Keywords

1 Introduction

A complex system is any system consisting of a great number of heterogeneous entities interacting with each other within an environment to shape an emerging behavior. Such emerging behaviour depends on a non-trivial space of correlations that derive from the interplay of agents entangled in loops of non-linear interactions. In the metaphor of the flock of starlings, any environmental change perceived by the starlings during their flight is visible in the formation of the flock shape due to their reaction. This implies that there is an underlying feedback loop between the agents and the global system.

Mastering the complexity of these systems has always been a challenge in almost every branch of science. In computer science, process-, actor- and agent-based models and languages have been developed for describing the behaviour of complex software systems [1, 2, 6, 11]. All these approaches require an a priori knowledge of the basic rules governing the dynamic of the system in order to define the behaviour of the components and of the environment. Unfortunately, for natural or social phenomena, it is quite impossible to have enough knowledge about the real interaction rules. However, global information about the system is hidden inside phenomenological data produced by the individual components. Thus, we need suitable methods to extract a specific model of interest.

Topological Data Analysis (TDA) is a relatively new field of study in which topology driven methods are used to analyse big collections of data [4, 5, 8]. The S[B] paradigm is a general framework of modelling in which a complex system is described as a pair of entangled systems: S, the global environment, and B, a set of interactive agents [13, 14]. Persistent Entropy (PE) is a Shannon-like entropic measure able to describe the global dynamics of a complex system [16]. PE has been used for studying complex phenomena in different fields [15,16,17]. As shown in [16], by analysing the trajectories of PE and its derived quantities, an automaton, called Persistent Entropy Automaton (PEA), which models the global dynamics of the system under study, can be manually devised.

The global financial system is one of the most important, human-made, complex systems. This system is composed of multiple interacting autonomous components or “selfish” agents, who - very often - act for their own benefit, and of complex interactions among those components. Each component behaves according to his/its own strategies, under the influence of the environment and interacting with other heterogeneous components. Classical tools for analysing and modelling such systems operate under a range of rather unrealistic assumptions. For example, interactions are normally abstracted with equations: this implies that the system reaches the equilibrium through non-linear optimisation methods rather than emerging from the agents interactions [12].

In this paper we use TDA to construct a data space from the components of the Dow Jones stock market index. The considered data set is the time series of the daily log-returns of Dow Jones’ components from 1987 to 2017. From the data space, we calculate the PE and we devise a PEA whose locations model global states of the stock market. We show that early warnings about the emergence of the already occurred financial crisis can be identified by the PEA.

2 Methods

TDA employs concepts and principles of the field of computational topology to reveal higher dimensional patterns hidden in big data sets [5, 8]. Computational topology studies invariants of shapes among which the so called Betti numbers, or barcodes, that characterise the existence of n-dimensional holes in the topological data space. TDA builds a discrete topological space, a simplicial complex, following a filtration procedure. In this work we use the Vietoris-Rips complex filtration that works on point clouds [8]. At each step of the filtration the persistent homology is computed yielding a collection of barcodes that indicate the life span of the topological invariants. PE is then computed from the barcodes [15].

The 24 time series of the considered Dow Jones components were mapped into a point cloud using a sliding window of 50 days and scrolling one point at a time with superposition of 49 points. Each window then produced 50 points in \(\mathbb {R}^{24}\). This technique has been demonstrate suitable for studying the time-varying properties of systems similar to the one we are studying [9, 15].

The mechanism driving critical transitions in complex systems is called tipping point, which is an abrupt qualitative change in the behaviour of a dynamical system when one or more control parameters change. In approaching a tipping point, a complex system shows a phenomenon called Critical Slowing Down (CSD), which can be considered an Early Warnings Signal (EWS) for the critical transition [3, 18]. Since PE describes a system globally, it contains a summary of the knowledge about the system. Moreover, it can be considered a time series itself and can be calculated for all the dimensions. Thus, we study the total PE time series (\(\mathrm {PE}_\mathrm {tot}\)) - calculated as the sum of PEs for all the dimensions - with an analytical approach. The goal is to identify EWSs about a crisis by detecting the occurrence of tipping points. The obtained \(\mathrm {PE}_\mathrm {tot}\) is shown in Fig. 1.

To delimit the CSD areas we used an adaptation of the \(W_2\) index, i.e. a combination of statistical indices (coefficient of variation, 1-lag autocorrelation, and kurtosis), described in [7]. \(W_2\) is computed from \(\mathrm {PE}_\mathrm {tot}\) with the R package “earlywarnings” using another sliding window of size 450. Thus, \(W_2\) is another time series and it is plotted along with its running average and \(2\sigma \) confidence bands in Fig. 1. Potential areas of CSD are identified by finding points where \(W_2 > \overline{W_2}+2\sigma \) [7]. These areas are shown in Fig. 1 with black bands and represent the EWSs in our system.

Identified CSD areas can be used to define PEA states. A PEA monitors the PE and derived functions w. r. t. equilibrium conditions that define its states [16]. It remains in a state s as long as the associated equilibrium condition ec(s) is satisfied. When it is violated, the PEA exits s and starts a non-instantaneous transition, which can be seen as an adaptation phase. This adaptation may end into an adjacent PEA state \(s'\) as soon as \(ec(s')\) is satisfied or may not terminate. This definition is based on the fact that PEA states are devised from the observed trajectories of PE and derived functions. Indeed, it may happen that the monitored functions exhibit evolutions that were not identified as equilibrium conditions. This is expected for natural complex systems for which “complete” models can not be established. The main difference between a PEA and a hybrid automaton, which is a top-down defined model not considering unknown evolutions [10], is essentially in this different perspective.

Fig. 1.
figure 1

Plot of \(\mathrm {PE}_\mathrm {tot}\) and its running average (above). Plot of \(W_2\) and its running average with confidence bands (below). The grey and black vertical bands correspond to the relative states of the automaton in Fig. 2. The thin coloured vertical bands correspond to financial crises. (Color figure online)

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3 Persistent Entropy Automaton of Dow Jones

In the S[B] paradigm the structural level S is a model of the global dynamics of the system and the behavioural level B is a model of the local interactions among the entities of the system [13]. In this study, the behavioural level B is represented by real human agents that produced the data that we use. The structural level S is defined as a PEA in the following.

We consider two financial crises, dot-Com and Lehman Brothers Crash, both represented with coloured bands in Fig. 1. We devise the PEA by monitoring the functions \(\mathrm {PE}_\mathrm {tot}(t)\), \(W_2(t)\), their running average \(\overline{\mathrm {PE}_\mathrm {tot}}(t)\), \(\overline{W_2}(t)\) together with their derivatives indicated with a dot over the symbol. Each discrete instant t corresponds to one day observation. The derived PEA is called \(\mathrm {PEA}_{W_2}\), is depicted in Fig. 2 and its states are described in the following. A stable state is characterised by the equilibrium condition \(|W_2-\overline{W_2}|<2\sigma \), that is \(W_2\) does not exceed the confidence bands. The state called \(\mathrm {Stable}\) is the initial one and holds this condition. As soon as the functions violate the stable condition, \(\mathrm {PEA}_{W_2}\) exits state \(\mathrm {Stable}\) and starts an adaptation. The only state in which the adaptation can end is the one called \(\mathrm {Grey}\), a state indicating that there was in the past at least one violation of the stable state condition. The equilibrium condition of \(\mathrm {Grey}\) is \(\dot{\overline{W_2}}\approx 0 \wedge \dot{\overline{\mathrm {PE}_\mathrm {tot}}}<0 \wedge W_2 < \overline{W_2}- 2\sigma \), which means that the running average of \(W_2\) has minimal oscillations, the running average of the total PE is decreasing and \(W_2\) exits the confidence band \(-2 \sigma \). Visually, the periods in which \(\mathrm {PEA}_{W_2}\) stays in this state are represented by the grey bands in Fig. 1. State \(\mathrm {White}\) corresponds visually to the period after a grey band, it has the stable equilibrium condition and records the fact that the system entered at least once state \(\mathrm {Grey}\). After \(\mathrm {White}\), another grey band can occur (in this case the PEA goes back to \(\mathrm {Grey}\)) or a black band occurs. A black band corresponds to state \(\mathrm {Black}\). This is the early warning state because from \(\mathrm {Black}\) the system can only evolve to state \(\mathrm {Tipping}\) that represents a tipping point. In \(\mathrm {Tipping}\) a crisis, represented by the dashed transition towards state \(\mathrm {Stable}\), can occur or the system can return to state \(\mathrm {Grey}\). The dashed transition can be interpreted as the occurrence of a phase transition of the system.

\(\mathrm {PEA}_{W_2}\) is then able to give a warning, in state \(\mathrm {Black}\), that a crisis may occur, without giving a prediction. However, if the current state is different from \(\mathrm {Black}\), the model says that a crisis cannot occur immediately: there must be at least one (or more) adaptations before the tipping point state is reached.

Fig. 2.
figure 2

\(\mathrm {PEA}_{W_2}\), describing the transitions among the global states of the system.

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4 Conclusions

In this work we have modelled the global dynamics of a complex system by manually devising a PEA. TDA has been used for analysing the phenomenological data of the system and PE has been calculated from a topological space derived from a data set. The application domain is the Dow Jones stock market. The derived automaton models the global behaviour of the market and is able to recognise a tipping point state in which a financial crisis may occur and previous states in which there is some degree of warning but there is not an immediate alarm because some other adaptations are required to reach the tipping point state. The transition that goes from the tipping point state to the stable state can be interpreted as a phase transition of the system.

Despite the encouraging result we are aware that the proposed analysis presents some limitations: one is about the peculiarity of the data set that does not allow one to set up a statistical validation of the results because of the unicity of the phenomenon under study, for which other instances do not exist. Another regards the computation of the indices for deriving the \(W_2\), for which it is necessary to try different lengths of the sliding windows.