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Demba Diallo - One of the best experts on this subject based on the ideXlab platform.

  • kullback Leibler Divergence for fault estimation and isolation application to gamma distributed data
    Mechanical Systems and Signal Processing, 2017
    Co-Authors: Claude Delpha, Demba Diallo, Abdulrahman Youssef
    Abstract:

    Abstract In this paper we develop a fault detection, isolation and estimation method based on data-driven approach. Data-driven methods are effective for feature extraction and feature analysis using statistical techniques. In the proposal, the Principal Component Analysis (PCA) method is used to extract the features and to reduce the data dimension. Then, the Kullback-Leibler Divergence (KLD) is used to detect the fault occurrence by comparing the Probability Density Function of the latent scores. To estimate the fault amplitude in case of Gamma distributed data, we have developed an analytical model that links the KLD to the fault severity, including the environmental noise conditions. In the Principal Component Analysis framework, the proposed model of the KLD has been analysed and compared to an estimated value of the KLD using the Monte-Carlo estimator. The results show that for incipient faults (  40 dB), the fault amplitude estimation is accurate enough with a relative error less than 1%. The proposed approach is experimentally verified with vibration signals used for monitoring bearings in electrical machines.

  • statistical approach for nondestructive incipient crack detection and characterization using kullback Leibler Divergence
    IEEE Transactions on Reliability, 2016
    Co-Authors: Jinane Harmouche, Claude Delpha, Demba Diallo, Yann Le Bihan
    Abstract:

    This paper is a contribution to the detection and characterisation of small cracks using Eddy Current Testing in the Non Destructive Evaluation framework. Small cracks are considered as incipient faults defined as gradual faults whose signature is weak and concealed by the noise. They are characterized by high signal to noise ratio and low fault to noise ratio. The detection and diagnosis of such faults is still an open challenge. For complex systems, model-based incipient fault detection and diagnosis (FDD) methods usually fail because of the inaccuracy of the model to describe all the phenomena and their interactions. Data-driven methods using statistical features are very promising as long as historical data are available. However in the case of incipient faults, there is not a significant variation of a single feature. The fault signature lies in the global variation of the signal properties. The proposed method relies on the Kullback-Leibler Divergence (KLD) as a nonparametric fault indicator. It measures the slight dissimilarities between the probability density functions of the current signal compared to the faultless or healthy one. Through experimental results, the KLD exhibits a higher sensitivity than the usual statistical features for the detection of small cracks (with dimensions in the order of 0.1 mm) realized in a nickel-based superalloy plate. Moreover, the detection is done with zero missed detection probability. Furthermore, the fault severity is assessed through the characteristics of the crack (surface, length, and depth). In the principal component analysis framework, the analysis of four statistical features (KLD, mean, variance, and maximum) dependency to the excitation frequency allows to discriminating among the cracks.

  • an optimal fault detection threshold for early detection using kullback Leibler Divergence for unknown distribution data
    Signal Processing, 2016
    Co-Authors: Abdulrahman Youssef, Claude Delpha, Demba Diallo
    Abstract:

    The incipient fault detection in industrial processes with unknown distribution of measurements signals and unknown changed parameters is an important problem which has received much attention these last decades. However most of the detection methods (online and offline) need a priori knowledge on the signal distribution, changed parameters, and the change amplitude (Likelihood ratio test, Cusum, etc.). In this paper, an incipient fault detection method that does not need any a priori knowledge on the signals distribution or the changed parameters is proposed. This method is based on the analysis of the Kullback-Leibler Divergence (KLD) of probability distribution functions. However, the performance of the technique is highly dependent on the setting of a detection threshold and the environment noise level described through Signal to Noise Ratio (SNR) and Fault to Noise Ratio (FNR). In this paper, we develop an analytical model of the fault detection performances (False Alarm Probability and Missed Detection Probability). Thanks to this model, an optimisation procedure is applied to optimally set the fault detection threshold depending on the SNR and the fault severity. Compared to the usual settings, through simulation results and experimental data, the optimised threshold leads to higher efficiency for incipient fault detection in noisy environment. HighlightsWe propose an incipient fault detection method that does not need any a priori information on the signals distribution or the changed parameters.We show that the performance of the technique is highly dependent on the setting of a detection threshold and the environment noise level.We develop an analytical model of the fault detection performances (False Alarm Probability and Missed Detection Probability).Based on the aforementioned model, an optimisation procedure is applied to optimally set the fault detection threshold depending on the noise and the fault severity.Compared to the usual settings, a performed validation of this approach with through simulation results and experimental data is given.

  • Statistical Approach for Nondestructive Incipient Crack Detection and Characterization Using Kullback-Leibler Divergence
    IEEE Transactions on Reliability, 2016
    Co-Authors: Jinane Harmouche, Claude Delpha, Demba Diallo, Yann Le Bihan
    Abstract:

    This paper is a contribution to the detection and characterisation of small cracks using Eddy Current Testing in the Non Destructive Evaluation framework. Small cracks are considered as incipient faults defined as gradual faults whose signature is weak and concealed by the noise. They are characterized by high signal to noise ratio and low fault to noise ratio. The detection and diagnosis of such faults is still an open challenge. For complex systems, model-based incipient fault detection and diagnosis (FDD) methods usually fail because of the inaccuracy of the model to describe all the phenomena and their interactions. Data-driven methods using statistical features are very promising as long as historical data are available. However in the case of incipient faults, there is not a significant variation of a single feature. The fault signature lies in the global variation of the signal properties. The proposed method relies on the Kullback-Leibler Divergence (KLD) as a nonparametric fault indicator. It measures the slight dissimilarities between the probability density functions of the current signal compared to the faultless or healthy one. Through experimental results, the KLD exhibits a higher sensitivity than the usual statistical features for the detection of small cracks (with dimensions in the order of 0.1 mm) realized in a nickel-based superalloy plate. Moreover, the detection is done with zero missed detection probability. Furthermore, the fault severity is assessed through the characteristics of the crack (surface, length, and depth). In the principal component analysis framework, the analysis of four statistical features (KLD, mean, variance, and maximum) dependency to the excitation frequency allows to discriminating among the cracks.

  • incipient fault detection and diagnosis based on kullback Leibler Divergence using principal component analysis
    Signal Processing, 2015
    Co-Authors: Jinane Harmouche, Claude Delpha, Demba Diallo
    Abstract:

    Most of fault indicators are devoted to detect deviations related to specific features but they fail to detect and estimate unpredictable slight distortions often caused by incipient faults. The Kullback-Leibler Divergence is characterised with a high sensitivity to incipient faults that cause unpredictable small changes in the process measurements. This work has two main objectives: first estimate the amplitude of incipient faults in multivariate processes based on the Divergence and second evaluate, through detection error probabilities, the performance of the Divergence in the detection of incipient faults in noisy environments.Throughout all the paper, the Fault-to-Noise Ratio (FNR) has been referred to as a comparative criterion between the fault level and noise; particularly the region around 0 dB of FNR is of interest in the evaluation. A theoretical study is developed to derive an analytical model of the Divergence that considers the presence of Gaussian noise and allows obtaining a theoretical estimate of the fault amplitude. After application on a simulated AR process, the fault amplitude estimate turns out to be an overestimation of the actual amplitude, therefore guaranteeing a safety margin for monitoring. Accurate fault severity estimation for an eddy currents application shows the effectiveness of this approach. HighlightsWe propose to enhance the fault detection approach based on the KLD modelling with the introduction of the noise.Based on the aforementioned model an estimator of the fault amplitude is developed and validated.The performances of the detection are studied in a noisy environment with the introduction of the Fault to Noise Ratio (FNR).The robustness of the proposed method is evaluated with the computation of the miss-detection and false alarms probabilities.A performed validation of this approach with a simulated AR model is given.

Claude Delpha - One of the best experts on this subject based on the ideXlab platform.

  • kullback Leibler Divergence for fault estimation and isolation application to gamma distributed data
    Mechanical Systems and Signal Processing, 2017
    Co-Authors: Claude Delpha, Demba Diallo, Abdulrahman Youssef
    Abstract:

    Abstract In this paper we develop a fault detection, isolation and estimation method based on data-driven approach. Data-driven methods are effective for feature extraction and feature analysis using statistical techniques. In the proposal, the Principal Component Analysis (PCA) method is used to extract the features and to reduce the data dimension. Then, the Kullback-Leibler Divergence (KLD) is used to detect the fault occurrence by comparing the Probability Density Function of the latent scores. To estimate the fault amplitude in case of Gamma distributed data, we have developed an analytical model that links the KLD to the fault severity, including the environmental noise conditions. In the Principal Component Analysis framework, the proposed model of the KLD has been analysed and compared to an estimated value of the KLD using the Monte-Carlo estimator. The results show that for incipient faults (  40 dB), the fault amplitude estimation is accurate enough with a relative error less than 1%. The proposed approach is experimentally verified with vibration signals used for monitoring bearings in electrical machines.

  • statistical approach for nondestructive incipient crack detection and characterization using kullback Leibler Divergence
    IEEE Transactions on Reliability, 2016
    Co-Authors: Jinane Harmouche, Claude Delpha, Demba Diallo, Yann Le Bihan
    Abstract:

    This paper is a contribution to the detection and characterisation of small cracks using Eddy Current Testing in the Non Destructive Evaluation framework. Small cracks are considered as incipient faults defined as gradual faults whose signature is weak and concealed by the noise. They are characterized by high signal to noise ratio and low fault to noise ratio. The detection and diagnosis of such faults is still an open challenge. For complex systems, model-based incipient fault detection and diagnosis (FDD) methods usually fail because of the inaccuracy of the model to describe all the phenomena and their interactions. Data-driven methods using statistical features are very promising as long as historical data are available. However in the case of incipient faults, there is not a significant variation of a single feature. The fault signature lies in the global variation of the signal properties. The proposed method relies on the Kullback-Leibler Divergence (KLD) as a nonparametric fault indicator. It measures the slight dissimilarities between the probability density functions of the current signal compared to the faultless or healthy one. Through experimental results, the KLD exhibits a higher sensitivity than the usual statistical features for the detection of small cracks (with dimensions in the order of 0.1 mm) realized in a nickel-based superalloy plate. Moreover, the detection is done with zero missed detection probability. Furthermore, the fault severity is assessed through the characteristics of the crack (surface, length, and depth). In the principal component analysis framework, the analysis of four statistical features (KLD, mean, variance, and maximum) dependency to the excitation frequency allows to discriminating among the cracks.

  • an optimal fault detection threshold for early detection using kullback Leibler Divergence for unknown distribution data
    Signal Processing, 2016
    Co-Authors: Abdulrahman Youssef, Claude Delpha, Demba Diallo
    Abstract:

    The incipient fault detection in industrial processes with unknown distribution of measurements signals and unknown changed parameters is an important problem which has received much attention these last decades. However most of the detection methods (online and offline) need a priori knowledge on the signal distribution, changed parameters, and the change amplitude (Likelihood ratio test, Cusum, etc.). In this paper, an incipient fault detection method that does not need any a priori knowledge on the signals distribution or the changed parameters is proposed. This method is based on the analysis of the Kullback-Leibler Divergence (KLD) of probability distribution functions. However, the performance of the technique is highly dependent on the setting of a detection threshold and the environment noise level described through Signal to Noise Ratio (SNR) and Fault to Noise Ratio (FNR). In this paper, we develop an analytical model of the fault detection performances (False Alarm Probability and Missed Detection Probability). Thanks to this model, an optimisation procedure is applied to optimally set the fault detection threshold depending on the SNR and the fault severity. Compared to the usual settings, through simulation results and experimental data, the optimised threshold leads to higher efficiency for incipient fault detection in noisy environment. HighlightsWe propose an incipient fault detection method that does not need any a priori information on the signals distribution or the changed parameters.We show that the performance of the technique is highly dependent on the setting of a detection threshold and the environment noise level.We develop an analytical model of the fault detection performances (False Alarm Probability and Missed Detection Probability).Based on the aforementioned model, an optimisation procedure is applied to optimally set the fault detection threshold depending on the noise and the fault severity.Compared to the usual settings, a performed validation of this approach with through simulation results and experimental data is given.

  • Statistical Approach for Nondestructive Incipient Crack Detection and Characterization Using Kullback-Leibler Divergence
    IEEE Transactions on Reliability, 2016
    Co-Authors: Jinane Harmouche, Claude Delpha, Demba Diallo, Yann Le Bihan
    Abstract:

    This paper is a contribution to the detection and characterisation of small cracks using Eddy Current Testing in the Non Destructive Evaluation framework. Small cracks are considered as incipient faults defined as gradual faults whose signature is weak and concealed by the noise. They are characterized by high signal to noise ratio and low fault to noise ratio. The detection and diagnosis of such faults is still an open challenge. For complex systems, model-based incipient fault detection and diagnosis (FDD) methods usually fail because of the inaccuracy of the model to describe all the phenomena and their interactions. Data-driven methods using statistical features are very promising as long as historical data are available. However in the case of incipient faults, there is not a significant variation of a single feature. The fault signature lies in the global variation of the signal properties. The proposed method relies on the Kullback-Leibler Divergence (KLD) as a nonparametric fault indicator. It measures the slight dissimilarities between the probability density functions of the current signal compared to the faultless or healthy one. Through experimental results, the KLD exhibits a higher sensitivity than the usual statistical features for the detection of small cracks (with dimensions in the order of 0.1 mm) realized in a nickel-based superalloy plate. Moreover, the detection is done with zero missed detection probability. Furthermore, the fault severity is assessed through the characteristics of the crack (surface, length, and depth). In the principal component analysis framework, the analysis of four statistical features (KLD, mean, variance, and maximum) dependency to the excitation frequency allows to discriminating among the cracks.

  • incipient fault detection and diagnosis based on kullback Leibler Divergence using principal component analysis
    Signal Processing, 2015
    Co-Authors: Jinane Harmouche, Claude Delpha, Demba Diallo
    Abstract:

    Most of fault indicators are devoted to detect deviations related to specific features but they fail to detect and estimate unpredictable slight distortions often caused by incipient faults. The Kullback-Leibler Divergence is characterised with a high sensitivity to incipient faults that cause unpredictable small changes in the process measurements. This work has two main objectives: first estimate the amplitude of incipient faults in multivariate processes based on the Divergence and second evaluate, through detection error probabilities, the performance of the Divergence in the detection of incipient faults in noisy environments.Throughout all the paper, the Fault-to-Noise Ratio (FNR) has been referred to as a comparative criterion between the fault level and noise; particularly the region around 0 dB of FNR is of interest in the evaluation. A theoretical study is developed to derive an analytical model of the Divergence that considers the presence of Gaussian noise and allows obtaining a theoretical estimate of the fault amplitude. After application on a simulated AR process, the fault amplitude estimate turns out to be an overestimation of the actual amplitude, therefore guaranteeing a safety margin for monitoring. Accurate fault severity estimation for an eddy currents application shows the effectiveness of this approach. HighlightsWe propose to enhance the fault detection approach based on the KLD modelling with the introduction of the noise.Based on the aforementioned model an estimator of the fault amplitude is developed and validated.The performances of the detection are studied in a noisy environment with the introduction of the Fault to Noise Ratio (FNR).The robustness of the proposed method is evaluated with the computation of the miss-detection and false alarms probabilities.A performed validation of this approach with a simulated AR model is given.

Abdkrim Seghouane - One of the best experts on this subject based on the ideXlab platform.

  • identification of directed influence granger causality kullback Leibler Divergence and complexity
    Neural Computation, 2012
    Co-Authors: Abdkrim Seghouane, Shunichi Amari
    Abstract:

    Detecting and characterizing causal interdependencies and couplings between different activated brain areas from functional neuroimage time series measurements of their activity constitutes a significant step toward understanding the process of brain functions. In this letter, we make the simple point that all current statistics used to make inferences about directed influences in functional neuroimage time series are variants of the same underlying quantity. This includes directed transfer entropy, transinformation, Kullback-Leibler formulations, conditional mutual information, and Granger causality. Crucially, in the case of autoregressive modeling, the underlying quantity is the likelihood ratio that compares models with and without directed influences from the past when modeling the influence of one time series on another. This framework is also used to derive the relation between these measures of directed influence and the complexity or the order of directed influence. These results provide a framework for unifying the Kullback-Leibler Divergence, Granger causality, and the complexity of directed influence.

  • a kullback Leibler Divergence approach to blind image restoration
    IEEE Transactions on Image Processing, 2011
    Co-Authors: Abdkrim Seghouane
    Abstract:

    A new algorithm for maximum-likelihood blind image restoration is presented in this paper. It is obtained by modeling the original image and the additive noise as multivariate Gaussian processes with unknown covariance matrices. The blurring process is specified by its point spread function, which is also unknown. Estimations of the original image and the blur are derived by alternating minimization of the Kullback-Leibler Divergence between a model family of probability distributions defined using the linear image degradation model and a desired family of probability distributions constrained to be concentrated on the observed data. The algorithm presents the advantage to provide closed form expressions for the parameters to be updated and to converge only after few iterations. A simulation example that illustrates the effectiveness of the proposed algorithm is presented.

  • the aic criterion and symmetrizing the kullback Leibler Divergence
    IEEE Transactions on Neural Networks, 2007
    Co-Authors: Abdkrim Seghouane, Shunichi Amari
    Abstract:

    The Akaike information criterion (AIC) is a widely used tool for model selection. AIC is derived as an asymptotically unbiased estimator of a function used for ranking candidate models which is a variant of the Kullback-Leibler Divergence between the true model and the approximating candidate model. Despite the Kullback-Leibler's computational and theoretical advantages, what can become inconvenient in model selection applications is their lack of symmetry. Simple examples can show that reversing the role of the arguments in the Kullback-Leibler Divergence can yield substantially different results. In this paper, three new functions for ranking candidate models are proposed. These functions are constructed by symmetrizing the Kullback-Leibler Divergence between the true model and the approximating candidate model. The operations used for symmetrizing are the average, geometric, and harmonic means. It is found that the original AIC criterion is an asymptotically unbiased estimator of these three different functions. Using one of these proposed ranking functions, an example of new bias correction to AIC is derived for univariate linear regression models. A simulation study based on polynomial regression is provided to compare the different proposed ranking functions with AIC and the new derived correction with AICc

Shunichi Amari - One of the best experts on this subject based on the ideXlab platform.

  • identification of directed influence granger causality kullback Leibler Divergence and complexity
    Neural Computation, 2012
    Co-Authors: Abdkrim Seghouane, Shunichi Amari
    Abstract:

    Detecting and characterizing causal interdependencies and couplings between different activated brain areas from functional neuroimage time series measurements of their activity constitutes a significant step toward understanding the process of brain functions. In this letter, we make the simple point that all current statistics used to make inferences about directed influences in functional neuroimage time series are variants of the same underlying quantity. This includes directed transfer entropy, transinformation, Kullback-Leibler formulations, conditional mutual information, and Granger causality. Crucially, in the case of autoregressive modeling, the underlying quantity is the likelihood ratio that compares models with and without directed influences from the past when modeling the influence of one time series on another. This framework is also used to derive the relation between these measures of directed influence and the complexity or the order of directed influence. These results provide a framework for unifying the Kullback-Leibler Divergence, Granger causality, and the complexity of directed influence.

  • the aic criterion and symmetrizing the kullback Leibler Divergence
    IEEE Transactions on Neural Networks, 2007
    Co-Authors: Abdkrim Seghouane, Shunichi Amari
    Abstract:

    The Akaike information criterion (AIC) is a widely used tool for model selection. AIC is derived as an asymptotically unbiased estimator of a function used for ranking candidate models which is a variant of the Kullback-Leibler Divergence between the true model and the approximating candidate model. Despite the Kullback-Leibler's computational and theoretical advantages, what can become inconvenient in model selection applications is their lack of symmetry. Simple examples can show that reversing the role of the arguments in the Kullback-Leibler Divergence can yield substantially different results. In this paper, three new functions for ranking candidate models are proposed. These functions are constructed by symmetrizing the Kullback-Leibler Divergence between the true model and the approximating candidate model. The operations used for symmetrizing are the average, geometric, and harmonic means. It is found that the original AIC criterion is an asymptotically unbiased estimator of these three different functions. Using one of these proposed ranking functions, an example of new bias correction to AIC is derived for univariate linear regression models. A simulation study based on polynomial regression is provided to compare the different proposed ranking functions with AIC and the new derived correction with AICc

Jinane Harmouche - One of the best experts on this subject based on the ideXlab platform.

  • statistical approach for nondestructive incipient crack detection and characterization using kullback Leibler Divergence
    IEEE Transactions on Reliability, 2016
    Co-Authors: Jinane Harmouche, Claude Delpha, Demba Diallo, Yann Le Bihan
    Abstract:

    This paper is a contribution to the detection and characterisation of small cracks using Eddy Current Testing in the Non Destructive Evaluation framework. Small cracks are considered as incipient faults defined as gradual faults whose signature is weak and concealed by the noise. They are characterized by high signal to noise ratio and low fault to noise ratio. The detection and diagnosis of such faults is still an open challenge. For complex systems, model-based incipient fault detection and diagnosis (FDD) methods usually fail because of the inaccuracy of the model to describe all the phenomena and their interactions. Data-driven methods using statistical features are very promising as long as historical data are available. However in the case of incipient faults, there is not a significant variation of a single feature. The fault signature lies in the global variation of the signal properties. The proposed method relies on the Kullback-Leibler Divergence (KLD) as a nonparametric fault indicator. It measures the slight dissimilarities between the probability density functions of the current signal compared to the faultless or healthy one. Through experimental results, the KLD exhibits a higher sensitivity than the usual statistical features for the detection of small cracks (with dimensions in the order of 0.1 mm) realized in a nickel-based superalloy plate. Moreover, the detection is done with zero missed detection probability. Furthermore, the fault severity is assessed through the characteristics of the crack (surface, length, and depth). In the principal component analysis framework, the analysis of four statistical features (KLD, mean, variance, and maximum) dependency to the excitation frequency allows to discriminating among the cracks.

  • Statistical Approach for Nondestructive Incipient Crack Detection and Characterization Using Kullback-Leibler Divergence
    IEEE Transactions on Reliability, 2016
    Co-Authors: Jinane Harmouche, Claude Delpha, Demba Diallo, Yann Le Bihan
    Abstract:

    This paper is a contribution to the detection and characterisation of small cracks using Eddy Current Testing in the Non Destructive Evaluation framework. Small cracks are considered as incipient faults defined as gradual faults whose signature is weak and concealed by the noise. They are characterized by high signal to noise ratio and low fault to noise ratio. The detection and diagnosis of such faults is still an open challenge. For complex systems, model-based incipient fault detection and diagnosis (FDD) methods usually fail because of the inaccuracy of the model to describe all the phenomena and their interactions. Data-driven methods using statistical features are very promising as long as historical data are available. However in the case of incipient faults, there is not a significant variation of a single feature. The fault signature lies in the global variation of the signal properties. The proposed method relies on the Kullback-Leibler Divergence (KLD) as a nonparametric fault indicator. It measures the slight dissimilarities between the probability density functions of the current signal compared to the faultless or healthy one. Through experimental results, the KLD exhibits a higher sensitivity than the usual statistical features for the detection of small cracks (with dimensions in the order of 0.1 mm) realized in a nickel-based superalloy plate. Moreover, the detection is done with zero missed detection probability. Furthermore, the fault severity is assessed through the characteristics of the crack (surface, length, and depth). In the principal component analysis framework, the analysis of four statistical features (KLD, mean, variance, and maximum) dependency to the excitation frequency allows to discriminating among the cracks.

  • incipient fault detection and diagnosis based on kullback Leibler Divergence using principal component analysis
    Signal Processing, 2015
    Co-Authors: Jinane Harmouche, Claude Delpha, Demba Diallo
    Abstract:

    Most of fault indicators are devoted to detect deviations related to specific features but they fail to detect and estimate unpredictable slight distortions often caused by incipient faults. The Kullback-Leibler Divergence is characterised with a high sensitivity to incipient faults that cause unpredictable small changes in the process measurements. This work has two main objectives: first estimate the amplitude of incipient faults in multivariate processes based on the Divergence and second evaluate, through detection error probabilities, the performance of the Divergence in the detection of incipient faults in noisy environments.Throughout all the paper, the Fault-to-Noise Ratio (FNR) has been referred to as a comparative criterion between the fault level and noise; particularly the region around 0 dB of FNR is of interest in the evaluation. A theoretical study is developed to derive an analytical model of the Divergence that considers the presence of Gaussian noise and allows obtaining a theoretical estimate of the fault amplitude. After application on a simulated AR process, the fault amplitude estimate turns out to be an overestimation of the actual amplitude, therefore guaranteeing a safety margin for monitoring. Accurate fault severity estimation for an eddy currents application shows the effectiveness of this approach. HighlightsWe propose to enhance the fault detection approach based on the KLD modelling with the introduction of the noise.Based on the aforementioned model an estimator of the fault amplitude is developed and validated.The performances of the detection are studied in a noisy environment with the introduction of the Fault to Noise Ratio (FNR).The robustness of the proposed method is evaluated with the computation of the miss-detection and false alarms probabilities.A performed validation of this approach with a simulated AR model is given.