# Lithium Battery SOH Estimation Based on Manifold Learning and LightGBM
## 5 Instance Validation
### L-ISOMAP Manifold Learning  
Manifold learning is used to recover the structure of low-dimensional stream shapes from high-dimensional sampled data, i.e., to find low-dimensional stream shapes in high-dimensional space and find the corresponding embedding mappings for dimensional simplification or data visualization. It is used to find the essence of things from observed phenomena and to find the inner laws that generate data.  
Figure 9: Autocorrelation coefficient and partial autocorrelation coefficient. (**a**) Partial autocorrelation coefficient; (**b**) autocorrelation coefficient.  
ISOMAP is an unsupervised streamlining method that can maintain global characteristics and is a generalization of the MDS (multi-dimensional scaling) algorithm for nonlinear feature extraction. The ISOMAP algorithm uses nonlinear geodesic distances instead of Euclidean distances as the similarity degree between sample points. The L-ISOMAP (landmark ISOMAP) algorithm is an improved algorithm based on ISOMAP, which only calculates the geodesic distances from each sample point to landmark points to generate the dimensional matrix and obtain the Euclidean embedding of the observed data. In this paper, we use L-ISOMAP to dimensionally approximate the results of feature extraction described in the previous paper, and, as such, achieve the purpose of solving the dimensional disaster and overfitting problems.  
As can be seen from Figure 10, firstly, after the landmark points are selected, the neighbor-joining graph is constructed based on the high-dimensional health factor sequences obtained in the previous section to obtain its representation in the low-dimensional space, and the six-dimensional feature subset is finally obtained after L-ISOMAP manifold learning. The Spearman correlation coefficient is then used to characterize the intrinsic correlation between the feature set and the capacity after feature screening, and for the variables \(x_{i}\), \(y_{i}\), its Spearman correlation coefficient is calculated as follows:  
\[\rho=1-\frac{6\sum\limits_{i=1}^{n}(R_{i}-Q_{i})}{n(n^{2}-1)} \tag{19}\]  
where represents the rank of \(R_{i}x_{i}\), \(Q_{i}\) represents the rank of \(y_{i}\), \(R_{i}-Q_{i}\) is the difference in the rank of the variables, \(x_{i}y_{i}\), and n denotes the number of samples. Figure 11 shows the Spearman correlation coefficient plot between the six-dimensional health factor and the capacity obtained after L-ISOMAP manifold learning.  
Figure 11: Spearman correlation coefficient between health factor and capacity.  
Figure 10: L-ISOMAP algorithm flow.  
A through F in the horizontal and vertical axes in Figure 11 characterize the six-dimensional health factors above, in that order. In addition, blue indicates that the correlation coefficient is less than 0 and red indicates that the correlation coefficient is greater than 0. The darker the color, the larger the value. From the definition of Spearman correlation coefficient, it is clear that the absolute value of the correlation coefficient is approximately close to 1, which represents a higher correlation, and 0 indicates no correlation. From Figure 11, it can be seen that the Spearman correlation coefficient values between the six-dimensional health factor sequences obtained by manifold learning and the capacity are larger, indicating that the correlation between them is higher, which is conducive to the subsequent input model and achieving accurate prediction of SOH.