SAM
https://sam.ensam.eu:443
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 23 May 2019 03:09:21 GMT2019-05-23T03:09:21ZProbabilistic Decision Trees using SVM for Multi-class Classification
http://hdl.handle.net/10985/7401
Probabilistic Decision Trees using SVM for Multi-class Classification
URIBE, Juan Sebastian; MECHBAL, Nazih; REBILLAT, Marc; BOUAMAMA, Karima; PENGOV, Marco
In the automotive repairing backdrop, retrieving from previously solved incident the database features that could support and speed up the diagnostic is of great usefulness. This decision helping process should give a fixed number of the more relevant diagnostic classified in a likelihood sense. It is a probabilistic multi-class classification problem. This paper describes an original classification technique, the Probabilistic Decision Tree (PDT) producing a posteriori probabilities in a multi-class context. It is based on a Binary Decision Tree (BDT) with Probabilistic Support Vector Machine classifier (PSVM). At each node of the tree, a bi-class SVM along with a sigmoid function are trained to give a probabilistic classification output. For each branch, the outputs of all the nodes composing the branch are combined to lead to a complete evaluation of the probability when reaching the final leaf (representing the class associated to the branch). To illustrate the effectiveness of PDTs, they are tested on benchmark datasets and results are compared with other existing approaches.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10985/74012013-01-01T00:00:00ZURIBE, Juan SebastianMECHBAL, NazihREBILLAT, MarcBOUAMAMA, KarimaPENGOV, MarcoIn the automotive repairing backdrop, retrieving from previously solved incident the database features that could support and speed up the diagnostic is of great usefulness. This decision helping process should give a fixed number of the more relevant diagnostic classified in a likelihood sense. It is a probabilistic multi-class classification problem. This paper describes an original classification technique, the Probabilistic Decision Tree (PDT) producing a posteriori probabilities in a multi-class context. It is based on a Binary Decision Tree (BDT) with Probabilistic Support Vector Machine classifier (PSVM). At each node of the tree, a bi-class SVM along with a sigmoid function are trained to give a probabilistic classification output. For each branch, the outputs of all the nodes composing the branch are combined to lead to a complete evaluation of the probability when reaching the final leaf (representing the class associated to the branch). To illustrate the effectiveness of PDTs, they are tested on benchmark datasets and results are compared with other existing approaches.A multi-sine sweep method for the characterization of weak non-linearities ; plant noise and variability estimation.
http://hdl.handle.net/10985/10288
A multi-sine sweep method for the characterization of weak non-linearities ; plant noise and variability estimation.
GALLO, Maxime; EGE, Kerem; REBILLAT, Marc; ANTONI, Jérôme
Weak non-linearities in vibrating structures can be characterized by a signal-model approach based on cascade of Hammerstein models. The experiment consists in exciting a device with a sine sweep at different levels, in order to assess the evolutions of non linearities on a wide frequency range. A method developed previously, based on exponential sine sweep, is able to give an approximative identification of the Hammerstein models, but cannot make the distinction between nonlinear distortion and stationary plant noise. Therefore, this paper proposes improvements on the method that provide a more precise estimation of the Hammerstein models through the cancellation of the plant noise: it relies on the repetition of the signal on a certain amount of periods (multi-sine sweeps) and then on the consideration of the synchronous average out of the different periods from the resulting signal. Mathematical foundations and practical implementation of the method are discussed. The second main point of improvement concerning the study of the vibrating device is the use of the Bootstrap analysis. By considering some periods randomly chosen among the multisine sweep, one can study the variability of the experiments. The method becomes more robust.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10985/102882015-01-01T00:00:00ZGALLO, MaximeEGE, KeremREBILLAT, MarcANTONI, JérômeWeak non-linearities in vibrating structures can be characterized by a signal-model approach based on cascade of Hammerstein models. The experiment consists in exciting a device with a sine sweep at different levels, in order to assess the evolutions of non linearities on a wide frequency range. A method developed previously, based on exponential sine sweep, is able to give an approximative identification of the Hammerstein models, but cannot make the distinction between nonlinear distortion and stationary plant noise. Therefore, this paper proposes improvements on the method that provide a more precise estimation of the Hammerstein models through the cancellation of the plant noise: it relies on the repetition of the signal on a certain amount of periods (multi-sine sweeps) and then on the consideration of the synchronous average out of the different periods from the resulting signal. Mathematical foundations and practical implementation of the method are discussed. The second main point of improvement concerning the study of the vibrating device is the use of the Bootstrap analysis. By considering some periods randomly chosen among the multisine sweep, one can study the variability of the experiments. The method becomes more robust.On-board Decision Making Platform for Structural Health Monitoring
http://hdl.handle.net/10985/12378
On-board Decision Making Platform for Structural Health Monitoring
BARTHES, Clément; MECHBAL, Nazih; MOSALAM, Khalid; REBILLAT, Marc
The ability to monitor the health of complex structures such as aeronautic or civil engineering structures in real time is becoming increasingly important. This process is referred to as structural health monitoring (SHM) and relies on onboard platforms comprising sensors, computational units, communication resources, and sometimes actuators. Many of such platforms have been developed within the last years but there is still a lack of structuration and knowledge exchange regarding the software and hardware architectures of such platforms. The aim of the present paper is to introduce an open hardware and open software platform dedicated to SHM within the fields of aeronautics and civil engineering. The platform presented here will be made available in an open hardware and open source framework to allow SHM researchers to run concurrent detection, localization, classification or quantification algorithms using simple interpreted languages such as Python.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/123782017-01-01T00:00:00ZBARTHES, ClémentMECHBAL, NazihMOSALAM, KhalidREBILLAT, MarcThe ability to monitor the health of complex structures such as aeronautic or civil engineering structures in real time is becoming increasingly important. This process is referred to as structural health monitoring (SHM) and relies on onboard platforms comprising sensors, computational units, communication resources, and sometimes actuators. Many of such platforms have been developed within the last years but there is still a lack of structuration and knowledge exchange regarding the software and hardware architectures of such platforms. The aim of the present paper is to introduce an open hardware and open software platform dedicated to SHM within the fields of aeronautics and civil engineering. The platform presented here will be made available in an open hardware and open source framework to allow SHM researchers to run concurrent detection, localization, classification or quantification algorithms using simple interpreted languages such as Python.Peaks Over Threshold Method for Structural Health Monitoring Detector Design
http://hdl.handle.net/10985/10377
Peaks Over Threshold Method for Structural Health Monitoring Detector Design
HMAD, Ouadie; MECHBAL, Nazih; REBILLAT, Marc
Structural Health Monitoring (SHM) system offers new approaches to interrogate the integrity of structures. The most critical step of such systems is the damage detection step since it is the first and because performances of the following steps (damage localization, severity estimation…) depend on it. Care has thus to be taken when designing the detector. The objective of this communication is to discuss issues related to the design of a detector for the structural health monitoring of composite structures. The structure under monitoring is a substructure of an aircraft nacelle. In the absence of damage, the detector principle is to statistically characterize the healthy behavior of the structure. This characterization is based on the availability of a decision statistics synthesized from a damage index. Airline business models rely on Probability of False Alarms (������) as main performance criterion. In general, the requirement on ������ is 10E-9 which is very small. To determine the decision threshold, the approach we consider, consists to model the tail of the decision statistics using the Peaks Over Threshold method extracted from the extreme value theory (EVT). This method has been applied for different configuration of learning sample and probability of false alarm. This approach of tail distribution estimation is interesting since it is not necessary to know the distribution of the decision statistic to develop a detector. However, its main drawback is that it is necessary to have very large databases to accurately estimate decision thresholds to then decide the presence or absence of damage.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10985/103772015-01-01T00:00:00ZHMAD, OuadieMECHBAL, NazihREBILLAT, MarcStructural Health Monitoring (SHM) system offers new approaches to interrogate the integrity of structures. The most critical step of such systems is the damage detection step since it is the first and because performances of the following steps (damage localization, severity estimation…) depend on it. Care has thus to be taken when designing the detector. The objective of this communication is to discuss issues related to the design of a detector for the structural health monitoring of composite structures. The structure under monitoring is a substructure of an aircraft nacelle. In the absence of damage, the detector principle is to statistically characterize the healthy behavior of the structure. This characterization is based on the availability of a decision statistics synthesized from a damage index. Airline business models rely on Probability of False Alarms (������) as main performance criterion. In general, the requirement on ������ is 10E-9 which is very small. To determine the decision threshold, the approach we consider, consists to model the tail of the decision statistics using the Peaks Over Threshold method extracted from the extreme value theory (EVT). This method has been applied for different configuration of learning sample and probability of false alarm. This approach of tail distribution estimation is interesting since it is not necessary to know the distribution of the decision statistic to develop a detector. However, its main drawback is that it is necessary to have very large databases to accurately estimate decision thresholds to then decide the presence or absence of damage.Estimation of the temperature field on a composite fan cowl using the static capacity of surface-mounted piezoceramic transducers
http://hdl.handle.net/10985/12044
Estimation of the temperature field on a composite fan cowl using the static capacity of surface-mounted piezoceramic transducers
LIZE, Emmanuel; REBILLAT, Marc; MECHBAL, Nazih
The influence of temperature on SHM (Structural Health Monitoring) systems using guided waves is a major problem for their industrial deployment. One of the most used and cheapest SHM process developed in aeronautic context is based on piezoelectric transducers mounted on the monitored structure. Several methods are then used to assess the presence of damage. A popular one is based on tracking changes in the static capacity of the transducers: it is an efficient damage indicator in close area surrounding the device and is often used as a fault diagnosis of the transducer itself. However, monitoring decision are robustified with temperature sensors also mounted on the structure, adding wires and signal processing post-treatments. In this article, the static capacity is used to determine the temperature on each lead zirconate titanate transducer (PZT). By imparting this additional functionality to PZT devices, supplementary instrumentation is not necessary and an estimation of the entire temperature field of the structure is obtained. The original proposed approach is tested experimentally on a on a small composite plate and then on a large real part of an A380 composite nacelle. The results show that the temperature field on the structure can be estimated with a precision of ±5 °C using a linear regression between static capacity and temperature
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/120442017-01-01T00:00:00ZLIZE, EmmanuelREBILLAT, MarcMECHBAL, NazihThe influence of temperature on SHM (Structural Health Monitoring) systems using guided waves is a major problem for their industrial deployment. One of the most used and cheapest SHM process developed in aeronautic context is based on piezoelectric transducers mounted on the monitored structure. Several methods are then used to assess the presence of damage. A popular one is based on tracking changes in the static capacity of the transducers: it is an efficient damage indicator in close area surrounding the device and is often used as a fault diagnosis of the transducer itself. However, monitoring decision are robustified with temperature sensors also mounted on the structure, adding wires and signal processing post-treatments. In this article, the static capacity is used to determine the temperature on each lead zirconate titanate transducer (PZT). By imparting this additional functionality to PZT devices, supplementary instrumentation is not necessary and an estimation of the entire temperature field of the structure is obtained. The original proposed approach is tested experimentally on a on a small composite plate and then on a large real part of an A380 composite nacelle. The results show that the temperature field on the structure can be estimated with a precision of ±5 °C using a linear regression between static capacity and temperatureParallel Hammerstein Models Identification using Sine Sweeps and the Welch Method
http://hdl.handle.net/10985/12377
Parallel Hammerstein Models Identification using Sine Sweeps and the Welch Method
BOUTILLON, Xavier; CORTEEL, Etienne; REBILLAT, Marc; ROGGERONE, Vincent
Linearity is a common assumption for many real life systems. But in many cases, the nonlinear behavior of systems cannot be ignored and has to be modeled and estimated. Among the various classes of nonlinear models present in the literature, Parallel Hammertein Models (PHM) are interesting as they are at the same time easy to understand as well as to estimate when using exponential sine sweeps (ESS) based methods. However, the classical EES- based estimation procedure for PHM relies on a very speci c input signal (ESS), which limits its use in practice. A method is proposed here based on the Welch method that allows for PHM estimation with arbitrary sine sweeps (ASS) which are a much broader class of input signals than ESS. Results show that for various ASS, the proposed method provides results that are in excellent agreement with the ones obtained with the classical ESS method.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/123772017-01-01T00:00:00ZBOUTILLON, XavierCORTEEL, EtienneREBILLAT, MarcROGGERONE, VincentLinearity is a common assumption for many real life systems. But in many cases, the nonlinear behavior of systems cannot be ignored and has to be modeled and estimated. Among the various classes of nonlinear models present in the literature, Parallel Hammertein Models (PHM) are interesting as they are at the same time easy to understand as well as to estimate when using exponential sine sweeps (ESS) based methods. However, the classical EES- based estimation procedure for PHM relies on a very speci c input signal (ESS), which limits its use in practice. A method is proposed here based on the Welch method that allows for PHM estimation with arbitrary sine sweeps (ASS) which are a much broader class of input signals than ESS. Results show that for various ASS, the proposed method provides results that are in excellent agreement with the ones obtained with the classical ESS method.Automatic Damage Quantification Using Signal Based And Nonlinear Model Based Damage Sensitive Features
http://hdl.handle.net/10985/12043
Automatic Damage Quantification Using Signal Based And Nonlinear Model Based Damage Sensitive Features
GHRIB, Meriem; REBILLAT, Marc; MECHBAL, Nazih; VERMOT DES ROCHES, Guillaume
Structural Health Monitoring (SHM) can be de ned as the process of acquiring and analyzing data from on-board sensors to evaluate the health of a structure. Classically, an SHM process can be performed in four steps: detection, localization, classi cation and quanti cation. This paper addresses damage quanti cation issue as a classi cation problem whereby each class corresponds to a certain damage extent. Starting from the assumption that damage causes a structure to exhibit nonlinear response, we investigate whether the use of nonlinear model based features increases classi cation performance. A support Vector Machine (SVM) is used to perform multi-class classi cation task. Two types of features are used as inputs to the SVM algorithm: Signal Based Features (SBF) and Nonlinear Model Based Features (NMBF). SBF are rooted in a direct use of response signals and do not consider any underlying model of the test structure. NMBF are computed based on parallel Hammerstein models which are identi ed with an Exponential Sine Sweep (ESS) signal. A study of the sensitivity of classi cation performance to the noise contained in output signals is also conducted. Dimension reduction of features vector using Principal Component Analysis (PCA) is carried out in order to nd out if it allows robustifying the quanti cation process suggested in this work. Simulation results on a cantilever beam with a bilinear torsion spring sti ness are considered for demonstration. Results show that by introducing NMBF, classi cation performance is improved. Furthermore, PCA allows for higher recognition rates while reducing features vector dimension. However, classi ers trained on NMBF or on principal components appear to be more sensitive to output noise than those trained on SBF.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/120432017-01-01T00:00:00ZGHRIB, MeriemREBILLAT, MarcMECHBAL, NazihVERMOT DES ROCHES, GuillaumeStructural Health Monitoring (SHM) can be de ned as the process of acquiring and analyzing data from on-board sensors to evaluate the health of a structure. Classically, an SHM process can be performed in four steps: detection, localization, classi cation and quanti cation. This paper addresses damage quanti cation issue as a classi cation problem whereby each class corresponds to a certain damage extent. Starting from the assumption that damage causes a structure to exhibit nonlinear response, we investigate whether the use of nonlinear model based features increases classi cation performance. A support Vector Machine (SVM) is used to perform multi-class classi cation task. Two types of features are used as inputs to the SVM algorithm: Signal Based Features (SBF) and Nonlinear Model Based Features (NMBF). SBF are rooted in a direct use of response signals and do not consider any underlying model of the test structure. NMBF are computed based on parallel Hammerstein models which are identi ed with an Exponential Sine Sweep (ESS) signal. A study of the sensitivity of classi cation performance to the noise contained in output signals is also conducted. Dimension reduction of features vector using Principal Component Analysis (PCA) is carried out in order to nd out if it allows robustifying the quanti cation process suggested in this work. Simulation results on a cantilever beam with a bilinear torsion spring sti ness are considered for demonstration. Results show that by introducing NMBF, classi cation performance is improved. Furthermore, PCA allows for higher recognition rates while reducing features vector dimension. However, classi ers trained on NMBF or on principal components appear to be more sensitive to output noise than those trained on SBF.Damage indexes comparison for the structural health monitoring of a stiffened composite plate
http://hdl.handle.net/10985/12042
Damage indexes comparison for the structural health monitoring of a stiffened composite plate
MECHBAL, Nazih; REBILLAT, Marc
Stiffened composite structures are very appealing in aeronautic applications due to their unique stiffness to mass ratio. However, they are also prone to various and complex damage scenario (stiffener debonding, impact damage...) and to complex wave propagation phenomena due to the presence of the stiffener. Consequently, autonomous monitoring of such structure is still a real issue. The process of monitoring in real-time a structure is referred to structural health monitoring (SHM) and consists of several steps: damage detection, localization, classification, and quantification. The focus is put here on the damage detection step of SHM. To detect damages, stiffened composite structures are equipped with piezoelectric elements that act both as sensors and actuators. A database at the unknown (and possibly damaged state) is then compared to a healthy reference database. Several damage indexes (DIs) designed for detection are extracted from this comparison. The SHM process classically relies on four sequential steps: damage detection, localization, classification, and quantification. The most critical step of such process is the damage detection step since it is the first one and because performances of the following steps depend on it. A common method to design such a detector consists in relying on a statistical characterization of the damage indexes available in the healthy behavior of the structure. On the basis of this information, a decision threshold can then be computed in order to achieve a desired probability of false alarm (PFA). In this paper, the performances of these DIs with respect to damage detection in a stiffened composite plate are studied. Results show that DIs based on energy consideration perform better than the ones based on cross-correlation. Furthermore Fourier-transform based DIs appear to be insensitive to the presence of damage in such structure.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/120422017-01-01T00:00:00ZMECHBAL, NazihREBILLAT, MarcStiffened composite structures are very appealing in aeronautic applications due to their unique stiffness to mass ratio. However, they are also prone to various and complex damage scenario (stiffener debonding, impact damage...) and to complex wave propagation phenomena due to the presence of the stiffener. Consequently, autonomous monitoring of such structure is still a real issue. The process of monitoring in real-time a structure is referred to structural health monitoring (SHM) and consists of several steps: damage detection, localization, classification, and quantification. The focus is put here on the damage detection step of SHM. To detect damages, stiffened composite structures are equipped with piezoelectric elements that act both as sensors and actuators. A database at the unknown (and possibly damaged state) is then compared to a healthy reference database. Several damage indexes (DIs) designed for detection are extracted from this comparison. The SHM process classically relies on four sequential steps: damage detection, localization, classification, and quantification. The most critical step of such process is the damage detection step since it is the first one and because performances of the following steps depend on it. A common method to design such a detector consists in relying on a statistical characterization of the damage indexes available in the healthy behavior of the structure. On the basis of this information, a decision threshold can then be computed in order to achieve a desired probability of false alarm (PFA). In this paper, the performances of these DIs with respect to damage detection in a stiffened composite plate are studied. Results show that DIs based on energy consideration perform better than the ones based on cross-correlation. Furthermore Fourier-transform based DIs appear to be insensitive to the presence of damage in such structure.A data-driven temperature compensation approach for Structural Health Monitoring using Lamb waves
http://hdl.handle.net/10985/11239
A data-driven temperature compensation approach for Structural Health Monitoring using Lamb waves
FENDZI, Claude; REBILLAT, Marc; MECHBAL, Nazih; GUSKOV, Mikhail; COFFIGNAL, Gérard
This paper presents a temperature compensation method for Lamb wave structural health monitoring. The proposed approach considers a representation of the piezo-sensor signal through its Hilbert transform that allows one to extract the amplitude factor and the phase shift in signals caused by temperature changes. An ordinary least square (OLS) algorithm is used to estimate these unknown parameters. After estimating these parameters at each temperature in the operating range, linear functional relationships between the temperature and the estimated parameters are derived using the least squares method. A temperature compensation model is developed based on this linear relationship that allows one to reconstruct sensor signals at any arbitrary temperature. The proposed approach is validated numerically and experimentally for an anisotropic composite plate at different temperatures ranging from Formula to Formula . A close match is found between the measured signals and the reconstructed ones. This approach is interesting as it needs only a limited set of piezo-sensor signals at different temperatures for model training and temperature compensation at any arbitrary temperature. Damage localization results after temperature compensation demonstrate its robustness and effectiveness.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/112392016-01-01T00:00:00ZFENDZI, ClaudeREBILLAT, MarcMECHBAL, NazihGUSKOV, MikhailCOFFIGNAL, GérardThis paper presents a temperature compensation method for Lamb wave structural health monitoring. The proposed approach considers a representation of the piezo-sensor signal through its Hilbert transform that allows one to extract the amplitude factor and the phase shift in signals caused by temperature changes. An ordinary least square (OLS) algorithm is used to estimate these unknown parameters. After estimating these parameters at each temperature in the operating range, linear functional relationships between the temperature and the estimated parameters are derived using the least squares method. A temperature compensation model is developed based on this linear relationship that allows one to reconstruct sensor signals at any arbitrary temperature. The proposed approach is validated numerically and experimentally for an anisotropic composite plate at different temperatures ranging from Formula to Formula . A close match is found between the measured signals and the reconstructed ones. This approach is interesting as it needs only a limited set of piezo-sensor signals at different temperatures for model training and temperature compensation at any arbitrary temperature. Damage localization results after temperature compensation demonstrate its robustness and effectiveness.A General Bayesian Framework for Ellipse-based and Hyperbola-based Damage Localisation in Anisotropic Composite Plates
http://hdl.handle.net/10985/9218
A General Bayesian Framework for Ellipse-based and Hyperbola-based Damage Localisation in Anisotropic Composite Plates
FENDZI, Claude; MECHBAL, Nazih; REBILLAT, Marc; GUSKOV, Mikhail
This paper focuses on Bayesian Lamb wave-based damage localization in structural health monitoring of anisotropic composite materials. A Bayesian framework is applied to take account for uncertainties from experimental time-of-flight measurements and angular dependent group velocity within the composite material. An original parametric analytical expression of the direction dependence of group velocity is proposed and validated numerically and experimentally for anisotropic composite and sandwich plates. This expression is incorporated into time-of-arrival (ToA: ellipse-based) and time-difference-of-arrival (TDoA: hyperbola-based) Bayesian damage localization algorithms. This way, the damage location as well as the group velocity profile are estimated jointly and a priori information taken into consideration. The proposed algorithm is general as it allows to take into account for uncertainties within a Bayesian framework, and to model effects of anisotropy on group velocity. Numerical and experimental results obtained with different damage sizes or locations and for different degrees of anisotropy validate the ability of the proposed algorithm to estimate both the damage location and the group velocity profile as well as the associated confidence intervals. Results highlight the need to consider for anisotropy in order to increase localization accuracy, and to use Bayesian analysis to quantify uncertainties in damage localization.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/92182016-01-01T00:00:00ZFENDZI, ClaudeMECHBAL, NazihREBILLAT, MarcGUSKOV, MikhailThis paper focuses on Bayesian Lamb wave-based damage localization in structural health monitoring of anisotropic composite materials. A Bayesian framework is applied to take account for uncertainties from experimental time-of-flight measurements and angular dependent group velocity within the composite material. An original parametric analytical expression of the direction dependence of group velocity is proposed and validated numerically and experimentally for anisotropic composite and sandwich plates. This expression is incorporated into time-of-arrival (ToA: ellipse-based) and time-difference-of-arrival (TDoA: hyperbola-based) Bayesian damage localization algorithms. This way, the damage location as well as the group velocity profile are estimated jointly and a priori information taken into consideration. The proposed algorithm is general as it allows to take into account for uncertainties within a Bayesian framework, and to model effects of anisotropy on group velocity. Numerical and experimental results obtained with different damage sizes or locations and for different degrees of anisotropy validate the ability of the proposed algorithm to estimate both the damage location and the group velocity profile as well as the associated confidence intervals. Results highlight the need to consider for anisotropy in order to increase localization accuracy, and to use Bayesian analysis to quantify uncertainties in damage localization.