*3.5. Modeling*

The generated database (inputs from Table 3 and outputs from Table 1) was modeled using the commercial software FormRules ® v4.03 (Intelligensys Ltd., Stokesley, UK) which is a hybrid system that combines Artificial Neural Networks (ANN) and fuzzy logic. Nozzle diameter, pressure and temperature were introduced as inputs, while percentage of fine particles, mean particle size and standard deviation were introduced as outputs. A separate model was developed for each output. These models are split into different submodels, when it is possible, to generate simple and understandable rules.

Among the fitness criteria included by FormRules® (cross validation, minimum description length, structural risk minimization, leave one out cross validation and Bayesian information criterion), minimum description length was selected because it gives the best R-squared as well as the simpler and more intelligible rules. Modeling was carried out using the parameters shown in Table 4.

**Table 4.** Training parameters setting with FormRules® v4.03.


Three sets of "IF...THEN" rules were subsequently generated to express the model, one set for each output. IF...THEN rules are made up of two parts: the initial one, which includes the input or inputs explaining a specific output, followed by the second part describing the output characteristics, which are defined by a word and its corresponding membership degree (Table A1) [36].

The predictability of the models was assessed using the determination coefficient (R2) defined by Equation (2):

$$R^2 = \left(1 - \frac{\sum\_{i=1}^{n} \left(y\_i - y\_i'\right)^2}{\sum\_{i=1}^{n} \left(y\_i - y\_i'\right)^2}\right) \times 100\tag{2}$$

where yi is the actual point in the data set, yi is the value calculated by the model and yi" is the mean of the dependent variable. Values of R<sup>2</sup> must be lower than 99.9%, otherwise there is a risk of overtraining the neural network [50]. The larger the value of the train set R2, the more the model captured the variation in the training data. Values for R<sup>2</sup> > 70% are indicative of reasonable model predictabilities.

The accuracy of the models was evaluated with the analysis of variance to compare predicted and experimental results, respectively. Computed f ratio values higher than critical f values for the degrees of freedom of the model, indicate no statistical significance between predicted and experimental results and hence, model accuracy.
