Prognostics and Health Management Approaches based on belief functions

Prognostics and Health Management Approaches based on belief functions FEMTO-ST institute / Dep. of Automation and Micromechatronics systems (AS2M), Besançon Emmanuel Ramasso Collaborated work with Dr. Lisa Serir, Pr. Michèle Rombaut, Pr. Thierry Denoeux, Dr. Rafael Gouriveau, Dr. Vincent Placet, Pr. Lamine Boubakar and Pr. Noureddine Zerhouni 2/12/212 Séminaire DIS GIPSA-lab

Overview Stating the context Prognostics and Health Management Belief functions Tools Evidential Evolving Systems Partially Hidden Markov Models Evidential Case-Based Reasoning Applications Bearings : Pronostia platform Turbofan : NASA Composites materials : Applied Mechanics dept. - FEMTO-ST 2

Part 1 : Stating the context 3

FEMTO-ST 65 people in 7 departments : Energie Informatique des Systèmes Complexes (DISC) Mécanique Appliquée Micro Nano Sciences et Systèmes (MN2S) Optique Temps Fréquence Automatique et Systèmes Micromécatroniques (AS2M) 4

AS2M Dept. ~8 people, 4 teams Biomedical Micronanorobotics Strategies of Perception at MicroNano scales Control and Design PHM (Prognostics and Health Management) 7 permanents + 6 PhD in 212 5

Prognostics and health management (PHM) Industrial context 6

Prognostics and health management (PHM) Maintenance activities 7

Prognostics and health management (PHM) Example : wind farm 8

Prognostics and health management (PHM) PHM decomposition 9

Prognostics and health management (PHM) Uncertainties in PHM [PhD Lisa Serir] 1

Prognostics and health management (PHM) Uncertainties in PHM [PhD Lisa Serir] 11

Prognostics and health management (PHM) The considered approach Data-driven PHM (opposed to model-based PHM) Three main steps : Uncertainty management using belief functions Condition assessement made by a classifier or a clustering algorithm 12

Belief functions Origin 13

Belief functions Main elements A variable taking values in a finite set Ω called frame of discernment A belief mass m is a mapping from the set of subsets 2Ω to [,1] with A Ω m(a)=1, and : m(a)> A is a focal set m(a)>, A >1 imprecise information Many useful tools for pattern recognition : Generalized Bayesian Theorem Conjunctive and disjunctive rules of combination Clustering algorithms 14

Belief functions Example on clustering Given data generated by accelerometers (x1 and x2) on a bearing, detection of faults is performed by a classifier that estimates the degree of belief to each possible functioning states. Objects & mass C1 C2 C12 C3 C13 C23 C123 m1 sure cluster 2 1 m2 doubt clusters 2-3 1 m3 full ignorance 1 m4 unknown 1.2.3.1.2.1.5.3.2.2.5.3 subsets m5 general case m6 probablity dist. 15

Belief functions Example on clustering One way to define the mass function : Where Aj is the j-th subset mij 1 A j d 2 ij dij is the distance between cluster j and data point i, Euclidean or Mahalanobis Let consider two cluster, with centers C1 (-3) and C2 (3), x varies between both of them C1 x C2-3 3 16

Part 2 : Tools 19

Evidential Evolving Systems Context One way is to learn the model again... another way, is to estimate the parameters recursively. In some cases, the system parameters may change, thus the system's model may be adapted to well fit current data. Work of Lisa Serir, PhD during 29-212 Evolving approaches well studied by P. Angelov (24-212). 21

Evidential Evolving Systems The method Prognostics performed in three phases : Detection of the current state : clustering Prediction of the degradation : regression Management of uncertainty : belief functions Evolving clustering : studied in «Evidential Evolving Gustafson-Kessel Algorithm For Online Data Streams Partitioning Using Belief Function Theory, L. Serir, E. Ramasso and N. Zerhouni, International Journal of Approximate Reasoning, 53(5):747-768, July 212». Prediction : studied in «L. Serir, E. Ramasso, P. Nectoux and N. Zerhouni, E2GKpro: An evidential evolving multi-modeling approach for system behavior prediction with applications, Journal of Mechanical Systems and Signal Processing, in press, 212.» 22

Evidential Evolving Systems Evolving Clustering Principle 23

Evidential Evolving Systems Evolving Clustering Computation 24

Evidential Evolving Systems Evolving Prediction Principle 25

Evidential Evolving Systems Evolving Prediction - Computation 26

Evidential Evolving Systems Results Clustering and model's adaptation in the following dataset (used in several papers) : 27

Evidential Evolving Systems Results 28

Evidential Evolving Systems Results PRONOSTIA's dataset Platform developed in FEMTO-ST / AS2M Study of bearings degradation 29

Evidential Evolving Systems Results Feature vector at time k: x(k) = [RMS(k) PSD(k)]' Goal: Predict at k + 1 : Regressors : xin = [ x(k-18) x(k-12) x(k-6) x(k)]' Output : xout = [xrms(k+1) xpsd (k+1)]' 3

Partially Hidden Markov Models Context A Hidden Markov model (HMM) is a Markov model in which the states are not directly visible. The system output, dependent on the state, is observed. A dynamical system can be represented by a statistical model assumed to be a first-order Markov process with unobserved (hidden) states. Joint collaboration with Pr. Thierry Denoeux, Heudiasyc lab., UTC, within a PEPS CNRS HMMS are widely applied to temporal pattern recognition problems such as: speech, DNA sequence analysis, machine condition monitoring, etc. 32

Partially Hidden Markov Models Motivation We considered problems in which the states are not completely hidden but partially observed, for instance : In speech recognition, partial information about phonemes may be provided by the analysis of lip motion In machine diagnosis, experts may express probability judgements on the machine condition at different time steps 33

Partially Hidden Markov Models Problem statement We focused on parameter learning based on observations of outputs : modeled by pz(.;θ), Z=(X,Y), θ unknown parameter partial information on states expressed in the belief function framework (m) The probability function pz(z;θ) and the mass function m represent two different pieces of knowledge: pz(z;θ) represents generic knowledge about the data generating process; it corresponds to random uncertainty m represents specific knowledge about a given realization z ; it captures epistemic uncertainty, i.e., uncertainty due to lack of knowledge 34

Partially Hidden Markov Models Estimation The algorithm iterates between : Estimation of the conditional expectation of the complete-data likelihood function Maximisation of this function with report to parameters of the model both given the prior information on states «m». Inference algorithms (classification phase in PHMM + Viterbi algorithm) have been modified in the same way to integrate partial knowledge m Proof in «Making use of partial knowledge about hidden states in HMMs: an approach based on belief functions, Emmanuel Ramasso, Thierry Denœux, Revised in Dec. 212 in IEEE Transactions on Fuzzy Systems» 35

Partially Hidden Markov Models Results Simulated data We considered data generated using a HMM with three states and three-dimensional Gaussian emission probability distributions pk(xt;φk) = N(μk,Σk) : 36

Partially Hidden Markov Models Results Setting partial knowledge The following parameters were studied : Imprecision of labels : experts may express doubt or may be more or less reliable, what is the sensitivity of the model? Labeling error : experts may sometimes be wrong and imprecise, what is the sensitivity of the model? Evaluation : the PHMM generates a sequence of states (using Viterbi alg.) which is a partition : it is compared with the ground truth using the Adjusted Rand Index (value between and 1) 37

Partially Hidden Markov Models Influence of labels error «Partially» supervised learning + «unsupervised» Viterbi algorithm applied to find the sequence, compared to the real one using ARI 38

Evidential Case-Based Reasoning Context In some cases, a statistical model can not represent all cases due to the lack of information. CBR approaches are then well suited. Given a library of possible degradation signals, CBR looks for similar cases and generates an estimate of the current and future health states. CBR approaches for PHM considered in T. Wang's PhD, now at General Electric Company. Joint collaboration with Pr. Michèle Rombaut, GIPSA-lab., Grenoble, within a PEPS CNRS 4

Evidential Case-Based Reasoning Motivation In usual CBR for PHM, the states are unknown. Only prediction followed by thresholding is performed. We considered the case where partial knowledge is available about the states, in the form of a belief function 41

Evidential Case-Based Reasoning Model Assume a library of trajectories (degradation signals) A «time window» defines a block of data 42

Evidential Case-Based Reasoning Prediction standard CBR Given the block of «observed» data, find the trajectories «close to» the observations, and use them to predict the future evolution 43

Evidential Case-Based Reasoning Prediction using belief functions Each piece of trajectories close to observations may have different length Each data may be accompanied by a belief function : fusion using the cautious rule 44

Evidential Case-Based Reasoning Algorithm Main contributions Classification of prediction : Does not make use of thresholds as usual PHM methods Works for multiple health indicators Fusion with direct prediction based on CBR Details in «Joint prediction of observations and states in time-series based on belief functions, IEEE Tr. on Systems, Man and Cybernetics - Part B: Cybernetics, E. Ramasso, M. Rombaut and N. Zerhouni» 45

Evidential Case-Based Reasoning Results (some) NASA's dataset (turbofan) Evaluation : Prediction between 25 and 6 time-units in advance, considered correct if falls in [-1,13] around the true prediction 46

Evidential Case-Based Reasoning Results (some) Ex : Prediction is precise from t=9, so 18 timesteps in advance Evolution of the error on prediction, and the length of prediction 47

Conclusion Prior information can be doubtful but still useful The use of belief functions is relevant to express «partial» knowledge» about the state of the world This knowledge is easier to obtain than precise knowledge (see the example on PHMM) Even corrupted prior can be useful in case doubt is present (see the example on PHMM) The use of fusion process in the belief function framework may improve detection and prediction (see the example on evidential CBR) The use of doubt in multimodeling : Decreases the number of cluster (compact models) Improves prediction (see the example on E2GKpro) 48

Future work Theoretical : Adaptation and improvements of previous models for enhanced prognostics including Decision-making process to help end-users Information fusion with physics-based prognostics Applied Composites materials (organic matrix) analysis based on acoustic emissions, jointly with the department of applied mechanics, LaBeX «ACTION» Project started in March 212 (one paper submitted in Journal of Composites Materials Part A + 2 conf.) One PhD (started on october 212) Still looking for a postdoc candidate (starting on 2/213) 49

Thanks for attending! 5