## 4 Results and Discussion
### Influences of Parameter Setting on Recognition Performance  
As indicated by the parameter analysis of the LRSC algorithm in Section 4.1, being different from other algorithms, its classification process depends little on parameter optimization. Theoretically, when the sparseness degree changes because of the parameter (such as the parameter \(T\)) setting, e.g., when the sparseness degree increases, perhaps only one or two non-zero solutions can be found. The positions of these solutions recognize the training samples to which the prediction samples are most similar, and the corresponding classification labels can still be found. In the case of the infrared spectral data under one sample division, in Figure 6a, \(T\) and \(\lambda\) were selected as the parameters for LRSC (\(T\): \(10^{-1}\), \(10^{-2}\), \(10^{-3}\), \(10^{-4}\), \(10^{-5}\), \(10^{-6}\), \(\lambda\): \(10^{-1}\), \(10^{-2}\), \(10^{-3}\), \(10^{-4}\), \(10^{-5}\), \(10^{-6}\), \(10^{-7}\), \(10^{-8}\)). With the grid method used to select the parameters, the recognition rates changed slightly, both reaching more than 90%. Similarly, in Figure 6b, different parameters were used, which, again, did not have a significant impact on the classification of the LIBS data. Even the value of \(\lambda\) made little contribution to the accuracy. For the optimal parameters of the 50 experiments of SIMCA, the parameter optimization is similar to that of principal component analysis, except that it gives a principal component coefficient by learning each class of samples. As verified by the calculation, the optimal values of SIMCA's five parameters varied from 2 to 10, with a relatively large variance in 50 experiments of random division samples. SIMCA's main problem is that when the number of classes increases, there are too many parameters. When the algorithm's parameters are uncertain, increasing the number of classes obviously increases the degrees of freedom and the uncertainty of the classification results. For example, under one specific division of IR samples, the five optimal parameters of SIMCA were 3, 6, 6, 8, and 10, respectively. The recognition rate of the prediction sample set, obtained with its optimal parameter,  
Figure 6: Influences of LRSC’s parameters on recognition effects of (**a**) the infrared spectral data and (**b**) LIBS data. was 92%. If the five optimal parameters had been 3, 7, 3, 2, and 3 under the previous division of samples, then the recognition rate would only have been 78%. Therefore, parameter selection has a great influence on multi-parameter algorithms such as SIMCA, with the difference of the recognition rate exceeding 10%. Unlike SIMCA, which learns different classes, PLSDA has only one parameter, although it is also a spatial-projection algorithm. Theoretically, a large parameter does not necessarily guarantee a good recognition effect. The known samples can be well-trained when the parameter is larger and the model is more complicated, and this is likely to cause over-fitting during prediction. As a matter of fact, for IR data, the number of optimal latent variables of PLSDA fluctuates between 7 and 11, with the average being 9.2 and the standard deviation being 1.3. Regarding LIBS data, the average number of optimal latent variables of PLSDA is 13.4 and the standard deviation is 2.6. In practice, when dealing with some prediction samples, the optimal latent variables of the prior cross-validation usually need to be selected. When the conditions are very close, the choice of number of principal components is non-negligible, which greatly affects the prediction results. Therefore, in the case of high dimensions, the larger the number of principal components, the broader the range of the optimal principal components. However, computational efficiency is drastically reduced during cross-validation to select the optimal parameters. Therefore, considering efficiency and stability, LRSC is obviously a better choice for parsimony.