# Fault Diagnosis Method Based on Time Series in Autonomous Unmanned System
## 5 Experimental Results and Analysis
### Model Evaluation Criteria  
(1) Data fitting effect evaluation index  
This paper uses the fitting error to measure the effect of time series coincidence. The segmentation model of time series is the set of elements composed of the start time and the end time of the subsequence, which is denoted as \(\mathrm{M}=\{(\mathrm{b}_{0},\mathrm{e}_{0}),(\mathrm{b}_{1},\mathrm{e}_{1} ),\ldots,(\mathrm{b}_{k},\mathrm{e}_{k})\}\). The element \((\mathrm{b}_{0},\mathrm{e}_{0})\) is a two-tuple, in which \(\mathrm{b}_{i}\) is the start time of the segmentation subsequence, and \(\mathrm{e}_{i}\) is the end time of the segmentation subsequence.  
The symbolization of time series can be regarded as the segmentation and description of time series. For time series \(\mathrm{S}=(\mathrm{x}_{1},\mathrm{t}_{1}),(\mathrm{x}_{2},\mathrm{t}_{2}), \ldots,(\mathrm{x}_{n},\mathrm{t}_{n})\), the segmentation mode \(\mathrm{M}\) is adopted, and the fitting error is:  
\[\mathrm{E}=\sqrt{\sum_{j=0}^{k}\sum_{i=\mathrm{b}_{j}}^{\mathrm{e}_{j}}\left( \mathrm{x}_{i}-\mathrm{avg}_{j}\right)^{2}} \tag{12}\]  
where \(\mathrm{avg}_{j}\) is the mean value of the \(i\)-th subsequence.  
Considering that the selected time series are not equally long, to measure the effect of time series symbolization better, we add a normalization factor to the fitting error to balance the length of time series, that is:  
\[\mathrm{E}\prime=\frac{1}{n}\sqrt{\sum_{j=0}^{k}\sum_{i=\mathrm{b}_{j}}^{ \mathrm{e}_{j}}\left(\mathrm{x}_{i}-\mathrm{avg}_{j}\right)^{2}} \tag{13}\]  
The smaller the fitting error, the better the effect of time series coincidence.  
(2) Clustering effect evaluation index  
Considering the data confidentiality and labeling difficulties, we use internal indexes of clustering to evaluate the clustering effect. In order to better describe the index, some function definitions are given.  
Function definition 1: Average distance between samples in the cluster:  
\[\mathrm{avg}(\mathrm{C})=\frac{2}{|\mathrm{C}|(|\mathrm{C}|-1)}\sum_{1\leq i \leq|\mathrm{C}|}\mathrm{dist}\left(\mathrm{x}_{i},\mathrm{y}_{j}\right)\]  
Function definition 2: Maximum distance between samples in the cluster:  
\[\mathrm{diam}(\mathrm{C})=\max_{1\leq i\leq|\mathrm{C}|}\mathrm{dist}\left( \mathrm{x}_{i},\mathrm{y}_{jFor cluster partition \(\mathrm{C}=\{\mathrm{C}_{1},\mathrm{C}_{2},\ldots,\mathrm{C}_{\mathrm{k}}\}\), there are several common internal indexes of clustering performance measurement as follows:  
**Internal index 1**: The Davies-Bouldin Index (DBI)  
\[\mathrm{DBI}=\frac{1}{\mathrm{k}}\sum_{i=1}^{\mathrm{k}}\max_{j\
eq i}\left( \frac{\mathrm{avg}(\mathrm{C}_{i})+\mathrm{avg}(\mathrm{C}_{j})}{\mathrm{d}_{ \mathrm{c}en}\big{(}\mathrm{C}_{i},\mathrm{C}_{j}\big{)}}\right) \tag{14}\]  
**Internal index 2**: Dunn Index (DI)  
\[\mathrm{DI}=\min_{1\leq i\leq k}\left\{\min_{j\
eq i}\left(\frac{\mathrm{d}_{ \mathrm{min}}\big{(}\mathrm{C}_{i},\mathrm{C}_{j}\big{)}}{\max_{1\leq i\leq k} \mathrm{diam}(\mathrm{C}_{i})}\right)\right\} \tag{15}\]  
For clustering, if the average value between samples of each cluster is smaller, the DB index is smaller. If the distance between the cluster center points is larger, the DB index is smaller. In contrast, the Dunn exponent characterizes the ratio between the minimum of the closest distance between any two clusters and the maximum of the distance between the two furthest points within any cluster. If the minimum value of the nearest distance between any two clusters is larger, the Dunn index is larger; if the maximum value of the distance between the two furthest points in any cluster is smaller, the Dunn index is larger.