time series.

In the problem setting considered in this paper, we have evaluated the potential use of sym-

bolic recurrence plots for the analysis of discrete-valued data, whose occurrence is by far notrestricted to the elds of production and logistics. With respect to the dynamics of manufac-turing networks, in future studies, a more systematic application of the described methods tosimulated as well as real-world observational time series of logistic observables (incorporatingdifferent network topologies, production strategies, order policies, and production parameters)may help to quantitatively evaluate potential mechanisms of instability, which affect the dynam-ics of the production processes. In this respect, our approach offers a novel tool for anticipatingthe complexity of temporal variations of logistic observables in such systems, which is a majorstep forward towards a more efficient and robust control in real-world systems.

As another possible closely related eld of applications, we would like to point out that thematerial ows in manufacturing systems have strong structural similarities to those in trafficnetworks. Indeed, recurrence plots have already been applied to study the short-term dynamicsof vehicular traffic [6163]. One has to mention that the corresponding applications have howeverconsidered ow rates (which can be assumed to be continuous) instead of discrete-valued single-vehicle counts. Our new approach of symbolic recurrence plots offers a possibility for moredetailed investigations of such discrete-valued time series from traffic ows and therefore toobtain results that go beyond those of recent studies.

This work has been supported by the German Research Foundation (projects no. He 2789/8-1 andScho 540/15-1), the Gottlieb Daimler and Karl Benz foundation, and the Volkswagen foundation(project no. I/82697). For the recurrence quantication analysis, we have used the MATLAB crossrecurrence plot toolbox and the commandline recurrence plots software provided by Norbert Marwan(http://www.agnld.uni-potsdam.de/ ~ marwan ). We thank N. Marwan, M. Thiel, M.C. Romano andK. Padberg as well as three anonymous referees of an earlier version of this paper for their helpfulcomments, and M. Winkelmann and C. Schicht for their assistance in systematically classifying thedynamic behaviour of the considered model system.

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