## 3 Results  
### Parameter Estimation of SOMRS Model with a Multivariate \(t\)-Distribution Approach  
Maximum likelihood estimation for the parameters of the SOMRS model with the EM algorithm is given in Theorem 1. **Theorem 1. **_Suppose it is known that the expected value of a SOMRS model with a multivariate \(t\)-distribution is \(E(\mathbf{Y}_{u})=\mathbf{B}^{\prime}\mathbf{g}(\mathbf{x}_{u})\) for \(u=1,2,\ldots,n\), with \(\mathbf{Y}_{u}\) is \(q\times 1\) response variable, \(\mathbf{B}\) is \(p\times q\) model parameter matrix, and \(\mathbf{g}(\mathbf{x}_{u})\) is a known vector function of the factor levels (according to the polynomial model order). If \(\mathbf{Y}_{u}\) has multivariate \(t\)-distribution with a scale parameter \(\ \uppsi\) and the degrees of freedom \(v\), then the maximum likelihood estimator for each parameter \(\ \mathbf{B}\) and \(\ \uppsi\), i.e.,_  
\[\mathbf{B}^{\overset{\boldsymbol{\alpha}}{\mathbf{B}}(h+1)}=\left(\mathbf{X}^ {\prime}\mathbf{\Omega}^{(h+1)}\mathbf{X}\right)^{-1}\mathbf{X}^{\prime} \mathbf{\Omega}^{(h+1)}\overset{\boldsymbol{\alpha}}{\mathbf{Y}}^{(h)}\]  
\[\overset{\boldsymbol{\alpha}}{\mathbf{Y}}^{(h+1)}=\frac{1}{n}\!\left( \overset{\boldsymbol{\alpha}}{\mathbf{Y}}^{(h+1)}-\overset{\boldsymbol{ \alpha}}{\mathbf{X}\mathbf{B}}\right)^{\prime}\mathbf{\Omega}^{(h+1)}\!\left( \overset{\boldsymbol{\alpha}}{\mathbf{Y}}^{(h+1)}-\overset{\boldsymbol{ \alpha}}{\mathbf{X}\mathbf{B}}^{(h+1)}\right)+\frac{1}{n}\!\sum\limits_{u=1}^{n }\uppsi^{(h)}\]  
**Proof of Theorem 1. ** The development of a SOMRS model with a multivariate t-distribution is based on the probability density function of the multivariate t-distribution. As Equation (5) shows that this model is \(E(\mathbf{Y}_{u})=\mathbf{B}^{\prime}\mathbf{g}(\mathbf{x}_{u})\) for \(u=1,2,\ldots,n\), with \(\mathbf{B}\), and \(\uppsi\) respectively according to that Equation. If a random variable \(\mathbf{Y}= \begin{bmatrix}Y_{1}&Y_{2}&\cdots&Y_{q}\end{bmatrix} ^{\prime}\) has a multivariate t-distribution, then \(n\) random samples \(\mathbf{y}_{u}= \begin{bmatrix}y_{u1}&y_{u2}&\cdots&y_{uq}\end{bmatrix} ^{\prime}\) for \(u=1,2,\ldots,n\) are \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No** & \multicolumn{3}{c}{**Input Variable**} & \multicolumn{3}{c}{**Response**} \\ \hline  & \(X_{1}\) & \(X_{2}\) & \(X_{3}\) & \(\begin{array}{c}Y_{1}\\ (\%)\end{array}\) & \(\begin{array}{c}Y_{2}\\ (N)\end{array}\) & \(\begin{array}{c}Y_{3}\\ (sec)\end{array}\) \\ \hline  
\hline \end{tabular}
\end{table} 
Table 3: Experiment Data Result using CCD. taken that are mutually independent and identical to the \(t_{q}\big{(}\textbf{B}^{{}^{\prime}}\textbf{g}(\textbf{x}),\psi,v\big{)}\) distribution. The joint probability density function of \(\textbf{y}_{u}\) is  
\[f(\textbf{y}_{u})=\frac{\Gamma\Big{(}\frac{v+q}{2}\Big{)}}{(\pi v)^{\frac{q}{2}} \Gamma\big{(}\frac{v}{2}\big{)}|\psi|^{\frac{1}{2}}}\cdot\left(1+\frac{\big{(} \textbf{y}_{u}-\textbf{B}^{{}^{\prime}}\textbf{g}(\textbf{x}_{u})\big{)}^{ \prime}\psi^{-1}\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}}\textbf{g}( \textbf{x}_{u})\big{)}}{v}\right)^{-\frac{v+q}{2}} \tag{6}\]  
for \(u=1,2,\ldots,n\). The likelihood function of the Equation (6) is  
\[ \begin{array}{l}L(\textbf{B},\psi,v)=\prod\limits_{u=1}^{n}f(\textbf{y}_{u} |\textbf{B},\psi,v)=\prod\limits_{u=1}^{n}f(\textbf{y}_{u})\\ \\ =\frac{\big{[}\Gamma\big{(}\frac{v+q}{2}\big{)}\big{]}^{n}}{(\pi v)^{\frac{q}{2 }}\big{[}\Gamma\big{(}\frac{v}{2}\big{)}\big{]}^{n}|\psi|^{\frac{1}{2}}}\cdot \prod\limits_{u=1}^{n}\left(1+\frac{\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{ \prime}}\textbf{g}(\textbf{x}_{u})\big{)}^{\prime}\psi^{-1}\big{(}\textbf{y}_{ u}-\textbf{B}^{{}^{\prime}}\textbf{g}(\textbf{x}_{u})\big{)}}{v}\right)^{- \frac{v+q}{2}}\end{array}  \tag{7}\]  
The log-likelihood function from Equation (7) is  
\[