## 2 The Proposed DCLNN Method for Failure Time Prediction
### Comprehensive Selection Indicator before Data Modeling  
In general, data sequences with high stationarity and low complexity can be modeled accurately by the behavior of mathematical modeling. The failure time has certain statistical rules, but with various reasons for failure, different escalators have different failure times. Those regular escalators whose failure time series are stationary and have lower complexity can be selected as a reference. For the same batch of equipment, other escalators have similar failure occurrence rules. In this paper, two stationary indicators (SIs) and two complexity indicators (CIs) are combined to form a comprehensive selection indicator to select excellent referenced time series. Here are details about four indicators:  
(1) Stationary indicator  
(a) Standard deviation  
\[STD=\sqrt{\frac{1}{N}{\sum_{i=1}^{N}{(x_{i}-\mu)^{2}}}} \tag{1}\]  
In statistics, the standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the time series tends to be more stationary.  
(b) \(p\)-value  
Figure 1: The flowchart of the method proposed in this paper.  
The \(p\)-value is the probability value in the statistical significance test, that is, the probability that the hypothesis test occurs. It is generally assumed that if a small probability event occurs, the assumption is true. In this paper, the data sequence is used as the hypothesis test for the stationary data, and the P-value can be calculated as the probability of the unit root test to judge whether the failure time series is stationary or not. A low P-value indicates that the time series tends to be stationary.  
\[P\_value=P\Big{(}x\text{ is a non - stationary sequence}\Big{|}x\text{ is a stationary sequence}\Big{)} \tag{2}\]  
(2) Complexity indicator  
(a) Lempel-ziv  
The Lempel-ziv indicator is used to characterize the rate at which new patterns appear in a time series. The higher the Lempel-Ziv complexity of the sequence, the greater the degree to which the signal sequence is near-random. The lower the Lempel-ziv indicator, the lower the complexity of the time series.  
\[C_{Lempel-ziv}=\frac{C_{N}(N)}{\underset{N\rightarrow\infty}{\lim}C_{N}(N)} \approx\frac{C_{N}(N)\times\log_{k}(N)}{N} \tag{3}\]  
\(C_{N}(N)\) is the Lempel-ziv complexity of the time series, and \(C_{Lempel-ziv}\) is the normalized complexity. Details about the Lempel-ziv algorithm can be found in Ref. [25].  
(b) Sample entropy  
The Sample Entropy (SampEn) measures the complexity of the time series by measuring the probability of generating a new pattern in the signal. The greater the probability of the new pattern, the greater the complexity of the sequence. The lower the sample entropy indicator, the lower the complexity of the time series.  
\[SampEn=-\ln\Big{\{}A^{k}(r)/B^{m}(r)\Big{\}} \tag{4}\]  
where \(A^{k}(r)\) and \(B^{m}(r)\) are the number of subsets of the original time series that satisfies the requirement within the similar distance \(r\) (usually \(m=2;\;k=m+1;\;r=0.1\sim 0.25\)).  
(3) Comprehensive selection indicator  
In order to describe the inherent characteristics of a data sequence, a comprehensive selection indication (CSI) that can describe both stationarity and complexity is constructed in this part to select an inherently excellent data sequence before data modeling. In addition, CSI can be given by the following equation:  
\[CSI=STD\times P\_value\times C_{Lempel-ziv}\times SampEn \tag{5}\]  
CSI can be used to compare the regularity itself between data sequences; the smaller the CSI, the better the data sequence itself and the more accurate the data modeling will be.