1 Introduction
Process monitoring is indispensable because it is the premise and guarantee for the safe and stable running of industrial systems [12, 2, 14, 3, 36, 30]. In recent decades, a large number of datadriven approaches have been proposed for process monitoring [11, 20, 5, 23, 10, 35, 29, 4, 33, 38]. However, most of them are highly based on continuous variables because they can’t avoid involving Euclidean or Mahalanobis distance and can’t be utilized for hybrid variables (containing continuous and binary variables) [33].
Among datadriven methods, principal component analysis (PCA) has received continuous attention once it was applied in process monitoring due to its effectiveness of data dimensionality reduction
[19, 11]. Based on PCA, dynamic PCA (DPCA) adopted the technology of time lag shift to construct augmented matrix to mine timerelated information [20]. Considering slowly changing in normal process, recursive PCA (RPCA) was proposed for adaptive process monitoring [24]. In order to capture nonlinear property, kernel PCA (KPCA) was developed [28, 5]. Unlike PCA, partial least squares (PLS) and its variants pay much attention to qualityrelated fault [26, 23]. To weaken the Gaussian hypothesis, independent component analysis (ICA) was proposed for process monitoring
[22]. The Mahalanobis distance (MD) can also be directly used for process monitoring [18]. As understanding of the fault initiation becomes more and more thorough, the moving window methods also be proposed for incipient fault detection [17, 29, 27]. Considering the practical applicability in industrial processes, a large number of improved methods have been developed for multimode and nonstationary monitoring [37, 39, 16, 34].The aforementioned methods have made remarkable achievements in process monitoring, but almost all methods are based on Euclidean or Mahalanobis distance and are highly dependent on continuous variables. However, the practical industrial processes sometimes have not only continuous variables, but also binary variables which may carry some useful information for process monitoring [33] and are usually deleted in the data preprocessing [13]. For hybrid variables, Langseth et al.
used hybrid Bayesian networks for estimating human reliability
[21]. Aguilera et al. developed the naïve Bayes (NB) and tree augmented naïve Bayes (TAN) models and applied to species distribution [1]. Zhu et al. considered the mixture of continuous and discrete variables in semantic model [40]. Talvitie et al. introduced a related model through employing an adaptive discretization approach for structure learning in Bayesian networks when there are both continuous and discrete variables [31]. Recently, Wang et al. utilized continuous and binary (twovalued) variables to detect the abnormalities of thermal power plant for the first time [33]. Then a more effective anomaly monitoring model named feature weighted mixed naive Bayes model (FWMNBM) was developed
[32].However, the hybrid variable approaches mentioned above are supervised methods and require both normal and fault data during training. Unfortunately, the systems in actual industrial processes are running without fault in most time, and the determination of fault samples requires repeated research and careful discussion by experts which is timeconsuming and costly. So that the healthy state samples are usually available and it is difficult to collect sufficient fault instances, which is one of the reasons why monitoring methods only based on normal working condition data, such as PCA, PLS, ICA et al., have attracted much attention. Process monitoring methods with hybrid variables only based on healthy state data are very urgently. Therefore, this paper focuses on hybrid variable process monitoring and proposes a novel unsupervised framework of process monitoring with hybrid variables named HVM which can simultaneously capture the process information of both continuous and binary variables. The main contributions are summarized as follows:

The article firstly focuses on hybrid variable monitoring only based on healthy state data. And a novel unsupervised framework of process monitoring with hybrid variables named HVM is proposed.

Under the unsupervised framework, the statistics and the monitoring strategy suitable for hybrid variables are firstly defined and the physical explanation behind the framework is elaborated. In addition, the estimation of parameters is derived in detail and the detectable condition is analyzed.

The effectiveness and efficiency of the proposed method is fully demonstrated first on a numerical simulation and then on a practical fan system of ultrasupercritical power plant.
The remainder of the paper is organized as follows. The problem formulation and motivation are described in detail in Section 2. The framework of hybrid variable monitoring is introduced in Section 3. In Section 4, parameters learning and corresponding derivation are described. The fault form of hybrid variables is defined and the detectable condition is analyzed in Section 5. In Section 6, the effectiveness and efficiency of proposed framework is verified. Finally, conclusions are given in Section 7.
2 Problem formulation and motivation
With industrial processes becoming more and more complex and integrated, binary variables also appear in monitoring variables. For example, in Zhejiang Zheneng Zhongmei Zhoushan Coal and Electricity Co., Ltd. (Zhoushan Power Plant), Zhejiang Province, China, the number of variables monitoring in the No.1 power unit is about , in which the number of binary variables among them is as many as [33]. In the fan system of No.1 power unit, continuous variables and binary variables are collected, where the number of binary variables is more than that of continuous variables [32]. The appearance of binary variables makes traditional monitoring approaches no longer applicable and process monitoring with hybrid variables more intractable. The binary variables are usually discarded during the data preprocessing because the traditional approaches mostly have applied Euclidean or Mahalanobis distance which can’t be used to describe binary variables [13]. However, binary variables may carry some useful information for process monitoring [33, 32].
The issue of supervised classification with hybrid variables has been paid attention to and investigated in other fields [21, 1, 40, 31]. In process monitoring, Wang et al.
have utilized continuous and binary variables for the anomaly detection of thermal power plant
[33, 32]. However, these approaches are supervised methods, which require both normal samples and fault instances to train the model. In practical processes, a lots of healthy state samples can be collected and it is difficult to obtain sufficient faulty samples. Therefore, this paper proposes a novel unsupervised framework of process monitoring with hybrid variables named HVM. HVM can simultaneously mine the information of both continuous and binary variables through a probabilistic framework. In HVM, the statistics of hybrid variables are computed with healthy state data and the control limit is determined by kernel density estimation (KDE)
[25]. Then for the arriving sample , the statistic can be computed with the same way of training. The state of can be determined through the monitoring strategy. Finally the superiority of HVM is demonstrated through a numerical simulation and an actual case in the fan system of a thermal power plant.3 Hybrid variable monitoring framework
3.1 Offline statistics
Training data are sampled under normal operating condition with samples. is the th instance and contains features where binary features and continuous features are respectively collected. Let be the th variable. When the system is running in a steady state, the monitoring data tends to be stationary and with no trends [38]. Then the following assumptions are introduced. If
is a continuous variable, we suppose it obeys Gaussian distribution under normal condition, that is
[33](1) 
where
is the probability density function (pdf) which is defined as
, andare the mean and corresponding variance of the
th variable.If
is a binary variable, the Bernoulli distribution is introduced as follows
[8]:(2) 
where is the distribution series (ds),
is the response probability which is defined as
.Definition 1.
The occurrence probability of under normal condition is defined as
(3) 
where and mean the th and th variable of binary variables and continuous variables respectively, means the weight of the corresponding variable.
Affected by noise, there may be some outliers in data sampled under normal operating condition. Then the probability that
belongs to can be obtained by(4) 
where is the prior normal probability, which represents the confidence level of the health state data and equals to , is the significance level [9].
Theorem 1.
, a positive decimal to satisfy .
Proof. For , suppose Assumption 3.1 holds, then
(5) 
Since the number of training samples is an integer less than infinity and Assumption 3.1 is introduced, we have
(6) 
Then for any . There must be a positive value that satisfies
(7) 
The prior normal probability , so that a positive decimal can be find to satisfy , where . ∎
Remark 1.
and
are probability distributions (pdf or ds), which are fitted by training data. Thus the more
deviates from the statistical characteristics of , the smaller is and the smaller is.When of is obtained, then is computed as
(8) 
where is the natural logarithmic function.
Proposition 1.
Compared to , obtained in equation (8) is more sensitive to faulty instance.
Proof. According to Theorem 1, there must be a lower bound that satisfies for normal data . When , the natural logarithmic function monotonically increases, the derivative always satisfy that . Fault data often deviates more from the statistical characteristics of . Then we have . The detection performance is mainly reflected in the recognition ability of fault in the neighborhood of , where . Since , so is more sensitive to faulty instance than . ∎
Let , it can be learned that
(10) 
Considering equation (2), we have
(11) 
where . According to equation (1), the following equation can be obtained that
(12) 
For the collected training samples , the monitoring statistics are computed as
(14) 
where is the statistic of .
3.2 Online monitoring strategy
4 Parameters learning
The model described in 3.1 mainly involves the estimation of parameters , , , and . , and can be obtained through maximum likelihood estimation (MLE) [6].
(16)  
(17)  
(18) 
The practical data is usually correlated and variables that are more related to other variables are usually more sensitive when abnormalities occur [33]. So each variable is assigned with the different weight . The calculation of involves mutual information (MI) of continuous and binary variables. In order to estimate MI of hybrid variables, is constructed as follows if is continuous variable.
Definition 2.
If is a continuous variable, is constructed as
(19) 
where is Iverson brackets. If the condition is true, it returns , otherwise it returns .
Definition 2 makes it possible to characterize the correlation between hybrid variables. Then instead of is used to compute MI. However, the process of equation (19) brings the calculation error of MI between continuous variables.
Lemma 1.
For continuous variables and , assume and obey Gaussian distributions and respectively , and are constructed by equation (19), is the Pearson correlation coefficient between and , the Pearson correlation coefficient between and is denoted as . Then
(20) 
Proof. For continuous variables and , the joint probability function of and is
(21) 
The expectation of is
(22) 
where and . Since
(23) 
Thus can be written to
(24) 
can also be obtained by
(25) 
The Pearson correlation coefficient between and is
(26) 
∎
Theorem 2.
If and are continuous variables, and obey Gaussian distributions and respectively, and are constructed by equation (19), the calculation error of MI is
(28) 
where .
Proof. According to Lemma 1, we have
(29) 
Substituting equation (29) into equation (27), it can be learned that
(30) 
where . Hence, Theorem 2 has been proved. ∎
With Definition 2, the MI computation of hybrid variables is transformed to that of binary variables (or constructed binary variables). is defined as
(31) 
where are binary variables or constructed binary variables. is the probability of , is the indicative coefficient ( when is being computed, and otherwise), is the joint probability of and . ( can be obtained in the same way) can be computed by
(32) 
where is the value at time of .
Proposition 2.
For binary variables and , can be denoted as
(33) 
where .
Proof. Let
(34) 
it has
(35)  
(36) 
Then we have
(37) 
Theorem 3.
For binary variables and , is obtained as
(39) 
where , .
Proof. Since
(40) 
The likelihood function is
(41) 
where
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