## 2 Materials and Methods
### Outlier Detection and Replacement
#### 2.3.2 MICE Theory  
This paper addresses outliers of small-sample data of many varieties and small batches. First, a box was used to eliminate outliers, and then the MICE method was used to fill in the outliers. The replacement of the abnormal outlier values should make the filling value close to the true value to avoid deviation of the analysis result of the original dataset when analyzing the filled dataset. At present, most of the filling methods use single-value interpolation, average, median, etc., but it is very likely that the original distribution of the data will be changed, which will have a greater impact on the mean and variance of the data and increase machine learning. The error cannot adequately reflect the uncertainty of the value that needs to be replaced and this affects the later prediction. MICE is proposed by [PERSON] to define a predictive model for each variable with missing data [37; 38]. The basic idea is as follows:  
* Use initial estimates to fill in missing values by randomly sampling from existing observations.
* For each missing value, use the observed parts of other variables as predictors to estimate the regression model.
* Replace the missing values with randomly drawn values from the posterior prediction distribution of the results. Using the observed and most recent estimates as the predicted value, this process is repeated for each variable with missing data.  
Given the following dataset with p incomplete variables \(X=X_{1},X_{2},\dots,X_{p}\), the observed part of \(X\) is \(X^{obs}=X^{obs}_{1},X^{obs}_{2},\dots,X^{obs}_{3}\), and the missing part of \(X\) is \(X^{miss}=X^{miss}_{1},X^{miss}_{2},\dots,X^{miss}_{p}\), where \(\theta_{1},\theta_{2},\dots,\theta_{p}\) describes the conditional density parameter of \(X_{1},X_{2},\dots X_{p}\). Interpolation for each incomplete variable in \((i=1,\ 2,\dots,\ p)\) is iteratively drawn. First, the random extraction of parameters is simulated, and then the random extraction of the missing values in the variables is simulated. The posterior  
Figure 2: Box plot: the green triangle is the sample median.  
distribution of the missing variable \(\theta\) is established by [PERSON] iterative sampling of the observation data. The variable \(X_{obs}\) is used as a covariate to perform regression modeling on other variables with missing values \(X_{miss}\). For all variables with missing values, this process is repeated, that is, \(X_{1}\) to \(X_{p}\) need to be interpolated for each iteration. At the end of each iteration, missing values are replaced by predicted values from the regression model. The description is shown in the equation:  
\[\begin{split}&\theta_{1}^{*(t)}\sim P(\theta_{1}|X_{1}^{obs},X_{2}^ {t-1},\ldots,X_{p}^{t-1})\\ & X_{1}^{*(t)}\sim P(\,X_{1}|X_{1}^{obs},X_{2}^{t-1},\ldots,X_{p}^ {t-1},\theta_{1}^{*(t)})\\ &\vdots\\ &\theta_{p}^{*(t)}\sim P(\theta_{1}|X_{p}^{obs},X_{1}^{t-1}, \ldots,X_{p-1}^{t-1})\\ & X_{p}^{*(t)}\sim P(\,X_{p}|X_{p}^{obs},X_{2}^{t-1},\ldots,X_{p- 1}^{t-1},\theta_{p}^{*(t)})\end{split}, \tag{1}\]  
where \(t\) is the number of iteration cycles \(t\in(1,2,\ldots,n)\), \(\theta_{1}^{*(t)},\theta_{2}^{*(t)},\ldots,\theta_{p}^{*(t)}\) is the random drawing parameter of \(\theta_{1},\theta_{2},\ldots,\theta_{p}\) during t iterations, and \(X_{1}^{*(t)},X_{2}^{*(t)},\ldots,X_{p}^{*(t)}\) is the estimated value of \(\theta_{1}^{miss},\theta_{2}^{miss},\ldots,\theta_{p}^{miss}\) in iteration t. The observed value \(X_{obs}\) will not change in the iterative update process, but the missing data \(X_{miss}\) will be updated in each iteration.