## 2 Materials and Methods
### Methods
#### 2.2.4 Phenological Normalization Based on Multilayer Perceptron  
As mentioned above, the changes that are caused by the growth of vegetation are the most typical nonlinear changes in the radiometric characteristics. In other words, a nonlinear relationship is required in order to perform normalization for the radiometric differences as well as the phenological differences [25; 26]. In this study, the nonlinear normalization is performed while using an MLP that is a representative algorithm for modeling nonlinear relationships [33; 48]. The architecture of MLP consists of three units: the input layer, output layer, and hidden layer. The input layer is the first layer that transfers the initial data into the network, the output layer is the last layer that collects and transmits the results, and the hidden layer is an intermediate layer between the input and output layers that processes the data [31]. Each layer is composed of several neurons, where every neuron is connected to all of the neurons in the next layer through weighting [33]. When the input data are \(\mathrm{x}~{}=~{}x_{1},~{}\cdots x_{n}\), a neuron \(k\) can be expressed, as in Equation (11):  
\[y_{k}~{}=~{}\varphi(\sum_{i~{}=~{}1}^{n}w_{ki}~{}x_{i}+~{}b) \tag{11}\]  
where \(w_{k1},~{}\cdots w_{kn}\) are the weights of neuron \(k\), \(b\) is the bias, \(\varphi\) is the activation function, and \(y_{k}\) is the output of the neuron \(k\). In this study, the input layer has two neurons, which represent the DN (or reflectance) and the optimal spectral index obtained above, and the output layer has one, which represents the normalized DN (or reflectance). Furthermore, the learning process is performed using a back-propagation algorithm that adjusts weights to minimize the error between the actual value and predicted value based on loss function. The process is continued until a predefined accuracy level or the maximum iterations are reached.  
At this point, hyperparameters, such as the number of hidden layers, the number of neurons for hidden layers, learning rates, epochs, activation function, loss function, or optimizer, must be set up in order to construct the applicable MLP model [30]. The hyperparameters in the proposed method are selected based on experiences and experiments. As for the number of hidden layers, an increasing number causes a loss in the generalization power of the network, which might cause overfitting [49]. Furthermore, it increases the overall complexity of the algorithm and the training time [50]. Therefore, after considering the nature of the normalization, with limits of training time, only one hidden layer is considered [21; 25]. As for the selection of the number of neurons for the hidden layer, the comparison experiments are conducted by setting the number of neurons to 2, 3, 4, 5, 6, 7, 8, 9, 10, and 15. The performance evaluation is conducted with normalization accuracy, normalized RMSE (NRMSE), which provides a lower value and better performance while taking the visual and quantitative aspects of the normalized image into account; Figure 5 shows the confirmatory results. The results show that, after three neurons, the time cost sharply increases, while the accuracy is not improved with the increase in neurons. Therefore, to balance the accuracy and time cost, three neurons for the hidden layer are selected in this experiment. As for the epochs and the learning rate, the figures 200 and 10\({}^{-4}\) are chosen, respectively, to prevent overfitting [34; 51]. Furthermore, the activation function, loss function, and optimizer are selected as the rectified linear unit (ReLU), squared error, and adaptive moment estimation (ADAM), respectively. Table 2 summarizes all of the hyperparameters of the MLP and the values used in the proposed method.  
Figure 5: Experiments regarding the number of neurons for the hidden layer: (**a**) accuracy by setting different neurons, (**b**) training time by setting different neurons.