A novel methodology based on hidden semi-Markov model for equipment health assessment

1.1. Hidden semi-Markov model (HSMM)

The Hidden Markov Model (HMM) is a probability model for describing the statistical properties of stochastic process [3]. In parallel with the extensive use of HMMs in applications [4] and with the related statistical inference work, a new type of model was derived, initially in the domain of speech recognition. A recent book that covers much work on the field is Barbu V. S. et al. (2008) [5]. Because the main drawback of hidden Markov models requires that the sojourn time in a state be geometrically or exponentially distributed, so Ferguson (1980) proposed a model called a hidden semi-Markov model that allows arbitrary sojourn time distributions for the hidden process.

Since then HSMM has been applied in many scientific and engineering areas, such as speech/handwriting recognition, gene identification, human activity prediction and network anomaly detection [6, 7]. An important example of practical interest of hidden semi-Markov models is GENSCAN [8], a program for gene identification which developed by Chris Burge at Stanford University. Although HMM and HSMM have been well studied and applied, there are few papers of HSMM in the state recognition and fault diagnosis fields, mainly represented by Shun Zheng Y., Dong Ming and David He. Shun Zheng Y. also pointed a new and computationally efficient forward–backward algorithm for HSMM with missing observations and multiple observation sequences. And an integrated platform based on HSMM for multi-sensor equipment diagnosis and life prognosis is presented [9]. Then Dong Ming [10, 11] proposed a segmental HSMMs based method for performing both diagnosis and prognosis in a unified framework. Furthermore, Peng Ying in his paper proposed three types of aging factors that discount the probabilities of staying at current state while increasing the probabilities of transitions to less healthy states [12]. An R package for analyzing hidden semi-Markov models can also be found in Bulla et al. [13].

2.1. Parameters of hidden semi-Markov model

Suppose that the equipment health state has been classified into $N$ discrete hidden states ${H=\{H}_1,H_2,\cdots ,H_N\}$ and the equipment health degradation changes with time. Although the equipment health states are hidden, there is often some physical signal attached to the health states of the model as shown in Fig. 1. The complete specification of an HSMM consists of the following elements, described as$\mathrm{}\lambda =\{A,B,D,\pi \}$.

The state transition probability distribution matrix is$\mathrm{}A={\left[a_{ij}\right]}_{N\times N}\text{,}$ where $a_{ij}=P(S_{t+1}=H_j|S_t=H_i)$, $1\le i$, $j\le N$, and $N$ is the number of health states and $S_t$ is the state at time $t$ in HSMM. The conditional probability distribution matrix of observing $O_t=k$ when given state $H_i$ is parameter $D$ represents the state duration distribution ${d(H}_j)$, $1\le j\le N$, written as $d\left(H_j\right)=P\left(d\left|H_j\right.\right)=P\left(\left.S_{t+d+1}\ne H_j,S_{t+d-u}=H_j,\mathrm{}\mathrm{}u=\mathrm{0,1},\cdots ,d-2\right|S_{t+1}=H_j,\mathrm{}\mathrm{}{S}_t\ne H_j\right)\text{,}$ where state $H_i$ lasts $d$ time units. And the initial state probability distribution is $\pi ={\pi }_i=P(S_1=H_i)$, $1\le i\le N$.

Fig. 1HSMM based health state transition

2.2. HSMM based three-step health assessment

The traditional health assessment based on HMM/HSMM all need different state data to get different training HMMs or HSMMs and input the test data to calculate the corresponding emission probabilities in these models. Then the hidden state is the one that generates the maximal emission probability. However, the required data is hard to capture and identify which state belonged in real applications. Therefore, we proposed an HSMM based method for equipment health assessment, which only required the health state data for training. This three-step health assessment based on HSMM includes training step, input step and deviation calculation step.

[Step 1] Learning/training step.

First all of the captured data should be analyzed for feature extraction, and a health standard HSMM $\lambda =\{A,B,D,\pi \}$ is trained and the parameters are estimated by using only health/normal state training data set.

[Step 2] Input step.

On the basis of this trained health standard HSMM $\lambda =\{A,B,D,\pi \}$, we can obtain observation probability of the trained health model $P\left(\left.O_{standard}\right|\lambda \right)$, where the other test data should be inputted to calculate the observation probabilities $P\left(\left.O_{test}\right|\lambda \right)$ associate with the test states.

[Step 3] Deviation calculation step.

With calculating the corresponding relative Kullback-Leibler Divergence between the health/normal state and the unknown state, the health index can be obtained for equipment health assessment.

Define Kullback-Leibler Divergence $d\left(p‖f\right)$ (also known as Relative Entropy) as the variance between two probability distributions $p\left(x\right)$, $f\left(x\right)$ [14]. With achieving the observation probability of the health state $P\left(\left.O_{standard}\right|\lambda \right)$ and the test state $P\left(\left.O_{test}\right|\lambda \right)$, the Kullback-Leibler Divergence $d_{kl}$ represents the deviation extent from the standard health state can be calculated by Eq. (1):

The smaller the $d_{kl}$, the higher equipment health level is, and vice versa. When $d_{kl}$ is greater than the threshold that depends on the performance requirements for equipment, the equipment has completely failed. Therefore, the equipment health index $h\in$[0, 1] can be obtained through normalization.

3.1. Forward/backward algorithm

Baum proposed the Forward/backward algorithm for the emission probability $P\left(\left.O\right|\lambda \right)$, and the partial forward probabilities ${\alpha }_t\left(i\right)$ and partial backward probabilities ${\beta }_t\left(i\right)$ are defined as follows.

1. The forward algorithm.

We define a forward variable ${\alpha }_t\left(i\right)$ described as the joint distribution of observation sequence $O_1,O_2,\cdots ,O_t$and health state $S_t$ at time $t$ when given the model $\lambda$.

where the emission observation of state $H_i$ is $O_1,O_2,\cdots ,O_{t-d}$ and state $H_j$ is $O_{t-d+1},O_{t-d+2},\dots ,O_t$, $H_{t-d+1:t}$ represent the state lasts from $t-d+1$ to $t$, and $D$ is the maximal duration time.

2. The backward algorithm.

Similar to forward variable, we define a backward variable ${\beta }_t\left(i\right)$ as Eq. (3):

3.2. Baum-Welch based re-estimation algorithm

The training process is the adjusting and re-estimation the HSMM parameters in order to maximize the observation sequence probability $P\left(\left.O\right|\lambda \right)$ when the parameters are unknown or inaccurate. There are many methods for learning such as Maximum Likelihood (ML), Maximum Mutual Information (MMI), Minimum Discriminate Information (MDI), and the most classical one is the Baum-Welch algorithm which is presented here.

In order to obtain the parameters of the hidden Markov model, note two variables first as following: We denote the posterior probability as ${\gamma }_{t,d}(i,j)$, shown in Eq. (4):

Let Baum-Welch variable be ${\xi }_{t,d}\left(i,j\right)$, expressed as Eq. (5):

The parameters $A$ and $\pi$ usually have little effects on determining the initial values, but the initial $B$, $D$ matrix are opposite and can be got by the K-means algorithm. Then the parameters $\lambda =\{A,B,D,\pi \}$ can be obtained by Baum-Welch algorithm based re-estimation formulas as Eq. (6) until the observation probability $P\left(\left.O_{standard}\right|\lambda \right)$ converges to a pre-determined interval: