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The final form of the FFBPNN model with the substitution of weights is given as

      (2.8)

      where Y = Yield per unit area, q = number of nodes in hidden layer, Vj = weight coefficient between j^th hidden and k^th output nodes, c = threshold of the output node.

       2.6.1.2 Model Training

The trained ANN model recognizes the functional relationship between input parameters and desired outputs. The network training started with a random initiation of weights and proceeds and optimizes an error function (RMSE) [72, 73]. The generated weights by neural network saves and also remembers this functional relationship for further calculations. Yield estimation models are developed at the regional scale for paddy crops in kharif season. The right time stopping of the training of neural network is called because early stopping is an important step to avoid over fitting. To achieve this, the training, validation, and test set data were used to adjust the weights of neuron and bias, to stop the training process, and for external prediction respectively. Initially, 75% of the samples are selected randomly for the training, and the remaining 25% are used for testing to evaluate the model performance.

      The data of different parameters have wide range of values. For uniformity and also to avoid the confusion of learning algorithm, all the input data are normalized before input layer to represent 0 as minimum and 1 as maximum values. The output results (yields) are converted back to the similar unit by a denormalization procedure. Learning rate, number of hidden nodes, and training tolerance were adjusted. The initial selected number of hidden nodes was equal to inputs +1.

       2.6.1.3 Model Validation

      Four statistical parameters are used for performance analysis of the developed FFBPNN models, namely R², RMSE, MAE, and the R_ratio. These parameters are calculated using the testing data for finding out to optimize neural network. The criteria for optimum neural network are minimum RMSE, minimum MAE (Should be optimally 0), and the value of coefficient of determination is nearer to 1. The R_ratio is used to explain the models underprediction or overprediction of the simulated yield values. R_ratio that is less than 1 indicates underestimation, R_ratio that is more than 1 indicates overestimation. The relative error for each data point was also calculated. Additionally, the simulated values were plotted against the observed values and tested the statistical significance of parameters of regression analysis.

      To achieve this, training, validation, and test set were used to adjust the weights and biases, to stop the training process and for external prediction, respectively. Initially, 75% of samples are randomly selected for training, and the remaining 25% are used for testing to evaluate the model performance. The selected four statistical parameters were used for performance analysis of the developed FFBPNN models. These parameters were calculated using the testing data for finding out optimized neural network. The normalized output results (yields) from the ANN model are converted into original values at the end. Relative error between the targeted and neural network model predicted yield values. All relative errors of the model obtained are smaller than 10% except for two readings. 85% of the relative errors between predicted and observed values are even smaller than 10%.

      2.6.2 Results and Conclusions

Relative error between the targeted and neural network model predicted yield values is shown in Figure 2.8 for paddy and Figure 2.9 for sugarcane crop. Relative errors between the actual and neural network model were calculated for all the values of observation. All relative errors of the model for paddy crop are smaller than 10% in case of paddy crop. Only two readings are above ±10% in the case of sugarcane crop. 85% of the relative errors between predicted and observed values are even smaller than 8%.

Figure 2.9 Relative error between observed and predicted crop yields of training and testing data during 2015 of sugarcane crop (original figure).

The normalized output results (yields) from the ANN model are converted into original values at the end. Mean Mandal wise average relative error between observed and model predicted crop yields are presented in Table 2.3. The highest mean error of a mandal was 6.166% (at Pedana). The lowest relative error was 0.133% (at Challapalle).

The statistical parameters of the training and testing for sugarcane were compared. The range of R² values were 0.946 and 0.967 for training and same for testing was 0.936 and 0.950 for paddy in Kharif and Rabi seasons (Table 2.4), whereas for sugarcane the values are 0.916 and 0.924 during testing and training, respectively. The highest MAE was 0.178 for Paddy (Rabi). The R_ratio values for paddy (kharif) crop were 1.063 and 1.065. The same for sugarcane, it is 1.006 and 0.556. The R_ratio values showed the underestimation of crop yield of sugarcane crop during testing. The RMSE values were 0.15 and 0.184 with sugarcane crop in the year 2015. The results indicated that the R² for testing is more than that for training, which means that the FFBPNN models performed better during testing. The simulations produced highly satisfactory output in all predictions. This indicates that a well-trained FFBPNN model can be successfully used for crop yield prediction.

Table 2.3 Sample and training result of FFBPNN yield prediction model of paddy crop in Kharif during 2015.

S. no.	Mandal name	Mean actual observed yield, kg/ha	FFBP NN predicted yield, kg/ha	Relative error (%)
1	Vijayawada rural	7440.96	7618.65	-2.388
2	Kankipadu	8028.28	8406.09	-4.706
3	Challapalle	6897.78	6906.95	-0.133
4	Pamarru	7617.60	7162.68	5.972
5	Vuyyuru	7595.52	7914.53	-4.20
6	Movva	7286.40	7623.40	-4.625
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