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0.363 0.844 0.723 0.725 7 Thotlavalluru 0.687 0.439 0.844 0.686 0.450 8 Avanigada 0.563 0.454 0.785 0.359 0.000 9 Pamidimukkala 0.575 0.687 0.791 0.302 0.900 10 Guduru 0.459 0.767 0.735 0.000 0.915 11 Penamaluru 0.478 0.706 0.744 0.063 0.525 12 Koduru 0.748 0.482 0.874 0.821 0.900 13 Pamarru 0.771 0.390 0.885 0.929 0.619 14 Machilipatnam 0.533 0.452 0.771 0.284 0.921 15 Pedana 0.491 0.203 0.750 0.266 0.957 16 Mopidevi 0.496 0.790 0.752 0.080 0.544 17 Nagayalanka 0.511 0.521 0.760 0.203 0.750

      Table 2.2 represents the maximum and minimum values of the normalized data. The average values of NDVI ranged from 0.459 to 0.687 for different mandals. Similarly, APAR and CWSI ranged from 0.750 to 0.874 and 0 to 0.929, respectively. Surface temperature and average yield were normalized. The normalized surface temperature was in the range of 0.148 for Kankipadu to 0.831 at Vijayawada rural. The highest normalized yield is at 0.957 at Pedana. The point wise normalized parameter values were used as input to the NN model.

Figure 2.6 Exporting the attribute values of the points to excel (original figure).

2.6 Neural Network Model Development, Calibration and Validation

      2.6.1 Materials and Methods

       2.6.1.1 ANN Model Design

The data extracted from the abovementioned maps are divided into two data sets for training and testing to analyze network results and testing the models. The typical architecture of three-layered MLFF perceptron used is shown in Figure 2.7. The derived five yield factors, such as NDVI, surface temperature, APAR, crop water stress index, and average yields are taken as neurons for input layer. The output layer is one neuron, i.e., yield.

      The hidden layer has a different number of hidden neurons and is tested for optimum number of neurons. The optimum number of neurons in hidden layer and parameters of the model is determined by trial and error method. W_ij is the connecting weight between i^th input layer neuron to the jth hidden layer neuron. The V_jk is weight between the j^th hidden layer neuron and the k^th output layer neuron (in this case k=1). Momentum and learning rate are two main parameters for training, which takes care of steepest-descent convergence [71]. The final weighting factors are used to simulate relationship between crop yield and corresponding crop growth factors. The final weighting factors generated by the network trained model are saved for estimation of new data. The hidden layer neurons were varied between 1 and 30 in the developed models. Sigmoidal transfer function and linear activation functions are in hidden output layers. The code to develop the neural network is written in MATLAB programming language package.

Figure 2.7 Architecture of the proposed FFBPNN model (original figure).

      The hidden layer receives data from the input neurons layer. In the hidden layer, inputs are multiplied by suitable weights and sums. The sigmoid transfer function was activated before the output layer. Mathematical expression linear transfer function is

      (2.5)

      The output y is expressed as:

      (2.6)

      where f is neuron activation or transfer function. The transfer function of each neuron in the network is a sigmoid transfer function and is given as

      (2.7)

Figure 2.8 Relative error between observed and predicted crop yields of training and testing data during 2015 of paddy crop in Kharif season (original figure).

The neuron activation function was shown in Figure

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