Simon Enjalbert

Livia Maria Gandini

Alexander Pereda Baños

Stefano Ricci
email: stefano.ricci@uniroma1.it

Frederic Vanderhaegen

Human Machine Interface: a review for future applications in trains' driving

The paper deals with the description of preliminary research work based on extended investigation focused on categorized Human Machines Interface (HMI), describing their technical approaches and emerging operational issues. The study aims at reviewing and analysing HMI, their design and control systems currently on the market or in experimental operation. The analysis acts as a review manual on HMI technologies to design customized HMI and analyse the technologies used or tested in the present technological scenario. The paper starts by addressing the systems developed by manufacturers of surface vehicles outside railway field: trucks, cars and ships. It follows with the performances of simulators developed for various transport systems: rail, cars, aviation and integrated solutions. Finally, the focus is on the comparative analysis of the performances of supporting tools for train driver's assistance, a synthetic analysis of results and the organization of the survey among drivers to check expectations and acceptance of potential solutions.

Introduction

The field of automation is gaining increasing interest from research in various application fields, along a path began in the 1920s, with the first radio-controlled vehicle. Today, there is an increasing interest of companies to invest towards large-scale adoption of self-driving vehicles. Studies on the user experience aboard autonomous or semi-autonomous vehicles are in close connection with the question of trust in the Human Machine Interface (HMI). In particular, in a semi-automatic vehicle, the design of HMI must give awareness of driving situation, to regain effectively the control, whenever called upon to intervene. Within the CARBODIN research project, co-funded under the Shift2Rail Programme, the aim is meeting the expectations of train drivers and staffs with HMI automatic functions to arrive to progress towards their introduction. The focus is on the identification of required inputs and available outputs of various systems and operator preferences, further investigated by dedicated surveys on potential HMI configurations to design new drivers' cabins and build a virtual mock-up with immersive technology using specific banks of sounds and gestures.

Path towards the future train drivers cabin

The railway transport systems in general are historically less sensible than others are to the pilot application of disruptive technologies. This is firstly due to the combination of the rigorous intrinsic safety requirements qualifying it and the general longer technical life of its components and the system itself. Therefore, in view of the progressive introduction of the innovations supporting the train driving functions, the CARBODIN project [START_REF] Shells | H2020 -Shift2Rail Joint Undertaking[END_REF] aims at joining the identification of the most promising technological driving supports with the expectations of train drivers or other railway operator staffs about HMI in future cabins. The basis of this are 360° surveys about potentially applicable mature technologies on one side as well as preferences and efficiencies of possible new innovative configurations on the other side, by taking into account the key role of human factors, particularly in the design of input and output sensors, gestures and supporting acoustic and visual signals.

State-of-the-art in driving automation

Within the framework of what discussed above, the progressive introduction of the innovations supporting the train driving functions, started from the analysis of systems developed by manufacturers of surface vehicles, namely trucks, cars and ships. The reasoning behind this choice is that for the application in railway operational environment, almost mono-dimensional due to the guidance of rails, the applications in other surface transport systems have a higher applicability potential than those tested or applied in typical three-dimensional operational environments, such as aviation and the same is for the speed ranges. Nevertheless, aeronautical applications remains in the scope of the study as potential sources of solutions for specific interfaces, such as sensors, sounds and gestures used in real driving systems and simulators.

A systematic review of potentially promising systems is in [START_REF] Enjalbert | Human-Machine Interface in Transport Systems: An Industrial Overview for More Extended Rail Applications -Machines[END_REF]. Table 1 provides with an overview of the automation functions performed by some systems on the market, in view of a potential use in the trains driving functions.

Table 1. Overview of investigated systems implemented in various typologies of vehicles

State-of-the-art in driving simulation

Another important source of technological solutions applicable to rail cabins comes from the professional simulators. Also in this field, surveys and analysis focuses on the surface transport systems and covers purely rail vehicles applications, road vehicles and integrated systems [START_REF] Vanderhaegen | MissRail: a platform dedicated to training and research in railway systems[END_REF] [4] [START_REF] Vanderhaegen | Pedagogical learning supports based on human-systems inclusion applied to rail flow control[END_REF], as summarised in Table 2.

Emerging gesture technologies

From the analysis on driving automation and simulation systems, as well as from a dedicated literature review, emerged a large set of technologies available on the market to control devices by gestures. A not comprehensive list of them is in Table 3.

Starting from analyses developed in a similar study [START_REF] Zhao | Gesture Control Technology: An investigation on the potential use in Higher Education -University of Birmingham[END_REF], Table 4 summarize the results of a comparative Strength, Weaknesses, Opportunities and Threats (SWOT) analysis focused on the most relevant features differentiating three of the most promising gesture control devices: KINECT (by Microsoft), LEAP MOTION (by Ultrahaptics) and MYO BRACELET (by Thalmic Labs). Due to the recurrence for all systems, the table does not include general threats, such as the lack of definition of a process for the recovery of gesture control intrusion. Moreover, all the investigated systems investigated have reached, in their respective fields, Technology Readiness Levels (TRL) variable among TRL7 (System prototype demonstration in operational environment), TRL8 (System complete and qualified) and TRL 9 (Actual system proven in operational environment).

Survey on train drivers acceptance

The eminent role of the human behaviour in the field requires integrating the ongoing activity on the promising technologies and solutions with a survey directly involving drivers about their expectations and acceptance of potential solutions.

The objective of this first survey is to select the key targets of the train cab of the future and the interactions should be possible in this driver's cab by involving the drivers themselves from the earliest stages of the creation of these new systems. The envisaged automation is willing to assist the driver in providing the best possible user experience, adapted to the constraints of the job and integrating the latest technologies for more comfort of use and efficiency.

The investigated technologies are applicable to various means of Action (A) and Information (I) that would enable the driver to keep informed of the state of the train, such as:

 AUDIO: Voice control of the system (A) and audible notification (I);  TOUCH SCREEN: Use of touch screens (smartphone, tablet, etc.) for entering commands (A) and reading information (I);  GESTURE: Gestures recognition to act on the system without contact (A);  VISUAL: Projection of information into the driver's field of vision and accessible through a Head-Up Display (HUD), without leaving the lane of sight (I).  HAPTICAL: Use of manipulator to add force feedback functions, assist the driver, accompanying or resisting the driver's movements for actions control (A) and information on speed (I). Table 5 summarizes the potentially covered drivers' tasks included into the survey.

Conclusions

All the investigated systems have reached in the respective application fields TRL7, TRL8 or TRL9. Therefore, they appears highly reliable in view of potential applications in the design of future trains' drivers' cabin. The experience matured in the analysis and selection process and in the survey with drivers demonstrated that many of them are well in line with the expectation setup within the CARBODIN and the related PIVOT2 Shift2Rail funded projects. They aim at reaching Technology Readiness Levels variable among TRL5 (Technology validated in relevant environment), TRL6 (Technology demonstrated in relevant environment) and TRL7 (System prototype demonstration in operational environment) [START_REF] Shells | H2020 -Shift2Rail Joint Undertaking[END_REF].

The next research developments, based on the results from the first round of surveys, are progressive refinements that will allow at: 1) eliminating systems and technologies not adapted for innovative driving cabin and selection, based on the needs from operators, 2) implementing promising HMI in virtual mock-up demonstrator for further testing by professional drivers on a simulator.

  -ready user resuming manual driving by steering, braking, accelerating or switching off manually Cars THE FALCO Full automation with ergonomic HMI, control levers and touch screens for call up and control Ships

  Trains SPICA RAIL Increasing disturbances to evaluate human behaviours and supervision platform Trains OKTAL SYDAC Replicas of the cab to ensure realistic driving experiences Trains IFFSTAR-RAIL Reproduction of driving and rail traffic supervision Trains IFFSTAR TS2 Impact of internal and external factors on driver behaviours and fixed-base cab HMI Cars NDIVIA DRIVE Interface among environment, vehicles and traffic scenario by open platform sensors Cars VRX-2019 Autonomous vehicle experiencing feelings of cockpit's HMI by advanced sensors Cars MISSRAIL Automated driving assistance combining accident scenarios including pedestrians, trains and cars with human factor control IntegratedSimulators represent a key starting point for the design of innovative driving functions, as well as for the safety of traffic supervision within railway systems[6] [7] [8] and interactions with other transport networks[START_REF] Fang | DADA: A Large-scale Benchmark and Model for Driver Attention Prediction in Accidental Scenarios[END_REF] [START_REF] Lin | KrNet: A kinetic real-time convolutional neural network for navigational assistance -Lecture Notes on Computer Science[END_REF]. Their benefits are the possibility to test, in advance, applicability and capability to generate improvements of operational performances, to obtain information about the future design and reduce the deployment costs.

Table 2 :

 2 Overview of investigated simulators

	Name	Developer	Operating features	Vehicle
	PSCHITT-RAIL		Movements capture by eyes trackers and
			physiological measurement sensors

Table 3 :

 3 Overview of control devices by gesture

	Name	Developer	Operating features
	DEPTHENSE CARLIB	Controlling movements of infotainments
		by hand
	EYEDRIVE GESTURE	Recognizing hand gestures while driving
	CONTROL	to control devices by infrared sensors
	HAPTIX	Recognizing classical hand movements
		and building a virtual mouse to control
		screen interface by webcams
	KINECT	Capturing body or hand movements to
		control devices by webcams
	LEAP MOTION	Recognizing hands or objects movements
		for virtual reality by infrared light, 3D
		micro-cameras
	MYO BRACELET	Detecting hands or fingers movements to
		control interfaces with electrical
		activation of muscles
	SOLI	Identification movements of fingers or
		body by mini-radar

SWIPE

Reading hand motions in front of a small contactless tablet for home automation purposes XPERIA TOUCH DEVICE Tracking proximate hand gesture via the smartphone camera

Table 4 :

 4 Differential items in SWOT analysis of selected control devices by gesture

	Name	Developer	Strengths	Weaknesses	Opportunities Threats
	KINECT		1) Identification	1) Operational	1) Combination	-
			of body motions	difficulties in	of gesture
				restricted space	control with
				2) Interference	facial/voice
				movements-	recognition
				sensor
	LEAP		1) Identification	1) Deep training	1) Combination	-
	MOTION		of hands/fingers	need	of infrared light
			movements	2) Interference	and cameras
			2) Low price	movements-
			3) Light weight	sensor
	MYO		1) Gesture	1) Deep training	1) Combination	-
	BRACELET		detection of	need	of physiological
			bracelet wearer	2) Few	detection with
			2) Light weight	identified	gesture control
				movements

Table 5 :

 5 Drivers' tasks potentially affected by the use of the selected innovative technologies

	Key elements	Auxiliary systems	Additional devices
	Groups	Tasks	Group	Task	Group	Task
	Situation	Dead Man's	Station stop	movement	Lighting	Instrument
	awareness	Switch		authorization		brightness
		Signalling		Stopping		Cabin lighting
	Speed	Train Protection		Reverser	Comfort	Sanitary
	regulation	& Warning			systems
		System
		Cruise control	Accessories	Sand		Audio
				spraying		Announcement
		Engine Throttle		Windshield		Passenger's
				cleaner		energy
		Brake		Wiper		Climate
						control
	Flow	Door / Bridging		Headlight		Lighting
	Management	plate
	Alarms	Whistle		Defrosting		Sound system
	systems	Emergency brake Management	Change	Configuration Start / Stop
			power	traction
				mode
		Emergency stop		Change		Train settings
		button		voltage
		Alarm systems		Pantograph
		Pantograph
		Passenger alarm
		signal
	The survey, carried out in the first months of 2021, involved 1689 drivers from 15 different
	European countries.

Statistical analyses will follow, developed by means of the following techniques:  Boxplot graphically depicting groups of numerical data through their quartiles, based on Action (A) / Information (I) and group of tasks depending on technologies);  Multiple Correspondence Analysis (MCA) to detect and represent underlying structures in a data set (Age, Profession, Gender, Nationality, Country Network, Train Type).  Hierarchical Clustering Analysis (HCA) to build a hierarchy of clusters.

Acknowledgments

The CARBODIN project has received funding from the Shift2Rail Joint Undertaking under the European Union's Horizon 2020 research and innovation programme under grant agreement No 881814.


Dyhia Bouhadjra

Angelo Alessandri

Patrizia Bagnerini

A high

A High-Gain Observer for Stage-Structured Susceptible-Infectious Epidemic Model with Linear Incidence Rate D. Bouhadjra1,2 , A. Alessandri 1 , P. Bagnerini 1 , A. Zemouche 2

Abstract-Epidemiological models play a vital role in understanding the spread and severity of a pandemic or epidemic caused by an infectious disease in a host population. The mathematical modeling of infectious diseases in the form of compartmental models are often employed in studying the probable outbreak growth. In this paper, we study the problem of state estimation for a stage-structured SI model with two classes of infected which can be considered as a simplified modelling approach to chronic diseases with progressive severity, as is the case with AIDS. Towards this end, a high-gain observer design based on system state augmentation approach is proposed to estimate the states of the SI model. Simulation results are reported with some comparisons to the standard high-gain observer.

Index Terms-Epidemiological modeling, Stage-structured models, SI model, High-gain observer.

I. INTRODUCTION

The spatial spread of infectious diseases, following their introduction at distinct locations, has always been a major concern for human populations due to their high death mortality in both developed and developing countries. Epidemiological models play an important role in analyzing the origins, dynamics and spread of such diseases. These models provide notation, concepts, intuition and disease-related factors such as the infectious agent, mode of transmission, latent and infectious periods, susceptibility and resistance which can be used to capture features that are most influential in the spread of diseases.

Epidemiological constraints, such as delays in symptom appearance and positive test confirmation (due to limited testing and detection resources), may limit the real-time use of epidemiological models [START_REF] Team | Ebola virus disease in west africa-the first 9 months of the epidemic and forward projections[END_REF], [START_REF] Kucharski | Early dynamics of transmission and control of COVID-19: a mathematical modelling study[END_REF]. In order to overcome such constraints, mathematical modeling of infectious diseases was employed in epidemiology, as recognized by WHO [START_REF] Egger | Developing who guidelines: time to formally include evidence from mathematical modelling studies[END_REF] and proven to be effective [START_REF] Jewell | Predictive mathematical models of the COVID-19 pandemic: underlying principles and value of projections[END_REF], [START_REF] Adam | Special report: The simulations driving the world's response to COVID-19[END_REF]. Compartmental modeling as a class of mathematical modeling are often employed in studying the probable outbreak growth [START_REF] Chen | Analyzing and modeling spatial and temporal dynamics of infectious diseases[END_REF]. These models consist of two parts: compartments and rules. The compartments divide the population into the different possible states with respect to the disease. The rules specify the proportion of individuals moving from one class to another.

Besides the various modelling assumptions underlying the derivation of these Compartmental models, their practical use rely on two other assumptions: the model parameters are known and an appropriate initial condition, i.e., the current state of the population is known [START_REF] Iggidr | State estimators for some epidemiological systems[END_REF]. The problem of estimating the model parameters is certainly important but will not be addressed in this work, one can see for example [START_REF] Audoly | Global identifiability of linear compartmental models-a computer algebra algorithm[END_REF], [START_REF] Jacquez | Numerical parameter identifiability and estimability: Integrating identifiability, estimability, and optimal sampling design[END_REF] and [START_REF] Toni | Approximate bayesian computation scheme for parameter inference and model selection in dynamical systems[END_REF]. Once the model parameters are known, we can identify the current population state. In most classical models the total population is conserved and this yields an estimate of one of the compartments in terms of estimates of the remaining ones.

Although the literature on the behaviors of epidemic models endowed with a treatment function is vast, there are fewer works on these models from the observer point which allows to predict and control the propagation of diseases and virus mutation [START_REF] Abdelhedi | Extended observer to estimate the spreading of contagious disease[END_REF]. Earlier work using observers in an epidemiological context dates back to the works in [START_REF] Alvarez-Ramirez | Feedback control of the chemotherapy of HIV[END_REF] and since 2012 there was a growing interest in the literature such as: the estimation of sequestered infected erythrocytes in Plasmodium falciparum malaria patients in [START_REF] Bichara | On the estimation of sequestered infected erythrocytes in plasmodium falciparum malaria patients[END_REF], parameters and states estimation for a SI-SI Dengue epidemic model [START_REF] Tami | Parameters and states estimation for Dengue epidemic model[END_REF], interval observer for uncertain SIR and SIIR-SI models [START_REF] Abdelhedi | Extended observer to estimate the spreading of contagious disease[END_REF], a state observer for a continuous and discrete time SEIR model whose values are then used to implement a vaccination strategy [START_REF] Ibeas | Stability analysis and observer design for discrete-time SEIR epidemic models[END_REF].

In this work, we analyze a class of stage-structured SI models with 2 infectious stages which can be considered as a simplified modelling approach to chronic diseases with progressive severity, as is the case with AIDS. Then, we propose a high-gain observer to track the states of the SI epidemic model in the presence of measurement uncertainty. The proposed observer is based on system state augmentation approach which transforms the original system of dimension n into an augmented system of dimension n + j s resulting in a new threshold for the observer parameter θ that guarantees the exponential convergence of the estimation error and reduces the value of the observer gain [START_REF] Bouhadjra | Highgain nonlinear observer using system state augmentation[END_REF]. The obtained results are compared with the standard high-gain observer to further show the efficiency and superiority of the proposed technique.

II. MATHEMATICAL MODELING

The SI model splits the population into two groups, the susceptible individuals who may contract the disease and the infected individuals who may spread the disease to the susceptible. Once a susceptible becomes infected, he or she moves into the infected group, increasing the size of the infected class and decreasing the size of the susceptible class as described in the flow diagram depicted in Figure 1. In SI modeling some assumptions are made about the population and disease. First, each person in the susceptible population is assumed to be equally likely to transmit the disease through contact with an infected individual and once a person is infected, they cannot recover; they remain in the infected class forever [START_REF] Allen | Some discrete-time SI, SIR, and SIS epidemic models[END_REF]. Second, the length of the disease outbreak is short compared with the average person's lifespan, so death is not a factor [START_REF] Smith | Modelling Disease Ecology with Mathematics[END_REF]. Therefore, this model can be applied to diseases for which individuals never recover and for which disease spread is relatively quick. During an epidemic, the population is divided into two compartments: the healthy individuals likely to catch the disease and the infected ones, denoted by S and I, respectively, the total population is represented by N .

Let us consider a class of stage-structured SI model with n infectious stages:

Ṡ = ∆ - n i=1 β i I i φ(S) -µ 0 S İ1 = n i=1 β i I i φ(S) -(µ 1 + γ 1 ) I 1 İ2 = γ 1 I 1 -(µ 2 + γ 2 ) I 2 . . . İn-1 = γ n-2 I n-2 -(µ n-1 + γ n-1 ) I n-1 İn = γ n-1 I n-1 -µ n I n ( 1
)
where ∆ is the recruitment and β i is the per capita contact in the compartment I i . The function φ is assumed to be continuous, positive and increasing that models the exposure of susceptible individuals to contacts with infectious ones, for instance, one can use φ(S) = S p or φ(S) = S 1+aS (with a > 0) to take into account saturation effects. The parameter µ 0 is the natural death rate of the susceptible individuals and µ i is the death rate of the infected individuals in stage i, in general µ i = µ 0 + d i with d i being the additional disease induced mortality rate and γ i is the transition rate from stage i to i + 1.

Let x(t) = S(t) I 1 (t) . . . I n (t) ⊤ be the state vector of the system (1) and suppose that we can only measure the level of infection in the last stage, i.e., the measurable output of the system is y(t) = I n (t). The aim of this work is to derive estimates Ŝ(t) and Îi (t) satisfying lim t→∞ Ŝ(t)-S(t) = 0 and lim t→∞ Î(t)-I(t) = 0 with only the knowledge of I n (t), ∀ t ≥ 0 using high-gain observer techniques.

III. HIGH-GAIN OBSERVER DESIGN A. Standard high-gain observer

Let us recall some basic concepts related to the standard high-gain observer. Given a system written under the following form:

ẋ = Ax + Bf (x) y = Cx , (2)
where

B = 0 . . . 0 1 T , C = 1 0 . . . 0 (3)
and the state matrix A is defined by

(A) i,j = 1 if j = i + 1 0 if j ̸ = i + 1 . (4)
f : R n → R is a nonlinear function satisfying the Lipschitz condition formulated under the following form:

f (x 1 + ∆ 1 , . . . , x n + ∆ n ) -f (x 1 , . . . , x n ) ≤ k f n j=1 |∆ j | . (5)
Then, the standard high-gain observer is defined as

ẋ = Ax + Bf (x) + L y -C x (6)
where, the observer gain L is written under the following form:

L := T(θ)K, θ ≥ 1 (7)
and T(θ) := diag θ, . . . , θ n with K ∈ R n×p .

Using a transformed estimation error

x := T -1 (θ)x, (8)
where x = x -x, the dynamics of the error x are given by:

ẋ = θ A -KC x + 1 θ n B∆f. (9)
From the Lipschitz condition ( 5) and the fact that θ ≥ 1, we can show as in [START_REF] Alessandri | Time-varying increasing-gain observers for nonlinear systems[END_REF] that there always exists a positive scalar constant k f , independent of θ, such that

∥T -1 (θ)B∆f ∥ ≤ k f ∥ x∥. ( 10
)
Following the high-gain methodology, the following theorem is derived. Theorem 1 ( [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF]): If there exist P > 0, λ > 0, Y of appropriate dimensions, such that

A T P + P A -C T Y -Y T C + λI < 0, (11)
then the observer converges exponentially to zero for

θ > max{1, 2k f λ max (P ) λ }, (12)
and

K = P -1 Y T
where λ max (P ) is the largest eigenvalue of the matrix P .

Proof: For more details about the proof of this theorem, we refer the reader to [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF], [START_REF] Alessandri | Time-varying increasing-gain observers for nonlinear systems[END_REF], [START_REF] Alessandri | Increasing-gain observers for nonlinear systems: stability and design[END_REF].

B. State augmentation based high-gain observer

This novel approach has been proposed in [START_REF] Bouhadjra | Highgain nonlinear observer using system state augmentation[END_REF], indeed, as demonstrated in this work, if the nonlinear function f (.) satisfies the condition

∂f ∂x j (x) ≡ 0, ∀ j > n -j s ( 13
)
for a given j s ≥ 0, then the Lipschitz inequality [START_REF] Toni | Approximate bayesian computation scheme for parameter inference and model selection in dynamical systems[END_REF] becomes

∥T -1 (θ)B∆f ∥ ≤ k f θ js ∥ x∥. ( 14
)
It follows that the high-gain inequality [START_REF] Alvarez-Ramirez | Feedback control of the chemotherapy of HIV[END_REF] becomes

θ > 2k f λ max (P ) λ 1 1+js ≜ θ 1 1+js 0 . ( 15
)
This new threshold on θ is significantly reduced due to the power 1 1+js . Hence, instead of T(θ) in L, we have T(θ)

1 1+js
. The main idea is to transform a system of dimension n, from its original coordinates, into a new system whose dimension is n + j s , where the new nonlinear function does not depend on j s last components of the new state. The following theorem is derived.

Theorem 2 ( [START_REF] Bouhadjra | Highgain nonlinear observer using system state augmentation[END_REF]): Let us consider the uniformly observable system: ẋ=ψ(x, u) y=ϕ(x, u)

Assume that there exists a state transformation given as:

Ψ : R n → R n+js x → z = Ψ(x) (17)
which transforms the system (16) into the following:

ż = A Ψ z + B Ψ f Ψ (z) y = C Ψ z (18)
where A Ψ , B Ψ , and C Ψ have the same structure than A, B, and C, respectively, but with dimension n + j s . We also have:

f Ψ (z) ≜ f Ψ (z 1 , . . . , z n ) ⇔ ∂f Ψ ∂z j (z) ≡ 0, ∀ j > n. ( 19
)
Consider the state observer described by ( 20)

ż = A Ψ x + B Ψ f Ψ (ẑ) + L Ψ y -C Ψ ẑ x = Φ(ẑ), (20)
where Φ is a continuous left invert of the embedding Ψ satisfying x = Φ(z) and L Ψ ≜ T Ψ (θ)K Ψ , with T Ψ (θ) ≜ diag(θ, . . . , θ n+js ). If there exist P > 0, λ > 0, Y , and θ ≥ 1 such that:

A ⊤ Ψ P + P A Ψ -C ⊤ Ψ Y -Y ⊤ C Ψ + λI < 0, (21)
K Ψ ≜ P -1 Y ⊤ , (22) θ > 1+js 2k fΨ λ max (P ) λ ≜ θ 1 1+js Ψ , (23)
then the estimation error x = x -x converges exponentially towards zero.

Proof: For the proof we refer the reader to [START_REF] Bouhadjra | Highgain nonlinear observer using system state augmentation[END_REF]. Remark 1: One way of transforming system (2) into a higher dimensional system is by adding a chain of intergrators (j s integrators) and keeping the properties stated in Theorem 2.

IV. ISS WITH RESPECT TO MEASUREMENT NOISE

In this section we compare the properties of the standard high-gain observer [START_REF] Chen | Analyzing and modeling spatial and temporal dynamics of infectious diseases[END_REF] and the proposed observer [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF] with respect to measurement noises. To this end, we consider the following system where the measurement is corrupted by a bounded disturbance:

ẋ = Ax + Bf (x) y = Cx + ν ( 24
)
where ν represents the disturbance affecting the measurement y. We will show that an upper bound on the estimation error, in a ISS sens with appropriate norms, can be ensured by the observers ( 6) and ( 20), respectively. However, we will demonstrate that the use of state augmentation approach can lead to a smaller bound on the estimation error, compared to the one we get using the standard high-gain observer.

A. ISS property with standard high-gain observer

Consider system (24) and the standard high-gain observer [START_REF] Chen | Analyzing and modeling spatial and temporal dynamics of infectious diseases[END_REF], then the transformed error dynamics system is given as

ẋ = θ A -KC A K x + 1 θ n B∆f -Kν. (25)
Therefore, the observer parameters designed by Theorem 1 ensure an ISS property as introduced in the following proposition.

Proposition 1: Assume that there exist a symmetric positive definite matrix P , a positive constant λ and a matrix Y of appropriate dimensions such that the inequalities ( 11)-( 12) hold. Then with the observer gain L given in [START_REF] Iggidr | State estimators for some epidemiological systems[END_REF], there exists a positive constant α such that the estimation error x(t) verifies the following ISS conditions:

∥x(t)∥ ≤ θ n-1 λ max (P ) λ min (P ) ∥x 0 ∥e -β 2 t + θ n γ(1 -e -βt ) βλ min (P ) sup s∈[0,t] ∥ν(s)∥, (26a)
lim t→+∞ ∥x(t)∥ ≤ θ n γ βλ min (P ) sup s∈[0, +∞] ∥ν(s)∥, (26b)
where

β = θλ -2k f λ max (P ) -α λ max (P ) , γ = ∥Y ∥ 2 α . (27)
Proof: The proof is omitted.

B. ISS property with the proposed high-gain observer

By a straightforward analogy, we know that we can apply the results of Proposition 1 on the augmented system [START_REF] Smith | Modelling Disease Ecology with Mathematics[END_REF]. That is the observer (20) ensures a similar ISS property than that in (26a)-(26b). However, the presence of the power 1 1+js in the case of augmented state based observer [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF], especially in the high-gain threshold condition (23), allows reducing significantly the values of the observer gain. For instance, for an ϵ > 0, if we take θ = θ 0 + ϵ in the standard high-gain, then according to (26b), the upper bound of the estimation error satisfies:

lim t→+∞ ∥x(t)∥ ≤ (θ 0 + ϵ) n γ βλ min (P ) sup s∈[0, +∞]
∥ν(s)∥

(28) while with the augmented approach for θ = θ ϵ Ψ ≜ θ Ψ + ϵ, we get

lim t→+∞ ∥x(t)∥ ≤ (θ ϵ Ψ ) n+js 1+js γ Ψ β Ψ λ min (P Ψ ) sup s∈[0, +∞] ∥ν(s)∥. (29)

V. APPLICATION TO A TWO-STAGE-STRUCTURED EPIDEMIC MODEL

In this section we study the special case where infected batch I is made up of two compartments, namely, the infected in the first stage of the disease and the infected in the terminal phase of the infection, denoted I 1 and I 2 , respectively. The corresponding SI model is given as follows,

   Ṡ = ∆ -(β 1 I 1 + β 2 I 2 )S -µ 0 S, İ1 = (β 1 I 1 + β 2 I 2 )S -(µ 1 + γ)I 1 , İ2 = γI 1 -µ 2 I 2 (30)
where -S, I 1 , I 2 represent the compartments of susceptible, first stage infected and second stage infected, respectively. -∆, µ, α, 1/γ are recruitment, mortality rate, recovery rate and time taken for an early-stage infected to become in the final phase of infection, respectively.

Let x(t) = S(t) I 1 (t) I 2 (t) ⊤ and y = I 2 ∈ R 3 , so the model given by (30) has the following form:

Γ : ẋ = f (x) y = h(x). (31)
When system (30) is observable, the map Ψ : x → Ψ(x) is a diffeomorphism and,

z = Ψ(x) =   I 2 γI 1 -µ 2 I 2 γ(β 1 I 1 + β 2 I 2 )S -γmI 1 + µ 2 2 I 2   , (32)
where

m = µ 1 + µ 2 + γ.
With the change of variable z = Ψ(S, I 1 , I 2 ), the expression of Ψ -1 is

Ψ -1 : z →                µ 2 (µ 1 + γ) z 1 + (µ 1 + µ 2 + γ) z 2 + z 3 (µ 2 β 1 + γβ 2 ) z 1 + β 1 z 2 , µ2 γ z 1 + 1 γ z 2 z 1 (
33) Then, system given by (30) is rewritten in the following triangular form:

Γ ′ :                    ż = F ′ (z) =   ż1 ż2 ż3   =   z 2 z 3 φ(z)   , y = Cz = 1 0 0   z 1 z 2 z 3   (34)
This latter form allows us to make use of the highgain observer techniques presented in the previous section to estimate the evolution of susceptible individuals S(t) and infected individuals I(t). The parameters used in these simulations are given as follows: ∆ = 40, β 1 = 0.01, β 2 = 0.15, µ 0 = µ 1 = 0.01, µ 2 = 0.025 and γ = 0.02.

In order to test the efficiency of the proposed observer given in [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF], we performed a batch of numerical simulations to compare with the standard high-gain observer in the presence of high-frequency measurement noise, numerically taken as Gaussian distributed noise with zero mean and standard deviation of 0.01. First, we design a standard highgain observer for the system (34) following Theorem 1, then by using Matlab & YALMIP, we can obtain the value of the observer gain K and the tuning parameter θ. Next, by augmenting the state of the system following the transformation described in Section III-B, we obtain the values for the corresponding gain K ψ and the new observer parameter θ ψ . The results are summarized in Table I where we can clearly notice that the proposed observer provides lower gain with smaller value of the tuning parameter compared to the standard high-gain observer. In Figure 2, the absolute value of the estimation error ∥x i,HG -x i ∥ and ∥x i1 -x i ∥ are plotted for i = 1 . . . 3, where xi,HG denotes the state estimate using the standard highgain observer and xi1 the state estimate using the proposed observer. An additional zero mean Gaussian disturbances with standard deviation of 0.01 are applied to the output measurements during time t= 5s and t=10s. It is clear that both observers converge towards the true state with roughly the same speed, however, our proposed observer has better transient response, in particular for the states x 2 and x 3 . We also note that our proposed observer is less sensitive to measurement noises since the peaking phenomenon that is typical of the standard high-gain observer is prevented using our proposed structure as shown particularly by the plot of ∥x 3 -x 3 ∥.

VI. CONCLUSION

In this paper, we considered a SI epidemiological model with two stages of infected individuals. For this class, we have proposed simple and easy implementable observer based on the high-gain approach. This proposed observer is characterized by two main features, namely better transient performances and less sensitivity to measurement noises with respect to the standard high-gain observer. The stability and robustness properties are established analytically using the theory of input-to-state stability. The efficiency of the obtained results is illustrated through numerical simulations.

Fig. 1 :

 1 Fig. 1: The transfer diagram for SI model.

Fig. 2 :

 2 Fig. 2: Absolute values of estimation errors with the presence of measurement noise.

TABLE I :

 I Simulation results for the two observers.

		Standard high-gain observer
	θ		K
	24.2308	8.8781 15.7945 7.9158
	State augmentation approach js = 1
	θ ψ		K ψ
	4.7056	6.0195 12.0353 11.4500 4.5601

DIME-University of Genoa, Via all'Opera Pia 15, 16145 Genoa, Italy (email: Bouhadjra@dime.unige.it,Alessandri@dime. unige.it,Bagnerini@dime.unige.it).

University of Lorraine, CRAN CNRS UMR 7039, 54400 Cosnes-et-Romain, France (email: ali.zemouche@univ-lorraine.fr).


Dyhia Bouhadjra

Angelo Alessandri

Patrizia Bagnerini

Ali Zemouche

High-gain estimation of mRNA and protein concentrations of a genetic regulatory network

High-Gain Estimation of mRNA and Protein Concentrations of a

Genetic Regulatory Network D. BOUHADJRA1,2 , A. ALESSANDRI 1 , P. BAGNERINI 1 , A. ZEMOUCHE 2

Abstract-In this paper, we investigate the problem of state estimation for a simple one-gene regulation dynamic process involving end-product activation to rebuild the non-measured concentrations of mRNA and the involved protein. We synthesize a convenient observer structure following the high-gain methodology by combining the observer proposed in [START_REF] Bouhadjra | Highgain nonlinear observer using system state augmentation[END_REF] based on the system state augmentation approach and the HG/LMI technique presented in [START_REF] Zemouche | High-gain nonlinear observer with lower tuning parameter[END_REF]. The proposed design reduces significantly the value of the tuning parameter and the observer gain along with improving its sensitivity to disturbances and measurement noise. The results are compared with the standard high-gain observer to evaluate the effectiveness of the proposed design.

Index Terms-Genetic regulatory network, Robust state estimation, high-gain methodology, Lipschitz systems.

I. INTRODUCTION

During gene expression, an enormous number of genes and proteins are either directly or indirectly interrelated with one another in living cells. The mechanisms which have progressed to control the expression of genes are known as gene regulatory networks (GRNs). GRNs describe gene expression as a function of regulatory inputs specified by the interactions between proteins and genes, that is, one gene can control another gene's expression through its product proteins called Transcriptional Factors [START_REF] Lockhart | Genomics, gene expression and dna arrays[END_REF].

For the purpose of gene identification and medical diagnosis or treatment, biologists and biomedical scientists are interested in knowing the exact states of GRNs [START_REF] Wang | Filtering for nonlinear genetic regulatory networks with stochastic disturbances[END_REF]. However, due to unavoidable complications such as transcription or translation delays and extrinsic/intrinsic noises, the available measurement outputs of GRNs might be different from their true states. As such, the state estimation or filtering problem for GRNs has become an important topic of research that has attracted many from the scientific community [START_REF] Balasubramaniam | Robust state estimation for discrete-time genetic regulatory network with random delays[END_REF], [START_REF] Sakthivel | Robust state estimation for discrete-time genetic regulatory networks with randomly occurring uncertainties[END_REF], [START_REF] Wang | Filtering for nonlinear genetic regulatory networks with stochastic disturbances[END_REF], [START_REF] Wang | Robust state estimation for discrete-time stochastic genetic regulatory networks with probabilistic measurement delays[END_REF], [START_REF] Zhang | Set-values filtering for discrete time-delay genetic regulatory networks with timevarying parameters[END_REF] and [START_REF] Zhang | State estimation for delayed genetic regulatory networks with reaction-diffusion terms[END_REF]. To the best of authors' knowledge, only few works have been reported in literature regarding the robust state estimation issue for GRNs. Taking this into account, we propose in this paper a robust observer that retains all the characteristics of the popular standard high-gain observers and provides additional robustness to measurements noise which can result in a significant improvement of the monitoring performances of the genetic regulation process.

In high-gain observers design methodology, the error trajectory has an exponential decay rate that can be imposed arbitrarily fast by acting on a design parameter, appearing in the observer structure, typically known as "high-gain parameter". Such observers are widely used in control contexts where fast observation is useful in order to prevent finite escape time of the closed-loop system. It was first introduced in [START_REF] Gauthier | A simple observer for nonlinear systems -application to bioreactors[END_REF], [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF], and since then it has attracted the attention of several researchers and a huge number of papers have been published on the topic (see for instance the survey [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] and references therein). Despite all the research activities about high-gain observer in output feedback control, this direction still remains active due to some open problems related to its sensitivity to output measurement noise because of the value of the tuning parameter which may be huge for higher dimensional systems having nonlinearities with large Lipschitz constants. Indeed, the higher the gain, the larger the peaking in the transient, which may cause the destabilization of the control loop if the high-gain observer is used in cascade with a feedback regulator. To reduce the peaking, various solutions have been proposed in the literature and for an overview we refer the reader to [START_REF] Ahrens | High-gain observers in the presence of measurement noise: a switched-gain approach[END_REF], [START_REF] Boizot | An adaptive high-gain observer for nonlinear systems[END_REF], [START_REF] Andrieu | High gain observers with updated gain and homogeneous correction terms[END_REF], [START_REF] Prieur | Hybrid high-gain observers without peaking for planar nonlinear systems[END_REF], [START_REF] Alessandri | Time-varying increasing-gain observers for nonlinear systems[END_REF], [START_REF]Increasing-gain observers for nonlinear systems: stability and design[END_REF], [START_REF] Oueder | A high gain observer with updated gain for a class of MIMO non-triangular systems[END_REF] and the references therein.

In this work we investigate a new strategy for highgain observer design. The proposed technique is based on a combination of the observer proposed in [START_REF] Bouhadjra | Highgain nonlinear observer using system state augmentation[END_REF] which is based on system state augmentation approach and the HG/LMI observer developed in [START_REF] Zemouche | High-gain nonlinear observer with lower tuning parameter[END_REF]. This new design technique provides a new threshold on the observer parameter θ that guarantees exponential convergence of the estimation error and reduces the value of the observer gain. The applicability and performance of the observer is explored through a simple genetic regulatory network with a comparison to the standard high-gain observer to clearly show the superiority of the proposed technique.

II. MATHEMATICAL MODELING OF GENETIC REGULATION PROCESS

There are different classes of genetic regulatory models, namely, thermodynamic, Boolean, Bayesian and differential equation-based models. In differential equation basedmodels, the regulatory networks can be modeled by differential equations, in which a set of molecules such as mRNAs and proteins interact by explicit rules defined in terms of rate equations. The concentrations of mRNAs, proteins and other elements of the system are are represented by time-dependent variables with values contained in the set of non-negative real numbers. Regulatory interactions take the form of functional and differential relations between the concentration variables. More specifically, gene regulation is modeled by reaction-rate equations expressing the rate of elements of the system in the following form:

dx i dt = f i (x), x i ≥ 0, 1 ≤ i ≤ n, (1)
where x is the vector of concentrations of proteins, mRNAs, or small metabolites, and f i a usually nonlinear function. Then, by specifying the function f we get equations of the form 2 [START_REF] Tyson | The dynamics of feedback control circuits in biochemical pathways[END_REF]:

dx1 dt = κ 1n H(x n ) -γ 1 x 1 , x 1 ≥ 0, dxi dt = κ i,i-1 x i-1 -γ i x i , x i ≥ 0, 1 ≤ i ≤ n (2)
The parameters κ 1n , κ 2n , • • • κ n,n-1 are all strictly positive and represent production constants and γ 1 , • • • γ n represent degradation constants and are also strictly positive. The reaction-rate equations express a balance between the number of molecules appearing and disappearing per unit time.

In the case of x 1 , the production term involves a nonlinear regulation function H : R → R ranging from 0 to 1. In common use, the function H is given as

H + (x j , θ ij , m) = x m j x m j + α m ij , H -(x j , θ ij , m) = 1 - x m j x m j + α m ij ,
corresponding to the activation and inhibition cases, respectively. The parameter α gives the threshold for the regulatory influence of the concentration of the metabolite on the target gene, whereas the steepness parameter m is a measure of the collective effect of groups of metabolite molecules defining the shape of the Hill curve.

A. Simple gene regulation process

A simple example of a kinetic model for the genetic regulation process is given in Figure 1, going back to the work by Goodwin [START_REF] Goodwin | Temporal organization in cells[END_REF] and [START_REF]Oscillatory behavior in enzymatic control processes[END_REF]. The end-product of a metabolic pathway co-inhibits the expression of a gene coding for an enzyme that catalyzes a reaction step in the pathway. This gives rise to a negative feedback loop involving mRNA concentration x 1 , the protein concentration x 2 , and the metabolic concentration x 3 . Let x 1 , x 2 and x 3 be the concentrations of the messenger RNA (mRNA) a, protein A and metabolite K, respectively. Then, the corresponding Tyson's model [START_REF] Tyson | The dynamics of feedback control circuits in biochemical pathways[END_REF] is written under the form:

Γ biology :        ẋ1 = κ 1 H(x 3 ) -γ 1 x 1 ẋ2 = κ 2 x 1 -γ 2 x 2 ẋ3 = κ 3 x 2 -γ 3 x 3 y = x 3 (3)
κ 1 , κ 2 and κ 3 are production constants, γ 1 , γ 2 and γ 3 degradation constants, and H is the nonlinear regulation function ranging from 0 to 1. System (3) is considered to be a good model for the simplest type of allosteric regulation in biochemistry, i.e., the inhibition or activation of an enzyme or protein by a small regulatory molecule that interacts with the enzyme at a site (allosteric site) other than the active site at which catalytic activity occurs.

III. BACKGROUND RESULTS ON HIGH-GAIN OBSERVER

In this section we introduce some background results on high-gain observer which will be essential in the development of the proposed technique.

A. Standard high-gain observer

Here, we recall the basic high-gain observer as in [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF]. We will consider the class of nonlinear systems in triangular form described by the following set of equations:

                       ẋ =           ẋ1 ẋ2 . . . ẋn-1 ẋn           =           x 2 x 3 . . . x n f (x)           y = x 1 (4)
where f : R n → R is a nonlinear function satisfying the Lipschitz property:

f (x 1 + ∆ 1 , . . . , x n + ∆ n ) -f (x 1 , . . . , x n ) ≤ γ f n j=1 |∆ j | . (5)
In a condensed form, system (4) can be rewritten as follows:

ẋ = Ax + Bf (x) y = Cx , (6)
where

B = 0 . . . 0 1 T , C = 1 0 . . . 0 (7)
and the state matrix A is defined by

(A) i,j = 1 if j = i + 1 0 if j ̸ = i + 1 . (8)
Consider the following Luenberger observer:

ẋ = Ax + Bf (x) + L y -C x . (9)
The dynamics of the estimation error x = x-x is then given by:

ẋ = A -LC x + B f (x) -f (x) . (10)
In the high-gain methodology, we write the observer gain L under the form:

L := T(θ)K, θ ≥ 1. ( 11
)
where

T(θ) := diag θ, . . . , θ n and K ∈ R n×p .
In addition, the high-gain methodology focuses on the transformed estimation error

x := T -1 (θ)x, (12)
where T -1 (θ) is the inverse of T(θ) given by

T -1 (θ) = diag 1 θ , . . . , 1 θ n .
It is well-known that the dynamics of the error x is given by

ẋ = θ A -KC x + 1 θ n B∆f, (13)
with ∆f := f (x) -f (x -T(θ) x).
From the Lipschitz condition (5) and the fact that θ ≥ 1, we can show as in [START_REF] Alessandri | Time-varying increasing-gain observers for nonlinear systems[END_REF] that there always exists a positive scalar constant k f , independent from θ, such that

∥T -1 (θ)B∆f ∥ ≤ k f ∥ x∥. ( 14
)
then, the following theorem is derived, Theorem 1 ( [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF]): If there exist P > 0, λ > 0, Y , and θ ≥ 1 such that

A T P + P A -C T Y -Y T C + λI < 0, (15)
θ > θ 0 = 2k f λ max (P ) λ , (16)
then the estimation error x is exponentially stable with

K = P -1 Y T ,
where λ max (P ) is the largest eigenvalue of the matrix P .

Proof: For more details about the proof of this theorem, we refer the reader to [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF], [START_REF] Alessandri | Time-varying increasing-gain observers for nonlinear systems[END_REF], [START_REF]Increasing-gain observers for nonlinear systems: stability and design[END_REF].

B. HG/LMI based observer

This technique follows the standard high-gain methodology with the same state observer structure of dimension n. However, by exploiting the LPV/LMI technique developed in [START_REF] Zemouche | On LMI conditions to design observers for Lipschitz nonlinear systems[END_REF], the values of tuning parameter and observer gain are decreased. Indeed, by introducing a "compromise index" j 0 , with 0 ≤ j 0 ≤ n, the power of the proposed high-gain is limited to j 0 with 2 j0 LMIs to solve instead of one single LMI as in standard high-gain observer. Lemma 1 introduced in the following is crucial for the HG/LMI observer design.

Lemma 1 ( [START_REF] Zemouche | On LMI conditions to design observers for Lipschitz nonlinear systems[END_REF]): Consider a function Ψ : R n -→ R n , then, the two following items are equivalent:

• Ψ is γ Ψ -Lipschitz with respect to its argument, i.e.:

Ψ(X) -Ψ(Z) ≤ γ Ψ X -Y , ∀ X, Z ∈ R n ; (17)
• for all i, j = 1, ..., n, there exist functions

ψ ij : R n × R n -→ R and constants γ ψij ≤ 0, γψij ≥ 0, so that ∀ X, Z ∈ R n , Ψ(X) -Ψ(Z) = i=n i=1 j=n j=1 ψ ij H ij X -Z , (18)
-γ Ψ ≤ γ ψij ≤ ψ ij ≤ γψij ≤ γ Ψ , (19)
and

ψ ij ≜ ψ ij X Zj-1 , X Zj and H ij = e n (i)e ⊤ n (j),
where, e n (j) is the j th vector of the canonical basis of R n . Since the nonlinear function f (.) in ( 6) is γ f -Lipschitz, then following Lemma 1 there are functions γ j : R n × R n -→ R and constants γ γj and γγj , so that

f (x) -f (x) = j=n j=1 ψ j e ⊤ n (j) x, (20)
and

γ γj ≤ ψ j ≤ γγj , (21)
where

ψ j ≜ ψ j x xj-1 k , x xj
is defined as in Lemma 1. For the sake of brevity, we use only ψ j instead of ψ j x xj-1 , x xj .

Following [START_REF] Tyson | The dynamics of feedback control circuits in biochemical pathways[END_REF], ∆f in (13) can be written under the following form

∆f = n-j0 j=1 ψ j xj ∆f1 + j0 j=1 ψ k(j) xk(j) , (22)
where k(j) = n -(j 0 -j),

0 ≤ j 0 ≤ n.
Hence, the error dynamics ( 13) is rewritten as follows:

ẋ = θ A(Ψ θ ) -KC x + 1 θ n B∆f 1 , (23)
where

A(Ψ θ ) = A + B j0 j=1 ψ θ j e ⊤ n (k(j)), (24)
Ψ θ =    ψ θ 1 . . . ψ θ j0    ∈ R j0 , ( 25
)
ψ θ j = ψ k(j) θ 1+(j0-j) . ( 26
)
Let us define the convex bounded set

H σ j0 = Φ ∈ R j0 : γ γ k(j) σ 1+(j0-j) ≤ Φ j ≤ γγ k(j) σ 1+(j0-j) (27)
for which the set of vertices is defined by

V H σ j 0 = Φ ∈ R j0 : Φ j ∈ γ γ k(j) σ 1+(j0-j) , γγ k(j)
σ 1+(j0-j) .

(28) Since γγ k(j) ≥ 0 and γ γ k(j)

≤ 0, then it is obvious that for two positive scalars σ 1 , σ 2 , we have the following implication

σ 1 < σ 2 =⇒ H σ1 j0 ⊃ H σ2 j0 . (29)
Moreover,

lim σ→+∞ H σ j0 = 0 R j 0 . ( 30
)
On the other hand, we can show that there exists a positive real number k j0 ≤ k f such that ∆f 1 satisfies

∥T -1 (θ)B∆f 1 ∥ ≤ k j0 θ j0 ∥ê∥. (31)
Hence, the following theorem is obtained.

Theorem 2 ( [2]

): If there exist P > 0, λ > 0, Y , and σ > 0 such that

A(Ψ σ ) T P + P A(Ψ σ ) -C T Y -Y T C + λI < 0, ∀Ψ σ ∈ V H σ j 0 , (32) θ > θ 1 1+j 0 j0 = 2k j0 λ max (P ) λ , (33)
then the estimation error x is exponentially stable with

L = T(θ) K P -1 Y T , θ ≥ max σ, θ 1 1+j 0 j0 .
Proof: For the proof, we refer the reader to [START_REF] Zemouche | High-gain nonlinear observer with lower tuning parameter[END_REF].

IV. IMPROVED HIGH-GAIN OBSERVER

In this section we propose a new high-gain observer strategy which follows the popular standard high-gain methodology presented in the previous section by exploiting the HG/LMI technique developed in [START_REF] Zemouche | High-gain nonlinear observer with lower tuning parameter[END_REF] and system state augmentation approach provided in [START_REF] Bouhadjra | Highgain nonlinear observer using system state augmentation[END_REF].

Our key idea lies on the following assumption. If the nonlinear function f (.) in ( 6) satisfies the condition

∂f ∂x j (x) ≡ 0, ∀ j > n -j s (34)
for a given j s ≥ 0, then the Lipschitz inequality ( 14) becomes

∥T -1 (θ)B∆f ∥ ≤ k f θ js ∥ x∥. (35)
It follows that the high-gain inequality ( 16) becomes

θ > 2k f λ max (P ) λ 1 1+js = θ 1 1+js 0 . (36)
This new threshold on θ is significantly reduced due to the power 1 1+js . Indeed, instead of T(θ) in L, we have T(θ) 1 1+js . Therefore, the aim of this work is to transform the original system of dimension n into a system of dimension n + j s , where the new nonlinear function does not depend on j s last components of the new state, we then construct a HG/LMI observer for the augmented system. This high-gain observer combination allows obtaining a new threshold on θ attenuated to the power 1 (1+js)(1+j0) . One of the natural solution to obtain a new system satisfying (34) is to add a set of integrators leading to the following transformation:

z =    z 1 . . . z n+js    = Γ(x) ≜              x 1 . . . x n f (x(t)) df (x(t)) dt . . . d (js -1) f (x(t)) dt (js-1)              . (37)
It is easy to see that z obeys to the following dynamics:

żi = z i+1 , for i = 1, . . . , n + j s -1, (38) żn+js
= d js f (x(t)) dt js ≜ f Ψ (z 1 , . . . , z n ), (39) x
=    z 1 . . . z n    = Φ I n 0 R n×js z (40)
where I n is the identity matrix of dimension n. It is clear to see that the new system (38)-(39) satisfies the condition (34). Hence, the idea consists in constructing a HG/LMI observer for system (38)-(39) and then we deduce an estimation x of x through (40), namely x = Φẑ. Let us summarize the strategy in the following proposition. Let us first rewrite (38)-(39) in the condensed form:

ż = A Γ z + B Γ f Γ (z, u) y = C Γ z (41)
where A Γ , B Γ , and C Γ have the same structure than A, B, and C, respectively, but with dimension n + j s . Now we consider the following state observer corresponding to (41):

ż = A Γ ẑ + B Γ f Γ (ẑ) + L Γ y -C Γ ẑ (42)
where L Γ ≜ T Γ (θ)K Γ , with T Γ (θ) ≜ diag(θ, . . . , θ n+js ).

The objective consists in determining K Γ and θ such that ẑ converges exponentially to z. Hence, the estimated state x(t) given by

x(t) = I n 0 R n×js ẑ(t) (43)
converges exponentially to x(t). Proposition 1: Assume there exist P Γ > 0, λ Γ > 0, Y Γ , and θ ≥ 1 such that the following conditions hold:

A Γ ( Ψσ ) T P Γ + P Γ A Γ ( Ψσ ) -C T Γ Y Γ -Y T Γ C Γ + λ Γ I < 0, ∀ Ψσ ∈ V H σ j 0 , (44)
K Γ ≜ P -1 Γ Y ⊤ Γ , (45)
θ > θ Γ ≜ 2k fΓ λ max (P Γ ) λ Γ 1 (1+js )(1+j 0 ) , (46)
where H σ j0 is as given in ( 27) with different vertices corresponding to the new nonlinear function f Γ . Then x given by (43) converges exponentially to x(t).

Proof: It is sufficient to prove that ẑ converges exponentially to z, which is ensured by the fact that we use the HG/LMI methodology on the transformed system (41). For more details on the proof we refer the reader to [START_REF] Zemouche | High-gain nonlinear observer with lower tuning parameter[END_REF].

V. SIMULATION RESULTS

The model [START_REF] Lockhart | Genomics, gene expression and dna arrays[END_REF] given in section II-A has the following form:

Γ y : ẋ = f (x) y = h(x) (47)
Using an appropriate change of coordinate, the system is transformed into a triangular form as given in (48):

Γ ′ y :                ξ = F ′ (ξ) =   ξ1 ξ2 ξ3   =   ξ 2 ξ 3 φ(ξ)   , y = Cξ = 1 0 0   ξ 1 ξ 2 ξ 3   (48)
The standard high-gain observer and the new proposed observer are designed to estimate the states of the system given by (48). For j s = 1, we transform the system given by (48) of dimension 3 to an augmented system of dimension 4 by adding an integrator as illustrated in section IV to obtain this fourth dimensional system:

           ż1 = z 2 ż2 = z 3 ż3 = z 4 ż4 = φ ′ (ξ) y = z 1 (49)
where φ ′ (.) is the new nonlinear function that does depend on the last component z 4 .

In order to carry out simulations, we have used the parameters given in Table I Figure 2 depicts the absolute values of estimation error (x i -xi ) for i = 1, . . . , 3 under the application of additional Gaussian disturbances with zero mean and standard deviation of 0.1 at t = 2s to t = 4s . We denote xi,SHG , the state estimate using the standard high-gain observer and xi,IHG , the estimate using the improved high-gain observer proposed in section IV. From the plots, we can clearly see the convergence of the estimation error towards zero for both observers, although the standard high-gain converges rapidly but the transient performance of our proposed observer is vastly superior in particular for the estimates x2 and x3 , since the smaller gain of our proposed observer makes the estimation less sensitive to the measurement noise.

VI. CONCLUSION

As evolution of the high-gain observer, we have presented an effective new design technique based on HG/LMI technique and system state augmentation approach. The obtained observer ensures stability in a noise-free setting with a smaller gain as compared to the standard high-gain observer in addition to a reduced sensitivity to noise measurements. The remarkable assets related to this new design have been successfully addressed through an application to a simple one-gene regulation dynamic process involving end-product activation, which aims to rebuild the non-measured concentrations of mRNA and the involved protein.

Fig. 1 :

 1 Fig.1: An example of a genetic regulatory system involving end-product inhibition. A and C represent proteins, F and K metabolites.

Fig. 2 :

 2 Fig. 2: Absolute values of estimation errors with additional measurement noise.

  which are not necessarily the experimental values but are consistent with the requirements of the model.

TABLE I :

 I Parameters of the GRNTableIIsummarizes the obtained values of the tuning parameter θ and the observer gain K for the standard highgain observer and the proposed observer, one can clearly see that the observer based on state augmentation approach and HG/LMI technique provides lower gain as compared to the classical high-gain observer.

	Symbol Meaning	Value(arb. units)
	κ 1	Production constant of mRNA	0.001
	κ 2	Production constant of protein	1.0
	κ 3	Production constant of metabolite	1.0
	γ 1	Degradation constant of mRNA	0.1
	γ 2	Degradation constant of protein	1.0
	γ 3	Degradation constant of metabolite 1.0
	α	Hill's threshold parameter	1.0

TABLE II :

 II Simulation results for the two high-gain observers.

		Standard high-gain observer
	θ		K
	25.4906	9.0771 16.1946 8.1170
	Improved high-gain observer (j 0 = js = 1)
	θ Γ		K Γ
	2.6023	3.4874 5.5409 6.3543 5.7634

DIME-University of Genova, Via all'Opera Pia 15, 16145 Genoa, Italy (email: Bouhadjra@dime.unige.it, Alessandri@dime.unige. it, Bagnerini@dime.unige.it).

University of Lorraine, CRAN CNRS UMR 7039, 54400 Cosnes-et-Romain, France (email: ali.zemouche@univ-lorraine.fr).


Grina Fares
email: grina.fares2@gmail.com

Zied Elouedi
email: zied.elouedi@gmx.fr

Eric Lefevre
email: eric.lefevre@univ-artois.fr

Uncertainty-aware resampling method for imbalanced classification using Evidence Theory

Keywords: Resampling, Imbalanced datasets, Evidence theory, Data uncertainty

Class imbalance is a common issue in many real world classification problems. It refers to situations where the number of observations in the training dataset significantly differs for each class. Ignoring this issue will make it more challenging for classifiers to properly learn data characteristics, which results in poor performance. Many strategies have been proposed to deal with this issue. The most common one is tackling the imbalance at the preprocessing level, by re-sampling the training set. However, imbalanced classification can be affected by other data factors, such as uncertainty, i.e., ambiguous samples and noise. In this paper, we propose an uncertainty-aware hybrid resampling technique based on the theory of evidence to tackle imbalanced binary datasets in the presence of aleatoric uncertainty. A soft evidential structure is assigned to each object in the training set, which is later used to clean the dataset out of overlapping and noisy majority samples, and then selectively generate synthetic minority objects using a modified SMOTE algorithm. Experimental results on benchmark imbalanced datasets have shown significant improvement over popular re-sampling techniques.

Introduction

Imbalanced classification is an active research topic in machine learning and data mining. It is a scenario in which class sizes are not equal making the class distribution imbalanced. Data imbalance exist in many real-life domains such as fraudulent credit card detection [START_REF] Makki | An experimental study with imbalanced classification approaches for credit card fraud detection[END_REF], medical diagnosis [START_REF] Bridge | Introducing the gev activation function for highly unbalanced data to develop covid-19 diagnostic models[END_REF], drug discovery [START_REF] Korkmaz | Deep learning-based imbalanced data classification for drug discovery[END_REF], etc. For instance, in imbalanced binary datasets, the class with the highest number of instances is referred to as the majority class, while the minority class is defined as the one with the fewest examples. In most cases, the minority class is more relevant than the majority one [START_REF] Chawla | Editorial : Special Issue on Learning from Imbalanced Data Sets[END_REF]. As an example, failing to detect intrusions in a company's network may result in huge financial losses.

A variety of variables may cause the class imbalance, such as the domain's nature (e.g. rare disease) or data collection factors (e.g. storage). Additionally, most classifier algorithms (such as decision trees, k-nearest neighbors, neural networks, etc) were designed with the presumption that training datasets have an even distribution, which reduces greatly their efficiency [START_REF] Haixiang | Learning from class-imbalanced data: Review of methods and applications[END_REF].

Many methods have been proposed over the years to cope with imbalanced datasets. Data resampling is one of the most efficient strategies for dealing with class imbalance [START_REF] Feng | Imbalanced classification: an objective-oriented review[END_REF]. This approach aims at fixing the uneven class distribution at the preprocessing level by re-balancing the training dataset. Being algorithmindependent, resampling is versatile and could be applied with any selected classifier.

In addition, recent findings show that class imbalance is not an issue in and of itself, but rather gets amplified by other data difficulties. Data uncertainty (can also be referred to as aleatoric uncertainty) refers to the imperfections present in the data. This type of uncertainty can include class overlapping and noise, which were proven to worsen the class imbalance issue [START_REF] Haixiang | Learning from class-imbalanced data: Review of methods and applications[END_REF].

To improve performance on imbalanced and uncertain binary datasets, we suggest an Uncertainty-Aware Hybrid reSampling (UAHS) method based on Evidence Theory, which was recently used for imbalanced classification [START_REF] Grina | A preprocessing approach for class-imbalanced data using smote and belief function theory[END_REF][START_REF] Grina | Evidential undersampling approach for imbalanced datasets with class-overlapping and noise[END_REF]. After creating soft evidential labels for each object, our method efficiently selects the majority instances to remove in an undersampling phase first, and the minority objects to focus on in the oversampling procedure lastly. The considered evidential label is appropriate for our goal, since it includes membership values towards single classes, in addition to a belief mass assigned to meta-classes (ambiguous region). This versatility allows us to create precise rules for the process of selecting undesirable samples in the undersampling phase, and intelligently select minority instances to generate new synthetic objects. It is important to note that our proposal is a hybrid resampling method, meaning that it performs both undersampling and oversampling, unlike [START_REF] Grina | A preprocessing approach for class-imbalanced data using smote and belief function theory[END_REF] and [START_REF] Grina | Evidential undersampling approach for imbalanced datasets with class-overlapping and noise[END_REF] which are respectively pure oversampling and undersampling approaches.

The remainder of this paper will be divided as follows. First, related work for resampling methods is presented in Section 2. Evidence Theory is recalled in Section 3. Section 4 details each step of our idea. Experimental evaluation is discussed in Section 5. Our paper ends with a conclusion and an outlook on future work in Section 6.

Related work

Data resampling is one of the most common approaches for dealing with imbalanced classification [START_REF] Haixiang | Learning from class-imbalanced data: Review of methods and applications[END_REF]. In fact, data resampling deals with class imbalance at the preprocessing level by changing the class distribution of the training set. As a result, it alleviates the effects of distribution skewness of the learning process. These methods can be further categorized into three groups, namely:

-Oversampling: These techniques introduce new minority synthetic samples to re-balance the dataset. The most straightforward method is random oversampling (ROS), which consists of selecting minority observations in the original data set and simply replicating them. Although it appears to be technically effective since the class balance is adjusted, it can lead to overfitting [START_REF] Japkowicz | Class imbalances: are we focusing on the right issue[END_REF]. To cope with overfitting, the Synthetic Minority Oversampling Technique (SMOTE) was suggested in [START_REF] Chawla | Smote: Synthetic minority over-sampling technique[END_REF]. Unlike ROS, SMOTE generates new synthetic samples by interpolating among several minority objects that are close to each other. However, many studies [START_REF] Douzas | Improving imbalanced learning through a heuristic oversampling method based on k-means and SMOTE[END_REF][START_REF] Sáez | SMOTE-IPF: Addressing the noisy and borderline examples problem in imbalanced classification by a resampling method with filtering[END_REF] have shown SMOTE's drawbacks which involve potential amplification of noise, overlap already present in the data. SMOTE's improvement include Borderline-SMOTE [START_REF] Han | Borderline-SMOTE: A new over-sampling method in imbalanced data sets learning[END_REF], which identifies borderline minority class examples to generate new samples. Clustering-based oversampling techniques were also proposed [START_REF] Douzas | Improving imbalanced learning through a heuristic oversampling method based on k-means and SMOTE[END_REF][START_REF] Ma | CURE-SMOTE algorithm and hybrid algorithm for feature selection and parameter optimization based on random forests[END_REF] to smartly select the regions where to generate new points. -Undersampling: These approaches create a subset of the original dataset by removing some majority class instances. Like random oversampling, the naive undersampling technique is to randomly remove majority objects, which may potentially remove meaningful information from the dataset. Therefore, other techniques have been suggested to smartly remove unwanted majority class instances. Commonly, traditional filtering techniques have been used to perform undersampling. For example, Neighborhood Cleaning Rule (NCL) discards majority class instances using the Edited Nearest Neighbors (ENN) introduced in [START_REF] Wilson | Asymptotic properties of nearest neighbor rules using edited data[END_REF]. Similarly, Tomek Links (TL) [START_REF] Ivan | Two modifications of cnn[END_REF] is occasionally used as an undersampling method. Clustering has also been used for undersampling in a number of occasions [START_REF] Ofek | Fast-CBUS: A fast clusteringbased undersampling method for addressing the class imbalance problem[END_REF][START_REF] Tsai | Under-sampling class imbalanced datasets by combining clustering analysis and instance selection[END_REF], to optimize the selection process of majority instances to eliminate. -Hybrid: This strategy combines both oversampling and undersampling in order to re-balance the dataset. Typically, SMOTE is paired with an undersampling procedure to fix its drawbacks. For instance, SMOTE-ENN and SMOTE-TL were suggested in [START_REF] Batista | A study of the behavior of several methods for balancing machine learning training data[END_REF] to combine SMOTE with ENN and TL respectively. SMOTE-RSB* [START_REF] Ramentol | SMOTE-RSB *: A hybrid preprocessing approach based on oversampling and undersampling for high imbalanced data-sets using SMOTE and rough sets theory[END_REF] is a method which combines SMOTE for oversampling with the Rough Set Theory [START_REF] Pawlak | Rough sets[END_REF] as a cleaning technique.

In SMOTE-IPF [START_REF] Khoshgoftaar | Improving software quality prediction by noise filtering techniques[END_REF], SMOTE is firstly executed, and then the Iterative-Partitioning Filter (IPF) [START_REF] Khoshgoftaar | Improving software quality prediction by noise filtering techniques[END_REF] is performed to remove noisy original examples, and those introduced by SMOTE. Authors in [START_REF] Koziarski | Combined cleaning and resampling algorithm for multi-class imbalanced data with label noise[END_REF] suggested a combination of a SMOTE-like algorithm with a cleaning procedure to reduce the effects of overlapping. Similarly, the class overlap issue is touched upon in [START_REF] Vuttipittayamongkol | Improved overlap-based undersampling for imbalanced dataset classification with application to epilepsy and parkinson's disease[END_REF] combining a soft clustering method with Borderline-SMOTE.

Theory of evidence

The theory of evidence [START_REF] Dempster | A generalization of bayesian inference[END_REF][START_REF] Shafer | A mathematical theory of evidence[END_REF][START_REF] Smets | The Transferable Belief Model for Quantified Belief Representation[END_REF], also referred to as Dempster-Shafer theory (DST) or belief function theory, is a flexible and well-founded framework for the representation and combination of uncertain knowledge. The frame of discernment defines a finite set of M exclusive possible events, e.g., possible labels for an object to classify, and is denoted as follows:

Ω = {ω 1 , ω 2 , ..., ω M } (1)
A basic belief assignment (bba) denotes the amount of belief stated by a source of evidence, committed to 2 Ω , i.e., all subsets of the frame including the whole frame itself. Precisely, a bba is represented by a mapping function m : 2 Ω → [0, 1] such that:

A∈2 Ω m(A) = 1 (2) m(∅) = 0 (3)
Each mass m(A) quantifies the amount of belief allocated to an event A of Ω. A bba is unnormalized when m(∅) > 0, and must be normalized under a closedworld assumption [START_REF] Smets | The nature of the unnormalized beliefs encountered in the transferable belief model[END_REF]. A focal element is a subset A ⊆ Ω where m(A) = 0.

The Plausibility function is another representation of knowledge defined by Shafer [START_REF] Shafer | A mathematical theory of evidence[END_REF] as follows:

P l(A) = B∩A =∅ m(B), ∀ A ∈ 2 Ω (4)
P l(A) represents the total possible support for A and its subsets.

4 Uncertainty-Aware Hybrid re-Sampling method (UAHS)

To tackle binary imbalanced datasets, we propose an Uncertainty-Aware Hybrid re-sampling method (UAHS). Observations are firstly assigned soft evidential labels (bbas) using the credal classification rule (CCR) introduced in [START_REF] Liu | Credal classification rule for uncertain data based on belief functions[END_REF]. CCR uses the centers of each class and meta-class as pieces of evidence for each example's membership, instead of using nearest neighbors as it has been employed in [START_REF] Grina | A preprocessing approach for class-imbalanced data using smote and belief function theory[END_REF]. Unlike [START_REF] Grina | A preprocessing approach for class-imbalanced data using smote and belief function theory[END_REF], our case deals with the soft labeling of both majority and minority classes. Thus, an evidential nearest neighbor-based approach might produce biased memberships towards the majority class, since the latter usually have a much higher density than the minority one.

The computed bba is later used for cleaning unwanted majority objects and selectively generating synthetic minority instances.

Each step is detailed in the following subsections.

Creating soft labels

UAHS proceeds by determining the centers of each class and meta-class (the overlapping region), then creating a bba based on the distance between the majority sample and each class center. The class centers are simply computed by the mean value of the training set in the corresponding class. For the meta-class U , representing the overlapping region, the center is defined by the barycenter of the involved class centers as follows:

C U = 1 |U | ωi∈U C i (5)
where ω i are the classes involved in U , C i is the corresponding center and U represents the meta-class.

Once the centers are created, the evidential soft label of each example is represented by a bba over the frame of discernment Ω = {ω 0 , ω 1 , ω 2 } where ω 1 and ω 2 represent respectively the majority and the minority class. The proposition ω 0 is included in the frame of discernment explicitly to represent the outlier class.

Let x s be a sample belonging to the training set. Each class center represents a piece of evidence to the evidential membership of the sample. The mass values in regard to the class memberships of x s should depend on d(x s , C), i.e., the distance between x s and the respective class center. The greater the distance, the lower the mass value. Consequently, the closer x s is to a specific class center, the more likely it belongs to the corresponding class. Hence, the initial unnormalized masses should be represented by decreasing distance based functions. We use the Mahalanobis distance [START_REF] Mahalanobis | On the generalized distance in statistics[END_REF], in this work, as recommended by [START_REF] Liu | Credal classification rule for uncertain data based on belief functions[END_REF] in order to deal with anisotropic datasets.

The unnormalized masses are calculated accordingly:

m({ω i }) = e -d(xs,Ci) , i ∈ {1, 2} (6) m
(U ) = e -γ λ d(xs,C U ) , U = {ω 1 , ω 2 } (7) m({ω 0 }) = e t (8)
where λ = β 2 α . A value of α = 1 is fixed as recommended to obtain good results on average, and β is a parameter such that 0 < β < 1. It is used to tune the number of objects committed to the overlapping region. The value of γ is equal to the ratio between the maximum distance of x s to the centers in U and the minimum distance. It is used to measure the degree of distinguishability among the majority and minority classes. The smaller γ indicates a poor distinguishability degree between the classes of U for x s . The outlier class ω 0 is taken into account in order to deal with objects far from both classes, and its mass value is calculated according to an outlier threshold t. As performed in [START_REF] Liu | Credal classification rule for uncertain data based on belief functions[END_REF], the unnormalized belief masses are finally normalized as follows:

m(A) = m(A) B⊆Ω m(B) (9)
As a result, a bba is created to formally represent each object's soft label.

Cleaning the majority class

As a result of bba creation, each majority object will have masses in 4 focal elements namely: m({ω 1 }) for the majority class, m({ω 2 }) for the minority class, m(U ) for the overlapping region U , and m({ω 0 }) for the outlier class. These masses are used to remove problematic samples from the majority class. There are different types of unwanted samples which could be removed namely:

-Overlapping: Ambiguous samples are usually located in regions where there is strong overlap between classes as seen in Figure 1a. This situation could correspond to what is called "conflict" in Evidence Theory. In our framework, this type of examples will have a high mass value in m(U ). Thus, majority instances whose bba has the maximum mass committed to m(U ) are considered as part of an overlapping region, and are automatically discarded. To avoid excessive elimination and allow tuning, it is also possible to tune the parameter β. The higher value of β will result in fewer objects committed to the overlapping region. As for majority instances whose highest mass is not committed to m(U ) (i.e. not in overlapping regions), the observation is necessarily committed to one of the singletons in Ω ({ω 1 }, {ω 2 }, or {ω 0 }). In this situation, we make use of the plausibility function defined in eq. ( 4) to make a decision of acceptance or rejection. Each majority instance x s is affected to the class with the maximum plausibility P l max = max ω∈Ω P l({ω}). -Label noise: Normally, majority observations should have the maximum plausibility committed to m({ω 1 }) which measures the membership value towards the majority class. By contrast, having P l max committed to m({ω 2 }) signify that they are located in a minority location, as illustrated in Figure 1c. Consequently, these objects are eliminated from the dataset. -Outlier: The final scenario occurs when the sample in question is located in a region far from both classes, as shown in Figure 1b. In our framework, this is characterized by the state of ignorance and could be discarded in the undersampling procedure. Hence, majority objects whose maximum plausibility P l max committed to m({ω 0 }) are considered as outliers and removed from the dataset.

(a) Ambiguous samples in an overlapping area.

(b) An outlier far from both classes.

(c) A sample that could be characterized as label noise.

Fig. 1: Illustrations describing the different data difficulty factors that could worsen class imbalance. Green and red colors respectively represent the majority class and the minority one.

Applying selective minority oversampling

Once the cleaning procedure (undersampling) is performed, we execute the oversampling phase. The created bbas at the first step are further exploited in this stage, to intelligently create new synthetic minority samples.

Similarly to the cleaning step, the minority objects are categorized into three possible difficulties: overlapping, label noise, or outlier. The object is considered as "safe" if it does not belong to any of the three types. Thus, it does not need to participate in the oversampling phase. The same goes for both label noise and outlier, since using these types to create synthetic data could result in overgeneralization, which is a major drawback for oversampling [START_REF] Han | Borderline-SMOTE: A new over-sampling method in imbalanced data sets learning[END_REF]. Although, samples belonging to overlapping signify that they are located at the borders of the minority class. Hence, those are the samples that we are most uncertain about, and should focus on in the oversampling phase. Much like other popular oversampling methods such as BorderlineSMOTE [START_REF] Han | Borderline-SMOTE: A new over-sampling method in imbalanced data sets learning[END_REF], our goal is to emphasize the borders of the minority class in order to further improve its visibility.

More formally, let P denote the set of minority examples whose highest mass is committed to m(U ) (the set of ambiguous minority objects). We firstly compute the k minority nearest neighbors for each object in P . In this step, we generate |P | * s synthetic minority points, where s is a value between 1 and k. In other words, for each minority instance in P , we randomly select s samples from its k minority nearest neighbors. Finally, s new synthetic points are generated anywhere on the lines joining the samples in P and its s randomly selected neighbors:

-

-→ new = - → a + w * ( - → b -- → a ) (10)
where -→ a is the sample in P , -→ b is a selected minority neighbor, and w represents a random value between 0 and 1. This procedure is repeated for each sample in P , similarly to the SMOTE algorithm (more details in [START_REF] Chawla | Smote: Synthetic minority over-sampling technique[END_REF]). Furthermore, we tested our resampling method on a two-dimensional imbalanced dataset in order to showcase the behavior of the algorithm at each step (see Figure 2).

Experimental study

In this section, we will describe firstly the setup of the conducted experiments in subsection 5.1. Lastly, we will present the results and discuss them in subsection 5.2.

Setup

Datasets For the purpose of this evaluation, we selected binary imbalanced datasets from the keel repository [START_REF] Alcala-Fdez | Keel data-mining software tool: Data set repository, integration of algorithms and experimental analysis framework[END_REF]. The datasets are further detailed in Table 1. The imbalance ratio was calculated as IR = #majority #minority . The variations of the different parameters (IR, features, and size) allowed for experimenting in different real world settings. Baseline classifier. As baseline, we use the decision tree classifier, more specifically CART [START_REF] Breiman | Classification and regression trees[END_REF]. The implementation provided in the scikit-learn machine learning python library [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] was used, with the default parameters.

Metrics and evaluation strategy To appropriately assess the methods in imbalanced scenarios, we use the G-Mean (GM) [START_REF] Barandela | New applications of ensembles of classifiers[END_REF]. The GM is a popular measure for evaluating classifiers in imbalanced settings. It is calculated as the geometric mean of sensitivity and specificity:

G-Mean = sensitivity × specif icity [START_REF] Grina | A preprocessing approach for class-imbalanced data using smote and belief function theory[END_REF] In order to ensure the fairness of the observed results, we adopt a 10-fold stratified cross validation to eliminate inconsistencies. The dataset is split into 10 parts taking into account the class distribution, 90% of which is the training set, the rest is the test set, and the average of G-mean is taken as the final result. It is worth noting that at each fold, resampling was performed only on the training set.

To better evaluate the significance of the results, statistical analysis was run using the Wilcoxon's signed rank tests [START_REF] Wilcoxon | Individual comparisons by ranking methods[END_REF] for the significance level of α = 0.05.

Reference methods and parameters. We compared our proposed method (UAHS) against 4 well known re-sampling methods, in addition to Baseline (BL). The compared methods are SMOTE [START_REF] Chawla | Smote: Synthetic minority over-sampling technique[END_REF], SMOTE-IPF [START_REF] Sáez | SMOTE-IPF: Addressing the noisy and borderline examples problem in imbalanced classification by a resampling method with filtering[END_REF], SMOTE-ENN [START_REF] Batista | A study of the behavior of several methods for balancing machine learning training data[END_REF], and SMOTE-TL [START_REF] Batista | A study of the behavior of several methods for balancing machine learning training data[END_REF]. The SMOTE-IPF implementation in Smote-variants [START_REF] Kovács | Smote-variants: A python implementation of 85 minority oversampling techniques[END_REF] was used. For the rest of the methods, the implementations provided by the python toolbox imbalanced-learn [START_REF] Lemaître | Imbalanced-learn: A python toolbox to tackle the curse of imbalanced datasets in machine learning[END_REF] were applied.

The following parameters were considered for our proposed method UAHS: α was set to 1 as recommended in [START_REF] Liu | Credal classification rule for uncertain data based on belief functions[END_REF], the tuning parameter t for m({ω 0 }) was fixed to 2 to obtain good results in average, and we tested three different values for β in {0.3, 0.5, 0.7} and selected the most performing one for each dataset, since the amount of class overlap differs in each case. For the other reference methods, we used the recommended parameters in their respective original papers.

Results discussion

Results on 15 binary imbalanced datasets are shown in Table 2. The best G-Mean value is marked in bold. Our proposed method UAHS achieved the top performances across 10 out of 15 datasets. It showed clear improvement over the compared resampling methods and baseline across complex datasets, especially with high imbalance degree and class overlapping. Furthermore, UAHS performed significantly better in cases where there is a high number of borderline instance. This confirms that our proposal succeeded at emphasizing on the visibility of the minority class, and improving its borders by cleaning the uncertain samples present in the overlapping and noisy regions of the majority class.

The results for Wilcoxon's pairwise test are shown in Table 3. R+ represents the sum of ranks in favor of UAHS, R-, the sum of ranks in favor of the reference methods, and exact p-values are calculated for each comparison. All comparisons

Conclusion

Solutions for imbalanced datasets are increasingly being applied to critical real world domains. In order to deal with such scenarios, the proposed methods should also handle the uncertainty in the data. In this work, we propose an Uncertainty-Aware Hybrid resampling which combines undersampling and oversampling phases to efficiently re-balance binary datasets. We use an evidential structure to represent soft labels for each sample in the dataset. These representations are later used to remove majority samples which are ambiguous and noisy, and to select minority observations at the borders to generate new minority points.

For future work, we plan to further optimize our method using heuristic methods in order to approximate the amount of instances which should be cleaned, and the number of instances to generate.

Fig. 2 :

 2 Fig. 2: An imbalanced binary example showing the behavior of our proposed algorithm at each step.

Table 1 :

 1 Description of the imbalanced datasets selected from the KEEL repository.

	Datasets	Imbalance ratios (IR)	Features	Samples
	wisconsin	1.860	9	683
	glass0	2.060	9	214
	vehicle3	2.990	18	846
	ecoli1	3.360	7	336
	yeast3	8.100	8	1484
	page-blocks0	8.790	10	5472
	ecoli-0-6-7 vs 3-5	9.090	7	222
	yeast-0-3-5-9 vs 7-8	9.120	8	506
	ecoli-0-2-6-7 vs 3-5	9.180	7	224
	ecoli-0-1-4-7 vs 2-3-5-6	10.590	7	336
	glass4	15.460	9	214
	yeast-2 vs 8	23.100	8	482
	yeast5	32.730	8	1484
	kr-vs-k-zero vs eight	53.070	6	1460
	abalone-20 vs 8-9-10	72.690	8	1916

Table 2 :

 2 G-Mean results for KEEL datasets using CART. can be considered as statistically significant under a level of 5% since all pvalues are lower than the threshold 0.05. This reveals statistically significant improvements by our method against SMOTE, SMOTE-IPF, SMOTE-ENN, SMOTE-TL, and baseline.

	Datasets	BL SMOTE SMOTE-IPF SMOTE-ENN SMOTE-TL UAHS
	wisconsin	0.934 0.928	0.924	0.950	0.932	0.964
	glass0	0.737 0.789	0.772	0.763	0.782	0.786
	vehicle3	0.689 0.660	0.682	0.727	0.679	0.673
	ecoli1	0.816 0.849	0.856	0.871	0.836	0.883
	yeast3	0.791 0.834	0.854	0.878	0.849	0.921
	page-blocks0	0.904 0.928	0.920	0.930	0.923	0.958
	ecoli-0-6-7 vs 3-5	0.764 0.721	0.769	0.772	0.782	0.790
	yeast-0-3-5-9 vs 7-8	0.532 0.620	0.606	0.677	0.582	0.613
	ecoli-0-2-6-7 vs 3-5	0.762 0.783	0.781	0.751	0.780	0.761
	ecoli-0-1-4-7 vs 2-3-5-6 0.735 0.797	0.878	0.857	0.861	0.835
	glass4	0.735 0.637	0.706	0.658	0.706	0.838
	yeast-2 vs 8	0.547 0.666	0.664	0.722	0.734	0.737
	yeast5	0.840 0.830	0.812	0.912	0.855	0.932
	kr-vs-k-zero vs eight	0.976 1.000	1.000	0.999	1.000	1.000
	abalone-20 vs 8-9-10	0.477 0.695	0.602	0.679	0.587	0.743

Table 3 :

 3 Wilcoxon's signed ranks test results comparing the G-Mean scores for CART.

	Comparisons	R+	R-	p-value
	UAHS vs BL	117.0	3.0	0.000153
	UAHS vs SMOTE	98.0	22.0	0.002364
	UAHS vs SMOTE-IPF	95.0	25.0	0.004187
	UAHS vs SMOTE-ENN	94.0	26.0	0.027679
	UAHS vs SMOTE-TL	83.5	36.5	0.004377