Realistic prognostic tools are essential for effective condition-based maintenance systems. In this paper, a Duration-Dependent Hidden Semi-Markov Model (DD-HSMM) is proposed, which overcomes the shortcomings of traditional Hidden Markov Models (HMM), including the Hidden Semi-Markov Model (HSMM): (1) it allows explicit modeling of state transition probabilities between the states; (2) it relaxes observations’ independence assumption by accommodating a connection between consecutive observations; and (3) it does not follow the unrealistic Markov chain’s memoryless assumption and therefore it provides a more powerful modeling and analysis capability for real world problems. To facilitate the computation of the proposed DD-HSMM methodology, new forward-backward algorithm is developed. The demonstration and evaluation of the proposed methodology is carried out through a case study. The experimental results show that the DD-HSMM methodology is effective for equipment health monitoring and management.
Fault is a change from the normal operating condition of a system to an abnormal condition, which occurs as a result of system performance degradation over time [
A CBM program can be used to do diagnostics or prognostics; however, regardless of the application, it follows three steps [
Data analysis for event data only is reliability analysis, which maps the event data over a time axis to determine the probability of events and uses the probability distribution to predict failures. On the other hand, data acquisition in CBM provides event and condition monitoring data. Therefore, it is more effective to combine events and conditions in a model in order to do diagnostics or prognostics. Hidden Markov model (HMM) is a technique for modeling and analyzing event and condition monitoring data together. It consists of two stochastic processes:
Researchers have proposed a number of techniques to address these limitations. Continuous variable duration HMM is adopted in the speech recognition. Compared to standard HMM, results show that the absence of a correct duration model increases the error rate by 50% [
Prognostic methods used in CBM are often a combination of statistical inference and machine learning methods [
The primary advantage of HMM is its robust mathematical foundation that can allow for many practical applications and different areas of use. An added benefit of employing HMMs is the ease of model interpretation in comparison with pure “black-box” modeling methods such as artificial neural networks that are often employed in advanced diagnostic models [
To overcome the limitations of HMM in prognosis, Dong and He [
This paper presents a new approach that expands the HSMM methodology [
A Hidden Semi-Markov Model (HSMM) is an extension of HMM by allowing the underlying process to be a semi-Markov chain with a variable duration or sojourn time for each state. The HSMM model is an ideal mathematical model for estimating the unobservable health states with observable sensor signal. For example, a small change in a bearing alignment could cause a small nick in the bearing, which could cause scratches in the bearing race and additional nicks, leading to complete bearing failure. This process can be well described by the HSMM. Let
In HSMM, there are
Although the distinct health-state transition
Similar to HMM, HSMM also has basic problems to deal with, that is, evaluation, recognition, and training problems: evaluation (also called classification): given the observation sequence decoding (also called recognition): given the observation sequence learning (also called training): how do we adjust the model parameters
Different algorithms have been developed for above three problems. The most straightforward way of solving the evaluation problem is enumerating every possible state sequence of length
Although HSMM has explicit state duration probability distribution
In DD-HSMM, the state transition probability distribution
Similar to HSMM, DD-HSMM also has basic problems to deal with, that is, evaluation, recognition, and training problems. To facilitate the computation in the proposed DD-HSMM-based health prediction model, in the following, new forward-backward variables are defined and modified forward-backward algorithm is developed.
A dynamic programming scheme is employed for the efficient computation of the inference procedures. To implement the inference procedures, a forward variable
Similar to the forward variable, the backward variable can be written as
For the backward probability, the initial conditions are set at time
For time
Then the total probability can be computed by
In order to give reestimation formulas for all variable of the DD-HSMM, one DD-HSMM-featured forward-backward variable is defined:
The forward-Backward algorithm computes the following probabilities.
The forward variable is shown as follows:
For
The backward variable is shown as follows:
For
The reestimation formula for initial state distribution is the probability that state
In this paper, state duration densities are modeled by single Gaussian distribution estimated from training data. The existing state duration estimation method is through the simultaneous training DD-HSMM and their duration densities. However, these techniques are inefficient because the training process requires huge storage and computational load. Therefore, a new approach is adopted for training state duration models. In this approach, state duration probabilities are estimated on the lattice (or trellis) of observations and states, which is obtained in the DD-HSMM training stage.
The mean
Many applications in the actuarial, econometric, engineering, and medical literature involve the use of the hazard rate function [
Let
Suppose that a machine will go through health states
In this case study, long-term wear experiments on rolling element bearings were conducted [
During the test running, under each condition, vibration signals were collected. These signals were extracted using a Mahalanobis-Taguchi System (MTS) based model in the original paper [ the bearing is operating normally; the bearing is operating and shows signs of deterioration; it is advisable to take some preventive action at the next planned maintenance; the bearing is operating but requires immediate attention; the bearing has failed.
In order to identify the accuracy of the operation state identification method proposed in this paper, experimental data with normal operating condition were obtained. The experimental data set included 50 samples for each state (denoted by 0, 1, 2, and 3). Of these data points, 20 of them were used to train the model, and the remaining 30 samples were used to validate the model.
In the DD-HSMM, mixture Gaussian distribution and the single Gaussian distribution were used to model the output probability distribution and the state duration densities separately, in which the number of states is 4. The maximum number of iterations in training process is set to 100 and the convergence error to 0.000001.
The DD-HSMM-based training model is shown as Figure
Training curve of the DD-HSMM model.
The classification results obtained on the remaining 30 data samples are shown in Table
Prognostics results.
System state | Normal 0 | Degradation level 1 | Degradation level 2 | Failure 3 | Recognition accuracy |
---|---|---|---|---|---|
Normal 0 | 29 | 1 | 0 | 0 | 96.7% |
Degradation 1 | 2 | 27 | 1 | 0 | 90% |
Degradation 2 | 0 | 1 | 28 | 1 | 93.3% |
Failure 3 | 0 | 0 | 1 | 29 | 96.7% |
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Total accuracy | 94.2% |
As described before, a four-state DD-HSMM prediction model is constructed. In the training process, even if the device is in the same running condition, the dwell time is different, transition probabilities between states and the mean or variance of duration in each state are not the same. Tables
State transition probability
System state | Normal 0 | Degradation level 1 | Degradation level 2 | Failure 3 |
---|---|---|---|---|
Normal 0 | 0.8813 | 0.0454 | 0.0732 | 0.0001 |
Degradation 1 | 0.0000 | 0.6231 | 0.3473 | 0.0296 |
Degradation 2 | 0.0000 | 0.0000 | 0.9263 | 0.0737 |
Failure 3 | 0.0000 | 0.0000 | 0.0000 | 1.0000 |
Mean and variance of duration in each state
System state | Normal 0 | Degradation level 1 | Degradation level 2 | Failure 3 |
---|---|---|---|---|
Mean | 7.9577 | 7.2105 | 7.1435 | 3.3435 |
Variance | 1.3953 | 0.7429 | 0.7924 | 0.5452 |
State transition probability
System state | Normal 0 | Degradation level 1 | Degradation level 2 | Failure 3 |
---|---|---|---|---|
Normal 0 | 0.8813 | 0.0454 | 0.0732 | 0.0001 |
Degradation 1 | 0.0000 | 0.4728 | 0.4154 | 0.1118 |
Degradation 2 | 0.0000 | 0.0000 | 0.9263 | 0.0737 |
Failure 3 | 0.0000 | 0.0000 | 0.0000 | 1.0000 |
Mean and variance of duration in each state
System state | Normal 0 | Degradation level 1 | Degradation level 2 | Failure 3 |
---|---|---|---|---|
Mean | 7.9577 | 5.3105 | 7.1435 | 3.3435 |
Variance | 1.3953 | 1.1328 | 0.7924 | 0.5452 |
First, the state
In order to compare the prognostic method based on the DD-HSMM with the prognostic method based on the HSMM, (
Table
Comparison of DD-HSMM versus HSMM.
Actual RUL | DD-HSMM model | HSMM model | ||
---|---|---|---|---|
Predicted RUL | Error (%) | Predicted RUL | Error (%) | |
27.0000 | 26.4027 | 2.212 | 26.9852 | 0.0548 |
24.0000 | 23.6596 | 1.418 | 12.438 | |
22.0000 | 22.0856 | 0.389 | 22.66 | |
17.0000 | 17.6142 | 3.613 | 17.0734 | 0.437 |
15.0000 | 15.7246 | 4.831 | 13.829 | |
12.0000 | 11.6945 | 2.546 | 10.5199 | 12.334 |
11.0000 | 10.0602 | 8.544 | 4.365 | |
9.0000 | 9.1944 | 2.16 | 16.888 | |
5.0000 | 5.1253 | 2.506 | 3.6104 | 27.792 |
3.0000 | 3.0122 | 0.407 | 20.347 |
This paper presents a Duration-Dependent Hidden Semi-Markov Model (DD-HSMM) for prognostics. As opposed to the Hidden Semi-Markov Model (HSMM), failure prediction capability of the DD-HSMM method uses state dependency and duration dependency. The two important aspects of equipment health monitoring, which are the stages and the rate of aging, are taken into consideration in an integrated manner in the proposed DD-HSMM model. The duration-dependent state transition probability in the Hidden Semi-Markov model makes the decision-making more relevant to real world applications.
In order to facilitate the computational procedure, a new forward-backward algorithm and reestimation approach are developed. By using autoregression, the interdependency between observations is established in the model. By incorporating an explicitly defined temporal structure into the model, the DD-HSMM is capable of predicting the remaining useful life of equipment more accurately.
The demonstration of the proposed model is carried out using experimental data on rolling element bearings. The proposed model provides a powerful state recognition capability and very accurate results in terms of remaining useful life prediction. In order to draw general conclusion on the capabilities of the proposed DD-HSMM, more experimental data in various prognostics areas are needed.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the financial support for this research from the National High Technology Research and Development Program of China (no. 2012AA040914), the National Natural Science Foundation of China (Grant no. 71101116), and the Basic Research Foundation of NPU (Grant no. JC20120228).