6.2 Training of ANN models for daily ETo estimation

All the ANN models were trained as per the procedure mentioned in methodology and after each training run; three performance indices (RMSE, R2 , and Rratio) were calculated, to find the optimum neural network. Several runs were used for determining the optimal number of hidden neurons with different architectural configurations. The optimum neural network was selected based on criteria such that the model has minimum RMSE and maximum R2 values. Here, it is worth to mention that the Rratio is used only to know whether the models overestimated or underestimated ETo values. Training with higher number of hidden nodes might increase the performance of ANN models. But training with a several number of hidden nodes requires more computation time and cause complexity in architecture as it has to complete number of epochs [7]. Therefore, to avoid the above difficulty, the selection of an optimum node was fixed with a trial run of 1–15 hidden nodes only (i.e., not tried beyond 15 hidden nodes). Figure 6 shows the relationship between RMSE and number of hidden nodes of ANN models for four locations (Parbhani, Hissar, Faizabad, and Dapoli) during training. These locations are chosen randomly from each agro-ecological region such that Parbhani, Hissar, Faizabad, and Dapoli represent semi-arid, arid, sub-humid, and humid climates, respectively.

For ANN models, the best network was resulted at a hidden node of i + 1 (where i = number of nodes in the input layer) for most of the locations. Thus, i + 1 hidden nodes are sufficient to model the ETo process using the ANN models [13–16, 44–46]. Table 3 shows the performance statistics of ANN models for 15 locations during training. The results pertaining to the optimal network structure of ANN models, resulted at i + 1 hidden nodes, are only summarized in Table 3 for 15 locations.

Figure 6. RMSE variations with number of hidden nodes for ANN models.

December 31, 2005) was collected from All India Coordinated Research Project on Agrometeorology (AICRPAM), Central Research Institute for Dryland Agriculture (CRIDA), Hyderabad, Telangana, India. These data were used for the development and testing of various ANN-based ETo models. Due to the unavailability of lysimeter measured ETo values for these stations, it is estimated by the FAO-56 PM method, which has been adopted as a standard equation for the computation of ETo and calibrating other Eqs. [10]. The normalization technique was applied to both the input and target data before training and testing such that all data points lies in between 0 and 1. The normalization process removes the cyclicity of the data. The following procedure was adopted for normalizing the input and output data sets. Each variable, Xi, in the data set was normalized (Xi, norm) between 0 and 1 by dividing its value by the upper limit of the data set, Xi, max. Resulting data was then

where Ti and Oi = target (FAO-56 PM ETo) and output (ETo resulted from MLR or ANN models) values at the ith step, respectively; n = number of data points, Tand O ¼ average of target (FAO-56 PM ETo) and output (ETo from

Evaluation criteria Formulae Root Mean Squared Error (RMSE) RMSE <sup>¼</sup>

Advanced Evapotranspiration Methods and Applications

Ratio of average output and target ETo values (R) <sup>R</sup> <sup>¼</sup> <sup>O</sup>

Coefficient of determination (R<sup>2</sup>

MLR or ANN models) values, respectively.

Performance evaluations of ANN and MLR models.

Table 2.

ANN simulated ETo was converted back to original form by denormalization procedure. The data from 2001 to 2005 was splitted into training (70% of 2001– 2004), validation (30% of 2001–2004), and testing (2005) sets. ANN models were trained with the LM algorithm consists of one hidden layer (sigmoid transfer function) and one output layer (linear transfer function). The parameters that were fixed after a number of trials include: RMSE = 0.0001, learning rate = 0.65, momentum rate = 0.5, epochs = 500, and initial weight range = �0.5 to 0.5. The developed various ANN models were compared with basic statistical MLR models. The developed ANN models were evaluated and compared based on different error functions described in Table 2. Training window of the model contains general information used for training the networks like, error tolerance, Levenberg parameter (lambda) and maximum cycles of simulation. For weights selection, two options are there, weights can be randomized or it can be read from an existing

ANN model with six climatic variables (Tmax, Tmin, RHmax, RHmin,WS, and Sra) were trained and tested to evaluate the feasibility of ANN models corresponding to FAO-56 PM conventional ETo method for 15 individual locations in India. In order

Xi,norm ¼ Xi=Xi,max (24)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>i</sup>¼<sup>1</sup>ð Þ Ti‐Oi <sup>2</sup>

1 <sup>n</sup> <sup>∑</sup><sup>n</sup>

q

∑<sup>n</sup> <sup>i</sup>¼<sup>1</sup> ð Þ Oi‐<sup>O</sup> <sup>2</sup> ∑<sup>n</sup> <sup>i</sup>¼<sup>1</sup> ð Þ Ti‐<sup>T</sup> <sup>2</sup>

T

) <sup>R</sup><sup>2</sup> <sup>¼</sup> <sup>∑</sup><sup>n</sup> ½ � <sup>i</sup>¼<sup>1</sup> ð Þ Oi‐<sup>O</sup> ð Þ Ti‐<sup>T</sup> <sup>2</sup>

used for mapping.

weight file of previous training.

6. Results and discussion

36

6.1 Development of ANN models for daily ETo estimation
