## 2 Materials and Methods
### Three-Dimensional Scatter Plot  
Three-dimensional scatter plots are an extension of two-dimensional scatter plots and are used to identify possible correlations between observations and variables in a data set on a single plot of three mutually perpendicular axes, where each variable is interpreted as a Cartesian coordinate vector. Cartesian coordinate vectors are defined such that each element of the matrix is the inner product of the vectors corresponding to its rows and columns [23].  
The analysis of a three-dimensional scatter plot goes beyond data visualization: it also involves the understanding of different statistical tools involved, such as three-dimensional Euclidean space and measures of central tendency. The exploration begins with the definition of a three-dimensional Euclidean space, with a set of the term \((x,y,z)\) of real numbers, formally defined as \(\mathbb{R}^{3}\). The observations represent the relationship between the three variables, where f is a function of two variables, \(z=f(x,y)\).  
The Euclidean distance between two observations (Equation (4)) indicates the distance between the influence of two different treatments \(P(p_{1},q_{1},r_{1})\) and \(Q(p_{1},q_{1},r_{1})\)[24].  
\[d(P,Q)=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left( z_{2}-z_{1}\right)^{2}} \tag{4}\]  
**Equation (4).** Euclidean distance fitted to a three-dimensional plane.  
The calculation of the mean of each variable (Equation (5)) allows us to understand the concentration of the data in each dimension and to detect possible biases in the data [25].  
Mean for variable \"\(xa\)\"  
\[\overline{a}=\frac{1}{n}\sum_{i=1}^{n}a_{i} \tag{5}\]  
**Equation (5).** Average of the variables in a three-dimensional plane.  
Where:  
* \(\overline{a}\) represents the mean of the variable.
* \(a_{i}\) represents each individual observation of the variable.
* \(n\) is the total number of observations.
* \(a\) can be \(x,y,z\) depending on the variable of interest.  
Covariance (Equation (6)) measures the linear relationship between two variables, indicating the degree to which these two change together. Given a positive covariance, the variables tend to increase; given a negative covariance, one variable tends to increase while the other variable tends to decrease and given a covariance close to zero, a weak linear relationship between the variables is inferred.  
Covariance between variable \"\(a\)\" and variable \"\(b\)\".  
\[Cov(a,b)=\frac{1}{n-1}\sum_{i=1}^{n}\left(a_{i}-\overline{a}\right)\left(b_{i }-\overline{b}\right) \tag{6}\]  
**Equation (6).** Calculation of covariance in a three-dimensional plane.  
Where:  
* \(\overline{a}\) and \(\overline{b}\) represents the mean of the variable.
* \(a_{i}\) and \(b_{i}\) are the individual observations of the variables \(a\) and \(b\), respectively.
* \(n\) is the total number of observations.
* \(a\) and \(b\) can be any pair of the variables \(x,y,\) or \(z\) to compute \(Cov(x,y)\), \(Cov(x,z)\) or \(Cov(y,z)\).  
[PERSON]'s correlation (Equation (7)) is another measure of association, which allows us to denote how related two variables are to each other [26]. For a three-dimensional plane, the Pearson correlation is calculated in pairs.  
Correlation between variable \"\(a\)\" and variable \"\(b\)\".  
\[r_{a,b}=\frac{Cov\left(a,b\right)}{\sigma_{a}\sigma_{b}} \tag{7}\]  
**Equation (7).** Calculation of [PERSON]'s correlation in a three-dimensional plane. Where:  
* \(r_{a,b}\) represents the correlation coefficient between the variables \(a\) and \(b\).
* \(Cov(a,b)\) is the covariance between the variables \(a\) and \(b\).
* \(n\) are the standard deviations of the variables \(a\) and \(b\), respectively.
* \(a\) and \(b\) can be any pair of the variables \(x,y\), or \(z\) to compute \(r_{x,y}\), \(r_{x,z}\) or \(r_{y,z}\).