D Faranda
email: davide.faranda@cea.fr

B Podvin

A Sergent

On reversals in 2D turbulent Rayleigh-Bénard convection: insights from embedding theory and comparison with Proper Orthogonal Decomposition analysis

Turbulent Rayleigh-Bénard convection in a 2D square cell is characterized by the existence of a large-scale circulation which varies intermittently. We focus on a range of Rayleigh numbers where the large-scale circulation experiences rapid non-trivial reversals from one quasi-steady (or metastable) state to another. In previous work (Podvin and Sergent JFM 2015, Podvin and Sergent PRE 2017), we applied Proper Orthogonal Decomposition (POD) to the joint temperature and velocity elds at a given Rayleigh number and the dynamics of the ow were characterized in a multi-dimensional POD space. Here, we show that several of those ndings, which required extensive data processing over a wide range of both spatial and temporal scales, can be reproduced, and possibly extended, by application of embedding theory to a single time series of the global angular momentum, which is equivalent here to the most energetic POD mode. Specically, embedding theory conrms that the switches among meta-stable states are uncorrelated. It also shows that, despite the large number of degrees of freedom of the turbulent Rayleigh Benard ow, a low dimensional description of its physics can be derived with low computational eorts, providing that a single global observable reecting the symmetry of the system is identied. A strong connection between the local stability properties of the reconstructed attractor and the characteristics of the reversals can also be established.

We use Rayleigh-Bénard turbulent convection simulations to compare the properties of quasistationary states and transitions (reversals) previously identied via Principal Orthogonal Decomposition (POD) to those obtained via embedding the global angular momentum of the system. Our main result is to show that we can map POD properties on the attractor reconstructed via embedding techniques. Specically, the embedding technique conrms that the switches among dierent metastable states are uncorrelated. Moreover, a local stability indicator can be used to distinguish and classify the dierent metastable states. The low computational costs of embedding analysis suggests to use this procedure whenever a global observable reecting the symmetry of the system can be identied, while the POD should be preferred when such information is not available.

INTRODUCTION

In the last decades, the study of complex systems has shifted from a pure dynamical systems-based approach to a mixture of statistical and statistical-mechanics techniques. This can be justied on both practical and theoretical levels. At the beginning of the 80's, the abundance of data from measurements as well as from numerical models was limited. The complex systems studied were mostly laboratory ows whose laminar or turbulent behavior depended on a control parameter [START_REF] Eckmann | Roads to turbulence in dissipative dynamical systems[END_REF][START_REF] Bergé | Order within chaos[END_REF]. The use of numerical weather forecasts was limited to a few days [START_REF] Mason | A model for assessment of weather forecasts[END_REF] and climate models were mostly conceptual [START_REF] Benzi | Stochastic resonance in climatic change[END_REF]. At that time, new results in dynamical systems theory seemed to convince the scientic community that a suciently long time series could be sucient to reconstruct the dynamics of the system via its attractor. This object is a geometric set towards which the system tends to evolve, independently on its initial conditions.

The Takens [START_REF] Takens | Detecting strange attractors in turbulence[END_REF][START_REF] Noakes | The takens embedding theorem[END_REF] reconstruction theorem was then used to determine the dynamics of climate attractors as well as of complex ows. Initially, the dimension of these objects was set to be extremely low, usually smaller than 10 [START_REF] Fraedrich | Estimating the dimensions of weather and climate attractors[END_REF]. However, from the beginning of the 90's, several authors [810] found out that those low dimensional estimates were wrong: in fact, despite the complexity of the systems, the reconstructions were mostly made using time series measured at specic points of the physical space. Moreover, the time series used were not suciently long. In parallel, computational power increased quickly [START_REF] Molteni | The ecmwf ensemble prediction system: Methodology and validation[END_REF] and measurement techniques opened new possibilities to sample the behavior of complex systems [START_REF] Daviaud | Subcritical transition to turbulence in plane couette ow[END_REF]. In experimental facilities, visualization techniques gave rapid access to eld measurements instead of local measurements [START_REF] Kd Hinsch | Particle image velocimetry[END_REF][START_REF] Juan G Santiago | A particle image velocimetry system for microuidics[END_REF] and numerical models were capable to simulate several years of Earth's climate [START_REF] Mjp Cullen | The unied forecast/climate model[END_REF], seconds of human brain activity [16], and molecules/proteins dynamics [17,[START_REF] Peter | Protein hydration elucidated by molecular dynamics simulation[END_REF].

The complexity of high-dimensional elds as opposed to single-point time series required new methodologies in order to assess the probability of each conguration and forecast the evolution of the system. For this purpose, one of the most popular techniques is the Proper Orthogonal Decomposition (POD). In POD, an orthogonal transformation is used to transform a dataset with correlated variables into linearly uncorrelated variables. Geometrically, this is like tting an n-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. Principal Component Analysis was invented by Karl Pearson [START_REF] Pearson | Principal components analysis[END_REF], as an analogue of the principal axis theorem in mechanics. In the 1930s it was developed independently also by Harold Hotelling [20]. Depending on the eld of application, it is also named the discrete Kosambi-Karhunen-Loève transform (KLT) in signal processing [START_REF] Dd Kosambi | Statistics in function space[END_REF], the Hotelling transform [20] in multivariate quality control, proper orthogonal decomposition (POD) in turbulence [START_REF] Podvin | A low-dimensional approach for the minimal ow unit[END_REF], empirical orthogonal functions (EOF) in meteorological science [START_REF] Edward | Empirical orthogonal functions and statistical weather prediction[END_REF], empirical eigenfunction decomposition [START_REF] Sirovich | Turbulence and the dynamics of coherent structures. i. coherent structures[END_REF], and it is also connected to singular value decomposition (SVD) of X [START_REF] Gene | Matrix computations, the john's hopkins univ[END_REF], eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis, Eckart-Young theorem [START_REF] Harry H Harman | Modern factor analysis[END_REF], or Schmidt-Mirsky theorem in psychometrics, quasiharmonic modes [START_REF] Bernard R Brooks | Harmonic analysis of large systems. i. methodology[END_REF], spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics [START_REF] Norden | The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis[END_REF].

The belief that embedding theorems could not be applied either to systems featuring a large number of degrees of freedom or to time series of high-dimensional vector elds, caused the data analysis scientic community to focus on POD-based approaches to study complex systems. Meanwhile, several theoretical developments in dynamical systems theory were aimed at understanding how systems with large numbers of degrees of freedom could be dealt with [START_REF] Bagley | Counting and classifying attractors in high dimensional dynamical systems[END_REF][START_REF] Temam | Innite-dimensional dynamical systems in mechanics and physics[END_REF].

In particular, the framework of stochastic dynamical systems [START_REF] Arnold | Random dynamical systems[END_REF][START_REF] Mickaël D Chekroun | Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I[END_REF] brings into view the idea that the dynamics of complex systems can be represented by a small number of variables if one lumps the small scales contributions into noise terms and choose as observables for the embedding procedure, global quantities tracking symmetry properties of the ow.

For turbulent ows, such as the von Karman swirling ow, or for atmospheric dynamics, this revised approach produces phase portraits which make it possible to dene low-dimensional models which capture the essential features of the dynamics [START_REF] Faranda | Stochastic chaos in a turbulent swirling ow[END_REF] in the fashion of the 1963 Lorenz equations [START_REF] Edward | Deterministic nonperiodic ow[END_REF]. The goal of the present paper is to examine how information provided by the embedding procedure compares with that obtained by a more extensive analysis such as the Proper Orthogonal Decomposition (POD). This will be done for the specic case of 2-D turbulent Rayleigh-Bénard convection, a ow which is characterized by a series of random transitions from a quasi-steady state to another. This system is an interesting platform for exploring the relation between POD and embedding methods for few reasons: i) It is a turbulent ow and this would prevent at least according Landau's conjecture [START_REF] Landau | On the problem of turbulence[END_REF] a projection of the dynamics onto a low dimensional space due to the large number of degrees of freedom introduced by small scales perturbations. ii) The dynamics features a non-trivial bifurcation structure. This is only observed in a limited range of Prandtl and Rayleigh numbers, for reasons that are not understood [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF]. iii) The dynamics for some values of the parameter analysed in the paper feature the switching among dierent metastable states, plus reversal with some hybrid states. The physical origin of the switching remains largely unknown. The paper is structured as follows: rst we will recap the methodologies used, with emphasis on the recent developments on the embedding strategies for complex systems. Then we will describe the dataset used and outline the results. Eventually we will discuss the advantage of a combined approach to complex systems analysis.

DYNAMICAL SYSTEMS METHODS

In this section we give some elements of dynamical systems theory and some guidelines derived from recent studies for reconstructing the attractors using global rather than local observables. We also introduce indicators of stability derived from the theory of stochastic processes.

Attractor Reconstruction via embedding methodologies

The aim of dynamical systems theory is to reconstruct the underlying attractor of a system and to assess its properties. The attractor is a compact geometrical object in phase space that hosts all the possible trajectories of a system. Once this object is reconstructed, it provides information on the (unstable) xed points (if the system is deterministic) or the metastable states (if the system is stochastic) and the transitions linking them. This description is useful when the attractor is low dimensional because its structure can be visualized and interpreted in an intuitive way. Up to recently it was believed that low dimensional attractors would not be suitable to describe complex geophysical and turbulent ow. However, it has beeen shown [START_REF] Faranda | Stochastic chaos in a turbulent swirling ow[END_REF] that it is possible to describe the large scale motion of a fully-developed turbulent ow with only a few degrees of freedom, if an appropriate observable reecting the ow symmetry is selected. The large embedding dimensions which prevented the applications of dynamical systems theory to turbulence arise from small scale disturbances that can be modeled in terms of stochastic perturbations. This picture reconciles the Landau [START_REF] Landau | On the problem of turbulence[END_REF] and Ruelle-Takens [START_REF] Ruelle | On the nature of turbulence[END_REF] descriptions of turbulence, the former being valid at small scales, and the latter describing the large scale motions. The observable can be derived as global mean of some local quantities in the ow (average energy or momentum), or it can be a single global output quantity measured in an experiment (such as for instance the frequency of rotation of Von Karman ow turbines, once a certain torque is imposed).

Once the observable is selected and a time series is obtained, dierent embedding procedures can be used to reconstruct the attractor from the signal. The rst thing to determine is the embedding dimension, i.e. the number of variables necessary for the attractor reconstruction. This can be done using the method by Cao [START_REF] Cao | Practical method for determining the minimum embedding dimension of a scalar time series[END_REF], or by trying to see how much information is added using another dimension. In what follows we will adopt the second strategy. After selecting the number of variables n, embedding consists in dening a n-dimensional state vector from the observable time series X i consisting i = 1, 2, ...t observations, as M i = (X i , X i+τ1 , X i+τ2 , . . .). The series can be embedded with the method of delays [START_REF] Takens | Detecting strange attractors in turbulence[END_REF], or with the local peaks procedure [START_REF] Norman H Packard | Geometry from a time series[END_REF]. In the method of delays, the value of τ i is kept constant, which is most eective when there is a precise time scale in the system. In contrast, the local peaks method consists in selecting τ i dynamically as the interval between two subsequent partial maxima (or minima), which is most eective when there are no well-identied time scales (e.g. switching frequencies between metastable states). In fact, local maxima (or minima) are robust features of the dynamics even in the presence of noise, because they track the position of metastable states [START_REF] Faranda | Stochastic chaos in a turbulent swirling ow[END_REF]. Since we deal with a turbulent system, we stick to the local peaks methodology and we refer the reader to [START_REF] Faranda | Stochastic chaos in a turbulent swirling ow[END_REF] for a comparison of the two methods.

Once the series of m partial maxima is obtained, the attractor is visualized by plotting in a n-dimensional phase space, M j , M j+1 , ..., M j+n , where j = 1, 2, ..., m. As mentioned earlier, the value of n, known as the embedding dimension, plays a crucial role in the applications of dynamical systems theory to real data [START_REF] Kantz | Nonlinear time series analysis[END_REF].

Stability indicator

To characterize the stability of the attractor, we will use the indicator Υ introduced in [START_REF] Faranda | Early warnings indicators of nancial crises via auto regressive moving average models[END_REF]. We dene our stability indicator by using a simple example: even for a complex system, the dynamics near a metastable state resembles that of a stochastic spring (or of a particle in a quadratic potential). The typical equation associated to these systems is the Langevin equation:

dX(t) dt = - k m X(t) + 1 m ξ(t)
where k is a frictional force (e.g. the Stokes' drag), m is the mass of the particle and ξ is a noise term modeling the random collisions the particle undergoes. The discretized equation then becomes:

X t = φX t-1 + t
This denes a linear process called an ARMA, an autoregressive moving-average model. These models have been widely used over the past decades, especially in econometrics and nance to forecast markets trends. An ARMA model is characterized by p autoregressive terms and q moving-average terms and is denoted ARMA(p, q). The equation above is therefore an ARMA(1,0). We briey present the normal form of an ARMA(p, q) model and some criteria used to t it to a time series (see [START_REF] Poon | Forecasting volatility in nancial markets: A review[END_REF] for a detailed review).

Let us consider a series X(t) of an observable with unknown underlying dynamics. We further assume that for a time scale τ of interest, the time series X t1 , X t2 , ..., X tτ represents a stationary phenomenon. Since X t is stationary, we may then model it by an ARMA(p, q) process such that for all t:

X t = p i=1 φ i X t-i + ε t + q j=1 θ j ε t-j with ε t ∼ W N (0, σ 2 )
-where W N stands for white noise -and the polynomials φ

(z) = 1 -φ 1 z t-1 -... -φ p z t-p θ(z) = 1 -θ 1 z t-1 -... -θ q z t
-q , with z ∈ C, have no common factors. Notice that, hereinafter, the noise term ε t is assumed to be a white noise. For a general stationary time series, this model is not unique. However there are several standard procedures for selecting the model which best ts the data. The one we exploit is the Bayesian information criteria [START_REF] Akaike | Time series analysis and control through parametric models[END_REF]. It is based on the Akaike information criteria (AIC) [START_REF] Akaike | Information theory and an extension of the maximum likelihood principle[END_REF] which was designed to be an approximately unbiased estimate of the KullbackLeibler index of the tted model relative to the true model. Assuming we know the likelihood estimators β and σ 2 of the tted model model thanks to an innovation algorithm, the best ARMA model is the one where p and q minimize

AIC(β) = -2 ln L X (β, σ 2 ) + 2(p + q + 1)
In order to correct the tendency of the AIC to prefer complex models, we use the BIC (Bayesian information criteria) which introduces a penalty for large-order models:

BIC = (τ -p -q) ln τ σ 2 τ -p -q + τ (1 + ln √ 2π) + (p + q) ln τ t=1 X 2 t -τ σ 2 /(p + q)
Intuitively, p and q are related to the memory lag of the process, while the coecients φ i and θ i represent the persistence: the higher their sum (in absolute value), the slower the system forgets its past history, the higher the correlations in the time series. Most of the time ARMA models are used in econometrics tting the whole time series and trying to forecast the future trend of the variable. This assumes a correlation with the past and provide some signicant results for the very near future. Our interest here is rather to use ARMA to detect the local stability. The procedure is the same as for tting the whole time series: after slicing the time series by intervals τ , we obtain a time series X t-τ , ..., X t-1 . We then t each ARMA(p, q) model until p ≤ p max and q ≤ q max assessing the best one (according the BIC criterion). We then compute the stability indicator Υ for the system at time t and then move to time t + 1 to perform the same analysis on the time series X t-τ +1 , ..., X t .

When the system is close to an unstable point, separating multiple basins of attraction, the behavior cannot be described by a Langevin equation as the underlying potential is not quadratic anymore. The change in the shape of the potential introduces new correlation in the time series resulting in higher order ARMA terms. The indicator is then dened as:

Υ = 1 -exp - |BIC(p, q) -BIC(1, 0)| τ
Thus, Υ gives us a normalized distance between the stablest state the particle could be in (Υ = 0) and the state where it really is. The limit Υ -→ 1 correspond to a very unstable state, where the particle is a the edge of a basin of attraction and the probability to jump to another connected basin is high. The only free parameter is the choice of τ . To understand its role, we revert to the spring example: the characteristic time scale of the problem is the relaxation time of the particle to the basin of attraction. This denes a typical time scale of the system. The ∆t between subsequent observation of the time series should be close to this quantity. Instead, τ must be a multiple of this quantity but should be smaller than the residence time in the basin of attraction. In previous works, some of the authors of this paper have shown the validity of the Υ indicator to study nancial [START_REF] Faranda | Early warnings indicators of nancial crises via auto regressive moving average models[END_REF] and climate time series [START_REF] Faranda | A wavelet-based approach to detect climate change on the coherent and turbulent component of the atmospheric circulation[END_REF].

POD ANALYSIS

The goal of the paper is to examine how embedding theory results compare with those from a data-intensive analysis technique such as Proper Orthogonal Decomposition (POD). Proper Orthogonal Decomposition extracts the most energetic spatial or temporal ow patterns from either time series or a collection of spatial elds such as the velocity and/or the temperature. Any physical eld q(x, t) can be decomposed into the superposition of an innity of spatial structures φ n (x), the amplitude of which, a n (t), varies in time:

q(x, t) = ∞ n=1 a n (t)φ n (x) (1)
where the φ n (x) are eigensolutions of the eigenvalue problem

R(x, x ).φ(x )dx = λ n φ(x) (2)
where R(x, x ) is the spatial autocorrelation tensor at zero time lag. The associated eigenvalues λ n =< (a n ) 2 >, where <> represents a temporal average, correspond to the energy level of the n-th spatial eigenfunction. By construction, the amplitudes a n are uncorrelated and the spatial eigenfunctions orthogonal. Since the eld at a given time can be expressed by the instantaneous amplitudes {a n (t)} of the corresponding spatial eigenfunctions, POD directly provides a dynamical systems representation of the ow in a natural manner. The representation is energetically optimal, to the extent that the rst N POD modes capture on average more energy than any other basis of size N [START_REF] Holmes | Turbulence, Coherent Structures, Dynamical Systems and Symmetry[END_REF]. The spatial eigenfunctions are extracted from the second-order statistics of q, so that Proper Orthogonal Decompo- sition requires an extensive knowledge of the fully resolved ow and acts as a powerful data reduction technique. It is therefore of interest to compare results from this approach with the embedding technique, which is based on limited data.

RAYLEIGH-BÉNARD CONVECTION

In this section we provide a description of the ow physics. The conguration consists of a Rayleigh-Bénard square (2-D) cell in the turbulent regime (see gure 1). The top and the bottom plate of the cell are maintained at dierent temperatures, with a hotter bottom plate and colder top one. This leads to the generation of temperature plumes along the plates, which detach from the boundary layer and are transported into the ow. Natural ow in the square cell is characterized by two parameters. One is the Prandtl number P r = ν/κ where ν is the kinematic viscosity and κ the thermal diusivity. Here we consider water (P r = 4.3). The other is the Rayleigh number Ra = αg∆T H 3 νκ which measures the ratio of buoyancy and diusive eects, with α the thermal expansion coecient, g the gravity, ∆T the temperature dierence between the plates, H the dimension of the cell. Adiabatic boundary conditions are imposed on the cell sides.

The ow is numerically simulated in the Boussinesq approximation using a spectral code (see [START_REF] Xin | 3d spectral parallel multi-domain computing for natural convection ows[END_REF] for details). The simulation parameters are given in table I. The equations are made nondimensional using as characteristic units the cell height H for length, the temperature dierence ∆T for temperature and the convective velocity Ra 1/2 κ/H for velocity. More details on the numerical simulation can be found in [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF]. Three dierent Rayleigh numbers Ra = 3 • 10 7 , Ra = 5 • 10 7 and Ra = 10 8 are considered in the present study. For each of these Rayleigh numbers, the small-scale plumes are advected by a large-scale circulation (or roll), which can take two dierent orientations (see gure 1). The orientation of the roll remains nearly constant for a large portion of the time, but may switch randomly from one state to another over a relatively short duration (transition or reversal). Figure 1 shows two typical ow realizations at Ra = 5 • 10 7 respectively preceding and following a reversal. The ow therefore displays random large-scale pattern changes in the presence of small-scale uctuations. The occurrence and frequency of reversals depends on the Rayleigh number and the Prandtl number as shown by Sugiyama et al [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF]. Reversal processes in 2-D or near 2-D cells have been the object of several investigations (Sugiyama et al. [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF], Chandra et al., [START_REF] Chandra | Flow reversals in turbulent convection via vortex reconnections[END_REF], Ni et al. [START_REF] Ni | Reversals of the large-scale circulation in quasi-2d rayleighbénard convection[END_REF]), while several models have also been derived for reversals occuring in a cylinder cell (Brown and Ahlers [START_REF] Brown | Reorientation of the large-scale circulation in turbulent rayleighbénard convection[END_REF], Sreenivasan et al. Araujo et al. [START_REF] Katepalli R Sreenivasan | Mean wind and its reversal in thermal convection[END_REF][START_REF] Francisco Fontenele Araujo | Wind reversals in turbulent rayleigh-bénard convection[END_REF], Benzi [START_REF] Benzi | Flow reversal in a simple dynamical model of turbulence[END_REF]). More recently, Podvin and Sergent [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF], [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF] have used Proper Orthogonal Decomposition (POD) to construct dynamical systems to reproduce reversals in the square 2-D cell at Ra = 5 10 7 . By use of a time rescaling of the temporal series, Castillo et al. [START_REF] Castillo-Castellanos | Reversal cycle in square rayleighbénard cells in turbulent regime[END_REF] have evidenced a generic reversal cycle consisting of three successive phases.

RESULTS

POD spectrum

In previous works [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF], [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF], we have used POD to investigate the large-scale structure of the ow at Ra = 5 • 10 7 . POD was applied to the joint temperature and velocity elds, which amounts to dening a POD "energy", corresponding to the sum of the kinetic energy and thermal energy in the nondimensional units dened in the previous section [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF]. In this work we show results for two other Rayleigh numbers: Ra = 3 • 10 7 and Ra = 10 8 . The number of snapshots and separation between snapshots used in the study is given in table I. We checked that at all Rayleigh numbers the rst four modes corresponded to the same spatial structures described in [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF]. The amplitude of the dominant mode a 1 , which corresponds to a large-scale circulation scaling with the cell size, is shown in Figure 2. It is shown that reversals are more frequent as Ra increases. Moreover the amplitude a 1 is almost entirely correlated with the global angular momentum L (with a correlation coecient larger than 0.95). This conrms that the choice of the global angular momentum L, which is directly proportional to a 1 , is relevant to carry out the analysis. In the remainder of the paper we will speak indierently of L or a 1 (we actually use the notation L for a 1 in [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF] and [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF]). We note that this is a very particular case as in general the chosen physical indicator may not coincide with a single POD mode. However it is reasonable to expect that a suitable global indicator could be well approximated with a limited combination of POD modes, so that the observations made here should remain relevant in the general case.

The rst four modes of the POD capture more than 85% of the total POD energy. The full eigenvalue spectrum is represented in Figure 3 and shows that the energy in the rst modes is similar for Ra = 5 • 10 7 and Ra = 10 8 and larger than at Ra = 3 • 10 7 . igure 3(right) shows that the third mode is the least important for Ra = 3 • 10 7 , which can also be seen in table II. It is eight times less energetic than for the two higher Rayleigh numbers. However, as can be seen in table III, the dierent behaviors for dierent Rayleigh numbers do not correspond to a global increase in the energy of small scales. This suggests that the increase of the reversal frequency is associated with a higher energy in the lowest order modes, and in particular mode 3.

Phase portraits

The ow dynamics can be characterized by the phase portraits of the POD amplitudes given in Figures 4 to 6 (the amplitudes in the gure a i N = a i /(λ i ) 1/2 are normalized). As expected, the system spends a large portion of the time near one of two quasi-stable states (corresponding to a large-scale circulation which can be oriented either clockwise or counterclockwise, as was evidenced in gure 1). Each of the quasi-stable states can be dened in the POD space as a ± eq = {±|a 1 eq |, |a 2 eq |, 0, -(±|a 4 eq |)}. At random times it makes an excursion far from the state. The excursion is generally brief, and typically ends up with the system approaching the opposite steady state, which constitutes a reversal. Again, despite similarities, dierences between the Rayleigh numbers are clear in the phase portraits. As the Rayleigh number increases, the second mode takes smaller positive and even occasionally negative values during excursions (Figure 4). The system appears to spend more time outside the quasi-stable regions for the two highest Rayleigh numbers. At Ra = 5 • 10 7 , the system appears to spend a good portion of that time near a 1 = 0, which corresponds to a state where the large-scale circulation has disappeared. This is less present at Ra = 10 8 , where the presence of a limit cycle appears most probable, as can be seen in particular by comparing the dierent phase portraits in the (a 1 , a 4 ) space.

The phase portraits can be compared with the embedding portraits represented for three dierent ∆t in Figure 7. The dynamics consists of two main xed points, symmetric and located at |L| 0.07, and two marginal xed points at |L| 0.02. They are connected by two limit cycles. By increasing the Rayleigh number it is evident that the stability of the xed points decreases and the dynamics consist of limit cycles rather than xed points. One can see that both limit cycles and xed points are identied with the POD representation (Figures 4 to 6). While at Ra = 3 • 10 7 , the two main xed points are identiable, their structure becomes more stretched and circular as Ra increases. It is interesting to relate these ndings to the predictions reported in [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF] and [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF] for two models derived from the POD at Ra = 5 • 10 7 with respective dimensions of 3 and 5. The dynamics of the three-dimensional model was characterized by heteroclinic connections ( [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF]), which formed a cycle in the presence of noise, while the ve-dimensional model (with two extra modes) displayed a limit cycle ( [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF]).

In the embedding method, we nd that the optimal embedding dimension for the attractor is 2, which implies that reversals from one state to another are independent from each other. This is an important result, which is in agreement with experimental measures of the inter-reversal period, which was found to be well modelled by a Poisson distribution in cylinder cells by Xi and Xia [START_REF] Heng | Flow mode transitions in turbulent thermal convection[END_REF] and Brown et al. [START_REF] Brown | Reorientation of the large-scale circulation in turbulent rayleighbénard convection[END_REF]. This is in contrast with the results found for the Von Karman ow [START_REF] Faranda | Stochastic chaos in a turbulent swirling ow[END_REF]. For this ow, the attractor is 3D which suggests that subsequent reversals are not independent. A possible explanation of this dierence is the role of the hysteresis observed in the von Karman ow.

Stability analysis

We compute the stability parameter Υ using the following procedure: i) we coarse grain the time series of L at the three dierent ∆t = 3, 6, 9 used for the attractor reconstruction, ii) for each of the value L(t) obtained, we used the time series L(t -τ ), L(t + ∆t -τ ), ...L(t) where t ≥ τ to compute Υ. Note that we x τ = 50 for the statistical needs outlined in [START_REF] Faranda | Early warnings indicators of nancial crises via auto regressive moving average models[END_REF].

The values of Υ for each point of the time series and dierent ∆t are reported in Figures 8910. The histogram of Υ and the bivariate histogram L, Υ are reported in gure 11. Overall, we remark that there is a general agreement among the results obtained at dierent ∆t. For Ra = 3 • 10 7 , most of the values of Υ are close to 0, as the dynamics is that of a noisy xed points. During the reversal and immediately after the values of Υ are higher, indicating the switching of the dynamics. This is why Υ histograms (Figure 11 left) show a principal mode for low value of Υ and a small mode at higher values. The dynamics is dierent for the cases Ra = 5 • 10 7 and Ra = 1 10 8 as for the presence of the limit cycle. The periodicity of the reversals and the sporadic presence of a noisy xed points dynamics leads to a general increase of Υ values.

The theory says that Υ should be large when rapid changes occur in the dynamics, i.e at the onset or at the end of reversals. This means that large values of Υ should coincide with the edges of the attractor. Interestingly, as the Rayleigh number increases, large values of Y are found for an increasing and continuous range of L values, which is not easy to understand with a purely one-dimensional representation. Indeed, the histogram of a 1 (L), shown in gure 12, does not provide a clear picture of the attractor. In particular, there does not seem to be a marked dierence between Ra = 10 8 and Ra = 5 • 10 7 . However, multi-dimensional histograms in the POD space, such as the joint histogram in the a 1 -a 2 plane shown in gure 13, show that the attractor is elongated in the a 1 direction for increasing Rayleigh numbers. The values of a 1 associated with the edge of these regions agree well with the values of L for which Υ is high. Lower values of L (a 1 ) over a relatively small extent are associated with the case Ra = 3 • 10 7 , while the values increase over a wider range for Ra = 5 • 10 7 and even more so for the case Ra = 10 8 .

Time between reversals

A characteristic of the ow reversals can be given by the time intervals T between zeros of the dominant amplitude a 1 . Histograms of the time intervals are represented in gure 14-a,b,c). The mean values of the separation times T , given in table III, are very similar at the two highest Rayleigh numbers, while the mean value is higher by a factor of 6 in the case Ra = 3 • 10 7 . To try and understand this from a dynamical systems perspective , let us consider model dynamics consisting of a heteroclinic cycle characterized by a single unstable eigenvalue σu and submitted to a noise of amplitude . In that case Holmes et al. [START_REF] Holmes | Turbulence, Coherent Structures, Dynamical Systems and Symmetry[END_REF] have shown that a mean period T HC can be dened for the heteroclinic cycle, where T HC is the average time the system needs to leave the attractor, move towards the other attractor and come back again (so with our notations we have T = 1 2 T HC ). This mean period can be expressed as

T HC ∼ 2(K 0 + 1 σu (K 1 + log( 1 )))), (3)
where K 0 and K 1 are constants.

A crude estimate of the noise level here can be given by the standard deviation of the distance d eq between the current vector state a i , 1 ≤ i ≤ 4 and the nearest quasi-steady states a ± eq :

d eq = min( a -a + eq , a -a - eq )
The estimate for the noise level d rms eq is reported in table III. At Ra = 5 • 10 7 and Ra = 10 8 , the noise level appears similar and the mean reversal rate 1/ T is about the same, which suggests that the unstable growth rate is also the same, which is consistent with similar mean values of Υ (0.59 for Ra = 10 8 and 0.56 for Ra = 5 • 10 7 ). In contrast, Υ is on average twice as small at Ra = 3 • 10 7 its value being 0.26, while the average reversal rate is six times smaller. However this dierence is consistent with the fact that the noise level is smaller by a factor of 4 at this Rayleigh number.

We now take a detailed look at the distribution of the times normalized by their time-averaged value, which is shown in Figure 14. Normalizing the time separation T by its mean value T can be seen as loosely equivalent to the normalization used in the denition of the stability indicator. Although strict equivalence is not possible, since the indicator varies only between 0 and 1, we have also represented in the middle row of gure 14 the distribution of the normalized separation times limited to the interval [0, 1]. It is of interest to compare the histograms of T / T , which are obtained from the full time-series and over long periods of time, with the distributions of the indicator Υ shown in Figure 11, which is extracted from application of the autoregressive model over a short period of time. The general shape of the time distribution is dierent for each Rayleigh number. The very low values of the separation times observed at all Rayleigh numbers correspond to times which are smaller than a full transition time (i.e a complete switch from one metastable state to another). This means that there is no real reversal corresponding to these separation times: either the ow switches back to its former state, or it remains around zero for a while, which corresponds to the temporary disappearance of a large-scale circulation (which is termed a cessation). The low values are frequent at Ra = 3 • 10 7 and Ra = 5 • 10 7 , but less so at Ra = 10 8 , in agreement with both embedding and POD phase portraits. The low reversal rate at Ra = 3 10 7 is consistent with a mildly unstable system, which agrees with the distribution of Υ in gure 11.

The distributions at Ra = 5 • 10 7 and Ra = 10 8 present two main modes (see gure 14-a,b,c). One mode is associated with the relatively low values described above. The other mode associated with a higher value corresponds to a long-lived switch from one metastable state to the other i.e a full reversal. There are more reversals and less cessations at Ra = 10 8 than at Ra = 5 10 7 . A restriction of the histograms of the normalized time separation to the period [0, 1] is represented in gure 14-d,e,f) for the dierent Rayleigh numbers. It is of interest to compare these histograms, which are obtained from the full time-series and over long periods of time, with the distributions of the indicator Υ shown in Figure 11, which is extracted from application of the autoregressive model over a short period of time. The strong similarities observed betwen the time histogram and the variations of Υ support the idea that the Υ indicator is able to discriminate between the true reversals of the ow (higher values) from shorter-lived transitions (lower values).

Attractors and representative ow patterns

We retain the local peaks method for reconstructing attractors. But taking advantage of previous knowledge about the physical response of the system [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF][START_REF] Castillo-Castellanos | Reversal cycle in square rayleighbénard cells in turbulent regime[END_REF], we tune the peak identication process accordingly with physical considerations. First we extract the partial maxima from the time series of the absolute value of the angular momentum |L|. In order to avoid pointing a local peak during the transition period τ d between two plateaus, we add a specic criterion: two successive maxima must be at least separated by twice the transition duration (see [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF] for denition), which diers with the Rayleigh number. The corresponding attractors are shown in Figure 15.

As expected, most of the points corresponding to the (M i-1 , M i ) pairs are located close to the main diagonal for large |L| values. This is particularly obvious at Ra = 3 • 10 7 where the L plateaus (i.e. xed points) are very extended in time. Additional points show the path (limit cycle) between the opposite plateaus and point out transitional states in the quadrants of opposite signs. Fewer points are plotted in the center of the gure. They seems approximately equally distributed on the four quadrants for smaller |L| values. For the two highest Rayleigh numbers attractors display more complex patterns. Transition path is still present. But points appears clustering around specic locations along the transition path as well as at the diagram center i.e. for low values of L.

Figure 16 shows a rough attempt of clustering the (M i-1 , M i ) pairs for Ra = 5 • 10 7 , as well as the representative ow patterns of the M i-1 and M i states, estimated from a conditional averaging over each cluster. Ten clusters (which can be related two by two by centrosymmetry) have been identied in the (M i-1 , M i and M i states exhibits the growth of the corner ows. This physical phenomenon has been already identied in the literature (for example see [START_REF] Sugiyama | Flow reversals in thermally driven turbulence[END_REF][START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF]) and associated to the accumulation phase of the generic reversal cycle in [START_REF] Castillo-Castellanos | Reversal cycle in square rayleighbénard cells in turbulent regime[END_REF]. This is clearly shown on gure 17, where examples of (M i-1 , M i ) pairs are marked for each cluster on a particular sequence of the angular momentum time series at Ra = 5 • 10 7 . The orange cluster highlights the rst part of the transition period (called the release phase in [START_REF] Castillo-Castellanos | Reversal cycle in square rayleighbénard cells in turbulent regime[END_REF]), as the M i states point out the rebound instant of the transition on gure 17. On the contrary red clusters encompass the whole release phase from the reversal phenomenon to the following step consisting of the ow pattern reorganisation into the main diagonal roll surrounding by two smaller corner ows. The next cluster (in green) corresponds to the acceleration phase of the reversal cycle, where the large diagonal roll is being strengthened. Finally central clusters (in magenta) reveal the relative importance of the cessation regime in comparison to the plateau regime in L time series. This regime is characterized by the disappearance of the main diagonal roll [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF]. Figure 17 shows that the magenta points match well with rapid oscillations of |L| peaks close to zero which is typical of cessations.

The clustering process has been applied similarly to both other attractors (gures [START_REF] Peter | Protein hydration elucidated by molecular dynamics simulation[END_REF][START_REF] Pearson | Principal components analysis[END_REF]. It is noteworthy that the cluster shapes are quite similar whatever the Rayleigh number, as well as the corresponding representative ow patterns. However it can be noted that most clusters at Ra = 3 • 10 7 does not contain a sucient number of points to obtain a clear convergence of the representative ow patterns. Nonetheless the relative density of the clusters illustrates the relative time duration of dierent phases (as seen in gure 20). It shows shorter plateaus regarding the transition duration τ d (stability decrease of the two main xed points) as Ra increases, or the infrequency of cessations at Ra = 3 • 10 7 and 10 8 . At all Ra, the M i-1 representative ow patterns of the clusters related to the release phase (in orange or red) point out a specic ow pattern where the corner ows occupy at least half the cavity width along the top or bottom wall. This feature is typical of the precursor event as noted in [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF][START_REF] Castillo-Castellanos | Reversal cycle in square rayleighbénard cells in turbulent regime[END_REF]. Consequently this methodology of snapshots classication appears to be able to discriminate the dierent regimes of reversal or cessation [START_REF] Podvin | A large-scale investigation of wind reversal in a square rayleighbénard cell[END_REF], the successive phases of standard reversals and specic events (rebound, precursor) [START_REF] Podvin | Precursor for wind reversal in a square Rayleigh-Bénard cell[END_REF][START_REF] Castillo-Castellanos | Reversal cycle in square rayleighbénard cells in turbulent regime[END_REF] as previously identied by using dierent techniques.

DISCUSSION

In this work embedding techniques and comparison with POD analysis have been carried out for the simulation of turbulent Rayleigh-Bénard convection in a 2D square cell for Pr=4.3. Three dierent Rayleigh numbers were considered. Results suggest that signicant information on the structure of the stationary states and their stability can be recovered by the embedding technique, which can be applied on-line to the time series of a single observable. The results are consistent with the characteristics of reversals provided by POD analysis.

The embedding theory displays a two-dimensional attractor, which is consistent with the idea that the reversals are independent from each other, at least at long-term as suggested by [START_REF] Katepalli R Sreenivasan | Mean wind and its reversal in thermal convection[END_REF]. The existence of a two-dimensional description of the eective turbulent dynamics of the Rayleigh-Bénard ow is the rst main result of this paper. In fact, contrary to Landau's conjecture [START_REF] Landau | On the problem of turbulence[END_REF], low dimensional descriptions of turbulent ow exist, providing that the right observable is embedded. This result also reinforces those found in [START_REF] Faranda | Stochastic chaos in a turbulent swirling ow[END_REF] for the von Karman turbulent ow and in [START_REF] Nevo | Statistical-mechanical approach to study the hydrodynamic stability of the stably stratied atmospheric boundary layer[END_REF] for the turbulent atmospheric boundary layer. Moreover, the position on the reconstructed attractor can be used as a classier of the dierent phases of the dynamics. The second main achievement of this paper is to show that the embedding/dynamical systems approach is still of interest for studying ow with a complex behavior. The novelty is to perform the embedding using the time series of a global observable tracing the symmetry of the ow, such as the angular momentum in the Rayleigh Bénard convection in the present case or the reduced frequency of rotation in the Von Karman ow [START_REF] Faranda | Stochastic chaos in a turbulent swirling ow[END_REF]. Once the observable is identied it is a matter of seconds to embed the data and only a few minutes of CPU time on a laptop is needed to compute the stability Υ. Finally, our study also suggests that in some phenomena, as those involving the motions of geophysical ows such as the atmosphere or the ocean, it can be dicult to directly identify a single symmetry parameter and in this case POD analysis can be helpful to identify the relevant variables to which the embedding technique could be applied. The use of Empirical Orthogonal Functions, which are analogous to POD modes, is already a common practice in the climate community. When possible, combining Proper Orthogonal Decomposition and embedding theory will therefore provide a robust description of the system.

  ) diagram: two xed points at |L| | L| + σ(|L|) ( | L| and σ(|L|) being the mean |L| value and its standard deviation over the whole time series), two marginal xed points at |L| | L| -σ(|L|), and six clusters describing particular steps during the cycle limit. As previously mentioned, the most dense clusters (in blue on the gure 16) which are located close to the main diagonal, are closely related to the two main xed points i.e. plateaus. The evolution of the ow pattern between M i-1

FIG. 1 .FIG. 2 .FIG. 4 .FIG. 5 .

 1245 FIG. 1. Instantaneous ow eld in the 2D Rayleigh-Bénard cell at Ra = 5 • 10 7 for two arbitrary times preceding (a) and following (b) a reversal; Flow streamlines and temperature isocontours.

FIG. 9 .

 9 FIG. 9. Rayleigh-Bénard time series obtained at Ra = 3 • 10 7 (a), Ra = 5 • 10 7 (b), Ra = 1 • 10 8 (c) for ∆t = 6. The colorscale represents the value of Υ

FIG. 12 .

 12 FIG. 12. Histogram of the POD mode a 1 . a) Ra = 3 • 10 7 , b) Ra = 5 • 10 7 , c) Ra = 10 8 .

FIG. 15 .

 15 FIG. 15. Rayleigh-Bénard attractors obtained at Ra = 3 • 10 7 (a), Ra = 5 • 10 7 (b), Ra = 10 8 (c) with a time between subsequent peaks larger than twice the transition time τ d (τ d ∼ 30, 12.5, 15 convective time units for Ra = 3 • 10 7 , 5 • 10 7 , 10 8 respectively). Yellow lines show the paths between two subsequent (Mi-1, Mi) points.

TABLE I .

 I Ra (Nx, Nz) ∆t N snapshots ∆T snapshots N total 3 • 10 7 (159,385)6 10 -4 Simulation and POD analysis characteristics ar dierent Rayleigh numbers : numerical resolution, time step, number of snapshots used to extract the POD eigenfunctions, time separation between snapshots, total number of snapshots considered in the time series Ra λ 1 λ 2 λ 3 λ 4 3 • 10 7 0.7 0.13 0.02 0.01 5 • 10 7 1.15 0.26 0.17 0.05 10 8 1.07 0.26 0.15 0.05 TABLE II. POD rst eigenvalues at dierent Rayleigh numbers.

			825	6	12000
	5 • 10 7 (159,385) 6 10 -4	825	6	12000
	10 8 (201,513) 3 10 -4	660	4.5	8000
	Ra	λ≥5 / λ T d rms eq
	3 • 10 7	0.09		1160 0.09
	5 • 10 7	0.12		200 0.38
	10 8	0.10		195 0.30

TABLE III .

 III POD Characteristics at dierent Rayleigh numbers. T is dened as the time-averaged value separating two zeros of a 1 and is expressed in convective time units. d rms eq is dened as the room mean square value of the minimal distance between the POD state vector {a i , 1 ≤ i ≤ 4} and one of the steady states.

FIG. 18. Clusters in Rayleigh-Bénard attractors at Ra = 10 8 and the related mean ow patterns. Same legend as in gure 16.

FIG. 19. Clusters in Rayleigh-Bénard attractors at Ra = 3 • 10 7 and the related mean ow patterns. Same legend as in gure 16.

ACKNOWLEDGEMENTS

This work was performed using HPC resources from GENCI-IDRIS (Grant 2a0326). DF was supported by ERC grant No. 338965. We thank the anonymous referees for their helpful suggestions.


Abera Bereket

Yannick Yilma

Hervé Naudet

Panetto

Bereket Abera

Yannick Naudet

Hervé Panetto

Introduction to personalisation in cyber-physical-social systems

Keywords: Personalisation, Recommender System, Cyber-Physical Social-Systems, Cobotics

published or not. The documents may come

Introduction to Personalisation in

Cyber-Physical-Social Systems

From a general perspective, CPSS is a system comprising three intertwining subsystems: the Cyber, the Physical and the Social systems. The European Commission contributed a major part for the emergence of the Internet of Things (IoT) paradigm from a computer science perspective, which played an indispensable role for the orchestration of the physical and cyber systems with the goal of making tools and services intelligent by connecting tools and electronic equipments to the Internet so as to develop a network of computers and objects that are capable of self configuration [START_REF] Nunes | A survey on human-in-the-loop applications towards an Internet of all[END_REF]. On the other hand the US National Science Foundation (NSF) initially supported the notion of CPS derived from an engineering perspective, with the objective of controlling and monitoring physical environments and phenomena via the integration of sensing, computing, and actuating devices [START_REF] Kouba | A vision of cyber-physical Internet[END_REF]. Despite their initial philosophical difference, IoT and CPS share many similarities hence they have been used interchangeably without a clearly defined demarcation. There is however a fundamental difference that should be highlighted: the fact that a CPS refers to a particular system explicitly, while IoT refers at the same time to the concept, the system formed by all the connected devices and a particular system of interconnected objects. As a system, a CPS typically collects and controls information about phenomena from the physical world through networks of interconnected devices, in order to achieve its objective. Originally, humans are assumed as external entities interacting with the system. Over the years the increasing use of smart phones and the tight link with their users has lead CPS systems to exploit the large number of such users as a multifaceted source of information, i.e. human sensors. Subsequent research studies have then recognized the importance and perspectives open by a new, Human-in-the-Loop (HitL) CPS paradigm [START_REF] De | CyberPhysicalSocial Frameworks for Urban Big Data Systems: A Survey[END_REF], where humans are intrinsic actors of the system. Different techniques have then been used to introduce human actors in CPS, paving the way for the foundation of Cyber-Physical-Social Systems(CPSS). The development of CPSS, where social refers to the human aspects, is still in its infancy, especially because it often faces inevitable system instabilities, mainly due to the fact that human's actions and behaviour are the result of individual preferences, cognition, motivation and other natural or/and environmental factors. Moreover, each person is unique and might not follow system's rules that are not aligned with his way of thinking, convictions, etc. Indeed, the human behaviour is driven by complex phenomena that we do not fully understand and are still difficult to predict and manage compared to machines and softwares which are made by humans and usually prone to errors. To ensure a good functioning of CPSS, means are needed, to some extent, to have more control on the human system, while keeping individuals' freedom to operate. We think that the personalisation might be a solution; personalisation of IT services and devices, or more generally of Human-Machine Interaction. Our postulate is that personalisation provides a way to have more control over CPSS in such a way human behaviour does not interfere with the system's functioning, that is optimized for individual's satisfaction by adapting to individual's characteristics, needs, capabilities and preferences. There are several opportunities where the CPSS would benefit from, both for a better functioning of the system (the CPSS) and the user experience. The purpose of this paper is to make a first step in introducing the notion of personalisation in Cyber-Physical-Social Systems by discussing its added values, capabilities, applications, and by examining the associated issues and scientific challenges. The rest of this paper is organized as follows. Section 2 covers a brief literature review on CPSS, highlighting the common views and the widely accepted current conceptualizations of a CPSS. Section 3 presents the concept of personalisation in CPSS, and the problems and opportunities that personalisation brings when applied to CPSS. Section 4 presents illustrative Use-cases, their associated issues and scientific challenges for personalisation in CPSS. Section 5 proposes a concluding discussion.

2 Cyber-Physical-Social Systems CPSS encompass Cyber, Physical and Social systems. Recent foundational works [START_REF] Wang | The emergence of intelligent enterprises: From CPS to CPSS[END_REF][START_REF] Gharib | Towards an Approach for Analyzing Trust in Cyber-Physical-Social Systems[END_REF] presented it as an evolution of the preexisting notion of Cyber-Physical system (CPS) where Socio-technical aspects are added. In the following section we present a brief overview of the main conceptualizations of CPSS that appear in the literature.

State-of-the-art

In this section we present a brief literature review on CPSS, organized in two main categories. In the first category we introduce the different ways researchers define CPSS, whereas in the second we present the two main views conceptualizing the social aspect in CPSS.

A common understanding, shared by most of the works in the literature or constituting at least a common ground, is that CPSS is a system comprising three intertwining subsystems (i) The human-based system which refers to the social system containing human actors and their interconnected devices/agents and/or social platforms providing human-based services, (ii) The software-based system that refers to the cyber world providing software-based services including the underlying infrastructures and platforms, either on-premise or in the Cloud and (iii) The thing-based systems referring to the physical world that includes sensors, actuators, gateways and the underlying infrastructures [START_REF] Liu | Cyber-Physical-Social Systems for Command and Control[END_REF][START_REF] Candra | Cyber-physical-social systems[END_REF][START_REF] Gharib | Towards an Approach for Analyzing Trust in Cyber-Physical-Social Systems[END_REF][START_REF] Smirnov | Ontology for Cyber-Physical-Social Systems Self-Organisation[END_REF][START_REF] Zhu | Cyber-physical-social thinking modeling and computing for geological information service system[END_REF][START_REF] Su | Incentive scheme for cyber physical social systems based on user behaviours[END_REF][START_REF] Wang | Service composition in cyber-physical-social systems[END_REF][START_REF] Huang | Towards Reliable Social Sensing in Cyber-Physical-Social Systems[END_REF].

Different definitions have also been proposed. Particularly, in [START_REF] Zeng | A survey cyber-physical-social systems and their system-level design methodology[END_REF] CPSS has been defined as the integration of CPS and CSS (Cyber-Social System) to enable smart interaction between cyber, physical and social spaces, where CPS includes communicators, multimedia entertainment and business processing devices, etc. and CSS refers to social networks such as Facebook, Twitter, Youtube, etc. In [START_REF] Zeng | A survey cyber-physical-social systems and their system-level design methodology[END_REF] CPSS has also been described as the extension of IoT/CPS in the presence of humans interacting with CPS and other users.

In [START_REF] Sheth | Physical-Cyber-Social Computing: An Early 21st Century Approach[END_REF] and [START_REF] Murakami | CPSS (Cyber-physical-social System) Initiative-Beyond CPS (Cyber-Physical System) for a Better Future, Keynote Speech[END_REF], the following definition is used: "CPSS is a system that captures a synergetic interaction between computing and human experience while providing holistic computational solutions encompassing the PCS (Physical, Cyber and Social) dimensions." Some works have tried to analyze the social aspect further by subdividing it further into Social space and thinking space. [START_REF] Zhu | Cyber-physical-social thinking modeling and computing for geological information service system[END_REF] introduces the acronym CPST (Cyber-Physical-Social-Thinking hyperspace), a system established through the emergence of the new dimension of thinking space in the CPS space. The thinking space is related to high-level thoughts or ideas raised during intellectual activities.

The way of defining CPSS is clearly use-case dependent and not homogeneous. However, despite the existence of various definitions, all agree on the presence of humans as an integral part of the system. Depending on the meaning they associate to the social aspect of CPSS, most works adopt one of the two views discussed below.

1. Human as a sensor : [START_REF] Su | Incentive scheme for cyber physical social systems based on user behaviours[END_REF][START_REF] Wang | The emergence of intelligent enterprises: From CPS to CPSS[END_REF][START_REF] Gharib | Towards an Approach for Analyzing Trust in Cyber-Physical-Social Systems[END_REF][START_REF] Huang | Towards Reliable Social Sensing in Cyber-Physical-Social Systems[END_REF] conceptualize the social aspect by considering humans as information sources, i.e. sensors. This view of conceptualization primarily focuses on fusing various information originating from the social space (humans and their observations) with cyber-systems and physicalsystems in order to accommodate various application needs.

2. Human as a system part: [1-3, 8, 12] On the other hand most of the recent works intend to conceptualize the social aspect of CPSS not only by considering humans as social sensors but also as co-creators being an integral part of the system. It is also known as the human-centric way of conceptualizing CPSS [START_REF] Sheth | Physical-Cyber-Social Computing: An Early 21st Century Approach[END_REF]. This way of conceptualization considers humans as full members of the CPSS, involving observations, experiences, background knowledge, society, culture and perceptions (i.e human intelligence and social organizations (e.g. Communities)) in order to co-create products and services together with the CPS. Here, humans play the role of resources in that they provide information, knowledge, services, etc., which at the same time they consume, thus becoming users of the CPSS. Hence, the human aspect is at the center of the system design, to provide user centered services.

Definitions

Following the state of the art analysis, we propose to adopt a definition and conceptualization for CPSS that is grounded on the generic framework provided by the theory of systems. From this systemic perspective, a CPSS can be understood as an environment, or system, where humans and machines evolve in both the physical and the virtual world, interacting all together. As illustrated by Figure 1, the elements of a CPSS are its three interconnected systems, the Cyber, the Physical and the Social, which we represent as CPS and Social System, which interact in an environment composed by the physical and the virtual spaces. In an effort to a generic and uniform understanding of CPSS, we propose the following three definitions. Definition 1: CPSS. A system strictly composed of a CPS, a Social System, a virtual space and a physical space, in which the systems components interact, where CPS and Social System are defined respectively as follows.

Definition 2: CPS . A system encompassing all the systems and subsystems of Cyber and Physical Systems, their components and the interaction between them, as well as integrations of computation with physical processes [START_REF] De | CyberPhysicalSocial Frameworks for Urban Big Data Systems: A Survey[END_REF][START_REF] Zeng | A survey cyber-physical-social systems and their system-level design methodology[END_REF][START_REF] Wang | The emergence of intelligent enterprises: From CPS to CPSS[END_REF][START_REF] Zhu | Cyber-physical-social thinking modeling and computing for geological information service system[END_REF].

Definition 3: Social System. A system that comprises interacting individuals, having each their own cognition, preferences, motivation and behaviour.

In the following subsection we briefly cover Crowd systems which to a certain extent can be considered as a CPSS.

CPSS and Crowd Systems

Crowd systems are systems where a large group of people interact with an information system either in virtual or physical spaces towards the fulfillment of the system's objective [START_REF] Lykourentzou | Improving museum visitors Quality of Experience through intelligent recommendations: A visiting style-based approach, In Museums As Intelligent Environments (MasIE)[END_REF]. As it is the case in CPSS, the complexity of crowd systems arises from the existence of a human factor which can never be fully controlled. As a result, certain inconsistencies and systemic instabilities always remain in crowd systems. Examples of such systems are crowd-sourcing platforms, but also Wikis, or museums equipped with smart guidance systems. As a CPSS some crowd systems also involve CPS and Human systems interacting in Virtual and physical spaces, therefore to a certain extent, crowd systems can be considered as CPSS. However due to the fact that the interaction do not always involve physical spaces, not all crowd systems can be considered as a CPSS. Nevertheless, personalisation has been recently proposed in [START_REF] Lykourentzou | Improving museum visitors Quality of Experience through intelligent recommendations: A visiting style-based approach, In Museums As Intelligent Environments (MasIE)[END_REF] as a tool to mitigate the inconsistencies arising from the human involvement in crowd systems, so as to improve system performance and provide better services. We build on this proposal to extend this idea to the domain of CPSS, since similar challenges are shared, but in a broader context involving physical spaces.

Personalisation in CPSS

Personalisation, broadly known as customization, refers to tailoring a service or a product in a way that it fits to specific individuals' preferences, cognition, needs or capabilities under a given context. In some cases, it also targets a group of people sharing a common context. Strategies of personalisation primarily depend on the a-priori knowledge and understanding of the target individual or group so as to build a user model and latter apply the related customization. We strongly believe that it can be used in CPSS to mitigate challenges imposed by the human behaviour, cognition and preferences. In the following, we provide a high-level formalization of personalisation in CPSS, defined as an optimization problem.

Problem formalization

In general inconsistencies imposed by user's behaviour and preferences, together with constraining factors and other co-existing possibly divergent systemic objectives make the CPSS rather complex. Therefore, efficient means are needed to keep the CPSS equilibrium by fulfilling objectives respecting its constraints while at the same time fulfilling the objectives and interests of humans inside. Personalisation in CPSS needs to take into account all the constraints from the environment but the main objectives are those of the user (e.g. matching interests or preferences) and of the application (i.e. personalisation provider). This particular scenario can be formalized as a constrained multi-objective optimization problem. Providing that a personalized service generally implies making the best possible trade-offs between user interests and the general objectives. A classical personalisation problem can be formalized [START_REF] Naudet | Personalisation in Cyber Physical and Social Systems: the Case of Recommendations in Cultural Heritage Spaces[END_REF] as a function of a user u, the IT application a which implements the specific personalisation objective pursued and a context c (a set external elements) related to u or a which can have an influence on the personalisation objective (e.g. rules regarding the kinds of users, regulations, etc.), written as P (classic) u = f (u, a, c). In a CPSS, there exist multiple coexisting objectives, influencing each other and constrained by different factors from the external environment and adding to the complexity of the personalisation problem. The user evolves in a physical space, with other persons. His behaviour is constrained by the first, while influencing and being influenced by the later. Additionally, the physical space has itself a given purpose, calling for expected specific behaviours of people inside. This particular context leads to formulate the personalisation problem differently, as a function of the user u, the physical space s, the crowd of other persons in the physical space cr , the IT application implementing personalized services a and the global context c, the latter being feed typically by sensors' observations written as:

P (CP SS) u = f (u, s, cr, a, c) (1)
To illustrate this function, we can take the case of Personalized recommendations, which is a particular case of personalisation. In CPSS, with equation ( 1) rewritten as P R (CP SS) u = f (u, I, s, cr, a, c), it can be defined as the problem of finding the subset Irec u ⊂ I that an application a can suggest to a user u, under the set of constraints CO = CO u ∪ CO s ∪ CO cr ∪ CO c induced by u, the space s in which he evolves, the crowd cr to which he belongs and the influencing context c, and fulfilling at best the set of objectives O = O u ∪ O s ∪ O a linked to u, s and a.

Illustrative Use-Cases

In the following sections we illustrate the notion of personalisation in CPSS through different use cases with the goal of opening a new perspective for building personalized IT services in future physical spaces designed as CPSS.

Personalized guidance in exhibition areas

Exhibition areas are normally composed of a large number of items to be visited commonly refereed as points of interest (POI). They exist in the physical space, containing visitors having different visiting style and preferences and carrying their sensor-enabled smart devices, all together in one environment: a CPSS . In such sites visitors usually miscalculate their available time and spend wondering around exhibition sites, sometimes following different recommendations biased by many sources without visiting items that would have been interesting to them.

Introducing personalized services in such environments guiding visitors to their presumed POIs delivers a great deal of benefits to both the exhibition center and the visitors. However doing so is far from trivial not only because of the inconsistencies imposed by humans behaviour and preferences but also the existence of the IT services and the visitors together in a physical environment by itself brings additional constraints such as congestion around popular exhibits, the size of the museum and the number of items to be visited which is normally subject to available time of visitors. Apart from exhibition center's general objective of reducing congestion and increasing visitor satisfaction there might also be an objective of making less popular items to be visible. Another important scenario in exhibition areas is the case of personalized recommendation for a group visit, where each group comprises individuals having their own interests, but behaving as a single entity, the group. This is the typical scenario of the Social Choice problem, that a group of people arrive at a saddle point where a group-wise recommendation has to be provided that satisfies some notion of consensus among the group. Incorporating all these aspects in making a personalized guidance is a complex problem which has not yet been fully addressed. Some experimental results from preceding works indicated that personalized path recommendations increased the perceived visitor satisfaction and reduced congestion [START_REF] Naudet | Gaming and cognitive profiles for recommendations in museums[END_REF][START_REF] Lykourentzou | Improving museum visitors Quality of Experience through intelligent recommendations: A visiting style-based approach, In Museums As Intelligent Environments (MasIE)[END_REF] in exhibition areas like Museums. Nevertheless personalisation in physical spaces is rare due to the existence of multiple objectives and numerous constraining factors. We believe the formulation of the problem as a constrained multi-objective optimization could open a new perspective and bring us a step forward in introducing personalisation to the domain of exhibition areas for providing a better visitor experience.

Personalisation in Cobotics

Industry 4.0, often refereed to as the 4 th industrial revolution is a terminology for the emerging advent in manufacturing and automation technologies which includes CPS and humans interacting in a physical environment which could be seen as a typical CPSS. Our interest in this particular domain is the increasing trend of using Cobots in such manufacturing and automation facilities. The term cobotics is formed by joining the terms collaborative and robotics to imply the direct interaction between humans and robots [START_REF] Surdilovic | Development of collaborative robots (cobots) for flexible human-integrated assembly automation[END_REF]. A Cobot is defined as a robot that has been designed and built to collaborate with humans in various application domains such as industrial, domestic, medical, military, etc. for performing specific tasks such as , moving and carrying objects, transporting, assembling, surface processing, welding, cutting engraving, etc. [START_REF] Seban | Classification of Cobotic Systems for Industrial Applications I towards a Framework for Joint Action[END_REF]. Even though the Cobots are designed to perform a desired task, their duty in collaborating with humans might not be a satisfying experience to the human users. Despite the collaborating humans are assumed to possess similar set of skills and qualification towards the desired goal, their individual preference, cognition, motivation and other natural factors could result in an unenjoyable collaboration experience. This could lead to reduced system performance and worst case scenario failure to achieve the desired objective. In this particular scenario the main source of systemic instability comes from the complexity imposed by human nature. In addition the various industrial objectives and their constraining factors add up to this complexity. We believe introducing personalisation in this domain could greatly benefit the overall system performance and also user experience by customizing the system according to one's preference, cognition and behaviours.

Issues and Challenges of personalisation in CPSS

Personalisation in CPSS is subject to a number of challenges arising from the four components of CPSS in addition to the classical issues like, e.g. cold start, user profiling and situation identification. Since it is a broad subject which covers and tries to bring together concepts from different research areas mainly personalisation, Social systems, Cognitive science, Recommender systems, CPSS and CPS, it is vulnerable to different issues and challenges originating from these fields. The orchestration of various concepts from different fields by itself constitute a challenge adding to the complexity of the problem. In the following, we highlight some of these issues and challenges specific to Personalisation in CPSS. Interoperability Challenges: As the personalisation in CPSS is a broad notion encompassing various systems, subsystems, their components and different devices, efficient interoperability is an indispensable requirement for satisfying the overall Personalisation objectives. Indeed, different challenges arise with the orchestration of these components. For instance communication related challenges, tool selection challenges and data related challenges are among the main ones. Personalized recommendations mainly relies on data gathering about the users of personalized services, which is collected through different sensing sources (devices). Furthermore the question of which tools or sensing sources to employ, which data features to use and which techniques (algorithms) to adopt in different use cases is left for the personalisation engineer to decide. This as clearly an impact on the personalisation quality. Hence at the heart of Personalisation in CPSS, smart and context-aware interoperability is a prerequisite. Privacy and compliance Challenges: Providing personalized services in CPSS essentially requires the acquisition of Personal data of users. However it is compulsory to follow privacy rules by being completely transparent. There are a number of regulatory restrictions on tracking user's behaviour and accessing personal data of users. In this regards different countries have different rules. Users of personalized services expect service providers to know their preferences and needs. Therefore introducing personalisation in CPSS also faces a challenge to find a proper balance between the satisfaction of users with personalized offerings and organizational objectives, while keeping data secure. The data protection regulations are an important component of the context in a personalisation problem (the generic context c in equation ( 1)), which generate constraints and objectives that need to be taken into account. Human complexity Challenges: Human behaviour is a complex piece of the puzzle that is not yet fully understood. Providing personalized offerings to individuals, as such, impacts the CPSS where the individuals evolve. This in turn drives or limits their actions. Additionally human's actions and behaviour are demonstrations induced by individual preferences, cognition, motivation and other natural or/and environmental factors. On top of this, social choice scenarios like the personalized group tour recommendation, where a group comprises individuals having their own interests, but behaving as a single entity requires making group-wise decision that satisfies some notion of consensus among the group through deep and complex analysis of different preferences. Personalisation requires the modelling of human behaviour, accounting for the multiple kinds of interactions people have with the components of a CPSS. This is the main prerequisite to be able to use personalisation as a tool for building more efficient, more stable or more smart CPSS.

Conclusion

In this introductory paper we proposed an approach to introduce the notion of personalisation in CPSS which opens an interesting perspectives for the evolution and improvement of CPSS. Our approach primarily intends to provide a way for controlling the complexities caused by human behaviour in CPSS. We formalized the notion as solving a constrained multi objective optimization problem for the future to work on modeling and experimentally analyzing use cases involving physical spaces designed as CPSS. The perspective contributes both to the fields of CPSS, and personalisation/User Modelling/Recommender Systems where application to the physical world have gained momentum. This also brings opportunities to contribute to new Crowd Management approaches, matching the objectives of both the environment and the individual users.

Fig. 1 .

 1 Fig. 1. Elements of a Cyber-Physical and Social System