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Optimal state selection and tuning parameters for a degradation model in bearings using Mel-Frequency Cepstral Coefficients and Hidden Markov Chains. Complejo Educativo La Julita. Pereira, Colombia. E-mail: mau. Preventive maintenance is a philosophy for assets management that aims to maximize operation through routine inspections with increasing frequency when no abnormalities are exhibit. This leads to an increase in the probability of failure due to the repetitive intervention and the inherent human error.
Recently, forecasting research, or predictive research, have been addressed in order to obtain effective maintenance strategies and evaluate and manage the residual risk in equipment susceptible to degradation. Predictive research is related to the estimation of an active's Remaining Useful Life RUL by predicting its health state through the progression of its degradation.
This article presents the development of an automated system that identifies types of faults in bearings, using Cepstral Coefficients on the Mel scale MFCC as the features set for description, and Hidden Markov Chains HMC with discrete observations as the classification method.
Features are discretized using clustering by k-means. Signals in this study are vibrations signals from the bearings in electrical machinery. The two databases used here are labeled with four different operation scenarios: normal, inner ring fault, outer ring fault, and rolling element fault.
One of the databases allows for differentiation in severity levels for each scenario. Keywords : Fault identification, feature extraction, Hidden Markov Chains, k-means clustering, Mel-Frequency Cepstral Coefficients, remaining useful life.
American manufacturers invested around USD bi-million in maintenance of critical systems in the early eighties. These expenses doubled within just 20 years, and an alarming half of money was spent on non-effective maintenance . However, the constant pressure on improving the reliability of assets remains intact . So much that investments on preventive and predictive actions become globally What is worrying is that industrial energy consumption grows in global participation, with With the purpose of developing efficient maintenance strategies, challenges related to the predictive research have been faced to manage the residual risk of failing equipment .
Predictive research in this context should be understood as the estimation of Remaining Useful Life RUL for an asset by the prediction of the progression of a diagnosed anomaly .
Nonetheless, the diagnosis is basically a classification problem resorting to many methodologies addressed in the literature, as opposed to prognosis . An efficient diagnosis, in order to make decisions about the structural health state, should deliver information regarding the location and severity following a detection procedure of events and statistical analysis of the observed characteristics .
Some normal strategies for fault diagnosis are based on visual observation of the spectrum with 2-D Fast Fourier Transform FFT and time-frequency contour graphs, which are unable to comply with the requirements of a prediction model .
The prediction models usually are trained with off-line approaches, hence the trained model only respond properly to fault processes with equal features to the training data. Consequently, these off-line models are not suitable to situations where features are changing. Moreover, the computational cost of training, prevent them from being used on-line for real time applications .
Therefore, the prediction models to date are theoretical and limited to small amounts of models and fault modes . Another traditional strategy for RUL prediction is to use approaches based on the physical model of the mechanism, but they require specific knowledge of the system and generally do not reflect a general model for all the fault modes and for the entire life cycle of the mechanism [5, ].
These models normally imply uncertainty from assumptions and simplifications due to the complexity and stochastic nature involved in the systems . Furthermore, it is impractical to create a prediction model for the entire life cycle .
General interest has been focused on non-invasive techniques for fault detection and prediction, thus, basically implying signals-based techniques . Parametric linear prediction techniques as autoregressive moving-average models ARMA usually works for short term predictions given the assumption of linearity of the process. Artificial Intelligence techniques has been used as well, as Artificial Neural Networks, however since they are black boxes and have slow convergence, they suffer from shortcomings such as difficulties of interpretation and structure.
Modal analysis has shown that modes can be insensitive to damage location and that changes in the modes forms can be unidentifiable due to noise and the limited number of recognizable modes forms . Some applications where prediction is key part of the work, shows a model-based perspective applied in the early design and development of the asset. The model is simulated in order to improve reliability and availability.
As an example of this case, in  the development of a model for a vehicle suspension is introduced. Diagnosis and RUL predictions are also included with projections of the system under different operation conditions.
Whether model-based or data-based, the general RUL inference methods rely on previous knowledge about how the system operates and the relative frequency of occurrence of the different kinds of defects. This perspective resembles the structure of a double stochastic process which must find, by probabilistic means, the total degradation state of a system and the probability of transition between states . To a great extent, the aforementioned model is similar to those used in speech processing, since both have signals with quasi-stationary nature and in both cases there are differences in the parameters of similar characters from different systems.
An example of such case is the variability in similar phonemes generated by the same vocal tract, or the variability in vibration signals from presumably similar machines under identical operation conditions. Several speech processing systems use Hidden Markov Chains HMC since they allow for the analysis of a dynamic random process [18, 20, 21]. Given HMC hold the advantages of easy interpretation and the ability of performance in competitive learning environments, they are introduced as a method for classification of the residual life in a degradation process with the goal of characterize the health state of a mechanism [5, 6, 15].
Mel-Frequency Cepstral Coefficients MFCC provide signal information from both the time and the frequency domain, and they also enable dynamic features extraction, whether they are linear or non-linear. These features are commonly used in vibration signals and speech processing [22, 23], and that i's the reason for us to use MFCC jointly with HMC in order to perform the bearings diagnosis.
Data Base We used two different data bases for the training and the validation of the proposed system. The one selected corresponds to a data acquisition of 12k samples per second and faults are induced through electric discharge of 0.
More details on other parameters from this data base can be found in . Faults are induced through mechanized action on the rolling element, the inner ring, and the outer ring. MFCC have two filter types, linearly distributed for frequencies below 1 kHz, and logarithmically distributed for frequencies above 1 kHz .
This distribution is referred to as the Mel scale. Figure 1. MFCC Implementation. The frames usually overlap with the adjacent. The response magnitude is equal to unity at the Center frequency and decrease linearly to zero at center frequency of its two adjacent filters. The result of the conversion is called Mel Frequency Cepstrum Coefficients.
To represent this, 13 velocity deltas and 39 double deltas or accelerations are added. The original procedure follows the guideline of defining k centroids in compliance to the existence of k clusters. The centroids are located so that they remain as far as possible from each other and every continuous observation point is associated to its nearest discrete centroid.
Subsequently, several iterations are run in order to relocate the centroids and hence minimize the distance between continuous observations and centroids. This algorithm may have some variations directed to improve the computational efficiency . The collection of centroids for different discretization processes are known as the Code Book. At discrete uniformly distributed times, the system suffers state transitions according to a set of transition probabilities, for a time t at the current state q t.
The chain is of first order if its transition probability only depends on the previous state, denoted as:. This model is known as an observable Markov Chain, since the process output is the same set of states at any time instant.
On the other hand, a Hidden Markov Chain is the extension of the observable model, where the outputs are probabilistic functions of the state, and thus the model is an embedded double stochastic process which is not directly observable, but indirectly, through the set of output sequences [21, 28]. HMC has the following characteristics:. With the assistance of HMC, three basic problems can be addressed, namely:.
A description on how to address each of the aforementioned problems can be obtained from [21, 28]. Labelled Data bases For training and validation, two data bases were used. For the Bearing Data Center database, the data was acquired at 12k samples per second, faults were induced in three different parts of the bearing inner ring, outer ring and rolling element with three levels of severity, one state of normal operation base signal and at four different velocities , , and RPMs.
With the purpose of creating a labeled database, labels are assigned as follows: "Inner" for the inner ring faults, "Outer" for the outer ring faults, "Normal" for the base signal, "N1", "N2" and "N3" for the 3 severity levels, N1 is the less severe and 3 is the most. For the database from the Mechanical Vibrations Laboratory, data were collected at 20k samples per second with induced faults in three parts of the bearing inner ring, outer ring, and rolling element , one state of normal operation base signal , and eighteen different operation velocities , , , , , , , , , , , , , , , , y RPMs.
The resulting basis comprises 4 folders: Normal, Ball, Inner, and Outer. Afterwards, a Hamming window is applied in order to adjust the frames and integrate the closest frequency lines. Subsequent stages are the coefficients extraction, pre-emphasis, Mel filtering, DCT and delta energy spectrum.
The preceding parameter values are based on data reported by previous studies, as in . The resulting vector represents a continuous scale for each possible characteristic. To train a HMC with discrete observations, a method to discretize features is required. The algorithm iterates in two main steps : In first place, observations that are close to each other are associated to means.
Then, an update is performed by estimating new measures for the centroids in each division:. Finally, the algorithm converges when no significant changes occur in the actualization step. Model Evaluation To evaluate the different HMC models that better represent the observations, we created Receiver Operating Characteristics ROC curves, which are broadly accepted as the de facto analysis and comparison method for diagnosis tests . Sensitivity and Specificity are the measures for the diagnostic accuracy of a test.
Sensitivity is defined as the rate of true positives and represents the proportion of observations that yield positive results on the test. On the other hand, specificity is defined as the rate of true negatives and represents the proportion of observations yielding negative results on the test. An ROC curve represents a graphic of sensitivity against 1-specificity.
The curve is obtained after performing a Montecarlo sampling on the solution set of the tuning parameters of the HMC model. When the model tuning distinguishes clearly the positive observations from the negative ones, the sensitivity will be 1 and the specificity 0 i. When tuning is unable to make distinctions, sensitivity will be 0 and specificity 1 i. The area under the ROC curve is used as measure of the diagnostic accuracy and as comparison criterion between the models [33, 35].
In the specific case of our HMC trained models, the number of states is the discriminant criteria, and the tuning parameters: the number of coefficients, the number of Mel filters, and the number of centroids used in the discretization.
The range of each variable includes the following values:. Coefficients: 1, 2, 3, 4, 8, 10, 12, 14, 16, Filters: 1, 2, 3, 4, 8, 10, 12, 14, 16, 20, Centroids: 2, 4, 6, 8, 10, 12, 14, 16, 20, Out of this total numbers of the parameters, it can be seen that a ROC is constructed with a total of points, each one corresponding to sensitivity and specificity values for a trained model with particular tuning parameters.
Model ocult de Màrkov
Optimal state selection and tuning parameters for a degradation model in bearings using Mel-Frequency Cepstral Coefficients and Hidden Markov Chains. Complejo Educativo La Julita. Pereira, Colombia. E-mail: mau. Preventive maintenance is a philosophy for assets management that aims to maximize operation through routine inspections with increasing frequency when no abnormalities are exhibit.
Modelo oculto de Markov