Chu Wang
email: chu.wang@etu.univ-amu.fr

Zhongliang Li

Rachid Outbib

Manfeng Dou

Dongdong Zhao

Symbolic Deep Learning Based Prognostics for Dynamic Operating Proton Exchange Membrane Fuel Cells

Keywords: Degenerative behavior model, Symbolic-based long short-term memory networks, Proton exchange membrane fuel cell, Dynamic operating conditions, Prognostic horizon

Fuel cell (FC) is a promising alternative energy source in a wide range of applications. Due to the unsatisfactory durability performance, FC has not yet been widely used. Prognostics and health management (PHM) has been demonstrated to be an effective solution to enhance the FC durability performance by predicting FC degradation characteristics and adopting health condition based control and maintenance. As the primary task of PHM, prognostics seeks to estimate the remaining useful life (RUL) of FC as early and accurately as possible. However, when FC faces dynamic operating conditions, its degradation characteristics are often hidden in the complex system dynamic behaviors, which makes prognostics challenging. To address this issue, a hybrid prognostics approach is proposed in this paper. Specifically, the health indicator of FC is extracted using a degradation behavior model and sliding-window model identification method. Subsequently, a symbolic-based long short-term memory networks (LSTM) is used to predict the health indicator degradation trend and estimate the RUL. The experimental and simulation results show that the proposed model is able to describe the dynamic behavior of the FC stack voltage and the extracted health indicator show a significant degradation trend. Moreover, health indicator prediction and RUL estimation performance can be improved by deploying the proposed symbolic-based LSTM prognostics model. The proposed approach provides a prognostic horizon approaching 50% of the FC life-cycle, and the average relative accuracy of estimated RUL is close to 90%.

Introduction

Fuel cells (FC) convert chemical energy directly into electrical energy with proven efficiency of 60-80% [START_REF] Mcqueen | Department of Energy Hydrogen Program Plan[END_REF][START_REF] Lohse-Busch | Technology Assessment of a Fuel Cell Vehicle: 2017 Toyota Mirai[END_REF]. In addition, when hydrogen is used as the FC fuel, zero onboard emissions can be achieved as only water is generated in FC working processes. Among different FC types, proton exchange membrane fuel cells (PEMFC) is the most widely used one thanks to its advantages, such as compact system architecture, high power density, relatively low start-up/running temperatures. These advantages make PEMFC one of the most promising alternatives for internal combustion engines in transportation applications. In terms of actual commercial applications, PEMFC does dominate.

In 2020, PEMFC shipments exceed 53,600 units and 1,030 MW, which account respectively 65% and nearly 80% of the global total [START_REF] Hart | The Fuel Cell Industry Review. E4tech[END_REF]. However, it cannot be ignored that the durability performance of PEMFC in most applications is still far from the target values [START_REF] Eudy | Fuel Cell Buses in U.S. Transit Fleets: Current Status 2018[END_REF][START_REF] Sutharssan | A review on prognostics and health monitoring of proton exchange membrane fuel cell[END_REF]. In particular, in transportation applications, FC often has to face complex operating conditions, which leads to low FC durability. For instance, the durability goal of PEMFC for long-haul trucks is 25,000 hours, but in reality, it is less than half of this goal in most practical cases [START_REF] Mcqueen | Department of Energy Hydrogen Program Plan[END_REF][START_REF] Sharaf | An overview of fuel cell technology: Fundamentals andapplications[END_REF].

Among different solutions, prognostics and health management (PHM) aims to improve the remaining useful life (RUL) of the PEMFC by providing appropriate maintenance recommendations to defer or reduce failures. In other terms, PHM is able to continuously monitor and detect FC systems, prognosticate the future State of health (SoH), and draw up timely control decisions to achieve the end of FC mission [START_REF] Jouin | Estimating the end-of-life of pem fuel cells: guidelines and metrics[END_REF]. As the main task of PHM, prognostics attempts to estimate the RUL of FC systems as early and accurately as possible. The PHM of PEMFC is divided into three phases: "Observe", "Analyze", and "Act" [START_REF] Jouin | Prognostics of Pem fuel cell in a particle filtering framework[END_REF][START_REF] Lechartier | Proton exchange membrane fuel cell behavioral model suitable for prognostics[END_REF]. Prognostics is the last step of "Analyze", which is based on the comprehensive analysis of system knowledge and historical data, and it also determines the performance of operational control and maintenance decisions in "Act". Therefore Prognostics is crucial for PHM.

In general, prognostics focuses on analyzing the characteristics associated with degenerative behavior, namely health indicators (HI). A common practice for selecting HI under constant load/operating conditions is to utilize the stack voltage. In fact, as demonstrated in [START_REF] Jouin | Prognostics of Pem fuel cell in a particle filtering framework[END_REF][START_REF] Ma | Data-driven proton exchange membrane fuel cell degradation predication through deep learning method[END_REF][START_REF] Wang | Proton exchange membrane fuel cells prognostic strategy based on navigation sequence driven long short-term memory networks[END_REF][START_REF] Wang | Bi-directional long short-term memory recurrent neural network with attention for stack voltage degradation from proton exchange membrane fuel cells[END_REF][START_REF] Javed | Prognostics of proton exchange membrane fuel cells stack using an ensemble of constraints based connectionist networks[END_REF][START_REF] Cheng | A hybrid remaining useful life prognostic method for proton exchange membrane fuel cell[END_REF], the stack voltage is suitable for prognostics of constant load FC.

Other than stack voltage, other variables can be considered as HI in load-varying applications. In [START_REF] Bressel | Extended Kalman filter for prognostic of proton exchange membrane fuel cell[END_REF], Bressel et al. propose to use the resistance and limiting current as HI and estimate the proposed HI trend using Extended Kalman Filter and static FC model. The HI extraction and prediction method is demonstrated to be suitable for combined heat and power applications with slow dynamics.

In 2020 alone, more than 75% of new FC shipments are used for transportation applications [START_REF] Hart | The Fuel Cell Industry Review. E4tech[END_REF]. In these application scenarios, the FC stack voltage is not only affected by FC degradation level, but also dependent on time-varying operating conditions and highly dynamic loads. Using the stack voltage directly as the HI is no longer effective in the case. On the other hand, in transportation applications, the dynamic operating conditions and load changes are often operator-dependent and stochastic. This makes the estimation of the internal parameters, such as resistance, more complex and complicated. To handle the dynamic operating conditions of an FC in a transportation application, in [START_REF] Jouin | Degradations analysis and aging modeling for health assessment and prognostics of Pemfc[END_REF] and [START_REF] Jouin | Estimating the end-of-life of pem fuel cells: guidelines and metrics[END_REF], the authors Jouin et al. discussed the possibility of power and cumulative energy as HI based on the polarization curves model. However, cumulative energy often carries the nuisance of error accumulation. Lechartier et al, as in [START_REF] Jouin | Estimating the end-of-life of pem fuel cells: guidelines and metrics[END_REF], proposed a model based on polarization curves and electrochemical impedance spectroscopy (EIS) for prognostics and the model owns the ability to restore the behavior of FC. Nevertheless, when deployed in practice, EIS has to be deployed when the FC is shutdown or operated with constant current for minutes, which will affect its generalization. In [START_REF] Li | Adaptive prognostic of fuel cells by implementing ensemble echo state networks in time-varying model space[END_REF], a black-box linear parameter-varying model is used for modeling the FC dynamics, and extracting the virtual voltage as HI. However, the generalization capability of the proposed method is constrained by the black-box model. Therefore, extracting HI is fraught with challenges when deploying prognostics for PEMFC under complex operating conditions and dynamic loads.

Based on an extracted HI series, the process of estimating RUL in prognostics can be classified into three categories: data-driven, model-driven, and hybrid approaches [START_REF] Sutharssan | A review on prognostics and health monitoring of proton exchange membrane fuel cell[END_REF]. Pure model-driven approaches are often accompanied by strong assumptions specific to FC technical parameters, which affect their generalization ability. Data-driven (such as [START_REF] Ma | Data-driven proton exchange membrane fuel cell degradation predication through deep learning method[END_REF][START_REF] Wang | Proton exchange membrane fuel cells prognostic strategy based on navigation sequence driven long short-term memory networks[END_REF][START_REF] Wang | Bi-directional long short-term memory recurrent neural network with attention for stack voltage degradation from proton exchange membrane fuel cells[END_REF][START_REF] Javed | Prognostics of proton exchange membrane fuel cells stack using an ensemble of constraints based connectionist networks[END_REF][START_REF] Mezzi | An echo state network for fuel cell lifetime prediction under a dynamic micro-cogeneration load profile[END_REF]) and hybrid approaches (such as [START_REF] Jouin | Estimating the end-of-life of pem fuel cells: guidelines and metrics[END_REF][START_REF] Jouin | Prognostics of Pem fuel cell in a particle filtering framework[END_REF][START_REF] Jouin | Degradations analysis and aging modeling for health assessment and prognostics of Pemfc[END_REF][START_REF] Mezzi | An echo state network for fuel cell lifetime prediction under a dynamic micro-cogeneration load profile[END_REF][START_REF] Cheng | A hybrid remaining useful life prognostic method for proton exchange membrane fuel cell[END_REF][START_REF] Liu | Remaining useful life estimation for proton exchange membrane fuel cells using a hybrid method[END_REF]) have been receiving more attention. In the data-driven prognostics category, long and short-term memory networks (LSTM), in deep-learning framework, possess powerful time series prediction capabilities, and the PEMFC prognostics based on the LSTM framework have received much attention in recent years [START_REF] Ma | Data-driven proton exchange membrane fuel cell degradation predication through deep learning method[END_REF][START_REF] Wang | Proton exchange membrane fuel cells prognostic strategy based on navigation sequence driven long short-term memory networks[END_REF][START_REF] Wang | Bi-directional long short-term memory recurrent neural network with attention for stack voltage degradation from proton exchange membrane fuel cells[END_REF]. However, the raw LSTM, called Vanilla-LSTM, strongly relies on the historical data (training set) and the precise prediction of the data beyond the training data range is considered as a hard task [START_REF] Elsworth | Time series forecasting using lstm networks: a symbolic approach[END_REF].

This leads to its inability to predict the degradation trend of HI in the long term.

The paper aims at proposing a hybrid prognostics approach to overcome the above issues. Specifically, to extract the HI in dynamic and complex operating conditions, a modified FC polarization curve function is built. Based on the function, HI can then be extracted with sliding-window model identification technique. Following that, to mitigate the challenge of reliable long-term prediction and improve the performance of the LSTM-based prognostics, a symbolic representation-based LSTM is developed to predict the HI evolution trend and realize RUL estimation. The proposed hybrid prognostics approach is validated using actual FC aging experimental data acquired in dynamic operating conditions.

To summarize, the contributions of this paper are twofold:

 An HI extraction method based on modified polarization function is proposed to extract reliable HI in dynamic operating conditions.  A symbolic representation-based LSTM prognostics method is proposed to improve the HI long-term prediction and RUL evaluation performance.

The rest of the paper is organized as follows. The hybrid approach, including the HI extraction method and prognostics method, is presented in Section 2. The aging experiment setting and experimental data are described in Section 3. The evaluation results, including the HI extraction and prognostics, are reported in Section 4. Section 5 concludes the paper.

Symbolic-based hybrid prognostics approach

FC degradation behavior model

The factors affecting FC stack degradation are complex and are coupled with operating conditions [START_REF] Jahnke | Performance and degradation of proton exchange membrane fuel cells: state of the art in modeling from atomistic to system scale[END_REF]. The polarization curve is often used to characterize FC steady-state performance and thus can be used for quantifying FC degradation level. However, the FC voltage dynamic behaviors cannot be treated properly in classic polarization function. In the paper, a modified polarization function is proposed to overcome this shortcoming and make the polarization curve suitable to extract the parameters able to reflect the degradation level in dynamic operating conditions. In polarization function, FC stack voltage is expressed as [START_REF] Larminie | Fuel cell systems explained[END_REF] ( Equations ( 2)-(4) quantify specifically the three voltage losses mentioned above.

( ) (2)
(3)

( ) (4)
In equations ( 2)-( 4), is Molar gas constant (8.3145 J/mol/K);

is Faraday constant (96485 C/mol);

is the charge transfer coefficient;

is the stack thermodynamic temperature (K);

is the stack current density (A/cm 2 );

is the crossover current density (A/cm 2 ); is the exchange current density (A/cm 2 );

is the stack equivalent internal area-specific resistance (Ωcm 2 ); is a concentration constant (V); is limiting current density (A/cm 2 ).

The internal resistance can be further decomposed into three parts, as

(5) where is the electrode resistance as shown in Fig. 1 (a), the electrode resistance comes from the electrode material; is the ionic resistance comes from the proton exchange membrane; is the equivalent static resistance, which consists of and combined; is the contact resistance, which is caused mainly by the gap existing between the exchange membrane and the electrode when they are affixed. The layer of charge on or near the electrode-membrane interface is a store of electrical charge and energy, and as such behaves like an electrical capacitor [START_REF] Larminie | Fuel cell systems explained[END_REF]. One way of modeling this is by using an equivalent circuit, with the charge double layer represented by an electrical capacitor ( ). According to Kirchhoff's law and Ohm's law, the circuit in Fig. 1 (b) can be represented as

{ ( ) (6)
where and are the voltage and current density of the contact capacitor , respectively. We can further obtain

( ) (7)
The Laplace transform of equation ( 7) yields equation [START_REF] Jouin | Prognostics of Pem fuel cell in a particle filtering framework[END_REF], which is further organized to obtain equation [START_REF] Lechartier | Proton exchange membrane fuel cell behavioral model suitable for prognostics[END_REF], and the individual variables are calculated by equation [START_REF] Ma | Data-driven proton exchange membrane fuel cell degradation predication through deep learning method[END_REF].

( )[ ( ) ] ( )[( ) ( ) ] (8)
( ) ( ) ( ) ( ) ( ) ( ) (9)
{ ⁄ ⁄ [ ( ⁄ )] ⁄ (10)
where ( ) is the system transfer function, ( ) is the output voltage, ( ) is the input current density, is the complex variable, and , , and are the system parameters to be estimated. So far, the degradation behavior model is formed by equations ( 1)-( 5) and [START_REF] Ma | Data-driven proton exchange membrane fuel cell degradation predication through deep learning method[END_REF] together. In this model, and are known constants; , and are inputs; and the remaining parameters are outputs. In the outputs, and are identified as constants, and the others will be HI candidates.

Remark:

The equivalent circuit above, especially the contact capacitor, is considered to be a comprehensive equivalent for the dynamic physical processes of the fuel cell.

Sliding-window health indicator extraction

To achieve FC prognostics, a sliding-window health indicator extraction method is proposed. The method can be divided into three steps: sliding window procedure, parameter identification and HI selection.

As in Fig. 2(a), historical data are separated into N segments by sliding a data window. The window size and single sliding length are set according to the characteristics of the input historical data. The performance degradation level is considered to be unvaried in a data segment [START_REF] Jahnke | Performance and degradation of proton exchange membrane fuel cells: state of the art in modeling from atomistic to system scale[END_REF][START_REF] Larminie | Fuel cell systems explained[END_REF]. The window size can be set to the hourly level. Meanwhile the sliding length can be set to a positive value smaller than the window size. For each data segment, the Levenberg-Marquardt algorithm [START_REF] Fletcher | Practical methods of optimization[END_REF] and the simplified refined instrumental variable method for continuous-time systems (SRIVC) [START_REF] Garnier | Identification of continuous-time models from sampled data[END_REF] are used to identify the parameters. Specially, and are identified within first segment.

Then, the segments and the identified and are sequentially fed to the behavior model to identify the other parameters. Finally, HI is selected by degradation characteristics analyzing. Typically, the appropriate HI exhibits significant and consistent trends and physicochemical interpretability.

Symbolic-based LSTM

In the fuel cell lifetime estimation stage, classical LSTM has been successfully applied for short-term prediction of degradation trends. However, for long-term prediction, the classic LSTM trained on historical time series data (e.g., fuel cell health indicator) has limitations, such as high sensitivity to hyperparameters and the inability to break the numerical scale in the prediction stage [START_REF] Elsworth | Time series forecasting using lstm networks: a symbolic approach[END_REF]. LSTM based on symbolic representation can effectively cope with the above issues. By dimensionality reduction of time series, linear trends are extracted and maintained while accelerating the LSTM training.

Symbolic-based fuel cell health indicator conversion

The fuel cell health indicator conversion method used in this paper, called adaptive Brownian bridge-based aggregation (ABBA), allows the representation and reconstruction of a health indicator. ABBA uses increments in time and value coordinates to capture data trends in health indicator without preprocessing [START_REF] Elsworth | Abba: adaptive brownian bridge-based symbolic aggregation of time series[END_REF]. The implementation of the representation consists of two stages: compression and digitization.

The core of compression is to divide the -dimensional health indicator ( ) into segments

( )
. Then the compression is achieved by approximating with straight polygonal chain. To do this, ABBA selects breakpoints ( ̃ ̃ ̃ ) in . The is the time index corresponding to the 𝑚-th (𝑚 ) breakpoint. So the -th ( ) segment can be written as

( ̃ ̃ ̃ ).
In this segment, time direction increment and numerical direction increment are 𝑙𝑒 and 𝑐 ̃ ̃ respectively. Further, the squared Euclidean distance of the values in from the polygonal chain is bounded by (𝑙𝑒 ) 𝑜𝑙 , as

∑ ( ̃ 𝑐 𝑙𝑒 ⏟ ⏟ ) (𝑙𝑒 ) 𝑜𝑙 (11)
From the inequation, the compression rate which is represented by 𝑙𝑒 can be regulated by setting the tolerance parameter 𝑜𝑙. Usually, this value ranges from 0.05 to 0.5 and a large 𝑜𝑙 indicates a coarser compression process [START_REF] Elsworth | Abba: adaptive brownian bridge-based symbolic aggregation of time series[END_REF].

Subsequently, the appropriate time and value increments form a tuple (𝑙𝑒 𝑐 ). Thus, the time series is represented by -dimensional tuples-set as

( ) [(𝑙𝑒 𝑐 ) (𝑙𝑒 𝑐 ) (𝑙𝑒 𝑐 )] (12)
Before implementing digitization, each tuple needs to be standardized and scaled. Specifically, the scaling factors and are introduced, which represent the length and incremental weights, respectively. Both can take values between 0 and 1 and satisfy . The scaled set of tuples can be calculated by

[( 𝑙𝑒 𝑐 ) ( 𝑙𝑒 𝑐 ) ( 𝑙𝑒 𝑐 )] (13)
where and are the standard deviation of the segment lengths and the segment increments, respectively.

The first step of digitization is to further assign to clusters ( ) by minimizing the within-cluster sum of squares (WCSS)

∑ ∑ ‖ ‖ (14)
where, the ( ) represent arguments for which ( ) attains its smallest value. The ‖ ‖ denotes the Euclidean norm, and is the center of the 𝑙-th (𝑙 ) cluster . Further, as in equation ( 15), letter symbols can be used to represent each cluster separately. In practical deployments, the value of can be specified manually, as will be discussed in detail in the subsequent numerical example.

( ) ( 𝑐 ) ⏟ (15)
The second step of the digitization is to use the letters in to replace the tuples in the corresponding clusters in so that a string can be obtained:

( ) (16)
where . So far ABBA converts an -dimensional health indicator into an -letter one. The digitalized data maintains meaningful information in the original health indicator and is more lightweight, which is beneficial for the rapid deployment of a neural network in the following step. On the other hand, in contrast to the representation, the reconstruction means that ABBA converts symbolic string back to a health indicator. Reconstruction can be considered as the inverse process of representation and applies a similar mathematical approach [START_REF] Elsworth | Abba: adaptive brownian bridge-based symbolic aggregation of time series[END_REF].

A numerical example is shown in Fig. 3, using ABBA to represent the fuel cell health indicator. 𝑜𝑙 is set in the compression phase and ABBA divides the health indicator into 21 segments. Subsequently, , i.e., the segment lengths and segment increments are weighted equally. Two clusters ( ) are used in Fig. 3 (a), and the obtained symbols are represented as "abbbbbbbbbbbbbbabbbbb". Then, three clusters ( ) are used in Fig. 3 (b), and the obtained symbols are represented as "cdedededdedededcdeded". Fig. 3 (c) shows the process of ABBA implementation when , where the generated training set is used for the neural network described in subsection 2.3.2. It is seen that more detailed features can be depicted with a bigger value.

ABBA-LSTM

The LSTM is a modified recurrent neural network (RNN) that effectively solves the gradient vanishing and gradient exploding issues of RNN [START_REF] Hochreiter | The vanishing gradient problem during learning recurrent neural nets and problem solutions[END_REF][START_REF] Hochreiter | Long short-term memory[END_REF]. The subtlety of the LSTM is that it can assist in maintaining and updating the activity of neurons through an intrinsic chain of state transfers. As in the left part of Fig. 4, in the hidden unit of a Vanilla-LSTM (raw LSTM), there exist input gate ( ), forget gate ( ), and output gate (𝑜 ) consisting of a sigmoid function ( ), in addition to an internal state unit (𝑐̃ ) consisting of tanh function. These can be deployed by equations ( 17)- [START_REF] Elsworth | Time series forecasting using lstm networks: a symbolic approach[END_REF], where ℎ denotes the number of hidden units in an LSTM layer. The output vector (ℎ ) and the unit state (𝑐 ), which are continuously passed between each hidden unit, are represented by equations ( 21)

and [START_REF] Larminie | Fuel cell systems explained[END_REF], respectively. It is with the collaboration of ℎ and 𝑐 that a special chain is formed between the individual hidden units. The mathematical realization of a LSTM unit can be summarized as

( ℎ ) (17)
( ℎ ) (18)
𝑜 ( ℎ ) (19)
𝑐̃ ( ℎ ) (20)
𝑐 𝑐 𝑐̃ (21)
ℎ 𝑜 (𝑐 ) (22)
where represents the -dimensional input, and represent the weight matrices of the forget gate, the input gate, the output gate and the internal state unit, respectively. represent the recurrent connection matrices in the forget gate, the input gate, the output gate and the internal state unit, respectively.

The operator " " represents the Hadamard product of the elements.

However, Vanilla-LSTM trained based on the historical values of the data can hardly predict results beyond the value range of training set [START_REF] Elsworth | Time series forecasting using lstm networks: a symbolic approach[END_REF]. This constrains the direct use of Vanilla-LSTM for some specific applications, especially for fuel cell prognostics where the predicted health indicator are supposed to be in different value interval from the training data. Using time series incremental information in the ABBA symbolic representation facilitates health indicator evolution trend capture, which promisingly gives ABBA-LSTM the ability to predict beyond the value range limits of the original training in fuel cell prognostics.

The structure of ABBA-LSTM is shown in Fig. 4, which contains five layers: one input layer, two LSTM layers, one fully connected layer and one output layer. Specifically, the ABBA-LSTM input is an -row -column Boolean matrix

. Consistent with the subsection 2.3.1, represents the number of compressed segments and represents the number of digitized symbols (number of clusters). The -th ( ( )) segment can be represented as a -dimensional Boolean row vector ( ). In this row vector, there is only one element corresponding to the segment symbol , set as (𝑙 ( )) and the rest of the elements are set to 0. Subsequently, the input will pass through two LSTM layers, each containing 50 hidden units. Following the LSTM layers is a fully connected layer containing plain neurons (with the equivalent function ( ) ). Finally, a softmax layer outputs a -row -column probability matrix, where is the number of predicted time steps. The highest probability in each row of the probability matrix is selected and its corresponding symbol is used as the output, and a string ( ) can be obtained as the prediction result. This prediction result, as final output, will be converted into a health indicator by the ABBA reconstruction.

Hybrid prognostics approach

The hybrid prognostic approach combining the HI extraction and the ABBA-LSTM prognostic model is shown in Fig. 5.

As mentioned in Subsection 2.2, HI is identified by using the proposed the degradation behavior model and the sliding window model identification procedure. Thanks to the low preprocessing requirements of the ABBA-LSTM model, the historical HI time series can be fed directly to the model as input, as in Fig. 5 (a).In this work, to cope with the model uncertainty, multiple ABBA-LSTM models are constructed by setting random initial weight matrices.

Once the models are properly trained, ABBA-LSTM models can realize one-step or few-step ahead prediction. The long-term HI prediction is achieved by iterating the prediction procedure using the last-step prediction result as input.

The predicted trends are further used to estimate the RULs, as in Fig. 5 (b). The prognostic starting point corresponds to the time . Given a failure threshold for HI, its intersection time with each predicted trend is the predicted end-of-life (EOL). The estimated RUL at is calculated as (

) 23
where the subscript denotes the index of ( ). The RULs further constitute the prediction distribution.

The highest peak of the distribution (not the mean) is chosen as the final RUL.

FC aging experiments under dynamic operating conditions

Generally, obtaining long-term lifetime data for fuel cell operation, especially under dynamic operating conditions, is fraught with challenges. A compact low-power scale FC stack with an open cathode and dead-end anode structure is focused on and used to test the proposed prognostics approach. The FC is designed with self-humidifying capability.

The specific technical parameters are summarized in Table 1. The stack is integrated with a 24V DC fan for oxygen supply and cooling. The stack operating temperature is controlled by regulating the air fan and the reference varies with temperature as [START_REF] Garnier | Identification of continuous-time models from sampled data[END_REF] where is the temperature control reference and is the operating current. In the concerned compact fuel cell system, stack temperature, current and stack voltage are measured and analyzed for prognostics. It is noted that, an identical air fan is used for both air supply and temperature regulation. The air fan speed is regulated to make the fuel cell temperature attain the predefined temperature. The air pressure and stoichiometry are therefore highly coupled to temperature. Hence, the temperature is a representative variable to describe the fuel cell dynamic behavior. More details about the experimental setting and the experimental dataset can be found in [START_REF] Li | Adaptive prognostic of fuel cells by implementing ensemble echo state networks in time-varying model space[END_REF].

This aging experiment lasted 1486 hours, and the vast majority of the load current densities are below 0.4 A/cm 2 , as in Fig. 6. When the FC is operated at a small current density, it suffers from little concentration loss [START_REF] Larminie | Fuel cell systems explained[END_REF]. Therefore, we assume that the concentration loss is combined with the linear ohmic resistance losses at low current density conditions.

Further, the degradation behavior model in this case is simplified to

( ) (25)

FC cyclical dynamics

The aging experiments aimed to restore the dynamic operating conditions of FC in a hybrid electric vehicle (HEV).

In the application, the FC stack coupled to lithium-ion batteries and functions as a range extender. Specifically, once the battery state of charge (SOC) drops to a preset threshold, the FC is activated to charge the battery with constant current. When the battery SOC reaches at the targeted level, the charging process is paused, and the FC system is kept as a standby. In this work, a programmable electronic load is used to realize the above mentioned cyclic start-up-standby process as Fig. 7. The operating current decreases from 8 A to 0 A for about 1 second, as in Fig. 7 (a).

Correspondingly, the voltage rise process lasts about 170 seconds, as in Fig. 7 (b). On the other hand, it takes about 30 seconds to raise the operating current from 0 A to 8 A, and the corresponding voltage drop process takes about 170 seconds. It is noteworthy that the figure shows the early stage of the FC stack aging experiment. As the aging experiment progresses, the operating voltage response time is slightly extended. The overshoot phenomena exhibited by the operating voltage during current transitions can be simulated and equated by a capacitor-like dynamic element as mentioned in Section 2.

Long-term evolution of operating conditions

In the aging experiment, the FC even has to cope with the long-term evolution of operating conditions. It is important to note that, unlike the cyclic dynamic loading mentioned in subsection 3.1, the long-term evolution of the operating conditions comes from the coupled influence of multi-scale factors. In other words, this long-term evolution reverts to a certain extent to the HEV diverse environmental effects. The Fig. 8 shows the long-term evolution of the voltage (e.g., (a), (b)) and current (e.g., (c), (d)) in the FC aging experiment. First, in order to better describe the long-term evolution of the operating conditions that occurred in this aging experiment, four characteristic time points were set: 122 h ( ),

h ( ), 702 h ( ), and 168 h (

). The locations where the polarization curve tests were performed are marked in Fig. 8 (c), which corresponds to the anomalies that appear every approximately 168 hours in the operational behavior.

The aging experimental process is then divided into four phases: Phase [START_REF] Mcqueen | Department of Energy Hydrogen Program Plan[END_REF], as in Fig. 8(d), is from hour 0 to point . The FC stack gradually reaches the optimal working condition after the first start, influenced by the internal catalyst activity and other factors. Phase (4), the remaining part after point . During this phase, the FC effective operating current remained essentially 8 A. After point , the stack was restarted and the incomplete 5th polarization test was repeated at hour 721.

It is worth noting that all the three stack serious faults (points , , and ) occurred while the polarization test was being performed or shortly after the test completed. Distinct from the load cycle variations mentioned in subsection 3.1, the 2nd-9th polarization curve tests require the consideration of eight different currents between 0 and 10 A. This means that the operating conditions of the stack during the polarization test are more complex and prone to induce faults. In addition, the effect of ambient temperature should not be ignored. During the aging experiments between 0 hours and point , occasional laboratory air conditioning system faults triggered ambient temperature fluctuations.

The stack-integrated DC fan regulated the temperature while also changing the oxygen supply to the cathode, which in turn affected the operating behavior of the stack. After the FC restart at point , the laboratory ambient temperature remained constant and the operating current became smooth thereafter. Correspondingly, the operating voltage after point also shows a smoother reduction behavior.

Performance assessment and results analysis

Assessment metric

In order to fully and appropriately evaluate the proposed approach, the performance of the degenerate behavior model and the RUL estimation are validated separately in this paper. Specifically, considering that the degenerate behavior model output is identified stack voltage, the relative accuracy ( ) achieved when extracting HI is evaluated by

( | | ) (26)
where is the identified stack voltage.

On the other hand, the performance of RUL estimation is evaluated from 3 aspects. The first is the prognostic horizon (PH) [START_REF] Jouin | Estimating the end-of-life of pem fuel cells: guidelines and metrics[END_REF][START_REF] Saxena | Metrics for evaluating performance of prognostic techniques[END_REF]. In simple terms, PH will be given an acceptable error zone ( ). By setting the upper and lower boundaries parallel to the actual RUL, the is defined as the enclosing region formed by the upper and lower boundaries together with the two coordinate axes, as

where is the actual end of life. and are accuracy modifiers that restrict the lower and upper bounds of , respectively, which in turn affect the tolerance of PH. For the FC prognostic, when the estimated RUL is biased, the result smaller than the actual RUL is usually more acceptable. On the contrary, when the estimated RUL is larger, it tends to lead maintenance plan lag, even the fatal failure early occurrence. In view of this, different upper and lower bounds are set, i.e., and , respectively. If an estimated RUL can fall into , the prognostic is considered reliable. When there is a breakpoint after which all estimated RULs are guaranteed to fall into . Then this point is defined as the starting time point , and at this time PH can be calculated by [START_REF] Saxena | Metrics for evaluating performance of prognostic techniques[END_REF] It is important to note that PH appears to be relatively lenient and it is a minimal evaluation metric. In general, for a prognostic approach, a longer PH means better performance. On the contrary, approaches that do not meet the PH metrics are not worth adopting.

Similar to , a more stringent acceptable error zone ( ) is introduced. Parameter , called window modifier is used to describe the position of the fatal failure in the PH [START_REF] Saxena | Metrics for evaluating performance of prognostic techniques[END_REF]. In this paper, is considered to be set to 1, i.e., the fatal failure occurs at the EOL. can be calculated by

( ) ( ) (29)
The reliability of the prognostic approach can be further assessed by examining how well the estimated RUL falls into , which namely performance.

Finally, in order to quantitatively assess the accuracy of the estimated RUL, the relative accuracy ( ) is also introduced, similar to , and can be calculated by

( | | ) (30)
where is the estimated RUL.

Performance of the degeneration behavior model

The first step in deploying the hybrid prognostic approach is the sliding-window HI extraction. Specifically, the window size and sliding length are set to 2 hours. Sliding split is performed on historical data lasting 1486 hours, and parameters are identified in each window. different results from the other tests. The remaining 8 tests showed an extremely strong correlation with ohmic overpotential. Among them, the 1st, 3rd and 4th polarization curves show an increasing trend, which is very interesting and comes from the change in operating conditions and the influence of ambient temperature mentioned in Section 3.

The 4th-6th polarization curves almost overlap, which is due to the effect of fault points and , and in fact, a reversible performance improvement occurred when the FC is restarted after experiencing a shutdown. The 6th-9th polarization curves show a steady trend of degradation (decline).

The parameter identification procedure first determined the and as in Table 2. Subsequently, as in Fig. 9 (b), the identified activation losses parameters and show significant changes at points , , and , and become stabilize after 702 hours. This indicates that these two parameters are sensitive to the operation changes of FC.

The identified Ohmic Losses parameters are shown in Fig. 10 and Table 2. Among them, and are relatively stable in the first half, while in the second half there are monotonic-like variations. Meanwhile and show a clear upward trend, and the represents whole stack being selected as HI. It is worth noting that some extracted resistances show negative values, which occur mainly before the 6th polarization test, especially between points and . This corresponds to the previous analysis of the activation losses parameters and , indicating that the internal resistance values are also sensitive to the operating condition changes. Even so, the extracted HI still provides trend features sufficient for prognostic purposes. Further, in Fig. 10 (e), by comparing with the stack power. The extracted HI not only shows changes in operating conditions, but also indicates faults of the stack. Specifically, when there are significant breaks or gouges in the HI, such as points , , and , it means that a change in operating conditions or a serious fault/shutdown has occurred at that time. When the HI shows a short term (about tens of hours) bump followed by a return to the original trend of change, this indicates a recoverable stack fault.

Remark: 1) The faults indicated by HI refer to abnormal behaviors of the fuel cell system, such as abnormal operation, peripheral device fault, abnormal shutdown, recoverable fault, etc. 2) the complete polarization curve test is not mandatory in HI extraction procedure. Nevertheless, it is recommended to do the initial polarization curve test to correctly identify the constant parameters and the reasonable parameter intervals in Table 2. Moreover, by bringing the above identified parameters into equation [START_REF] Elsworth | Abba: adaptive brownian bridge-based symbolic aggregation of time series[END_REF], the identified stack voltage can be obtained. Then the degradation behavior model is quantitatively evaluated using equation [START_REF] Hochreiter | The vanishing gradient problem during learning recurrent neural nets and problem solutions[END_REF]. 11 (b), is higher than 90% for almost all data points except for few outliers. In fact, the average calculated over all 1486 hours is 99.33%. It is worth mentioning that the sampling rate of the aging experimental data is 1 Hz, which means that the above average accuracy comes from millions of data.

Performance of RUL estimation

The second step in deploying the hybrid prognostic approach is the RUL estimation. As shown in Fig. 12 (a), considering the particularly unusual trend change in HI at point , the first 122 hours HI should be ignored. A breakpoint is set every 74 hours (5% of the total duration), and totally 10 breakpoints are set from hour 594 to hour 1262. The proposed hybrid prognostic approach is implemented at each breakpoint separately.

Specifically, the main parameters involved in ABBA are set as follows: 𝑜𝑙 , , . The three hyperparameters are set in a trial-and-error way. On the one aspect, the parameters should be assigned to achieve satisfactory ABBA representation. On the other aspect, the precision of RUL estimation should also be regarded. For Additionally, thanks to the symbolic-based representation, the training computing burden can be reduced. In our case, ABBA-LSTM takes 667.96 seconds while Vanilla-LSTM takes 768.43 seconds.

In assessing the performance of the estimated RUL, the PH was first verified. As in Fig. 13 Further, as in Fig. 14, the proposed method was compared with the autoregressive integrated moving average model (ARIMA) and echo state network (ESN). In the comparison experiments, the input data of all three remain consistent with the previous, as in Fig. 12 (a). The comparative results show that the proposed method can provide longer PH.

Within the respective PH interval, the proposed method achieves higher and fits the more closely. This also indicates that the proposed method has a credible long-term prognostic capability despite the lack of data pre-processing.

Discussion

The fuel cell prognostics in dynamic operating condition which represents one of the most difficult conditions is treated in this paper. More particularly, the prognostics is realized based only on the historical data of the concerned fuel cell stack. The prognostics model training and testing could be realized by using the data from a single long-term fuel cell test.

In this paper, a low-power stack with simple system design is focused on. The strategy could be generally applied to fuel cells of different power levels or current densities. However, it should be noted that fuel cells operated in different current density regions could demonstrate different dynamic behaviors coming from not only the fuel cell stacks, but also the auxiliary components and their control methods. In these conditions, the modified polarization function should be adjusted to the specific fuel cell system.

Conclusion

In this paper, a hybrid prognostic approach is proposed for estimating the RUL of PEMFC. A degradation behavior model is constructed and the health indicator associated with FC degradation is extracted through it. The degradation trend of health indicator is subsequently predicted with the help of ABBA-LSTM and RUL is estimated. The prognostic approach is evaluated using data from the FC aging experiment, and its performance can be summarized as  The ABBA-LSTM is capable of capturing the trend information of health indicator and the prediction of the data in different data range can be achieved.

 The proposed approach provides a prognostic horizon approaching 50% of the FC life-cycle, and the average relative accuracy of estimated RUL is close to 90%.

The proposed approach has the ability to provide prognostic results with a wide horizon and high confidence, allowing more ample time for maintenance plan developments. However, it cannot be neglected that the degradation behavior hidden in FC systems is complex and difficult to be fully extracted. In the future work, efforts will be made to study FC intrinsic degradation phenomena at multiple scales and the influence of recovery faults.

Fig. 1 (Fig. 1 .

 11 Fig. 1 (b) shows the equivalent circuit of the resistance within the FC.

Fig. 2 .

 2 Sliding-window HI extraction procedure, (a) historical data sliding-window separation; (b) parameters identification and HI selection.

Fig. 3 .

 3 inc 1 ) = (2, -0.01602) (len 2 , inc 2 ) = (1, 0.4169) (len 3 , inc 3 ) = (1, -0.52007) ⁞ (len 21 , inc 21 ) = (1, 1.27819) S = [c, d, e] Symbolic representation of the time series, (a) setting to k=2 (2 symbols), the time series representation is "abbbbbbbbbbbbbbabbbbb"; (b) setting to k=3 (3 symbols), the time series representation is "cdedededdedededcdeded"; (c) the implementation of ABBA at k = 3.

Fig. 4 .

 4 ABBA-LSTM hidden unitct-1 ht-1

Fig. 5 .

 5 ABBA-LSTM2

Fig. 6 .Fig. 7 .

 67 Fig. 6. Voltage-current density Curve from the 1486 hours FC aging experiment.

Fig. 8 .

 8 Evolution of operating conditions, (a) operating voltage; (b) operating voltage in detail; (c) operating current; (d) operating current in detail;

Phase ( 2 )

 2 , from point to point . At the beginning of the 2nd test (hour 168.5), a peripheral fault occurred, represented by point , which caused a sudden drop in the stack voltage but did not cause a FC shutdown. After the 4th polarization test, an abnormal FC shutdown occurred at point .Phase (3), pointsto . During the 5th polarization test, the abnormal FC shutdown shown at point occurred.

9 .

 9 The polarization curves fitted by the degradation behavior model are shown in Fig.9(a), and the fitted results match well with the experimental data. The 2nd test was affected by a peripheral device fault (point ) and presented very (a) (b) Fig. Polarization test and activation losses parameters identification, (a) the 1st-9th polarization curves fitting; (b) charge transfer coefficient and exchange current density.

Fig. 10 Fig. 10 .

 1010 Ohmic Losses parameters identification, (a) ionic and electrode resistance; (b) contact resistance; (c) health indicator (equivalent internal resistance); (d) equivalent capacitance. (e) Indication of dynamic operating conditions and faults via HI.

Fig. 11 (

 11 a) shows the identification results of the stack voltage for all 1486 hours, with a zoom-in display of the results from hour 1219 to hour 1231. The results show that the model output voltage can identify the cyclical dynamic load changes and the operating condition long-term evolutions well. The proposed model can capture the voltage overshoot behavior in transition periods. Further, the model identification performance can be described quantitatively using . As in Fig.

Fig. 11 .

 11 ABBA-LSTM compared with Vanilla-LSTM with the same LSTM configuration at the hour 742 breakpoint. It is seen that the prediction results of Vanilla-LSTM hardly exceed the historical value range. In contrast, the features represented by ABBA effectively reduce the anomalous fluctuations while maintaining the degradation trend.

Fig. 12 .Fig. 13 .

 1213 (a), the light pink zone represents the . When the estimated RUL falls into this zone, it is considered a largely plausible result. Starting from hour 668, all subsequent results fall into . The actual EOL is 1356 hours, and the PH can be calculated as 688 hours. Next, the light blue zone in Fig. 13 (a) represents , which is used to match with the estimated RUL in Deployment of RUL estimation, (a) HI without outliers and breakpoints setting; (b) at hour 742, predicted degradation trends by ABBA-LSTM; (c) at hour 742, predicted degradation trends by Vanilla-LSTM. RUL assessing: (a) prognostic horizon and α-λ performance; (b) relative accuracy.the PH interval. The results show that the estimated RUL is well matched, and only two outliers appear at hour 742 and hour 1262, as in Fig.13 (b). Specifically, at hour 742, it is due to the abnormal shutdown of point . And at hour 1262, it is due to a recoverable fault near hour 1225. The average in the PH interval is subsequently calculated as 88.62% using equation (30).

Fig. 14 .

 14 Prognostic comparative results: (a) prognostic horizon and α-λ performance; (b) relative accuracy.follows. The proposed degradation model has the ability to identify the stack voltage dynamic behavior, and the average relative accuracy of the identification results reaches more than 99%. The health indicator extracted by the proposed model shows a clear degradation trend. Also, the health indicator has potentially the ability to indicate different types of faults.

Table 1

 1

	Aging Experiment Operating Condition
	FC type	Open cathode/Dead-end anode
	Active surface	33.63 cm 2
	Nominal pressure at hydrogen inlet	1.35 bar
	Nominal pressure at oxygen inlet	normal atmospheric pressure
	Number of cells	15
	Nominal output power	73.5 W
	Operating temperature	corresponding to current
	Maximum temperature	75 °C
	Maximum current	13.45 A
	Lowest permitted stack voltage	7.5 V

Table 2

 2

		Model Parameters Identification
	Parameter	Value (or range)	Unit	Note
		13.37	V	Identified in first
		1	mA/cm 2	sliding-window
		0.004 to 0.13 0.6 to 4	-mA/cm 2	Fig. 9 (b)
		-7 to 14	Ωcm 2

Acknowledgement

This work was supported in part by the China Scholarship Council (CSC) under Grant [grant number 201906290107].


Jean-Michel Couturier
email: couturier@lma.cnrs-mrs.fr

ETHERACTION: PLAYING A MUSICAL PIECE USING GRAPHICAL INTERFACES

This paper introduces the use of graphical interfaces to interpret an electroacoustic piece, Etheraction. Electroacoustic pieces, commonly created for tape, can now be interpreted in live performance with dedicated interactive systems; the interaction between the performers and these systems can use graphical interfaces, largely implemented in nowadays computers. When using graphical interfaces for real time sound control, the tasks consist in controlling sound parameters through the manipulation of graphical objects, using pointing techniques or direct control with additional devices. The paper presents how I have designed two interactive systems dedicated to interpret in live Etheraction, a multichannel piece I have initially composed for tape. The piece is based on the motion of physical models of strings that control sound parameters. The two devices control both synthesis parameters and spatialisation parameters, are based on interactions with graphical interfaces, and use specific controllers.

INTRODUCTION

With progress of technique, more and more electroacoustic pieces, usually created for tape, are interpreted in live performance. The interpretation devices will depend on the nature of the piece and on the choice made by the performers. In case the piece contains a lot of synthetic sounds, those sounds can be played in real time using existing synthesizers, or using a specific device, especially designed for the piece. The modularity and the flexibility of digital and electronic tools enable to build a device dedicated to one musical piece. But designing such a device is not always easy; one must take into account which parameters one wants to control in each part of the piece: which ones can be fixed or driven by an automatic process, if some high level parameters can be defined. In a second step, one has to choose which gesture controllers to associate with the parameters.

The problematic is not the same than in designing a digital musical instrument: if in both cases, the work consists of linking controllers to synthesis parameters (mapping, [START_REF] Wanderley | Performer-Instrument Interaction: Applications to Gestural Control of Music[END_REF] [START_REF] Arfib | Strategies of Mapping between gesture data and synthesis parameters using perceptual spaces[END_REF]) there are lots of differences. A musical instrument is generally built to be used in several musical pieces, and those pieces are conceived while the instrument already exists; dedicated devices are built either after the piece or simultaneously to the piece creation. They are only used to play the piece. Another difference is that in a musical piece, sound processes can differ along the piece; the performer can choose if he wants to use different devices for each part or to use a unique device for the entire piece. In both cases, a lot of parameters have to be manipulated, and not necessarily all at the same time.

This paper introduces a new way of designing device that can be used for a piece interpretation, manipulating specific graphical interfaces. Graphical interfaces enable to display a lot of graphical objects that we can manipulate with the same controller; each graphical object is linked to synthesis parameters. The shapes of the graphical interface and its objects have no physical constraints; this gives more freedom to the designer. This paper introduces how I have designed devices with graphical interfaces to interpret a specific piece, called Etheraction. Section 2 introduces the use of graphical interface in music performance; section 3 describes the Etheraction musical piece, and section 4 the design of the interfaces and their use in live performance.

USING GRAPHICAL INTERFACES IN THE CONTROL

The graphical interfaces are not essential in a computerbased musical device, unlike in many computer applications, but they can provide a high level of interactivity in the performance. This section introduces the use of graphical interfaces in the context of real time performance: how to act in the interface, with which controller, what are the advantages.

Controlling graphical objects

Commonly implemented in nowadays computers, the graphical interfaces are often used in music softwares. In case of real time sound control, the tasks consist in controlling sound parameters through the manipulation of graphical objects, according to the direct manipulation principles [START_REF] Shneiderman | Direct Manipulation: a Step Beyond Programming Languages[END_REF]. All sound parameters are controllable via graphical objects that generally represent real objects like piano keyboards, faders, buttons, etc. The graphical interfaces tend to reproduce on screen an interaction area that is close to a real one, like front panels of electronic instruments. The aim of such interfaces is to make the user feel he has real objects in front of him. Generally, the use of graphical interface needs a pointing device, like the mouse, that controls the position of a graphical pointer displayed onscreen. In order to be manipulated, graphical objects need to be activated by the user, with the pointing device. To activate a graphical object, the user has to put the pointer over the object (pointing task) and to press a button (clicking task). Once activated, the data of the pointing device are linked to the graphical object in a way predefined in the software. The graphical object is inactivated when the user releases the button (unclick task). This interaction technique enables to use only one controller, the pointing device, to manipulate all the graphical objects. This technique is implemented in today computers and used to control WIMP (Windows, Icons, Menus and Pointing) interfaces, with a single mouse as pointing device. Nevertheless, complex musical tasks, with numerous parameters to control simultaneously, cannot be performed in real time with a single pointing device. Performers must use additional controllers or advanced interaction techniques, as we will see in the following paragraphs.

To have a better control on sound, music softwares usually use specific controllers, like MIDI ones, to control the graphical objects of the interface, and their associated sound parameters. Those controllers are directly mapped to the corresponding graphical objects, giving by this way a more direct access to the graphical object: there are no pointing and clicking tasks (activation task) in the interface, and several graphical objects can be manipulated simultaneously. This second interaction technique, which could be called direct mapping technique, seems better adapted to real time sound control, but is more expensive in hardware and less flexible than the pointing technique. The Etheraction devices, as it shows in section 4, use the two interaction techniques complementarily.

Beyond the single pointing technique and the direct mapping technique, advanced interfaces have been developed in the field of HCI (Human Computer Interaction). Those interfaces are more efficient than the traditional WIMP interfaces; some of them use bimanual gestures [START_REF] Guiard | Asymmetric Division of Labor in Human Skilled Bimanual Action: The Kinematic Chain as a Model[END_REF], mix real and graphical objects (tangible interfaces: Audiopad [START_REF] Patten | Audiopad: A Tagbased Interface for Musical Performance[END_REF], ReacTable [START_REF] Kaltenbrunner | Dynamic Patches for Live Musical Performance[END_REF]) or use 3D graphics [START_REF] Mulder | Design of Virtual 3D Instruments for Musical Interaction[END_REF]. At NIME 2003, Daniel Arfib and I introduced the pointing finger device [START_REF] Couturier | Pointing Fingers: Using Multiple Direct Interactions with Visual Objects to Perform Music[END_REF] (figure 4, 5 th picture), a multifingers' touchscreen-like device. This type of system provides the most direct and intuitive possible interaction: one can, with his fingers, manipulate graphical objects as if they were real objects. The pointing fingers use six DOF (degrees of freedom) sensors attached to four fingers and switch buttons on fingertips; this device gives the position of four fingers regarding a screen. A special program manages the graphical objects and disables conflicts between the different pointers. We design two musical instruments with this device, one scanned synthesis instrument and one photosonic instrument. This interface uses the pointing technique with four pointers, which allows the control of numerous parameters simultaneously.

Pointing devices and additional controllers

To manipulate a 2D graphical interface with a pointing device, one should use at least one controller that enables a pointer to move on a 2D plane; at least we need XY and a button. To drive several pointers, one can use several controllers or use one controller that gives several 2D coordinates. In some cases, one wants to use a specific controller (for example a piano keyboard or a foot controller) beside the pointing device: in this case, if necessary, the graphical interface can contain a graphical element linked to the controller. Then, the musical device can be described according to two points of view: in the first case, the controller manipulates the graphical element that is linked to the sound parameters (like in figure 3); in the second case, the controller directly manipulates the sound parameters and the graphical element is just a visual feedback. The first point of view is pertinent if the graphical element behavior is very close to the gestures that are done on the controller; in this case, when the performer uses the controller, he really has the impression that he manipulates the graphical interface (in fact the graphical element). This feeling will improve the user's immersion in the device and the interaction. Moreover, adding extra controllers will extend the amount of parameters controllable simultaneously.

Graphical interfaces and interpretation

If the graphical interfaces help to build digital musical instrument, they can be even more efficient to interpret a musical piece, especially if a lot of parameters have to be manipulated but not all at the same time, as when sound processes differ along the piece. Numerous objects can be designed, from current graphical interface buttons and sliders to specific synthesis objects (like interacting with a string through the pointing fingers device). Graphical interfaces enable to display all the parameters and the associated graphical objects in a same area, and enable to manipulate them with the same controllers; if we use other controllers, the graphical interface could integrate them and give good visual feedback. It provides an important freedom in the design of real-time sound systems.

The next sections will introduce a musical piece, Etheraction, and the control device I have built to interpret it.

ETHERACTION: INTERPRETATION OF AN ELECTROACOUSTIC PIECE

Etheraction is a musical piece I have composed in early 2004. The first version of the piece was a recorded one; once this version was completed, I have built devices with computer, graphical interfaces and controllers to interpret the piece in live situation. The recorded version was diffused on March 9 th 2004 at the GMEM, Marseilles, and the live version was performed on April 7

th 2004, at musée Ziem, Martigues, France.

Etheraction recorded version

Etheraction was composed in early 2004 as an 8-channel recording. This piece is based on the motion of physical model of strings that control synthesis parameters and uses digital sounds produced with Max/MSP patches. Most of the sounds of the recorded version were generated using gestures: the synthesis parameters were driven by gesture picked up by different controllers: graphical tablet, touch surface, joystick. I have built this piece in several steps: I have first played one by one the different elements and then I have spatialized them one by one (group 2 excepted) with a custom version of Holospat from GMEM (the sound was spatialised in real time with gestures). All the gesture data were recorded (a sort of automation); then I was able to modify and adjust some parameters afterwards, without replaying all the sequence. At the end, all the parts were mixed together in Logic Audio Environment. The piece is divided in three parts and uses different sound process, as shown in figure 5: Group 1 uses scanned synthesis method [START_REF] Verplank | Scanned Synthesis[END_REF] [5] with spectral aliasing. Sounds of group 2 come from 8 filters banks (with 8 filters each); the input sound is first a recorded sound and next a pink noise. Each filter banks is manipulated by a physical model of slow moving string: the string contains 8 masses and each mass position controls the gain of one filter. Each filter bank is associated to one channel (speaker). Group 3 is a mere texture of loud sound that only changes in dynamics. Sounds of group 4 come from filtering string instrument [START_REF] Arfib | Gestural strategies for specific filtering processes[END_REF], with some improvements. Sounds of Group 5 are short sounds from the filtering string instrument. They are mixed to create precise responses to some group 4 events. Group 6 is very close to group 4, but has different presets, that enable the production of very different sounds.

Etheraction live version

The more important difficulty was to build devices that enable to play all the elements of the piece, from the beginning to the end: in the recorded version, all events were created one by one, spatialised one by one, and then mixed together. To interpret the piece, a lot of sound processes have to be controlled and spatialised simultaneously. For the live version, the spatialisation has been made on four speakers to reduce the computation. All the devices have been built to keep the general spirit of the original version.

To design the devices that enable to perform Etheraction in live, I have first analyzed all the sound processes that were used in the recorded version, and have tried to see what processes could be grouped together and played with the same device. The idea was to share out the group processes in a minimum number of devices, keeping the cohesion of each element. Each device has been built to be controlled by one performer: he has to be able to control alone all the parameters. All devices are based on interaction with graphical interfaces, following the method we are applying to design digital musical instruments [START_REF] Arfib | Design and Use of Some Digital Musical Instruments[END_REF], adapted to the use of graphical interface. Starting from sound process, I try to define a minimum set of control parameters, then, I try to find which parameters could be controlled trough the graphical interface and which have to be controlled by an "external" controller. The graphical objects are defined according to the nature of the parameters they are controlling; when it was necessary, I have integrated in the graphical interface some visual feedbacks of the gesture done on the "external" controllers.

As seen in figure 5, there are five groups that intervene at different parts of the piece. Those groups have been displayed on two devices; those devices are described in the next section.

DESCRIPTION OF THE DEVICES

As seen before, all sound processes can be controlled by only two devices; this section introduces those devices. All graphical interfaces were created with Jitter, the video complement of Max/MSP. The synthesis technique and the mapping were implemented in Max/MSP.

First device

The graphical interface of the first device is controlled by an interactive pen display, a mouse and a 2D pedal (a modified joystick). The 2D pedal is dedicated to only one part of the interface, the spatialisation; the control of spatialisation consists in the displacement of a point in a 2D space. The other graphical objects are controlled by the pen and the mouse; to manage the graphical objects and to enable the use of several pointers in a same graphical interface, I use a specific Max/MSP object that I have designed for the Pointing Fingers device (section 2.1). The circular control of the "extra" parameter (a deformation of the force profile applied on the string) is incremental: turn clockwise will increase the parameter, and inversely; this enables a precise control on a very large scale. The string damping and centering stiffness are controlled by a 2D slider; to help the performer, four presets of these parameters can be loaded thanks to buttons. The loud texture of group 3 is controlled by a slider, and a visual feedback of the string shape is displayed at the bottom right.

Second device

This device is much more complex, because the sound processes are not the same along the piece, and there are numerous parameters to drive simultaneously. To lighten the graphical interface, I have created three different graphical interfaces, one for each part of the piece (figure This device uses three different controllers: a graphical tablet (which gives the angular position of the pen: tilt), which is used to control the graphical interface and the spatialisation; a Tactex [START_REF]Tactex, touch surfaces[END_REF] multi-finger touch surface that controls the forces applied on the different strings; and a foot controller with switches and two expression pedals. The pedals control two string parameters and a visual feedback of the pedal positions is displayed on the interface; the switches are used to choose one of the three interfaces and lock or unlock the pen on the control of the spatialisation. The spatialisation control uses the pen tip coordinates and the tilt: the displacement of the two extremities of the pen (perpendicularly to the tablet) controls the position of the two channels that are spatialised.

Use in live performance

I have used those graphical interfaces on stage with Denis Brun, Electroacoustic student, at the "Concert Emergence" at Martigues, in April 2004. For this first live performance, the devices were not as developed as described in previous sections. In the first device, the mouse was not used, a joystick was used instead of the 2D pedal and the string was not displayed. In the second device, only one graphical interface (instead of three) had been used. The learning period of the devices have been shortened thanks to the use of graphical interfaces. Displaying the shapes of strings increases the feeling of immersion in the device: we don't have a set of complex parameters to control but a physical object which we are interacting with. The position of the pedals and the pressure of the pen tip are difficult to evaluate; seeing them on screen provides a great help for their manipulation.

Nevertheless, with the second interface, I have encountered some difficulties to control both sound and spatialisation, in spite of the help of the graphical interface; whatever the quality of the musical device, a complete learning phase is always necessary to play the device.

CONCLUSION

With the live version of Etheraction, I have tried to experiment the use of graphical interface to interpret an electroacoustic piece. Etheraction uses complex sound processes with a lot of parameters to control. The graphical interfaces gives a visual representation of the sound processes as well as a control area adapted to them, making possible their control and their spatialisation in real time.

This experience shows me that a lot of strategies can be used to create an interface, according to the constraints and the musicians (composers and interprets) preferences. Creating such interface is very different than creating a digital musical instrument: here the interface has to be adapted to the music.

ACKNOWLEDGMENTS

I want to thanks Denis Brun for testing and performing the interface with me on stage and Magnolya Roy for the graphic design of the interfaces.

Figure 1 .

 1 Figure1. The usual Mapping chain links gesture data to sound parameters; with the graphical interface, there is an additional step in the mapping: graphical objects are linked to sound parameters and the gesture device can control any graphical objects.

Figure 2 .

 2 Figure 2. Different tasks in the control of graphical objects using a pointing device: activation (a, pointing task and b, clicking task), manipulation (c) and inactivation (d).

Figure 3 .

 3 Figure 3. An example of direct control of graphical interface through a specific controller. The graphical sliders displayed on screen are permanently linked with the physical sliders of the controller.

Figure 4 .

 4 Figure 4. Different controllers that can move one or several pointers in a graphical interface. They can be bimanual, with an object to hold, with a screen, multi-fingers, invasive / non invasive, …

Figure 5 .

 5 Figure 5. Etheraction overview. Each group corresponds to a different sound process. For the live version, the different groups of sound processes are dispatched on two devices (gray, white); Group 5 elements are generated auto-matically and are functions of group 4 interactions.

Figure 6 .

 6 Figure 6. Graphical interface of the first device. The graphical objects can be manipulated with an interactive pen display and a mouse; the spatialisation is controlled by a 2D pedal. The pen tip pressure controls the amount of force applied on the string and corresponds to the radius of the circle in the middle of the pen tip pointer. (the mouse pointer is not on the figure).

  the performer can switch between the interfaces using tabs.

Figure 7 .

 7 Figure 7. Second control unit. This device uses three different graphical interfaces, corresponding to the 3 parts of the piece. The performer can switch between the different interfaces using tabs. In all the parts, the strings are exited by forces controlled by a multi-finger touchpad.

Figure 8 .

 8 Figure 8. Martigues' concert, the two Etheraction devices and the performers in action.


domain: shs qfin

text:
Olivier Godechot

Paula Apascaritei

István Boza

Martin Hallsten

Lasse Henriksen

Are Hermansen

Feng Hou

Jiwook Jung

Alena Křížková

Zoltán Lippényi

Elvira Marta

Silvia Maja Melzer

Eunmi Mun

Halil Sabanci

Naomi Kodama

Size and evolution of the financial wage premium

Size and evolution of the financial wage premium.

It is no secret that salaries in finance are high. The financial press in the 2000s was full of examples of bonuses paid in London or on Wall Street. That financial remuneration has strongly contributed to the rise of contemporary inequalities is almost common knowledge. On the one hand, we hear the golden legend of the self-adjusting labor market: high demand for exceptional skills. On the other, the black legend of the dictatorship of financial markets resonates: traders manipulating prices, abusing clients and profiting from the misfortunes of the rest of the economy. Between the two, an intriguing and often misunderstood phenomenon: a persistent wage rent in the financial professions. We propose to evaluate and explain it by pointing out the phenomena of appropriation of economic activity by financial operators.

Persistence of high salaries in finance and contribution to rising inequality

The excesses of the 2000s and the brutal collapse of the financial market in 2008 produced a first wave of largely convergent work showing the contribution of finance to the increase in economic inequality, notably in France [START_REF] Godechot | Is finance responsible for the rise in wage inequality in France?[END_REF], the United States (Philippon and Resheff, 2012), the United Kingdom, or in OECD countries [START_REF] Kus | Financialisation and income inequality in OECD nations: 1995-2007[END_REF][START_REF] Godechot | Financialization is marketization! A study of the respective impacts of various dimensions of financialization on the increase in global inequality[END_REF].

To complement this finding we use the very detailed administrative wage data collected by the COIN group (Tomaskovic-Devey et al. 2020) for 13 countries.2 Figure 1 presents the wage gap between employees in finance and those in other sectors and allows us to draw several conclusions. First, wages are higher in finance than elsewhere, with a ratio ranging from 1.2 times higher in South Korea to 2.3 times higher in Spain. Second, in most countries, this gap is more pronounced at the top of the distribution. In Canada and Sweden, for example, the top percentile of finance salaries is, in 2006, 2.5 times higher than the top percentile of other sectors, while the ratio of the average of these two sectors was 1.4. 3 Third, the wage gap between finance and other sectors increased over the period in almost all countries (with the exception of the Czechia, the Netherlands and South Korea). The pace of growth is more pronounced before the crisis, especially at the top 1% threshold, and it slows down (for the average) or stabilizes (for the 1%) after 2008. Despite a major financial crisis and the beginnings of bonus regulation in Europe, there has been no significant decline in compensation in the financial sector.

Figure 1. Wage gap between the finance and insurance sector and other sectors

Note: In 2015, in France, average salaries were 1.6 times and the top percentile 2.1 times higher in the finance-insurance sector than elsewhere.

Although the size of the financial sector remains modest (between 2 and 6% of the workforce), this high wage gap is not without consequences for inequality.

Previous work has shown that, in the decade before the crisis, 40% of the increase in the top 1% share of the wage bill in France, and 70% in the UK, benefited members of the national top 1% working in finance [START_REF] Godechot | Is finance responsible for the rise in wage inequality in France?[END_REF][START_REF] Bell | Bankers and their bonuses[END_REF]. Table 1 replicates this analysis over our group of 13 countries for about 20 years. With the exception of Japan and the Czechia, finance is a strong contributor to rising inequality, and more often than not by more than 30 percent, especially before the crisis. Note: The share of the national top 1% in Sweden increased from 3.7% to 4.5% of the national wage bill between 1991 and 2011, an increase of +0.04 percentage points per year. 49% of this increase went to members of the national top 1% working in finance. To mitigate volatility, we smooth the series with a three-year moving average. Insignificant changes are shown in parentheses.

The wage rent in finance

The wage gap between finance and other sectors is not sufficient in itself to characterize finance salaries as rent. 4 It could be that these exceptional salaries are the product of rare skills. Indeed, the financial sector underwent a major revolution in the 1980s, with the mathematization and computerization of securities portfolio management, leading to the recruitment of highly qualified people trained in mathematics, physics, economics and computer science [START_REF] Godechot | Les traders: essai de sociologie des marchés financiers[END_REF][START_REF] Ho K | Liquidated: an ethnography of Wall Street[END_REF]. [START_REF] Célérier | Returns to talent and the finance wage premium[END_REF] attempt to demonstrate that high salaries in finance are solely due to "talent", which they approach through the rankings of French engineering schools. The return on "talent" is certainly much higher in finance than in other sectors [START_REF] Célérier | Returns to talent and the finance wage premium[END_REF]. But this fact shows above all that the financial rent is captured mainly by employees from the best grandes écoles. Unless one is invoking a rent phenomenon, the article does not explain why alumni from the same grandes écoles would earn less in other sectors. Moreover, the much better careers achieved by business school cohorts who entered the labor market during the financial boom, compared to those who entered during the financial crisis, offer a quasi-experimental evidence that finance offers wage rents [START_REF] Oyer | The making of an investment banker: Stock market shocks, career choice, and lifetime income[END_REF]. Indeed, cohorts in bad years are no less "talented" than those in good years. They are just less likely to start their careers in finance and stay there.

The existence of a financial wage premium is now well documented. [START_REF] Philippon | Wages and human capital in the US finance industry: 1909-2006[END_REF] show, taking into account skill differentials, how finance in the United States paid its employees more than other sectors throughout the twentieth century. The wage premium peaked at 40 percent in the early 1930s, then fell to 0 percent between 1945 and 1980, before rising sharply again to +50 percent in the mid-2000s. Using data from the U.S. CPS Employment Survey, Lin and Tobias Neely (2019) further investigate the heterogeneity of the wage gap and its evolution between finance and other sectors. In the early 1970s, employees at the bottom of the wage distribution benefited more strongly from a job in finance (+35% more pay for the top decile in finance compared to the top decile in other sectors) than those at the top (+20% for the top 5% threshold). 5 In contrast, a few decades later, employment in finance favors the top (+60%) more than the bottom (+10%). Finally, [START_REF] Denk | Financial Sector Pay and Labour Income Inequality: Evidence from Europe[END_REF] finds that the financial wage premium in 2010 averaged +28% in European countries.

In France, when controlling in the EDP-DADS panel for degree and standard demographic variables, the "financial wage premium" varies between +10% and +40% (Figure 2). While it fluctuated between +8% and +15% in the 1980s, it rose considerably in the 1990s to reach a high plateau in the 2000s between +25% and +30%. This premium has heterogeneous effects on the different regions of the wage distribution. At the beginning of the period, the finance effect is maximal at the level of the third quartile. From the mid-1990s onwards, the effect becomes maximal for the top of the wage hierarchy, in particular for the top 5% of earners. We then see their remuneration vary according to the rhythm of the economic situation: booms in 2001 and 2008 rewarding the good years of the stock market in 2000 and 2007, declines in 2003 and 2009 following the bursting of the internet bubble and the subprime crisis.

The salary premium for the "financial market managers" occupations, with the same controls, is even more spectacular. These occupations earn between 40 and 64% more than managers and professionals in other sectors. Moreover, the distribution of these salaries is very unequal. The bottom quartile is not much better off in these occupations than in others. On the other hand, the advantage is considerable for the 90th percentile, which is paid three times more than elsewhere.

Figure 2. Evolution of the wage premium in finance in France

Note: We calculate the wage premium for working either in the financial sector or as a financial markets manager controlling for gender, age, seniority, social category, working in the Île-de-France region, and degree. The average effect is obtained with an ordinary least squares regression. We use quantile (unconditional) regression models to obtain the effect at the 25, 90 or 95 percentile level. Sources: EDP-DADS panel.6

The appropriation of the activity

We propose to explain this financial rent on the basis of a "hold-up" mechanism, i.e. a situation in which one party obtains an economic advantage by threatening the other with a contractual breach that would devalue its investments. More than elsewhere, certain employees, such as traders, can appropriate the key assets of financial companies, in particular intangible assets that the company cannot easily protect, such as human capital (knowledge, know-how, etc.) and social capital (clients, teams). They can then move or threaten to move these assets to a competing company in exchange for higher salaries. For example, the head of a trading room and his deputy managed to obtain 10 and 7 million euros in bonuses respectively in 2001 after threatening to move their entire teams, and thus the core of the firm's financial activity, to another bank [START_REF] Godechot | Wages, bonuses and appropriation of profit in the financial industry: the working rich[END_REF]. The "hold-up" mechanism thus offers a more realistic explanation for some very high salaries than the superstar theory [START_REF] Rosen | The economics of superstars[END_REF][START_REF] Célérier | Returns to talent and the finance wage premium[END_REF]. According to the latter, small initial differences in talent are multiplied by the size of the market (or stock portfolio) to which they are applied. The leverage effect of size may have some relevance. But above all, we propose to extend the notion of "talent" not only to "innate" talent or talent acquired during education, but also to "talent" acquired in the workplace and more generally to all the resources accumulated (including social resources) in the financial firm and that can be transported elsewhere. In a work in progress, we are trying to quantify the benefits of shifting activity, in particular the wage gains obtained by employees during a full shift migration to a competing firm. For example, financial market executives increase their wages by 12% more when moving as a team rather than individually.

Conclusion

The appropriation of financial activity thus provides a strong lead to explain the distribution of income in finance and to understand why, at "market equilibrium", compensation in this sector is higher than elsewhere. It shows how the ordinary functioning of the financial labor market can contribute to global inequality.

Table 1 . Evolution of wage inequality in 13 countries and contribution of the finance-insurance sector to this increase

 1

		Top 1 % share			Pre-crisis
				Yearly	Finance	Yearly	Finance
	Country	Start End	evolution	contribution	evolution	contribution
	Germany (1991-2014)	3.3%	3.8%	+0.023%	14%	+0.034%	26%
	Canada (1990-2012)	6.5%	8.4%	+0.084%	47%	+0.207%	33%
	South Korea (1990-2011)	4.0%	4.2%	(+0.012%)	(76%)	(+0.024%)	(126%)
	Denmark (1995-2014)	3.7%	4.8%	+0.060%	38%	+0.050%	45%
	Spain (2007-2015)	6.3%	6.9%	+0.071%	50%	/	/
	United-States (CPS) (1990-2014) 6.9%	11.3%	+0.186%	4%	+0.105%	19%
	France (1994-2014)	5.5%	6.5%	+0.050%	66%	+0.108%	49%
	Hungary (2004-2010)	8.7%	8.7%	(-0.002%)	(1705%)	(-0.019%)	(-79%)
	Japan (1990-2012)	3.5%	3.8%	(+0.015%)	(-117%)	(+0.012%)	(-207%)
	Norway (1997-2013)	3.7%	4.7%	+0.062%	30%	+0.109%	19%
	Netherlands (2007-2017)	6.6%	7.1%	+0.046%	27%	/	/
	Sweden (1991-2011)	3.7%	4.5%	+0.042%	49%	+0.060%	52%
	Czechia (2003-2015)	5.7%	6.2%	+0.043%	-41%	+0.166%	-6%

These data are exhaustive for France, Canada, the Scandinavian countries, and the Netherlands, almost exhaustive for Hungary and the Czech Republic, and represent 4 to 8 percent of female employees in Germany, Spain, South Korea, and Japan. Finally, we add data from the US employment survey (CPS) based on a more modest population of 50,000 to 100,000 workers.

Recall that the top percentile of wages is the wage threshold above which the top 1% of earners are found. The wages reported in the German and US databases are respectively capped around the top decile (Germany) and P97 (US). Our method for "de-capping" them probably underestimates the differential in these two countries between the top percentile in finance and that in other sectors.

Rent can defined as a difference in the remuneration of a given factor of production compared with the situation that would prevail in a perfect market and with perfect factor mobility. If human capital were perfectly mobile, it should be remunerated at the same price in different sectors. The difference with respect to this reference is therefore a rent.

Recall that the first decile is the wage threshold below which the lowest paid 10% are found.

The EDP and DADS data were accessed through the CASD (Centre d'accès sécurisé aux données) after authorization from the Comité du secret statistique.


Fabienne Venant
email: fabienne.venant@loria.fr

Semantic visualization and meaning computation

. It allows a semantic visualization that can be used to compute the meaning of a lexical unit in a given context

Introduction

Meaning construction -how words take meaning in a sentence or in a text -i s an important issue for computational linguistics. The major problem we have to deal with is that each word can have a lot of different meanings, depending on context. This phenomenon -called polysemy -is constitutive of language, and is the basis of its richness. However it is quite difficult to formalize. In most models of language, polysemy is considered as a kind of artefact. In these models, polysemy amounts to very little: a choice in a list of pre-existing meanings. However the omnipresence of polysemy always leads this kind of computation to combinatorial explosions. To avoid this problem, we want to give a central place in meaning construction to polysemy: that is why we define our model within the framework of continuous mathematics. This model was first proposed by [START_REF] Victorri | La polysémie, construction dynamique du sens[END_REF]. It is deeply rooted in Gestalttheorie [START_REF] Guillaume | La psychologie de la forme[END_REF]. Each linguistic unit is associated with a semantic space, where its different meanings are organized according to semantic proximity. The other units of the utterance define a potential function, which c 2008.

Licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported license (http://creativecommons.org/licenses/by-nc-sa/3.0/). Some rights reserved. allows us to determinate the region of the semantic space corresponding to the meaning of the unit studied within the utterance.

Figure 1: Model

We present here VISUSYN, the software we developed in order to implement this model. VIS-USYN is still a work in progress. The working version was developed in Matlab. A Java version will be developed as soon as possible.

Presentation of VISUSYN

VISUSYN makes two kinds of operations, corresponding to the two componants of the model: it constructs semantics spaces and then computes meaning functions defined on these semantic spaces.

The algorithm for building semantic spaces relies on the analysis of a graph of synonyms. The aim of the exploration is to reveal the structure of the lexicon modelled by the graph so an automatic system can reach the information it contains. VIS-USYN can construct local spaces, representing the semantic of a given unit, as well as global spaces, representing a lexical paradigm in its whole (for French adjectives, or adverbs for instance). Local spaces are used to compute meaning of the unit under study when accompanied by a given word in a sentence. To compute the corresponding potential function, VISUSYN uses cooccurence data from large scale corpora.

3 Small-world graphs It has been discovered recently that most lexical graphs belong to the class of small-world graphs. This term denotes graphs where most nodes are neighbors of one another, but every node can be reached from other by a small number of hops or steps. Small-world networks were defined by [START_REF] Watts | Collective dynamics of small-world networks[END_REF]. They noted that graphs could be classified according to their clustering coefficient (C) and characteristic path length (L). Additionally, a third property can be associated with small-world networks even though it is not required for that classification. Specifically, if a small-world network has a degree-distribution which can be fitted with a power law distribution, it is taken as a sign that the network is smallworld. These networks are known as scale-free networks. [START_REF] Ravasz | Hierarchical organization in complex networks[END_REF] showed that a high clustering coefficient with scale free topology determines an original combination of modularity and hierarchical organization. It is not a simply pyramidal organization. The structure is made of groups of nodes, with small clusters at the bottom and very large groups at the top. Moreover, groups of nodes may overlap at any level. This self-similar nesting of different groups or module into each other forces a strict fine structure on real networks [START_REF] Ravasz | Hierarchical organization in complex networks[END_REF].

Building semantic spaces

The algorithm will be here illustraded on the French adjective lexicon. The graph under study, called Synadj, is a graph of synonymy with 3,699 vertices and 22,568 links. We verified the smallword structure of this graph (C, L and degree distribution). This small-world structure led us to use the cliques1 of the graph as a tool for building the semantic space. A clique in a graph is a maximal set of pairwise adjacent vertices, or -in other words -an induced subgraph which is a maximal complete graph. In the present case, a clique is made of adjectives which are all synonyms in a one to one relationship. By virtue of the definition, small-world networks will inevitably have high representation of cliques, and subgraphs that are a few edges shy of being cliques, i.e. smallworld networks will have sub-networks that are characterized by the presence of connections between almost any two nodes within them. This follows from the requirement of a high cluster coefficient. We can consider as a first approximation that the cliques define very precise meanings that can be considered as the intersection of the meanings of all the units belonging to the clique. We thus define the semantic space as the euclidian space generated by the vertices of the graph (the adjectives, here). Each clique of the graph is associated with a point of this space, which coordinates depend on wich vertices belong to the clique. VISUSYN uses the chi-square distance to compute the distances between the cliques. Then a principal component analysis is applied to reduce the dimensionality of the space. In order to build a local semantic space (for example associated with a given word), we select a sub-graph, made only by the word under study and all its synonyms. The local space only contains the cliques of this subgraph2 . Figure 2 shows a visualization of the semantic space associated with the French adjective sec (dry, severe, brusque...). It accounts for the six main meanings we can find in a dictionary. very centre of the space only contains intensive meanings like authentique; certain; evident; incontestable (authentic; certain; evident; incontestable) or agréable; charmant; enivrant; ravissant; séduisant (agreeable; delightful; exciting; attractive). These meanings are very general meanings and can apply to any nouns.

There are now many semantic branches more or less long growing out from the central core. These branches are very dense near the centre and then go in all the directions becoming sparser and sparser. They are homogenous from a semantic viewpoint. Each branch only contains one sort of adjectival meaning: relational, qualificative, or intensive. VISUSYN had also been used to explore other global lexicons, like French verbs (Figure 4 - [START_REF] Gaume | Hierarchy in lexical organisation of natural languages[END_REF]) or French adverbs ending in -ment (more or less corresponding to english adverbs ending in -ly).

Meaning Computation

Let's go on with French adjectives. An attributive adjective is always linked to a noun. It is this noun which mostly constrains the meaning of this adjective, even if other units like the article can play a role.

We show here how Visynsyn can be used to automatically find which synonyms match better the meaning of an adjective (here the French adjective sec) when used with a given noun. In order to do this, VISUSYN associates a characteristic func-Figure 4: Global semantic space: French verbs (3D) tion with each synonym. To compute the value of the function in a given point, it evaluates wether the synonym belongs to the corresponding clique or not. The basins of this function represent the meaning zones of the semantic space in which the synonymy between the word and the given synonym is relevant.

To compute the meaning of sec when used with a given noun, VISUSYN associates a potential function with the noun. The value of this function in each point depends on the frequencies of cooccurrence of the noun with the adjectives of the corresponding clique. The basins of the function determine the zone of the semantic space corresponding to the meaning of sec when used with the associated noun. Figure 6 shows the potential function associated with fleur (flower). In this case the noun fleur forces sec to take a precise meaning in the zone 'lack of water'. The method of disambiguation consists in comparing the function of a synonym to the function of the noun under focus. The more the functions overlap, the more you can replace sec with its synonym without changing the meaning of the syntagm. This method has been used to compute the overlap rate for 20 nouns among the most frequently used with sec in a large corpus (Frantext, http://atilf.atilf.fr/frantext.htm). The results automatically processed were compared with the human answers on the same task. The rate of success is 79% [START_REF] Venant | A geometric approach to graph exploration[END_REF].

This works for a fined grained disambiguation. VISUSYN can also deal with a more macrocopic partition of the semanctic space. Figure 2 shows a division of the semantic space of sec in 6 zones. VISUSYN can decide automatically which of these six zones correspond to the meaning of sec , when used with a given noun. The method is similar to the previous: It now associates each zone (and not only each synonym) with a characteristic function. This function is computed according to the cliques belonging to the zone. For each noun, VISUSYN compares its potential function with that of each zone. It thus computes an affinity rate between each noun and each zone. This method gives better results than the previous

Conclusion

Although VISUSYN is still a prototype, it seems very promising. Our basic result is that the kind of visualization presented here displays the structure of lexical graphs. It also constitutes an original tools to explore the structure of small-world graphs. As the algorithm is independent of the nature of the relation modelized by the graph, it could be used to explore other small-world graphs It offers interesting possibilities of using semantic informations in computational linguistics, especially in the domain of automatic disambiguation. (We also evaluated its results on verbs and noun). From a more theoretical point of view, it accounts for the validity of the underlying model. Of course this model has some limits and the work is still in progress. We have to develop the system. However, this work shows how continuous mathematics can be relevant for semantic modelization and encourages us in the challenge of using continuity for corpus linguistics.

Figure 2 :

 2 Figure 2: Local space associated with the French adjectives sec

Figure 3 :

 3 Figure 3: Global semantic space: French adjectives (2D)

Figure 5 :

 5 Figure 5: Characteristic function associated with the adjective brusque

Figure 6 :

 6 Figure 6: Potential function associated with fleur (flower)

Following the idea first proposed by[START_REF] Ploux | Construction déspaces sémantiques á láide de dictionnaires de synonymes[END_REF]

[START_REF] Ploux | Construction déspaces sémantiques á láide de dictionnaires de synonymes[END_REF] first proposed the algorithm and built the lexical data. Local semantic spaces can be consulted at http://elsap1.unicaen.fr/dicosyn.html or http://dico.isc.cnrs.fr/fr/index.html.