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On the Use of Transfer Entropy to Investigate the Time Horizon of Causal Influences Between Signals

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On the Use of Transfer Entropy to Investigate the Time Horizon of Causal Influences Between Signals entropy Article On the Use of Transfer Entropy to Investigate the Time Horizon of Causal Influences between Signals Andrea Murari 1, Michele Lungaroni 2 ID , Emmanuele Peluso 2,*, Pasquale Gaudio 2 ID , Ernesto Lerche 3,4, Luca Garzotti 3, Michela Gelfusa 2 ID and JET Contributors † 1 Consorzio RFX (CNR, ENEA, INFN, Universita’ di Padova, Acciaierie Venete SpA), I-35127 Padova, Italy; [email protected] 2 Associazione EUROfusion—University of Rome “Tor Vergata”, Via Orazio Raimondo, 18, 00173 Roma, Italy; [email protected] (M.L.); [email protected] (P.G.); [email protected] (M.G.) 3 EUROfusion Programme Management Unit, JET, Culham Science Centre, Abingdon OX14 3DB, UK; [email protected] (E.L.); [email protected] (L.G.) 4 LPP-ERM/KMS, Association EUROFUSION-Belgian State, TEC partner, Brussels 1000, Belgium * Correspondence: [email protected]; Tel.: +39-06-7259-7196 † See the author list of “X. Litaudon et al., 2017 Nucl. Fusion 57 102001. Received: 28 June 2018; Accepted: 15 August 2018; Published: 22 August 2018 Abstract: Understanding the details of the correlation between time series is an essential step on the route to assessing the causal relation between systems. Traditional statistical indicators, such as the Pearson correlation coefficient and the mutual information, have some significant limitations. More recently, transfer entropy has been proposed as a powerful tool to understand the flow of information between signals. In this paper, the comparative advantages of transfer entropy, for determining the time horizon of causal influence, are illustrated with the help of synthetic data. The technique has been specifically revised for the analysis of synchronization experiments. The investigation of experimental data from thermonuclear plasma diagnostics proves the potential and limitations of the developed approach. Keywords: transfer entropy; mutual information; Pearson correlation coefficient; time series; causality detection; sawteeth; pacing; ELMs; pellets 1. Detecting Causal Influence between Time Series In many fields of science, the causal relations between systems have to be investigated. Very often, the information about the phenomena, whose causal mutual influence is to be studied, is available in the form of time series. Of course, a general definition of causality is not available. On the other hand, in recent years, the concept of information flow has been used as an indicator of causal influence between systems [1]. In this framework, the phenomena under study are represented by stochastic variables and the information flow between them is quantified with tools, such as the Pearson correlation coefficient (Pcc) and the mutual information (MI). More recently a quite powerful tool, transfer entropy (TE), has been introduced [2] and its use is spreading quite fast, complementing or superseding alternative approaches such as Granger causality (GC) [3]. In this paper, the advantages and limitations of transfer entropy, in the analysis of the coupling between time series, are illustrated with some numerical examples. Then the issue of determining the time horizon in synchronization experiments is addressed. In particular, the coupling in pacing experiments is investigated with specific attention to real life examples from high temperature plasma physics. Entropy 2018, 20, 627; doi:10.3390/e20090627 www.mdpi.com/journal/entropy Entropy 2018, 20, 627 2 of 14 Our human environment is full of oscillating systems, both natural and man-made. Electromagnetic waves, circulation systems in mammals, and some chemical reactions all proceed at a periodic or quasi-periodic, oscillating pace. Most of these systems are not isolated and, sometimes, even minor interactions can affect their rhythms and bring them in synchrony. Recently, synchronization by external forcing has attracted a lot of attention as a method to influence delicate systems. In some cases, the best way to control the dynamics is some sort of soft interaction, to bring the target system in synchrony with some external oscillation. The control of chaotic systems is a good example. In some areas, such as thermonuclear fusion, the assessment of the actual effectiveness of an external perturbation is quite complex. Poor accessibility for measurements is a typical issue. Additionally, in some cases, there is a need to influence quasiperiodic systems whose behaviour needs to be only slightly changed. The degree of influence actually exerted by the external driver can therefore be quite delicate to assess. In some fields, physics models of the dynamics are not available and, therefore, the quantification of the driver efficiency can be performed only in a statistical way, over many examples on not on an individual basis. In particular, it can become essential to determine the “time horizon” of the causal influence, within which the driving system actually exerts an effect on the target system. This aspect is crucial to quantify the performance of the driving schemes, because synchronization beyond that time horizon has to be considered the result of random coincidences (see Section4). In this perspective, the power of TE can be essential. The paper is structured as follows. The definition of transfer entropy is introduced in the next section, together with the Pearson correlation coefficient and the mutual information. The added value of transfer entropy and some of its deficiencies are illustrated with the help of numerical examples in Section3, focussing on the determination of the causal influence time horizon. Examples of the application to synchronization experiments in tokamaks are presented in Section4. Discussion and future developments are the subject of the last section. 2. Pearson Correlation Coefficient, Mutual Information, and Transfer Entropy This section provides and introduction to the concept of transfer entropy, in the context of the other main statistical indicators, normally deployed to investigate the correlation between time series. The Pearson coefficient r is an indicator typically used to quantify the linear relation between two stochastic variables, X and Y. The mathematical definition of the Pearson coefficient r is: E[(X − mX)(Y − mY)] rX,Y = (1) sXsY where mX is the mean of the variable X, mY is the mean of the variable Y, sX is the standard deviation of X and sY is the standard deviation of Y. The correlation coefficient can range in value between −1 and +1 and reveals two aspects about the linear relation between variables: its strength and direction. The direction is determined by the sign, which indicates whether the variables are positive correlated or anti-correlated. With the appropriate reservations, the sign of the Pearson coefficient r can be interpreted as the direction of causal influence. The main limitation of the Pearson coefficient is its linear character. A traditional alternative quantifier, to take into consideration also the nonlinear influence between variables, is the mutual information. Intuitively, the mutual information I(X;Y) measures the quantity of information that the two variables X and Y have in common. Probably the best starting point, to understand the meaning of the mutual information and then of the transfer entropy, is the Kullback–Leibler Divergence (DKL). The DKL is an integral, which quantifies the difference between two probability distribution functions p(x) and q(x). The DKL is defined as: Z p(x) D (PjjQ) = p(x)· ln dx (2) KL q(x) Entropy 2018, 20, 627 3 of 14 The Kullback–Leibler Divergence assumes positive values and is zero only when the two probability distribution functions (pdfs), p and q, are exactly the same. Therefore, the smaller the DKL is, the more similar the two pdfs are. In our application, p and q are the pdfs of the time series X and Y. The mutual information is meant to quantify the information that two random variables Y and X have in common and it can be expressed in a form interpretable in terms of the DKL. Indeed, a typical definition of the mutual information is: p(x, y) I(X; Y ) = p(x, y) log (3) ∑ ∑ ( ) ( ) y2Y x2X p x p y where p(x,y) is the joint probability function of X and Y, p(x) and p(y) are the marginal probability distribution functions of X and Y, respectively. Thus, the higher the mutual information, the more different their joint pdf is from the case of independence, when the joint pdf factorizes into the product of the two individual pdfs. MI includes nonlinear interactions but is affected by the obvious limitation that it cannot provide any information about the directionality of the causal relation. Moreover, the MI does not take into account the memory effects of the signals (see the following). Almost two decades ago, the notion of transfer entropy was formalised, to provide a new and more powerful tool to investigate causal effects [2]. The main motivation, behind the development of the TE formalism, resides in the observation that the lagged mutual information I(Yt;Xt−s) is not adequate to properly quantify the strength of the causality between the time series X and Y. The lagged mutual information is unsatisfactory because it does not take into account the target history. Basically I(Yt;Xt−s) cannot discriminate the actual mutual causal effect, between the signals of interest, from the memory of the system Y on itself. Transfer entropy has been specifically developed to take into account the effect of the history of the signals. In this context, its most useful definition is the following: n m n m p(yt+1jyt , xt ) TX!Y = ∑ p(yt+1, yt , xt ) log n (4) n m p(yt+1jyt ) yt+1,yt ,xt n m In Equation (4), yt = (yt, yt−1, ... , yt−n+1) and xt = (xt, xt−1, ... , xt−m+1) are the n and m orders of the Markov processes X and Y. TE can, therefore, be interpreted as the degree of uncertainty in the present Y given the past state of Y and X, in excess of the degree of uncertainties resolved by Y’s own past.
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