# Analysis of Fluid Velocity inside an Agricultural Sprayer Using Generalized Linear Mixed Models
## 2 Material and Methods
### Useful Sequencing of the Best Model  
In this study, for the formulation and selection of the best model in terms of the number and nature of the fixed and random effects (Tables 2-4), a canonical link function was assumed under the premise of a normal distribution of the errors of the model and the target variable, fluid velocity (FV). The Kolmogorov-Smirnov (K-S) test was used to evaluated conformance with the normal distribution (theoretical curve shown in Figure 3). Once the best model was selected and the FV showed positive scale values that deviated towards larger positive values, it was tested until the combination \"distribution form-linking function\", providing a better fit, was obtained. With respect to the variance-covariance matrix, the basic rationale was that \"parsimony means power\", that is, to obtain the most efficient inferences about the mean model, one selects the most parsimonious covariance structure possible, that continues to reasonably fit the data. It is, thus, recommended, in each case, to test both the formats with the least number of parameters and heterogeneous structures [30]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**N** & **Model** & **AICc** & **BIC** \\ \hline
1 & FV = f(none variable) & 46,439.09 & 46,445.98 \\
2 & FV = f(FE(LT)) & 45,613.44 & 45,620.32 \\
3 & FV = f(FE(NN)) & 46,318.66 & 46,325.54 \\
4 & FV = f(FE(P)) & 46,406.36 & 46,413.24 \\
5 & FV = f(FE(LT, NN)) & 45,478.07 & 45,484.95 \\
6 & FV = f(FE(LT, P)) & 45,576.50 & 45,583.38 \\
7 & FV = f(FE(NN, P)) & 46,285.34 & 46,292.22 \\
8 & FV = f(FE(LT, NN, P)) & 45,440.38 & 45,447.26 \\
9 & FV = f(FE(LT, NN, P, SC)) & 44,656.55 & 44,663.43 \\
10 & FV = f(FE(LT, NN, P) + RE(H, S)) & 44,314.36 & 44,328.12 \\
11 & FV = f(FE(LT, NN, P) + RE(SC)) & 43,681.51 & 43,695.27 \\
12 & FV = f(FE(LT, NN, P) + RE(SC, H)) & 43,684.51 & 43,698.27 \\
13 & FV = f(FE(LT, NN, P) + RE(SC, H, S)) & 43,687.52 & 43,701.28 \\
14 & FV = f(FE(LT, NN, P) + RE(H, S)) & 45,137.79 & 45,151.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Corrected Akaike (AICc) and Bayesian (BIC) information criteria for proposed and numbered models (N). FE: fixed effects. RE: random effects. LT: liquid level in the tank. NN: number of active nozzles. P: circuit pressure. SC: section of the measured points. S: side. H: height. The first model is null. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & \(\Delta_{\textbf{i}}\) = \(\textbf{AIC}_{\textbf{i}}-\textbf{AIC}_{\textbf{min}}\) & \(\textbf{w}_{\textbf{i}}\) = \(\textbf{exp}(-0.5\cdot\Delta_{\textbf{i}})/\textbf{\Sigma}(\textbf{exp}(-0.5 \cdot\Delta_{\textbf{i}}))\) \\ \hline
1 & 2757.59 & \(0.00\times 10^{0}\) \\
2 & 1931.94 & \(0.00\times 10^{0}\) \\
3 & 2637.15 & \(0.00\times 10^{0}\) \\
4 & 2724.86 & \(0.00\times 10^{0}\) \\
5 & 1796.56 & \(0.00\times 10^{0}\) \\
6 & 1895.00 & \(0.00\times 10^{0}\) \\
7 & 2603.83 & \(0.00\times 10^{0}\) \\
8 & 1758.88 & \(0.00\times 10^{0}\) \\
9 & 975.04 & \(1.47\times 10^{-212}\) \\
10 & 632.85 & \(2.97\times 10^{-138}\) \\
11 & 0.00 & \(7.86\times 10^{-1}\) \\
12 & 3.00 & \(1.75\times 10^{-1}\) \\
13 & 6.02 & \(3.88\times 10^{-2}\) \\
14 & 1456.29 & \(0.00\times 10^{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Increments of the Akaike information criterion corrected for each model with respect to the best model (minimum), and the relative likelihood of each model expressed through its relative Akaike weight (\(w_{\textbf{i}}\)). Table 5 shows estimates of mean and variance of the Pearson's residues according to the different covariance matrix structures tested. A linear regression was performed between the predicted and observed values (Table 6), assessing their significance through an analysis of variance of the regression model, the constant, and the slope.