## 3 Mathematical Formulation of Our Proposed Models  
Our goal is to propose three models to evaluate QoE. Three types of video were reviewed: one slow, one fast and one moderately slow; and a model was proposed for each type of video according to the spatio-temporal characteristics. We require the values of the factors (QoS parameters) that enable the procurement of a response (QoE associated to video). The selection of the type of mathematical model influences the experimental design and the number of treatments to be performed in order to obtain the model. The most effective experimental design type is the Methodology of Surfaces of Response (MSR). MSR is defined as a set of mathematical and statistical techniques used for solving problems where a response of interest (video QoE) is influenced by several quantitative factors (QoS parameters) [30]. Using laboratory tests, the ranges of the parameters that affect the quality of the video were defined, and based on those ranges, we obtained a second-order hierarchical model for each type of video [6]. A system is stable when it only depends on the QoS parameters that are measured on the network--that is, it does not depend on subjective parameters that can vary the results. The behavior of this type of model is more stable, thus, allowing an exploration of the surface it represents. In addition, it allows no significant terms to remain in the model to achieve hierarchy. The models are represented by Equation (1), where \(X_{i}\) and the coefficients represent the factors (QoS parameters: Delay, Jitter and Packet loss), \(\beta_{i}\) represents the regression coefficients, and \(Y\) represents the response of interest (QoE) and \(\varepsilon\) is the random error. The entire detailed mathematical process is presented in [7]:  
\[Y=\beta_{o}+\sum_{i=1}^{k}\beta_{i}X_{i}\ +\sum_{i=1}^{k}\beta_{i}X_{i}^{2}+ \sum_{i=1}^{k}\sum_{j=1}^{k}\beta_{ij}X_{i}X_{j}+\varepsilon \tag{1}\]  
To estimate the coefficients (\(\beta_{i}\)) in Equation (1), we use the least squares method. Suppose \(n>k\) runs are available and let \(X_{ij}\) denote the \(j\)th value of factor \(X_{i}\). The run data is shown in Table 1. This estimation procedure requires the random error component to have E(\(\varepsilon\)) = 0 and V(\(\varepsilon\)) = \(\sigma^{2}\) and \(\{\varepsilon\}\) to be uncorrelated [31]. Model Equation (1) is written in terms of the data presented in Table 1 as follows:  
The least-squares method selects the \(\beta\) coefficients in Equation (2) so that the sum of squares of the errors (\(\epsilon_{i}\)) is minimized. \[Y_{i}=\ \beta_{o}+\beta_{1}X_{i1}+\beta_{2}X_{i2}+\cdots+\beta_{k}X_{ik}+ \epsilon_{i}=\beta_{o}+\sum_{j=1}^{k}\beta_{j}X_{ij}+\epsilon_{i}\ \ \ i=1,\ 2,\ \ldots,\ n \tag{2}\]  
Thus, the least-squares function is defined as Equation (3). \[S=\sum_{i=1}^{k}\epsilon_{i}^{2}=\sum_{i=1}^{n}\left[Y_{i}-\beta_{o}-\sum_{j= 1}^{k}\beta_{j}X_{ij}\right]^{2} \tag{3}\]  
Simplifying this, we obtain a system of \(k+1\) simultaneous equations, and the solution of such a system is the set of least-squares estimators, \(\beta_{j}\). It is more convenient to solve this system by using a matrix form. Thus, the model illustrated in Equation (2) is expressed in matrix form as:  
\[y=X\beta+\epsilon \tag{4}\]  
where  
\[y=\left[ \begin{array}{c}y_{1}\\ y_{2}\\ \vdots\\ y_{n}\end{array} \right],\ X=\left[ \begin{array}{cccc}1&x_{11}&x_{12}& \ldots&x_{1k}\\ 1&x_{21}&x_{22}&\ldots&x_{2k}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 1&x_{n1}&x_{n2}&\ldots&x_{nk}\end{array} \right], \tag{5}\]  
\[\beta=\left[ \begin{array}{c}\beta_{0}\\ \beta_{1}\\ \vdots\\ \beta_{k}\end{array} \right]\text{and}\ \varepsilon=\left[ \begin{array}{c} \varepsilon_{1}\\ \varepsilon_{2}\\ \vdots\\ \varepsilon_{n}\end{array}