# Analysis of Fluid Velocity inside an Agricultural Sprayer Using Generalized Linear Mixed Models
## 3 Results and Discussion
### Best Model Description  
The data was structured from control points, as observation units that were independent of each other. The measurements were carried out according to different levels of the FE, such as liquid level in the tank, pressure, and the number of active nozzles, in four sections and at three heights and distances from the perimeter (sides). The target variable (dependent) is the fluid velocity inside the tank. As previously warned, considering the form of the distribution of the target variable and the outcome of the various tests carried out, the combination that best matched what was observed was a Gamma-type distribution [31] with a logarithmic link function (f(x) = log(x)). The values of the information criteria after the assessment of the different forms of the variance-covariance matrix (Table 5) show that, in the case of the AICc, the structures that best fit the model are the First-order autoregressive moving-average, followed by the Toeplitz, while with the BIC, the best fit model was the identity model followed by the first-order autoregressive, and then the first-order autoregressive moving-average model, which corresponded to their respective relative weights (w\({}_{i}\) AICc or BIC). Among these, the model that recorded a minor statistical significance for \(\hat{\sigma}^{2}\) SC was the Compound-symmetry covariance structure, being the one that most closely related the observations to the assumption of independence [19; 44]. Despite having a large number of observations, and only one RE, it was difficult to choose the most appropriate covariance structure based only on the information criteria shown in Table 5[45]. The degree of adjustment of the model's predictions to the observed data was, thus, analyzed according to the different covariance structures (Table 6) [27; 28]. The accordance between the observed and predicted values of the model with regard to the determination coefficient, based on the structure of the covariance matrix (Table 6), revealed that the best fit corresponded to Unstructured and Toeplitz. The smallest standard error in the estimation was provided by First-order autoregressive moving-average, followed by the Identity. As for the estimation of the variance of Pearson's residues, the values closest to unity came from Unstructured, followed by the Toeplitz. Pearson's average residue estimate was very close to zero for all structures. Through the AICc and BIC information criteria (Table 5), and with the coefficients of fit between what was observed and what was predicted by the model (Table 6), it seems that the Toeplitz covariance structure could be the most appropriate for the case under consideration (Equation (6)), obtaining the following p-values: \(p(\sigma^{2})=0.001;p(\sigma_{1})=0.308;p(\sigma_{2})=0.367\); and \(p(\sigma_{3})=0.001\). This shows that the variability of the fluid velocity within each level of SC was significant, as was the difference in the observed variability between the first and last sections of the tank. \[ \begin{split}\text{{Toeplitz covariance structure}}=\left[ \begin{array}{cccc}\sigma^{2}&\sigma_{1}&\sigma_{2}&\sigma_{3}\\ &\sigma^{2}&\sigma_{1}&\sigma_{2}\\ &&\sigma^{2}&\sigma_{1}\\ &&&\sigma^{2}\end{array}\right]=\left[\begin{array}{cccc}0.060&0.011&-0.015& 0.036\\ &0.060&0.011&-0.015\\ &&0.060&0.011\\ &&&0.060\\ \end{array}\right]\end{split}  \tag{6}\]  
The number of samples in this test was high, and only one RE was included, which made it difficult to bias the iterative linear methods that maximize the PQL. The Laplace and Gauss-Hermite quadrature approaches were tested, resulting in higher (worse) values in the AICc and BIC information criteria (41,652.69). The MQL method was also tested, obtaining lower (better) values in the information criteria with respect to the PQL. A lower adjustment, however, between what was inferred and what was observed, as well as a clear bias due to excess in the estimation of the FE was observed.