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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
7	Thotlavalluru	6624.00	6485.03	2.098
8	Avanigada	5453.76	5374.90	1.446
9	Pamidimukkala	8765.76	8352.45	4.715
10	Guduru	8964.48	8513.84	5.027
11	Penamaluru	6853.62	6931.55	-1.137
12	Koduru	8854.08	8403.05	5.094
13	Pamarru	8589.12	9081.36	-5.731
14	Machilipatnam	8824.32	8582.00	2.746
15	Pedana	8311.36	7798.88	6.166
16	Mopidevi	6586.24	6354.08	3.525
17	Nagayalanka	7904.64	7890.10	0.184

Scatter plots of predicted yield and observed yield are shown in Figures 2.10 and 2.11. This shows that the points are equally distributed over 1:1 line and also in close agreement with an R² value of 0.9681 for paddy. There was wide scattering of yield points in case of sugarcane. There was an under estimation of yield in some cases of cane yield prediction. Although there was a deviation in crop yield prediction in few observations, the overall accuracy of the model prediction was high. The R² value was high for paddy compared with sugarcane crop. However, a slight underestimation of yield of the sugarcane crop indicates that sensitivity of yield algorithms to crop input parameters may be further improved by changing the input parameters.

Table 2.4 Statistical analysis of neural network training and testing of season in different years.

	Training	Testing
Year	RMSE	R ratio	MAE	R2	RMSE	R ratio	MAE	R2
Paddy (Kharif)	0.117	1.063	0.095	0.946	0.108	1.065	0.085	0.936
Paddy (Rabi)	0.125	0.987	0.108	0.967	0.317	0.620	0.178	0.950
Sugarcane	0.150	1.006	0.119	0.916	0.184	0.556	0.143	0.924

It is also observed during the study that for the model, the RMSE decreased with increased number of hidden nodes from 1 to [i + 1], where “i” is the number of input layer nodes (i = 5). Further, R² increased and MAE decreased with the increased the number of hidden nodes from 1 to [i + 1], in all the years. The results are in accordance with researchers [74–76]. After 20 hidden nodes, the trails are conducted at a step of 10. After a number of trials with 100 to 10,000 epochs with each step of 100 up to 2000 and a step of 1000 up to 10000, better results are found at 1000 epochs for most of the cases. Although the performance increased after i+1, the computation time increased with increase in the number of nodes and epochs. The smaller the data sets, the lesser hidden nodes requirement and lower learning rates in the optimized model is observed [62]. The best performance of the models was observed at i+1 and i+2 hidden nodes. Statistical analysis revealed that the reliability of the model in crop yield estimation. The final predicted yield map of paddy and sugarcane during 2015 are shown in Figures 2.12 and 2.13. Paddy yields in the study area varied from 3.25 t/ha to 6.6 t/ha. The sugarcane yields were ranged between 70000 kg/ha and 125,000 kg/ha.

Figure 2.10 Scatter plots of actual and FFBP NN model predicted yield of sugarcane during 2015 (original figure).

Figure 2.11 Scatter plots of actual and FFBP NN model predicted yield of sugarcane during 2015 (original figure).

There was an underestimation of yield in some cases of cane yield prediction. Although there was a deviation in crop yield prediction in few observations, the overall accuracy of the model prediction was high. The model prediction accuracy may be further improved by changing the input parameters. Sugarcane crop is sensitive to leaf area index (LAI), and number of stalks per meter [77]. A stronger relationship exists between sugarcane yield and rainfall. Total soil available water is an important indicator of yield. Another important point, which differs yield prediction, is input parameter as average yield. In case of sugarcane, the input is given as average of plant and ratoon for the 3 years. The improvement of model was not attempted because of the nonavailability of the data on sugarcane crop-sensitive parameters, like the number of stalks per meter and total soil available water.

Figure 2.12 Final predicted yield map of paddy during 2015 (original figure).

Figure 2.13 Final predicted yield map of sugarcane during 2015 (original figure).