Peter Martey Addo
email: peter.addo@univ-paris1.fr

Philippe De Peretti

Hayette Gatfaoui

Jakob Runge

The kiss of information theory that captures systemic risk

Keywords: Systemic risk, Causal dependencies, Financial institutions, Linkages, Sovereign Debt JEL: C40, C32, C51, G12, G29

We provide a new approach to understanding systemic risk by analysing complex linkages in finance and insurance sectors. The analysis is achieved by using a recently proposed method for quantifying causal coupling strength, which identifies the existence of causal dependencies between two components of a multivariate time series and assesses the strength of their association by defining a meaningful coupling strength. The measure of association is general, causal and lag-specific, reflecting a well interpretable notion of coupling strength and is practically computable. A comprehensive analysis of the feasibility of this approach is provided via simulated and real data.

Introduction

The recent financial crisis of 2007-2009 has led policymakers to search for new tools to better understand systemic risk, i.e. contagion, and to try to develop early warning indicators. As the events following the turmoil in financial markets unfolded, it became evident that modern financial systems exhibit a high degree of interdependence making it difficult in predicting the consequences of such an intertwined system. To understand the interdependence of the system taken as a whole entity, network analysis has emerged as a leading econometric tool. For instance, the nature of connections between financial institutions usually results from both the asset and the liability side of their balance sheet. The structure of such connections between financial institutions can be captured by employing a network representation of financial systems (Allen and Babus (2008); [START_REF] Addo | Understanding exchange rate dynamics[END_REF]; Bisias et al. (2012)).

Network analysis is advancing as a common tool for assessing dynamics within the various sectors (nodes) of the financial system; it reveals that a truly systemic perspective needs to combine the focus on various sections of the financial sector with an analysis of the interlinkages among them incorporating the interaction with the real economy (Bisias et al. (2012)). Intuitively, a network structure describes a collection of nodes and the links between them [START_REF] Lauritzen | Graphical models[END_REF]; Dahlhaus (2000); Eichler (2012)). The nodes can be individuals, institutions or countries, financial assets, or even collections of such entities. The direct relations between these entities of interest are usually referred to as links. These links could be directed or undirected. Network theories can be used to monitor and assess systemic risks, contagion, linkages and vulnerabilities in the financial system. The nature of networks permit us to picture beyond the immediate "point of impact" of a shock and enhances the analysis of spillovers likely to arise from interlinkages. Thus, network perspective would not only account for the various connections that exist within the financial sector or between financial sector and other sectors, but would also consider the quality, in terms of strength, of these links.

Recent works documenting the applications of network analysis to financial systems concentrates on issues such as financial stability and contagion. The literature mostly investigates how financial institutions (or entities) are interconnected and how different network structures react to the breakdown of a single (or an ensemble) institution in order to identify which ones are more fragile (Allen and Babus (2008); [START_REF] Addo | Understanding exchange rate dynamics[END_REF]). To our best of knowledge, existing research focused on the network topology of asset returns of financial systems are the works of [START_REF] Addo | Understanding exchange rate dynamics[END_REF] and Bisias et al. (2012). Most of these works that study interconnections between subprocess of multivariate process make use of lagged cross-correlation and regression analysis. These concepts are commonly used to gain insights into interaction mechanisms between multivariate processes, in particular to assess time delays and to quantify the strength of a connection. For instance, the noteworthy work of [START_REF] Addo | Understanding exchange rate dynamics[END_REF] aims to capture the interconnectedness based on principal components analysis and Granger-causality tests. A drawback of these concepts related to Granger causality network is that a strong dependence on the influence of serial dependencies or autocorrelation is usually observe, which can lead to misleading conclusions about time delays and also obscures a quantification of the interconnections. The question that still stands is how do we create a formal measure of systemic risk that adequately captures complex linkages in the financial system?

Our main aim is to offer a novel information theoretic approach to understanding systemic risk by identify linkages, and assess the strength of such connections. Our analysis exploits the concept of quantifying causal coupling strength, which identifies the existence of causal associations between two components of a multivariate time series and assesses the strength of their association by defining a meaningful coupling strength using time series graphs and information theory. In otherwords, we offer an approach which is based on time series graphs and information theory, which is able to detect and quantify causal dependencies from multivariate time series. The measure introduced in this work gives similar scores to equally noisy dependencies, which is essential for comparisions and ranking of strength of dependencies within subprocesses of the multivariate time series. Unlike network of Granger-causal relations proposed in Billio et al. (2012), we provide a measure that is lag-specific and gives non-zero value only to the dependency between lagged components of a multivariate process that are not independent conditional on the remaining process. The advantage of our approach over [START_REF] Addo | Understanding exchange rate dynamics[END_REF] is that the study of interconnectedness is uniquely determined by the interaction of the two components alone and in a way autonomous of their interaction with the remaining process since the misleading influence of autodependency within a process is excluded. In addition, our approach only require that the multivariate time series be stationary. This measure has been successfully use in quantifying the strength and delay of climatic interactions (Runge et al. (2014)). In this work, we provide a comprehensive feasibility of the use of information theory and time series graphs 2 Documents de Travail du Centre d'Economie de la Sorbonne -2014.69R (Version révisée)

to capture complex linkages in financial institutions.

Our first empirical illustration of the approach considers the dataset used in [START_REF] Addo | Understanding exchange rate dynamics[END_REF], which corresponds to monthly asset-and market-capitalization-weighted return indexes of hedge funds, brokers/dealers, banks, and insurers. These financial institutions represents the financial and insurance sectors as detailed in [START_REF] Addo | Understanding exchange rate dynamics[END_REF]. Our empirical findings show that there is high coupling strength of linkages between the four sectors, increasing the channels through which shocks can be transmitted throughout the financial system. The results point to important asymmetry in the causal dependencies in the insurance sector and brokers prior to the financial crisis of 2007-2009. Considering the whole time period, we observe that most directed links observed originates from insurers (IN) and in this case a failure of the insurers could cause a significant impact in the financial system. Moreover, the given the high coupling strength between banks and insurers, a failure in the banking system could significantly unstabilize the financial system. Based on our findings, we claim that the banks were more central to systemic risk. In addition, due to the high coupling strength of the contemporaneous links between the three sectors: banks (BK), Broker/dealers (PB) and insurers (IN), any large shock in one will spread quickly through the system. We find that the returns of insurers have significant impact on returns of brokers, where the time delay of impact is shorter during the crisis. Over time, we find that banks were always contemporaneously linked to brokers with a higher coupling strength observed before year 2000. In addition, our results indicate that insurers did not, in many cases, directly impact hedge funds significantly. After year 2000, we find contemporaneous links between banks and insurers, and also between brokers and hedge funds, where the former exhibits a higher coupling strength. The aftermath of 1998 collapse of the $5 billion hedge fund Long Term Capital Management (LTCM) could have lead to the contemporaneous link between brokers and hedge funds. In general, we find that banks were not directly connected to hedge funds. However, the two sectors were indirectly linked via a contemporaneous chain of dependencies from banks to brokers and then to hedge funds. We remark that just prior to the financial crisis of 2007-2009, the hedge funds did have a significant negative impact on banks. Interestingly, most connections were observe for the time period (1994M01-2007M12), where hedge funds had a negative impact on the financial system especially on the banks and brokers. This increase in size and causal dependencies between these sectors could serve as a significant systemic risk indicator.

In the second empirical application, we focus on investigating the transmissions of the European Sovereign debt crisis by examining the behavior of the 10-year government bond yields of eleven European Monetary Union (EMU 11) countries. The EMU 11 considered in our application is composed of the periphery group of countries: Portugal (Por), Ireland (Ire), Italy (Ita), Greece (Gre), Spain (Spa); and the central group of countries: Austria (AU), Germany (Ger), The Netherlands (Ne), Belgium (Bel), France (Fra), and Finland (Fin). Our findings shows an increase in the number of causal dependencies in the bond yields among the countries prior and even much more during the crisis. We remark that the crisis were marked by an increase in negative coupling strengths among sovereigns, which could serve as an indicator of systemic risk. The finding shows that government bond yield of the Netherlands were not linked to any of the periphery countries severely hit by the sovereign debt crisis. In particular, we find that the sovereign bond yields of Germany, the Netherlands, Finland and Austria were more stable compared to other core countries like France and Belgium, which were linked to at least two periphery countries. Our results point to France not been severely hit by the sovereign debt crisis, could be due to its high causal dependence with Germany, the Netherlands, Austria and Finland, although France did have a contagion spillover effects mainly from Ireland and Italy. Based on 3 Documents de Travail du Centre d'Economie de la Sorbonne -2014.69R (Version révisée)

our findings, the European Monetary Union was still in crisis as at the end of 2014. In order to safeguard the stability of the european financial system, we suggest that much importance be placed on reducing these causal dependencies and contagion among member countries. The paper is organised as follows: In Section 2, an overview of time series graphs and information theoretic measure is provided. An outline of the estimation procedure of the approach is also provided for practical purposes. In Section 3 & Section 4, a comprehensive analysis of the feasibility of this approach to simulated and real data is provided. Section 5 concludes.

Measure of causal dependence and coupling strength

In this section, we provide an overview of an approach which makes use of time series graphs and information theory. We will start by providing an insight to Granger-causality [START_REF] Granger | Investigating causal relations by econometric models and cross-spectral methods[END_REF]), the definition of time series graph and then the information theoretic measure. Let A and B be stationary stochastic processes, U - t = (U t-1 , U t-2 , • • • ) be the information in the entire universe until the present. Now let A - t = (A t-1 , A t-2 , • • • ) denotes the past until present and σ 2 (•) denote variance.

Definition 1 (Granger Causality). We say that A Granger-causes B if

σ 2 (B|U -) < σ 2 (B|U -\A -).
This implies that the error variance of predicting the process B given U - t is less than the error variance of predicting B when A -is excluded [START_REF] Granger | Investigating causal relations by econometric models and cross-spectral methods[END_REF]). We remark that although the prediction method to determine σ 2 could be linear or nonlinear, the use of σ 2 restricts the notion of causality and does not takes into account of the whole distribution of the error. As such, the need for information theory framework since it does considers the whole distribution of the residual (Runge et al. (2012a)).

Time series graphs

Like graphical models [START_REF] Lauritzen | Graphical models[END_REF]), time series graphs are based on the concept of conditional independence and the time-dependence in the time series is used to define directional links in the graph.

Definition 2 (Time Series Graph). Let X be a stationary multivariate time series with a set of subprocesses V at each time t ∈ Z and directional links be defined in E. Then

G = (V × Z, E)
is the time series graph of X, where the set of nodes in the graph are made up of V.

Here each node in the graph represents a single random variable i.e. a subprocess at a certain time t. Given a stationary multivariate discrete-time series stochastic process X, where X - t = (X t-1 , X t-2 , • • • ) denotes the past until present. We say that the nodes A t-τ ∈ G and B t ∈ G are connected by a lag-specific directed link "A t -→ τ B t " pointing forward in time if and only if τ > 0 and

I A link -→B (τ) ≡ I A t-τ ; B t X -\{A t-τ } > 0.
(1)

Thus, A t-τ and B t are connected if they are not independent conditionally on the past of the whole process excluding {A t-τ } (denoted by the symbol \) which implies a lag-specific causality with respect to X.

I A link -B ≡ I A t ; B t X - t+1 \{A t , B t } > 0. ( 2
)
Notice that the contemporaneous present

X t \{A t , B t } is included in the condition. Thus X t \{A t , B t } ⊂ X - t+1 \{A t , B t }.
In the case of the multivariate autoregressive process, this corresponds to non-zero entries in the inverse covariance matrix of the innovation terms. The equations 1 & 2 involve infinite dimensional vectors and can not be directly computed. However, using the Markov property, this issue can be circumvented by introducing the notion of parents and neighbors of subprocesses. Let

P B t ≡ {A t-τ : A ∈ X, τ > 0, A t-τ -→ B t } (3)
be a finite set of parents of B t which separates B t from the past of the whole process X - t \P B . The past lags of the subprocess B in X can be part of the parents. The neighbors of a process B t is defined as

N B t ≡ {A t : A ∈ X, A t -B t } (4)
The parents of all subprocesses in X together with the contemporaneous links forms the time series graph. Any node B t ∈ G in the time series graph is conditionally independent1 of X - t \P B t given its parents P B t .

It is noteworthy that the measure of connectedness based on Granger causality as proposed Billio et al. ( 2012) is similar to the information theory formulation in equations 1 & 2. However, the drawback here is that it is not uniquely determined by the interactions of the two components alone and does not exclude the misleading influence of autodependencies within processes. In the section that follows, we define the information theoretic measure which is lag-specific, and able to detect and quantify causal relationships between two subprocesses in a way autonomous of their interaction with the remaining process.

Information theoretic measure

In line with our objective to study the channels through which shocks can be transmitted between financial institutions, we provide in this section an information theoretic measure of causal dependencies and coupling strength of such causal relationships. The definitions of links based on information theoretic together with time series graph will be used to capture complex linkages in the financial system. Definition 3 (Momentary Information Transfer (MIT) Links). For two subprocesses A, B of a stationary multivariate discrete time process X with parents P A t and P B t in the associated time series graph and τ > 0, the general information theoretic measure MIT (Runge et al. (2012a)) between A t-τ and B t is given by

I A MIT -→B (τ) ≡ I A t-τ ; B t P B t \{A t-τ }, P A t-τ > 0 = H B t P B t \{A t-τ }, P A t-τ -H B t P B t
and contemporaneous MIT defined by

I A MIT -B ≡ I A t ; B t P B t , P A t , N A t \{B t }, N B t \{A t }, P N At \{B t } , P N Bt \{A t } .
(

) 5
where H(X) is Shannon's entropy and H(X|Y) denotes conditional Shannon's entropy (Shannon (1948)).

Definition 4 (MIT Coupling strength). The partial correlation MIT measure, denoted ρ MIT , associated with equation ( 5), for the strength of coupling mechanism between A t-τ and B t is given by

ρ A MIT -→B (τ) ≡ ρ A t-τ ; B t P B t \{A t-τ }, P A t-τ . (6)
The measure ρ MIT quantifies how much the variability in A at the exact lag τ directly influences B, irrespective of the past of A t-τ and B t . ρ MIT is the cross-correlation of the residual after A t-τ and B t have been regressed on both the parents of A t-τ and B t . In economics, the detection of causal relationships among variables are of great importance [START_REF] Granger | Investigating causal relations by econometric models and cross-spectral methods[END_REF]; Billio et al. ( 2012)). These variables can be viewed as nodes of graph where the links correspond to interactions. Unlike classical statistics, interactions in the framework of information theory are viewed as transfers of information.

The partial correlation MIT is a time-delayed conditional mutual information and a measure of association that is general in the sense that it does not assume a certain model class underlying the process that generates the time series. This measure of association will be useful in understanding the interdependence in multivariate economic and financial time series analysis as the process underlying the generated signals are a priori unknown [START_REF] Addo | Understanding exchange rate dynamics[END_REF]). In addition, the general framework of graphical models2 [START_REF] Lauritzen | Graphical models[END_REF]; Dahlhaus (2000); Eichler ( 2012)) makes MIT causal as it gives a non-zero value only to lagged components that are not independent conditional on the remaining process. This measure in many cases is able to exclude the misleading influence of autodependency within a process in an information-theoretic way. Runge et al. (2014) successfully used this approach in quantifying the strength and delay of climatic interactions. In this work, we will make use of the partial correlation MIT in quantifying the strength and delay of interactions between multivariate economic and financial time series. As documented in literature (Runge et al. (2012a(Runge et al. ( , 2014))), measures of multivariate dependence should usually fulfill the following properties:

1. Causality -the measure should give non-zero value only to the dependency between lagged components of a multivariate process that are not independent conditional on the remaining process.

2. Lag-specific -it should quantify dependence between lagged time series. This enhances better interpretation. 3. Coupling strength autonomy -given dependent components, the measure should provide a causal notion of coupling strength that is well interpretable. In otherwords, it is uniquely determined by the interaction of the two components alone and in a way autonomous of their interaction with the remaining process. 4. Practical computability -estimation does not for instance require somewhat arbitrary truncation as in the case of other methods like transfer entropy.

These properties are well filfulled by the MIT measure, which allows to reconstruct interaction networks where not only the links are causal, but also meaningfully weighted and have the attribute of coupling delayed. To our best of knowledge, other methods such as the transfer entropy which is the information-theoretic analogue of Granger causality do not support these properties.

For instance, the transfer entropy is not uniquely determined by the interaction of the two components alone and depends on misleading effects such as autodependency and interaction with other process. This voilates the coupling strength autonomy. In addition, aside that the transfer entropy requires arbitrary truncation during estimation, this measure can lead to false interpretation since it is not lag-specific. To enhance proper interpretation of results due to links in the time series graph, caution should be taken when interpreting contemporaneous links as it is not clear which direction they correspond to or whether they are entirely due to a common driver.

It is worth pointing out that the analytical distribution of the partial is known and equivalent to the distribution of cross-correlation with the degree of freedom reduced by the cardinality of the set of conditions (Fisher (1924)). Thus, statistical inference on the measure ρ MIT is straightforward (Runge et al. ( 2014)). In the empirical application, Section (4), of the approach, we present results that corresponds to a 95% significance level of a two-sided test on ρ MIT .

Estimation

To apply the above methodology in practice, we outline a "two-steps" approach for the detection and quantification of causal dependencies in multivariate time series:

1. A modified PC algorithm (Spirtes et al. (2000); Runge et al. (2012a)) is used to estimate the parents of each process. This step can be viewed as a variable selection method. Unlike graphical models, only undirected links are inferred and the second step of the original PC algorithm is omitted (Spirtes et al. (2000)). This step determines the existence or absence of a link, which also provide useful information on the causality between lagged components of the multivariate process X. The estimation of these time series graphs is obtained by iteratively inferring parents (Runge et al. ( 2012b)).

2. Next, MIT link definitions are used and all possible links are tested again since some links might still turn out to be non-significant. In this step, the problem of serial dependencies is drastically reduced and the estimation of the coupling measure serves as a meaningful weight for every existing link in the graph (Runge et al. ( 2012a)).

In estimating the conditional mutual information (CMI), the most commonly used approach is the binning method. In this method, the joint and marginal distributions are estimated and then plugged into the entropy formula to compute the CMI. The estimation is achieved by binned histograms with equiquantile (or some pre-defined) bins such that the marginals distributions are uniform (Paluš (1996); Darbellay and Vajda (1999) ,a) can be used. The section that follows will be dedicated to the empirical illustration of the feasibility of the approach on simulated and real data.

Simulations experiment

In this section, we provide a simulation study on the identification of interdependences and strength in a simulated multivariate process with a priori knowledge on the network structure. In this respect, consider a simulated 1000 points of a stationary multivariate autoregressive process (displayed in Figure ( 1a)) made up of four subprocesses {X t , Y t , Z t , W t } defined by

X t = aX t-1 + cZ t-4 + ε x (7) Y t = kX t-1 + hY t-1 + ε y (8) Z t = dY t-2 + bZ t-1 + f W t-1 + ε z (9) W t = eY t-3 + gW t-1 + ε w (10)
and the innovation covariance matrix given by

Σ ε =               1 0 d 0 0 1 0 d d 0 1 0 0 d 0 1               , where a = 0.6, b = 0.4, c = 0.3, d = -0.3, e = -0.6, f = 0.2, g = 0.
4, h = 0.6 and k = 0.3. Notice that the lagged causal chain for this process is X -→ 1 Y -→ 2 Z with feedback Z -→ 4 X, and Y -→ 3 W -→ 1 Z, plus contemporaneous links X -Z and Y -W. The results of the time series graph via MIT approach is provided in Figures (1b & 1c). We obtained that the causal dependencies between the subprocesses are well identified and coincide with our a priori knowledge on the network structure. We remark that not only are the causal lags of dependencies between subprocesses identified, the magnitude and sign of the MIT coupling strength of linkages is also obtained (see Figure (1c)). In checking for robustness, we perform the analysis on sliding windows of length 200 and step size of 100. The results on the ensemble graph for windows are displayed in Figure (2). Here the link width denotes the percentage of windows that contain this link in the time series graphs. In otherwords, robust links have a larger link width. We observe that the priori knowledge on causal dependencies among these subprocesses are well captured by the approach.

In the section that follows, we will provide an application of this approach to analysing systemic risk.

Empirical application

In this section, we provide two empirical applications of the approach to the financial context. The first illustration will focus on studying linkages in four financial and insurance sectors of the economy as done in [START_REF] Addo | Understanding exchange rate dynamics[END_REF]. The second application explores causal linkages in eleven European Monetary Union (EMU) countries before and during the european sovereign debt crisis. In both illustrations, we apply the proposed methodology to full sample period, expanding windows and then on sliding windows for proper robust conclusions. (c) The plot of significant lags for simulated process associated with the MIT plot.

Figure 1: The detection & quantification of causal dependencies in a simulated multivariate autoregressive process described in equations (7-10) via the MIT approach. The results indicates a lagged causal chain as

X -→ 1 Y -→ 2 Z with feedback Z -→ 4 X, and Y -→ 3 W -→ 1 Z, plus contemporaneous links X -Z and Y -W. In Figure (1c)
, the existence of a dot at a particular lag τ of a subprocess ξ on the plots located on the diagonals indicates an autoregressive link of the form ξ t-τ -→ ξ t , which we denote as "ξ t -→ τ ξ t ". Contemporaneous links can be read from the plots by observing dots at where τ = 0. The vertical axes represent the sign and magnitude of the MIT coupling strength (ρ MIT ) of the linkages, which corresponds to the coefficients of the simulated autoregressive process.

.2 0.4 0.6 X → X X → X X → X X → X X → X X → X X → X X → X → Y → Y → Y → Y → Y → Y → Y → Y → Z → Z → Z → Z → Z → Z → Z → Z → W → W → W → W → W → W → W → W 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 Y Y Y Y Y Y Y Y 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 Z Z Z Z Z Z Z Z

Financial and insurance sector

The recent global financial crisis (2007)(2008)(2009) has clearly demonstrated the need for the financial industry, regulators and policymakers to develop a better understanding of systemic risk. In particular, measuring the financial systemic importance of financial institutions is crucial in identifying linkages and the potentially destabilizing constituents of the global financial system. In this section, we consider the dataset used in Billio et al. ( 2012), with special attention on identifying the linkages, lags of linkages and to assess the strength of such linkages among the four main sectors: hedge funds (HR), banks (BK), Broker/dealers (PB) and insurers (IN). These financial institutions represents the financial and insurance sectors described in more detail by [START_REF] Addo | Understanding exchange rate dynamics[END_REF]. In this respect, we make use of the monthly asset-and market-capitalizationweighted return indexes of these sectors as inputs for the MIT approach in analysing systemic risk. The time period for the data set in Jan 1994-Dec 2008. Our analysis is performed on the full sample period and also considers the following different time windows : 1994M01-2004M12, 1994M01-2005M12, 1994M01-2006M12, 1994M01-2007M12, to study possible changes in the systemic risk network structure.

The results of the proposed MIT approach when applied to the multivariate process {HF t , PB t , BK t , IN t } for the full time period is shown in Figure (3). Lag-specific causal dependencies between the four sectors can be easily visualized in Figures (3a & 3b). The sign and magnitude of the coupling strength can be read from The existence of a dot at a particular lag τ of sector ξ on the plots located on the diagonals indicates an autoregressive link of the form ξ t-τ -→ ξ t , which we denote as "ξ t -→ τ ξ t ".

links between the three sectors: banks (BK), Broker/dealers (PB) and insurers (IN), any large shock in one will spread quickly through the system.

We now provide results for four time periods: (1994M01-2004M12, 1994M01-2005M12, 1994M01-2006M12, 1994M01-2007M12), in Figures (4& 5). We make use of expanding window size to allow for the detection of changes in network structure across time. For all the time periods considered, we identified three contemporaneous links: BK -PB, PB -HF and BK -IN with the latter link having a higher coupling strength ρ MIT 0.65 as shown in Figure (5). Moreover, there exists a causal autodependency in insurers at lag τ = 5 (IN t -→ 5 IN t ) and a direct link IN t -→ 2 HF t with |ρ MIT | 0.25. We obtain that hedge funds were only linked with banks (HF t -→ 2 BK t ) prior (1994M01-2006M12,-2007M12) to and during the financial crisis of 2007-2009, which was not evident at the end of the full sample period. Notice that right before and during the financial crisis of 2007-2009 (1994M01-2007M12), the interlinkages between the sectors increased. One of the lagged causal chain for this sample period is given by HF t -→ 2 PB t -→ 5 BK t , HF t -→ 2 BK t , and IN t -→ 6 PB t . We find that the lag of the direct link between insurers and brokers observed for time period (1994M01-2006M12,-2007M12) changed from τ = 6 to τ = 1 at the end of full sample period. This means that it did take less time for shocks in the insurance sector to be transmitted to brokers during the crisis. We also find that time series graph was unchanged for the time period (1994M01-2004M12) and (1994M01-2005M12) as shown in Figures (4a& 4b) except for the exclusion of IN t -→ 6 PB t in the latter.

In order to account for the possibility of nonlinearity in the form of breaks or rare events in the multivariate signal, we make use of the unthresholded recurrence plot, which is well documented in literature [START_REF] Iwanski | Recurrence plots of experimental data: To embed or not to embed?[END_REF] 8). For the time period (1994M01-1998M08 and 1994M01-2000M03), we obtain the same network structure and coupling strengths. The time series graph for this period is composed of a high coupling strength of a contemporaneous link between between brokers and banks (ρ MIT 0.75), and IN t -→ 6 HF t (ρ MIT = -0.4). Finally, we perform analysis on the whole sample period over a sliding window of length 36 months and a step size of 12 months to check for robustness on the linkages among these four institutions. The results on the ensemble graph and significant lag plots is reported in Figure ( 9). This figure shows the overall significant causal dependence among the four institutions for all 13 Documents de Travail du Centre d'Economie de la Sorbonne -2014.69R (Version révisée) time series graphs over the sliding windows. Our analysis reveals three contemporaneous links: BK -PB, BK -IN and PB -HF, that were previous identified in Figures (4& 5). We remark that the fraction of graphs for all windows that contained the contemporaneous link between brokers and hedge funds was less in comparision with the other two links. This accounts for the difference in link width in Figure (9a). Contrary to the view expressed in Billio et al. ( 2012), our findings provide support to claim that both banks and the so-called shadow banking system were more central to systemic risk. In particular, due to the high coupling strength of contemporaneous links between banks, Broker/dealers and insurers, any large shock in one will spread quickly through the system.

In general, our results point to important asymmetry in the causal dependencies in the insurance sector and brokers prior to the financial crisis of 2007-2009. We find that the returns of insurers have significant impact on returns of brokers, where the time delay of impact is shorter during the crisis (see Figures (4c &4d) and Figure (3a)). For all time periods considered, we find that banks were always contemporaneously linked to brokers with a higher coupling strength observed before year 2000. In addition, our results indicate that insurers did not, in many cases, directly impact hedge funds significantly. After the year 2000 (see Figure (7b)), we find contemporaneous link between banks and insurers, and also between brokers and hedge funds, where the former is with a higher coupling strength. The aftermath of 1998 collapse of the $5 billion hedge fund Long Term Capital Management (LTCM) could have lead to the contemporaneous link between brokers and hedge funds. In general, we find that banks were not directly connected to hedge funds. However, the two sectors were indirectly linked via a contemporaneous chain of dependencies from banks to hedge funds: BK -PB -HF, as seen Figure ( 4). We remark that Figure 7: A directed link "A -→ τ B" means that B is influenced by A at lag(s) τ. The existence of a dot at a particular lag τ of sector ξ on the plots located on the diagonals indicates an autoregressive link of the form ξ t-τ -→ ξ t , which we denote as "ξ t -→ τ ξ t ". Contemporaneous links can be read from the plots by observing dots at where τ = 0. The vertical axes represent the MIT coupling strength of the linkages. We define time windows: 1994M01-1998M08, 1994M01-2000M03, 1998M08-2008M11, based on the distance matrix plot in Figure ( 6).

HF

just prior to the financial crisis of 2007-2009, the hedge funds did have a significant negative impact on banks (see Figures (5c &5d)). In this period, as seen in Figures (4d &5d), hedge funds had a significant bilateral relationships with insurers, brokers and banks. We remark that most connections were observe for the time period (1994M01-2007M12), shown in Figure (4d), were hedge funds had a negative impact on the financial system especially on the banks and brokers. This increase in number of connections and causal dependencies between these sectors just before and in the beginning of the recent crisis could serve as a significant systemic risk indicator. Our results have showed that there is high coupling strength of linkages between the 15 four sectors, increasing the channels through which shocks can be transmitted throughout the financial system.

Documents

European Sovereign debt crisis

The data used in our empirical analysis consist of daily 10-year government bond yields for eleven European Monetary Union (EMU 11) countries. The EMU 11 used in this work is composed of the periphery group of countries: Portugal (Por), Ireland (Ire), Italy (Ita), Greece (Gre), Spain (Spa); and the central group of countries: Austria (AU), Germany (Ger), The Netherlands (Ne), Belgium (Bel), France (Fra), and Finland (Fin). The idea is to empirical investigate (b) The MIT lag plot for sliding window analysis on the full sample period.

→ HF → PB → PB → PB → PB → PB → PB → PB → PB → PB → PB → PB → PB → BK → BK → BK → BK → BK → BK → BK → BK → BK → BK → BK → BK → IN → IN → IN → IN → IN → IN → IN → IN → IN → IN → IN → IN 0.
Figure 9: Sliding window analysis on the full sample period: Jan 1994-Dec 2008. The sliding window is of length 36 months and a step size of 12 months. The link width denotes the percentage of windows that contain this link in the time series graph.

transmissions of the sovereign debt crisis by examining the behavior of the government bond yields of these countries for the period (01/01/2007-31/12/2014). Our daily data comes from the Bloomberg. The start date of the sample is chosen in order to study dynamics prior and during the Sovereign debt crisis. Given that the 10-year bond yield a country is denoted B t . In order to ensure stationarity, we work with the time series B t = log(B t )log(B t-1 ), which correspond to the growth rate of the sovereign bond yields, see Figure (10).

The time series graph for the year 2007 is shown in Figure (11). In this graph, we identify causal linkages during the start of the 2007-2009 global financial crisis. At the end of the global financial crisis in 2009 (see Figure (12c)), we observe an increase in the number of causal connections, with most previous links mentained. In general, the number of linkages did start increasing from the year 2007 to the end of 2009. Figures (12 & 13) provide information on the evolution of the causal dependencies among countries across the time frame before and during the European sovereign crisis. The significant lag plots of the time series are presented in Figures (A.15-A.22). The number of linkages among the sovereign bond yields increases across time as the window expands from 2007. We also observe more linkages with negative flows over the expanding windows at the start and during the sovereign debt crisis, as shown in Figure ( 13). It is clearly seen that there are more connections and negative flows at the end of the 2014, which signals that the EMU 11 is not yet out of the sovereign debt crisis. In Figure (12d), we find that the Netherlands were not linkaged to any of the periphery countries at the start of the sovereign debt crisis. The sovereign bond yields of the Netherlands and Germany were highly coupled during the crisis, see Figure (13).

Finally, we perform our analysis on sliding windows to enhance robust conclusions on the identification and quantification of causal dependencies among sovereign bond yields for the whole sample period. We present in Figure ( 14) the results for sliding window of length 1 year with a step size of 6 months, and the sliding window of length 1 year with a step size of 1 year. 17 ). We remark that our findings do not differ when we use sliding windows of length 2 years with step the results with sizes 1 year or 2 years. The results for these sliding window analysis can be made available upon request from the authors. The link width for the sliding window analysis denotes the percentage of windows that contain this link in the time series graph. In general, the graphs point to higher coupling between the Netherlands and Germany, the Netherlands and France, France and Germany, Belgium and Austria, and then Italy and Spain. Our finding shows that government bond yield of the Netherlands were not linked to any of the periphery countries severely hit by the sovereign debt crisis. In particular, the sovereign bond yields of Germany, the Netherlands, Finland and Austria were more stable compared to other core countries like France and Belgium, which were linked to at least two periphery countries. spillover effects mainly from Ireland and Italy. However, it's causal dependence with Germany, the Netherlands, Austria and Finland, could probably explain why France was not severely hit by the sovereign debt crisis. The graphs in Figure ( 14) points to transmissions of risk effects from Greece to the periphery countries hit by the crisis.

Concluding Remarks

In this study we have introduced and provided a novel methodology for analysing complex linkages among financial institutions. In particular, we have presented the drawback of using the original framework of Granger causality (linear or nonlinear) tests in measuring connectedness among subprocesses of a multivariate process. We offer a novel information theoretic approach for capturing and studying complex linkages in the financial system based on time series graphs 19 and information theory. In particular, we show that this approach identifies linkages, transmission channels, lags of linkages and assess the strengths of such connections. Unlike existing methodologies with assumptions on stationarity of time series, model structure and parameter space, our approach only requires that the multivariate time series be stationary. Our approach excludes the misleading influence of autodependency within a process by assigning a non-zero value only to lagged components that are not independent conditional on the remaining process.

Documents
We have applied this methodology to simulated data and presented a comprehensive feasibility of the approach in capturing lag-specific causal dependencies in multivariate process. We have also applied this approach to real data to analyse interconnectedness among financial institutions and also to study the European Sovereign debt crisis considering different time frames and sliding windows. Further studies should look into enhancing this methodology by defining a nonlinear equivalent measure of the coupling strength of interconnectedness. Spa 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

→ AU → Bel → Fin → Fra → Ger → Gre → Ire → Ita → Ne → Por → Spa 0.2 0.
ρ MIT lag τ [units] Spa 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Spa 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 ρ MIT lag τ [units] Spa 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 ρ MIT lag τ [units] Spa 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 ρ MIT lag τ [units] Spa 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 ρ MIT lag τ [units] Marwan, N., Romano, M. C., Thiel, M., Kurths, J., 2007. Recurrence plots for the analysis of complex systems. Physics Reports 438 (5-6), 237-329. Paluš, M., 1996. Coarse-grained entropy rates for characterization of complex time series. In: Physica D: Nonlinear Phenomena 93 (1-2), 64-77. Runge, J., Heitzig, J., Marwan, N., Kurths, J., 2012a. Quantifying causal coupling strength: A lag-specific measure for multivariate time series related to transfer entropy. Physical Review E 86, 061121. Runge, J., Heitzig, J., Petoukhov, V., Kurths, J., 2012b Por 0 1 2 3 4 5 6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 Spa 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 ρ MIT lag τ [units]

8

  Documents de Travail du Centre d'Economie de la Sorbonne -2014.69R (Version révisée) Plot of four subprocesses of a stationary multivariate autoregressive process 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Cross-MIT (b) MIT plot for the simulated process.

9

  Documents de Travail du Centre d'Economie de la Sorbonne -2014.69R (Version révisée) An ensemble of MIT plot for the simulated process on a sliding window.

  The plot of significant lags for sliding window analysis.

Figure 2 :

 2 Figure 2: An ensemble of MIT plot for the simulated process on a sliding window of length 200 and step size of 100. The link width denotes the percentage of windows that contain this link in the time series graph. The result coincide with our a priori knowledge on the causal chains among the variables and do not differ from the results obtained in Figure (1).

  Figure 3: The MIT Plots showing both the linkages among the four sectors and the strength of such connections for the full study period, Jan 1994-Dec 2008. The second representation Figure (3b) of the MIT plots provides additional information on autoregressive linkages by inspecting the diagonals. The existence of a dot at a particular lag τ of sector ξ on the plots located on the diagonals indicates an autoregressive link of the form ξ t-τ -→ ξ t , which we denote as "ξ t -→ τ ξ t ".

  Figure4: The MIT Plots displays how the four sectors are linked and the strength of the linkages. It shows autoregressive links, contemporaneous links and directed (i.e across sectors) links among sectors. The numbers at some ends of directed arrows (links) corresponds to the significant lag of the linkage. Thus, a directed link "A -→ τ B" means that B is influenced by A at lag(s) τ. Contemporaneous links between two sectors: A and B, is denoted "A -B".

Figure 5 :

 5 Figure5: This is an alternative representation of the MIT Plots in Figure4. A directed link "A -→ τ B" means that B is influenced by A at lag(s) τ. The existence of a dot at a particular lag τ of sector ξ on the plots located on the diagonals indicates an autoregressive link of the form ξ t-τ -→ ξ t , which we denote as "ξ t -→ τ ξ t ". Contemporaneous links can be read from the plots by observing dots at where τ = 0. The vertical axes represent the MIT coupling strength of the linkages.

Figure 6 :

 6 Figure 6: A contour plot (heat map) of the unthresholded recurrence matrix, which is also referred to as the distance matrix in Addo et al. (2014). Considering the entire sample, we identify breaks at the dates: 1998M08, 2000M03 and 2008M11. We define time windows: 1994M01-1998M08, 1994M01-2000M03, 1998M08-2008M11, based on these dates for the MIT analysis.

  MIT plot for full Sample: Jan 1994-Dec 2008

Figure 8 :

 8 Figure 8: The lag plot indicating significant causal dependencies and coupling strength between subprocesses for the time windows: 1994M01-1998M08, 1994M01-2000M03, 1998M08-2008M11. In otherwords, this is an alternative of Figure (7) with information on the magnitude and sign of the coupling strength.

  An ensemble of MIT plot for full sample: Jan 1994-Dec 2008 on a sliding window.

Figure 10 :

 10 Figure10: Time series plot of the daily 10-year European Sovereign bond yields, B t , the period from 1/01/2007-31/12/2014. The time indexes correspond to the following dates: "1"= 1/01/2007, "500"= 11/28/2008, "1000" = 10/29/2010, "1500"= 9/28/2012, and "2000"= 8/29/2014.

Figure 11 :

 11 Figure 11: Time series graph of the 10-year European Sovereign bond yields for 2007. The year "2007" denotes the period from 1/01/2007-31/12/2007.

Figure 12 :

 12 Figure12: Time series graph of daily 10-year European Sovereign bond yields over expanding windows. The year "2007" means the period from 1/01/2007-31/12/2007. The time window of the format "20AA"-"20BB" denotes the period from the start date of year "20AA" to the end date of year "20BB".

Figure A. 15 :

 15 Figure A.15: Significant lag plots for 10-year European Sovereign bond yields for 2007. This corresponds to the time series graph in Figure (12a).

Figure A. 17 :

 17 Figure A.17: Significant lag plots for 10-year European Sovereign bond yields for 2007-2009. This corresponds to the time series graph in Figure (12c).

Figure A. 19 :

 19 Figure A.19: Significant lag plots for 10-year European Sovereign bond yields for 2007-2011. This corresponds to the time series graph in Figure (13a).

Figure A. 20 :

 20 Figure A.20: Significant lag plots for 10-year European Sovereign bond yields for 2007-2012. This corresponds to the time series graph in Figure (13b).

Figure A. 21 :

 21 Figure A.21: Significant lag plots for 10-year European Sovereign bond yields for 2007-2013. This corresponds to the time series graph in Figure (13c).

Figure A. 22 :

 22 Figure A.22: Significant lag plots for 10-year European Sovereign bond yields for 2007-2014. This corresponds to the time series graph in Figure (13d).

  Bel → Fin → Fra → Ger → Gre → Ire → Ita → Ne → Por → Spa

	AU Bel	0.6 0.4 0.2 0.0 0.2 0.4 0.6 → AU → 0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Fin	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Fra	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Ger	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Gre	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Ire	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Ita	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Ne	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Por	0.6 0.4 0.2 0.0 0.6 0.4 0.2
		0.6 0.4 0.2 0.0 0.2 0.4 0.6
		0 1 2 3 4 5 6

  Figure A.16: Significant lag plots for 10-year European Sovereign bond yields for 2007-2008. This corresponds to the time series graph in Figure (12b). Bel → Fin → Fra → Ger → Gre → Ire → Ita → Ne → Por → Spa

	AU Bel	0.6 0.4 0.2 0.0 0.2 0.4 0.6 → AU → 0.6 0.4 0.2 0.0 0.6 0.4 0.2	ρ MIT
	Fin	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Fra	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Ger	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Gre	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Ire	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Ita	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Ne	0.6 0.4 0.2 0.0 0.6 0.4 0.2
	Por	0.6 0.4 0.2 0.0 0.6 0.4 0.2
		0.6 0.4 0.2 0.0 0.2 0.4 0.6
		0 1 2 3 4 5 6
		lag τ [units]
	Appendix B. Significant lag plots for window analysis on Bond yields

24 Documents de Travail du Centre d'Economie de la Sorbonne -2014.69R (Version révisée)

  Bel → Fin → Fra → Ger → Gre → Ire → Ita → Ne → Por → Spa

	AU Bel	0.4 0.2 0.0 0.2 0.4 0.6 0.8 → AU → 0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Fin	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Fra	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Ger	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Gre	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Ire	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Ita	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Ne	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Por	0.4 0.2 0.0 0.2 0.8 0.6 0.4
		0.4 0.2 0.0 0.2 0.4 0.6 0.8
		0 1 2 3 4 5 6

  AU → Bel → Fin → Fra → Ger → Gre → Ire → Ita → Ne → Por → Spa

	AU	0.2 0.0 0.2 0.8 0.6 0.4
	Bel	0.2 0.0 0.2 0.8 0.6 0.4
	Fin	0.2 0.0 0.2 0.8 0.6 0.4
	Fra	0.2 0.0 0.2 0.8 0.6 0.4
	Ger	0.2 0.0 0.2 0.8 0.6 0.4
	Gre	0.2 0.0 0.2 0.8 0.6 0.4
	Ire	0.2 0.0 0.2 0.8 0.6 0.4
	Ita	0.2 0.0 0.2 0.8 0.6 0.4
	Ne	0.2 0.0 0.2 0.8 0.6 0.4
	Por	0.2 0.0 0.2 0.8 0.6 0.4
		0.2 0.0 0.2 0.4 0.6 0.8
		0 1 2 3 4 5 6

→

  AU → Bel → Fin → Fra → Ger → Gre → Ire → Ita → Ne → Por → Spa

	AU	0.2 0.0 0.2 0.8 0.6 0.4
	Bel	0.2 0.0 0.2 0.8 0.6 0.4
	Fin	0.2 0.0 0.2 0.8 0.6 0.4
	Fra	0.2 0.0 0.2 0.8 0.6 0.4
	Ger	0.2 0.0 0.2 0.8 0.6 0.4
	Gre	0.2 0.0 0.2 0.8 0.6 0.4
	Ire	0.2 0.0 0.2 0.8 0.6 0.4
	Ita	0.2 0.0 0.2 0.8 0.6 0.4
	Ne	0.2 0.0 0.2 0.8 0.6 0.4
	Por	0.2 0.0 0.2 0.8 0.6 0.4
		0.2 0.0 0.2 0.4 0.6 0.8
		0 1 2 3 4 5 6

→

  . Escaping the curse of dimensionality in estimating multivariate 29 Documents de Travail du Centre d'Economie de la Sorbonne -2014.69R (Version révisée) AU → Bel → Fin → Fra → Ger → Gre → Ire → Ita → Ne → Por → Spa

	AU	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Bel	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Fin	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Fra	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Ger	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Gre	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Ire	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Ita	0.4 0.2 0.0 0.2 0.8 0.6 0.4
	Ne	0.4 0.2 0.0 0.2 0.8 0.6 0.4
		0.4 0.2 0.0 0.2 0.4 0.6 0.8

→

This is the causal Markov condition. 5 Documents de Travail du Centre d'Economie de la Sorbonne -2014.69R (Version révisée)

Graphical models provide a framework to distinguish direct from indirect interactions between and within subprocesses of a multivariate process[START_REF] Lauritzen | Graphical models[END_REF]).

Documents de Travail du Centre d'Economie de la Sorbonne -2014.69R (Version révisée)

Acknowledgement

The authors are grateful to participants of the 2014 Meeting of the Consortium for Systemic Risk Analytics (CSRA) held at Massachusetts Institute of Technology (MIT), for their valuable suggestions, and comments that have significantly improved this contribution. This project has received funding from the European Union's Seventh Framework Programme (FP7-SSH/20072013) for research, technological development and demonstration under grant agreement n o 320270 (SYRTO). This documents reflects only the author's view. The European Union is not liable for any use that may be made of the information contained therein.

Documents de Travail du Centre d69R (Version révisée)

 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5AU Spa Spa Spa Spa Spa Spa Spa Spa Spa Spa Spa Spa Spa Spa Spa 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Spa Spa Spa Spa Spa Spa Spa 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6


R Bertoni

M Lorenc

J Laisney

A Tissot

A Moréac

S F Matar

M.-L Boillot

E Collet
email: eric.collet@univ-rennes1.fr

Femtosecond spin-state photo-switching dynamics in an Fe III spin crossover solid accompanied by coherent structural vibrations

We investigate light-induced excited spin-state trapping (LIESST) dynamics of an Fe III spin-crossover material from low (S=1/2) to high (S=5/2) spin states. Our results show that this process occurs only at the molecular level as evidenced by the linear dependence of the fraction of photo-switched molecules with the excitation density as well as with the initial fraction of low spin molecules. The inter-system crossing from photoexcited LS (S=1/2) to HS (S=5/2) occurs within 200 fs and is accompanied by coherent non-equilibrium vibrational relaxation in the photo-induced HS state. These results reveal similar dynamical features to those already reported for LIESST in Fe II systems. The activation of coherent molecular vibrations is essential for reaching rapidly the HS potential on the timescale of molecular motions, whereas their fast damping allows an efficient trapping in the HS potential. The observed coherent oscillations are attributed to photoinduced molecules in the HS states, as supported by Raman spectroscopy at thermal equilibrium, and DFT analyses of molecular vibrations and TD-DFT calculations of optical absorption.

Introduction

In molecular science, light is known to be an efficient trigger for driving chemical reactions, breaking bonds or changing electronic molecular states 1 . Reaction coordinates connecting initial, photoexcited, and final photoinduced states are not simple to describe. Furthermore, the pathway followed on the potential energy surface is complex and numerous studies investigated how fast such processes are and how interconversion of molecular states proceeds. In the solid state too, light can switch molecules between different electronic states. Charge-transfer processes induced by light can lead to photo-induced phase transition from insulating to metallic 2 or from neutral to ionic 3 phases for example. Spin-crossover (SCO) materials are bistable molecular systems able to switch between low spin (LS) and high spin (HS) states under the effect of different control parameters such as temperature or pressure 4

. They are also prototype photoactive systems as they can be interconverted through the light-induced excited spin-state trapping (LIESST) process from LS to HS state or conversely from HS to LS state (reverse-LIESST). LIESST in crystals was discovered more than thirty years ago 5 : by irradiating SCO solids with cw light at low temperature complete conversion of molecular states can be generated at the macroscopic scale of the material and the photo-induced state can be metastable up to a critical temperature referred to as T(LIESST) 6 . In crystals, molecules may respond cooperatively to light excitation 2,3 . In addition, light can control order between the individual molecules of the materials and generate new phases which cannot be reached at thermal equilibrium 7 . This underlines the ability of molecular-based materials to show response to light different from the response of the individual constituting molecules. There are also other examples in the literature of molecular materials showing cooperative response to light excitation, for which a single photon transforms several molecules and the response of the molecular material is more than the sum of individual molecular events 2,3

. Spin-crossover has become a topical research area over the last three decades, with many research groups contributing to its advancement 4 . The very high efficiency and speed of the photo-induced spin state switching between two states that completely differ in physical properties (color, volume, magnetic susceptibility) represent a conceptual challenge 8 . For Fe II systems the role of ligand-field states was deeply investigated 9

. The study of reverse-LIESST revealed a mechanism involving two main inter-system crossing (ISC) and therefore true intermediate electronic states (such as S=1) between the photoexcited HS (S=2) and the photoinduced LS (S=0) state, which is reached within 40 ps. But for LIESST, the irradiation into metal-ligand charge transfer (MLCT) or d-d bands of S=0 species promotes the system to the S=2 state within 160 fs. This timescale was determined with ultrafast optical spectroscopy, a powerful technique for investigating the photo-physical dynamics of SCO compounds 10 . Complementary ultrafast X-ray absorption spectroscopy studies revealed the structural dynamics related to ligand-cage dilation, which leads to the structural trapping of the newly formed HS electronic state 11 . On the pathway from MLCT to HS states, other not structurally relaxed electronic states serve as intermediates. Indeed, these states are populated on a timescale shorter than an oscillation period in the potential 12 . For better understanding this process, it seems necessary to track the electronic and structural pathway followed by the system with the time resolution of elementary electronic and molecular events. Recent theoretical works suggested that such speed and efficiency of LIESST results from the instantaneous activation of molecular phonons after photoexcitation 13 . This is due to the important displacive nature of the inter-system crossing, which moves the system rapidly into the HS potential with a 0.2 Å elongation of Fe-N bonds. The experimental proof of such a conjecture was reported recently for an Fe II system 14 . This study revealed that the structural molecular reorganization, trapping the photoinduced electronic state, occurs in a two-step sequence: the molecule elongates first (within 170 femtosecond) and bends afterwards, via a coherent vibrational energy transfer between these two main structural modes.

Results and discussion

Thermal spin conversion. -a high spin one (HS) with S= 5/2 corresponding to the electronic configuration t 2g 3 e g 2 (or d π 3 d σ 2 ).

In the form of single crystals, this compound shows a firstorder phase transition from LS to HS phases at thermal equilibrium 18,19 . Nano-crystals of this material dispersed in Here we present new results and detailed analysis of the spin state photo-switching dynamics by means of femtosecond pump-probe optical spectroscopy. It is easier to investigate an assembly of nano-crystals dispersed in a PVP polymer film than single crystals: the low optical density of the nano-crystals allows to perform a detailed study over a broad spectral range from UV to IR. Single crystals restrict the range of wavelengths for spectroscopic probing due to the higher optical density (OD) 8a,16a .

Fig. 1 shows the variation of optical density OD measured on the films of nano-crystals, between pure LS state (16 K) and pure HS state (296 K). It reveals phenolate-to-iron chargetransfer bands (CT) in the near-infrared (NIR) -visible (VIS) with distinct features for LS and HS states. With respect to that of LS state, the OD of the HS state is higher in VIS and lower in NIR parts of the spectrum, with a well-defined isosbestic point around 650 nm. The thermal fraction of HS molecules X HS (T) is obtained by scaling the change of optical density OD(T) at a given temperature T to the change of optical density between LS (OD LS ) and HS (OD HS ) states: In the following, we exploit these spectroscopic fingerprints, to study the ultrafast spin-state photo-switching and the resulting energy dissipation into the material.

Inter-System Crossing and vibrational cooling

The spin-state photo-switching is known to proceed via transient intermediate states, as the direct low energy excitation from LS to HS state is forbidden by spin parity. In the case of Fe II systems, the photo-switching at the molecular level results from the promotion by light of an electron from the metal center to the ligand orbitals (MLCT excited state). For this Fe III [Fe(3-MeO-SalEen) 2 ]PF 6 system, we performed DFT calculations with Gaussian 09 package to establish the molecular orbital energy diagram. Fig. 2 shows that in the LS state the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbitals (LUMO), both of Fe d predominant character, are separated by a 1.6 eV energy gap. We found that the low energy excitation at 850 nm (1.45 eV) therefore may promote an electron from a few symmetryadapted and doubly-occupied molecular orbitals (centered on the phenol group, noted L) to the HOMO with a strong weight on the d orbital of the metal center (M). This LMCT state corresponds then to a L 1 d π 6 d σ 0 electronic configurations. By using femtosecond optical spectroscopy (see experimental section), it is possible to study the photo-switching dynamics since the different molecular states have peculiar optical spectra (Fig. 1). Selected time traces of transient OD measurements performed at different probing wavelengths are plotted in Fig. 345. The OD increase in the VIS part and bleaching in the NIR part (at the ps timescale), are characteristic of the formation of the HS state, as observed during the thermal LS to HS conversion (Fig. 1) and previously reported on similar samples 8a,16 . Around time zero, the OD spectrum differs significantly from those of the LS ground state and of the HS photoinduced state, in terms of intensities or spectral range, where OD increases or decreases. This is the optical fingerprint of the LMCT state. The kinetic evolution of the optical density around the isosbestic point at 625 nm, where the OD of LS and photoinduced HS states are the same 8a , allows to selectively follow the dynamics of this intermediate (Franck-Condon) state. The inter-system crossing (ISC) sequence from LMCT to HS is well described at 625 nm by a single exponential decay. The fit shown in Fig. 4 gives  ISC = 200 ± 20 fs. For other measurements with probe wavelengths away from the isosbestic point, the single exponential model fails to reproduce the observed dynamics and slower components are observed on the ps time-scale, as shown at 550 nm for example. This is especially true in the IR region (950 nm, Fig. 5), more sensitive to vibrational cooling since lower energetic levels are probed. Thus we used a two-exponential decay model for fitting both dynamics, the first time constant describing the ISC being fixed to  ISC = 200 fs. Depending on the probe wavelength the vibrational cooling time constant  vc varies from 470 (50) fs at 550 nm and up to 1.6 (0.2) ps at 950 nm. At 730 and 850 nm the probe is sensitive both to vibrational cooling but also to LS state bleaching, thus making the interpretation more difficult. These measurements in the visible and infrared region reveal therefore that even if the system has reached the HS potential within 200 (20) fs, more than one picosecond is needed for the HS molecule to thermalize with its environment. The  vc 1.6 ps observed here corresponds to the transfer of the excess energy deposited on the molecule by the laser to the lattice. This timescale is shorter than that reported for Fe II molecules in solution (up to 10 ps) with infrared experiments 10,11 . This may be due to the fact that in crystals the excited molecules interact efficiently with the lattice system via phonon-phonon coupling.

Coherent structural oscillations

A closer inspection of the OD time traces in Fig. 3-4 allows the observation of weak oscillating components. Fig. 6 (a-c) shows the deviation of the OD data from the exponential fits for different probe wavelengths. The corresponding Fourier transforms of the residual parts corresponding to positive delay are presented in Fig. 6 (d-f). At 550 nm, a prevailing mode is observed around 85 cm -1 (similar oscillations are observed at 525 and 475 nm in Fig. 3). At the wavelength on the isobestic point (625 nm) another mode is observed around 56 cm -1 . These modes are equally observed at 700 nm, where a third mode around 35 cm -1 also appears.

A similar vibrational coherence in the photoinduced high-spin state was reported for Fe II SCO molecules in solution 21 ) associated with the elongation of the Fe-N bonds was clearly identified 14 : the instantaneous activation of this molecular breathing phonon drives the system from the MLCT to the HS potential within 170 fs, i.e. on the timescale of a half period of such molecular vibration. The fast damping of the key molecular phonon modes leads to the efficient trapping the HS state. It may also result from intermolecular contacts. . We thus performed Raman measurements at room temperature, as well as calculations of molecular vibrations on the [Fe(3-MeO-SalEen)]PF 6 compound for gaining insights on the frequencies and nature of such modes.

Raman and DFT studies of the HS phase

We used a single crystal for finding the crystal axis orientation with x-ray diffraction and checking polarization effect on Raman signal. Fig. 7 shows the single crystal Raman spectra obtained at room temperature. Since the crystalline space group is and the molecules are in general positions in the unit cell 16a , the modes split in two categories: the totally symmetric ones (A g and Raman active) where molecules oscillate in phase with respect to inversion symmetry and the antisymmetric ones (A u and IR active). The Raman spectra reveal several low frequency modes, especially around 65, 85, 100, 145, 172, 181, 240 and 260 cm -1

. In this frequency range, the modes may be the skeletal modes of the [FeN 4 O 2 ] core, but may also correspond to ligand distortions. Because of the C i point symmetry of the crystal, there is no polarization selection: only the Raman intensity may depend on the sample orientation with respect to the laser polarization. For understanding better the nature of the molecular vibrations involved during LIESST, we carried out molecular vibration frequencies calculations with DFT methods for an isolated [Fe(3-MeO-SalEen) 2 ]PF 6 complex, after geometry optimization, by using hybrid B3LYP functional with triple-ζ 6-311g (d,p) basis set within Gaussian09 code 23,24 . In Fig. 8 we compare the calculated Raman spectrum with the experimental results from Fig. 7, averaged over the different orientations measured on a single crystal for better comparison with calculations. The overall calculations reproduce quite well the observed Raman spectrum. A complete analysis of the modes and the nature of the vibrations will be published elsewhere and in this paper we focus on the modes observed with time-resolved spectroscopy. In the low frequency region, the calculated modes at 88, 56 and 33 cm The obtained natural transition orbitals (NTO) taking into account the hole-particle pairs resulting from light absorption can then be used to interpret our data. Fig. 9 shows the NTO corresponding to the hole-particle transition state around 500, 617 and 700 nm. The strong weight of the particle orbital on the Fe-N (and Fe-O) bonds for the transition at 617 nm explains the sensitivity to the breathing mode of the pump-probe data at 625 nm. The transition state around 700 or 500 nm are characterized by a strong weight of the particle orbital mainly located on one ligand and explain the sensitivity to ligand torsion.

Ultrafast LIESST dynamics in Fe III systems

Our investigation gives important insights on the ultrafast mechanism of LIESST in a Fe III system, schematically represented in Fig. 10. Molecular orbital calculations (Fig. 2) indicate that the photo-excitation of the LS (L . However, since the HS potential is reached within 200 fs, these electronic states only serve as mediators and are dynamically mixed. The different photoexcited molecules arrive "simultaneously" on the HS potential, where they oscillate. The global coherent oscillations experimentally observed underline that the phase is mainly kept between the different photo-excited molecules. Indeed, the observation of vibrational coherence in the photoinduced HS state with oscillation around 85 cm -1 (392 fs period), requires that the photoexcited molecules reach the HS potential on a timescale shorter than a half oscillation period to avoid dephasing. This is in agreement with  ISC = 200 ± 20 fs observed by time-resolved spectroscopy (Fig. 4). This very short time-scale is similar to the one reported for LIESST in Fe II SCO systems 10,14 and also indicates that intermediates other than LMCT and HS states mainly serve as mediators.

The instantaneous activation of several phonons of the HS state results from the important displacive nature of the mechanism. However, the structural change between LS and HS structures involves several degrees of freedom such as molecular breathing, torsion..., which may be activated with different timescales. Therefore, the transformation pathway connecting the initial photo-excited state to the photoinduced HS state is only roughly described by the classical single coordinate picture employed here in Fig. 10. A more complex multi-dimensional energy surface description has to be used for a more detailed analysis. However, these results indicate that on such short time-scale the wave functions describing the electronic and structural degrees of freedom are strongly coupled during the process. This LS-to-HS photoswitching dynamics is therefore very different in nature from the reverse HS-to-LS one observed recently in a Fe II SCO material, where a true (vibrationally cooled) triplet state is populated and depopulated within 10's ps 9

. Indeed, such slow reverse-LIESST dynamics obey the Born-Oppenheimer approximation.

Local nature of ultrafast LIESST

Another important question associated with LIESST in crystals is the possibility of cooperative response to light excitation. We look into possible cooperativity on ps timescale by following the fraction X HS of molecules converted after 5 ps from LS to HS state. It is monitored by femtosecond optical spectroscopy for different excitation densities and for different temperature (and therefore different initial fractions of molecules in LS states as shown in Fig. 1). Fig. 11 shows how X HS changes with the pump laser fluence F (i.e. on the excitation density). It is clear that X HS depends linearly on F, i.e. there is no amplification on such ultra-short time scale. The observed slopes in Fig. 11 correspond to the photo-response and are given here in X HS /F. The slopes measured for different temperatures, proportional to the number of photoswitched molecules per incident photon, change with temperature. This is not surprising because the fraction of molecules in the LS state decreases above 120 K (Fig. 1). Therefore the fraction of molecules switched from LS to HS states, should be weighted both by X LS , the initial fraction of molecules in the LS state at a given temperature (X LS = 1-X HS ), and by the laser fluence F.

X HS  F  X LS or X HS / F  X LS Fig 11 (b) shows that the photo-response X HS /F depends linearly on X LS . Incidentally, based on such plot we can also conclude on the absence of back conversion with the same pump wavelength (from HS to LS), since in such a case the photo-response would be negative as X LS will tend to 0. These important results underline the local nature of the photoswitching process occurring within a picosecond. Each individual photo-switching process can be regarded as an isolated molecular event. It is only on longer time-scale that macroscopic conversion occur, driven by elastic and thermal mechanisms 18 .

Conclusions

This detailed study allows to draw a complete picture of the LIESST photo-switching process in Fe III systems in the solid state. First of all, the photo-response is local on ps timescale, as characterized by the linear dependence with the number of photons and with the initial fraction of molecules in the LS state. This confirms that the initial mechanism might be seen as a local molecular event. coordination sphere makes such identification more difficult and several modes may involve Fe-N elongation (as it is the case of the 56 cm -1 mode). In addition, the breathing frequencies depend on the ligand (mass) and on the strength of its bonding to the metal. However, our results indicate that the mechanism behind the ultrafast intersystem crossing in Fe II and other transition-metal complexes 13,14 , described in terms of the dephasing of the photoexcited state to the phonon continuum of the photoinduced (here HS) state with significantly different structure, is also valid for describing the photo-switching mechanism in Fe III systems. The observed structural coherence underlines the dynamical nature of the light-induced excited spin-state trapping process, as opposed to the so far prevailing kinetic description based on simple exponential decay from photo-excited to photoinduced states.

Experimental Sample preparation

The molecular compound [Fe(3-MeO-SalEen) 2 ]PF 6 was synthesized in the form of a microcrystalline powder and single-crystals as previously described 16a,18 . Nanocrystals 16c were prepared by rapidly adding a concentrated solution of salt (10 mg) in acetone (1 mL) into a large volume (20 mL) of butan-1-ol maintained at -40 °C under a vigorous stirring. After 15 minutes, the particles were isolated by centrifugation, rinsed with pentane, dried. The properties of needle-shaped nanocrystals (typical dimensions of 900( 230)x285( 65)x64( 18) nm 3 from TEM) were checked with powder X-ray diffraction, IR and magnetism 16c . The nanocrystals were processed in PVP polymeric films. 5 mg of nanocrystals were added to a viscous solution of 100 mg of PVP (MM = 45000 gmol -1 ) in 500 L of butan-1-ol ; the mixture was sonicated (30 min), then spincoated on glass substrates (1000 rpm) and dried at air before use.

Femtosecond optical spectroscopy

For tracking the photo-induced spin state switching dynamics in real time, we employ femtosecond pump probe transmission measurements 17a . The laser beam was provided from the output of a femtosecond regenerative amplifier (Legend Elite, Coherent) and then split in two beams, each of them seeding an optical parametric amplifier (TOPAS, Light Conversion) allowing to cover spectral range from UV to infrared. The experimental time resolution is 140 fs. The pump wavelength was set at 850 nm corresponding to the LMCT absorption band. The photo-excited HS state relaxes towards the LS state in less than 1 ms, allowing the 1 kHz repetition rate of the experiment. The sample was cooled down using a nitrogen cryostream. The one colour probe beam, tuned across a broad spectral range (480-950 nm) measured transient optical density. We could measure the variation of the fraction of HS molecules X HS (t) by scaling OD(t), the change of OD in time between LS and photo-excited HS state, to the OD change between LS and HS states at thermal equilibrium.

Raman spectroscopy

Raman spectra were collected in the 55-1200 cm -1 low frequency range using a LabRAM-HR Raman spectrometer (Horiba / Jobin Yvon). A 10030030µm 3 single crystal has been studied for different polarisation configurations. The 632.818 nm line of a 15 mW He-Ne laser was used as the excitation source. The exciting radiation was directed through a neutral density filter (optical density 2) to avoid sample heating problems and was focused on the sample via a x100 long working-distance objective. The scattered light was collected in backscattering configuration and the Rayleigh scattering was removed by means of a dielectric edge filter.

DFT calculations

Molecular vibration frequencies calculations were carried out for [Fe(3-MeO-SalEen)]PF 6 after geometry optimization, by using hybrid B3LYP functional with triple-ζ 6-311g (d,p) basis set within Gaussian09 code

23

. Frequencies are determined from the second derivatives of the energy with respect to the atomic positions and then operating transformation to massweighted coordinates. Exploring the results especially for the vibrations and their animations with screen captures was done with Gaussview annex module to Gaussian. TD-DFT as implemented in the Gaussian 09 package 23 was applied for obtaining the NTO of the HS [Fe(3-MeO-SalEen)]PF 6 system, starting from preliminary geometry optimized molecule with B3LYP/6-311g (d,p) functional-basis set.

Fig. 1

 1 Fig.1 Difference of optical density in the (350-925 nm) spectral window between HS state (296 K) and LS state (16 K) measured on the nanocrystals sample (a). Photo-excitation is performed at 850 nm in the tail of the LS absorption band. OD at 550 nm (left axis) of the nano-crystal assembly during the crossover scaled to XHS (right axis) (b). polymeric matrix lose the 1 st order transition and show a more gradual spin conversion 16 , as a result of both the pressure effects and anchoring processes (physisorption, capillarity) due to the polymers 20 . It was recently demonstrated by timeresolved studies that these Fe III nanocrystals exhibit photoswitching toward a transient HS state by photo-excitation of the LS state 8a . Here we present new results and detailed

Fig. 2 .

 2 Fig. 2. LS schematic energy diagram and frontier molecular orbitals obtained from DFT calculations. Red: d and d metal-like orbitals, blue : ligand-like orbitals. The HOMO-LUMO gap (1.6 eV) is larger than the pump excitation at 1.45 eV (850 nm). With respect to the HOMO (strong weight on a Fe d orbital), a few doubly occupied molecular orbitals (centered on phenol groups) falls within the energy range corresponding to the photo-excitation.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Change of optical density at different probe wavelengths after fs excitation at 850 nm measured on [Fe III (3-MeO-SalEen)2]PF6 nanocrystals. Measurements are performed at 100K in pure LS state.

Fig. 5 .

 5 Fig. 5. Fit of the OD traces at 730, 850 & 950 nm with bi-exponential functions describing ISC and vibrational cooling.

Fig. 6 .

 6 Fig. 6. (a-c), Oscillating component of OD at 550, 625, and 700 nm. (d-f) corresponding FFT of the experimental data, showing the activation of different modes around 85, 56 et 35 cm -1 .

Fig. 7 .

 7 Fig. 7. Raman spectra obtained on single crystals for different polarization (1), (2) and (3) shown in the photographs on the top.

Fig. 8 .

 8 Fig. 8. Comparison of the calculated and experimental (averaged of the different polarizations) Raman spectra.

25 .Fig. 9 .

 259 Fig. 9. Natural transition orbitals of hole (left) and particle (right) calculated in the case of HS species at 698, 617 and 488 nm, i.e. close to the experimental probes at 700, 625 and 550-475 nm. α and β designate spin ↑ and spin ↓. The calculated absorbance spectrum (top) underline the strong weight of these transitions.

Fig. 10 .

 10 Fig. 10. Schematic representation of the photoswitching trajectory along a simple reaction coordinate.

Fig. 11 .

 11 Fig. 11. (a) Temperature dependence of the fraction of molecules photo-switched to the HS state (XHS) for different pump fluencies. (b) Dependence of the photo-response (XHS/F) with the initial fraction of LS molecules (XLS).

1 d π 5 d σ 1 or L 1 d π 4 d σ 2

	generates a LMCT state (L	2 d π 5 d σ 0 ). Time-resolved spectroscopy 0 ) 6 d σ 1 d π
	data show that the LMCT state decays to the HS state
	(L	2 d π 3 d σ	2 ), possibly through other intermediate electronic
	states such as L

  Our new data in the present Fe

		III
	system also reveal that during the LIESST process several key
	modes are coherently activated. They are associated with
	changes of relevant reaction coordinates accompanying
	change of spin state (breathing, torsion...). The Fe-N
	lengthening is shorter in Fe III (0.15 Å) than in Fe	II (0.20 Å) and
	the frequency of the breathing modes differ. If a clear
	breathing mode (in phase elongation of the 6 Fe-N bonds) is
	easily identified in Fe lower symmetry of the present compound with Fe -1 ) 14 the II N 6 systems (around 113 cm III N 4 O 2

The thermal behavior of X HS (T) obtained in this way is displayed in Fig.3(b) and indicates a gradual conversion spanning from 140 K to 220 K, in agreement with magnetic measurements

Acknowledgements

This work was supported by the Institut Universitaire de France, Rennes Métropole, Région Bretagne (CREATE 4146), ANR (ANR-13-BS04-0002) and Fonds Européen de Développement Régional (FEDER). Raman measurements have been performed on the SIR raman system from the ScanMAT platform of Rennes 1 University.