## 2 Small-Sample Adaptive Residual Life Prediction Model Based on Feature Matching Preprocessor-LSTM
### A Preprocessor Model for Adaptive Matching of Mechanical Product Life Characteristics  
Assume that a full life-cycle complete lifetime profile contains data sampled in \(Q\) sampling intervals, denoted as total sample \(r_{1}(t),r_{2}(t),\cdots,r_{Q}(t)\), total number of samples \(Q\), sampling time \(t\in[t_{i},t_{i}+T],i=1,2,\cdots,Q\), and \(T\) is the length of the sampling interval. The life cycle characteristics are absolutized in order to highlight the trend of the overall data. Calculate the failure rate \(H(t)\) and the standard deviation \(S(t)\) of  
Figure 2: Standard deviation curve of general life span characteristics.  
the lifetime characteristics according to Equations (6)-(8), and find the mean value of \(H_{i}(t),i=1,2,\cdots,Q\) or \(S_{i}(t),i=1,2,\cdots,Q\) in each sampling interval \(a_{1},a_{2},a_{3},\cdots a_{Q}\). Randomly extract small samples of total N to match the mean value of the characteristic samples \(a_{c+i}\), where \(\left\{\begin{array}{l}0\leq c+i\leq Q\\ 1\leq i\leq N\end{array}\right..\)  
Calculate the mean \(\overline{P}_{c}\) of the randomly extracted \(N\) small samples \(a_{c+i}\) with the following formula.  
\[\overline{P}_{c}=\frac{1}{N}\sum_{i=1}^{N}a_{c+i} \tag{9}\]  
where \(c\) is taken as \(0,1,2,\cdots,Q-N\), and \(\overline{P}_{0},\overline{P}_{1},\overline{P}_{2},\cdots,\overline{P}_{Q-N}\) is obtained as in Equation (10).  
\[\begin{array}{l}\overline{P}_{0}=\frac{1}{N}(a_{1}+a_{2}+\cdots+a_{N})\\ \overline{P}_{1}=\frac{1}{N}(a_{2}+a_{3}+\cdots+a_{N+1})\\ \cdots\\ \overline{P}_{Q-N}=\frac{1}{N}(a_{Q-N+1}+a_{Q-N+2}+\cdots+a_{Q})\end{array} \tag{10}\]  
From the failure rate model and standard deviation model and the trend of the standard deviation of mechanical product life characteristics, it can be seen that \(\overline{P}_{0},\overline{P}_{1},\overline{P}_{2},\cdots,\overline{P}_{Q-N}\) has a monotonically increasing tendency, that is  
\[\overline{P}_{0}\leq\overline{P}_{1}\leq\overline{P}_{2},\cdots,\leq \overline{P}_{Q-N} \tag{11}\]  
With the small-sample life characteristics \(\hat{r}_{i}(t),i=1,2,\cdots,N\) to be predicted, \(\hat{H}_{i}(t)\), \(i=1,2,\cdots,N\), \(\hat{S}_{i}(t),i=1,2,\cdots,N\) and \(\hat{a}_{i},i=1,2,\cdots,N\) are calculated in turn, then \(\hat{P}=\frac{1}{N}(\sum\limits_{i=1}^{N}a_{i})\) is to be matched with the characteristics, and the adaptive life matching model is established as follows using the obtained series of RV reducer instantaneous failure rate and standard deviation matching criteria \(\overline{P}_{0},\overline{P}_{1},\overline{P}_{2},\cdots,\overline{P}_{Q-N}\).  
When \(\overline{P}_{c}\leq\hat{P}\leq\overline{P}_{c+1}\), matching yields a range of lifetime \(\hat{b}\) (becomes a linear model)  
\[\begin{array}{l}\frac{a_{c}-a_{\min}}{a_{\max}-a_{\min}}\leq\hat{b}\leq \frac{a_{c+1}-a_{\min}}{a_{\max}-a_{\min}},a_{c}<a_{c+1}\\ \frac{a_{c+1}-a_{\min}}{a_{\max}-a_{\min}}\leq\hat{b}\leq\frac{a_{c}-a_{\min}}{ a_{\max}-a_{\min}},a_{c+1}<a_{c}\end{array} \tag{12}\]  
and standard remaining life  
\[\begin{array}{l}=\frac{(a-a_{c})_{c+1}-(a_{c+1}-a)_{c}}{(2a-a_{c}-a_{c+1})} \\ l_{c}=\frac{a_{\max}-a_{c}}{a_{\max}-a_{\min}}\\ l_{c+1}=\frac{a_{\max}-a_{c+1}}{a_{\max}-a_{\min}}\end{array} \tag{13}\]  
Rectify to get  
\[l=\frac{2a_{\max}-(a_{c}+a_{c+1})(\hat{a}+a_{\max})+2a_{c}a_{c+1}}{(2\hat{a}-a _{c}-a_{c+1})(a_{\max}-a_{\min})} \tag{14}\]  
After completing the feature matching preprocessing, the input feature instantaneous failure rate \(\hat{H}_{c}(t)\) and the standard deviation \(\hat{S}_{c}(t)\) of the life feature and the standard remaining life \(l(t)\) of the output feature are obtained for the small-sample adaptive remaining life prediction model [27; 28; 29].