## 2 Materials and Methods
### Analytical Methods  
Chlorophyll fluorescence is closely related to LNC, and the bands were mainly located in the red and far-red regions. Then, some fluorescence intensity ratio indices were proposed to analyze LNC changes. Therefore, a fluorescence index was introduced for LNC monitoring based on the slope characteristics of fluorescence spectrum. The proposed fluorescence index was presented as follows:  
\[K=\frac{F(\lambda_{i1},\lambda_{ex})-F(\lambda_{i2},\lambda_{ex})}{\lambda_{i1 }-\lambda_{i2}}/F(\lambda_{i3},\lambda_{ex}) \tag{1}\]  
where \(\lambda_{ex}\) is the excitation light wavelength, and \(F(\lambda_{i1},\lambda_{ex})\), \(F(\lambda_{i2},\lambda_{ex})\), and \(F(\lambda_{i3},\lambda_{ex})\) denote the fluorescence intensities at fluorescence emission wavelengths \(\lambda_{i1}\), \(\lambda_{i2}\), and \(\lambda_{i3}\), respectively. According to previous researches, fluorescence characteristic peaks at 685 nm were mainly contributed by chlorophyll a of Photosystem II, whereas those at 740 nm were contributed by the antenna chlorophyll of Photosystems I and II. The fluorescence characteristics peaks were positively correlated to LNC. In addition, the fluorescence characteristics at 460 nm were mainly denoted by the nicotinamide adenine dinucleotide [23; 30]. Therefore, \(\lambda_{i3}\) was selected at 460 nm, and the fluorescence peak characteristics were selected at 740 nm and 685 nm. In addition, a reference fluorescence signal was selected at 640 nm. Then, a total of three fluorescence indices were calculated as follows:  
\[K1=\frac{F740-F685}{\Delta\lambda_{1}}/F460 \tag{2}\]  
\[K2=\frac{F740-F640}{\Delta\lambda_{2}}/F460 \tag{3}\]  
\[K3=\frac{F685-F640}{\Delta\lambda_{3}}/F460 \tag{4}\]  
Then, the performance of the proposed fluorescence indices was analyzed based on the back-propagation neural network (BPNN) model for LNC estimation. The results were then compared with the fluorescence ratios or extracted fluorescence characteristics through principal component analysis (PCA) analyzed fluorescence spectrum. BPNN serves as a feedforward network and has been widely used to solve non-linear problems due to its self-learning and self-adaption abilities [31; 32]. A typical BPNN model comprises three layers including input, output, and hidden layers, each with a few neurons. The BPNN model can modify weights of the neurons in response to the errors between the target output values and truth-value until it is optimal. The detailed description of the BPNN model can be found in this reference [32; 33]. In addition, the effect of training function and hidden layer size on the accuracy of LNC estimation was discussed. In this investigation, training functions were \"traingd,\" \"traingdm,\" \"traingda,\" \"traingdx,\" \"trainrp,\" \"traincgf,\" \"traincgp,\" \"traincgb,\" \"trainscg,\" \"trainbfg,\" \"trainln,\" and \"trainbr,\" which corresponded to the training methods \"gradient descent,\" \"gradient descent with momentum,\" \"gradient descent with adaptive learning rate,\" \"gradient descent with momentum and adaptive learning rate,\" \"resilient gradient backpropagation,\" \"conjugate gradient backpropagationwith Fletcher-Reeves restarts,\" \"conjugate gradient with Ploak-Ribiere restarts,\" \"conjugate gradient with Powell-Beale restarts,\" \"scaled conjugate gradient,\" \"Broyden-Fletcher-Goldfarb-Shanno quasi-Newton,\" \"one step secant,\" \"Levenberg-Marquardt,\" and \"Bayesian regulation\", respectively. In the BPNN model, the fluorescence characteristics were divided into two datasets: 70% as the training dataset to train the BPNN model and the remaining 30% as the testing dataset to validate the BPNN model. The coefficient of determination (\(R^{2}\)) between the predicted and measured LNC and standard deviation (SD) were used to assess the performance of the proposed fluorescence index for LNC estimation and the effect of internal parameter of the BPNN model on the accuracy of LNC estimation.