Transformation between type-2 TSK fuzzy systems and an uncertain Gaussian mixture model

Abstract

In this paper, an interval extension of the Gaussian mixture model called uncertain Gaussian mixture model (UGMM) is proposed and its transformation into the additive type-2 TSK fuzzy systems is presented. The conditions under which a UGMM becomes a corresponding type-2 TSK fuzzy system are derived theoretically. Furthermore, the mathematical equivalence between the conditional mean of a UGMM and the defuzzified output of a type-2 TSK fuzzy system is proved. Our results provide a new perspective for type-2 TSK fuzzy systems, i.e., interpreting them from a probabilistic viewpoint. Thus, instead of directly estimating the parameters of the fuzzy rules in a type-2 TSK fuzzy system, we can first estimate the parameters of the corresponding UGMM using any popular density estimation algorithm like the expectation maximization (EM) algorithm. Our experimental results clearly indicate that a type-2 fuzzy system trained in such a new way has higher approximation accuracy and stronger robustness than current type-2 fuzzy systems.

Appendices

Appendix 1

Equation 18 can be derived using the following derivations.

$$ \begin{aligned} \int\limits_{R} {y\tilde{G}_{c} (x,y)dy} & = \int\limits_{R} {y{\frac{{p_{c} }}{{2\pi \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\exp \left\{ { - {\frac{1}{2}}\left[ {x - \tilde{\mu }_{{cx}} ,y - \tilde{\mu }_{{cy}} } \right] \times \left[ {\begin{array}{*{20}c} {\tilde{\sigma }_{{cxx}} } & {\tilde{\sigma }_{{cxy}} } \\ {\tilde{\sigma }_{{cyx}} } & {\tilde{\sigma }_{{cyy}} } \\ \end{array} } \right]^{{ - 1}} \times \left[ {\begin{array}{*{20}c} {x - \tilde{\mu }_{{cx}} } \\ {y - \tilde{\mu }_{{cy}} } \\ \end{array} } \right]} \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{2\pi \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\int\limits_{R} {y\exp \left\{ { - {\frac{1}{{2\left| {\tilde{\Upsigma }_{c} } \right|}}}\left[ {x - \tilde{\mu }_{{cx}} ,y - \tilde{\mu }_{{cy}} } \right] \times \left[ {\begin{array}{*{20}c} {\tilde{\sigma }_{{cyy}} } & { - \tilde{\sigma }_{{cyx}} } \\ { - \tilde{\sigma }_{{cxy}} } & {\tilde{\sigma }_{{cxx}} } \\ \end{array} } \right] \times \left[ {\begin{array}{*{20}c} {x - \tilde{\mu }_{{cx}} } \\ {y - \tilde{\mu }_{{cy}} } \\ \end{array} } \right]} \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{2\pi \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\int\limits_{R} {y\exp \left\{ { - {\frac{1}{{2\left| {\tilde{\Upsigma }_{c} } \right|}}}\left( {\left( {x - \tilde{\mu }_{{cx}} } \right)^{2} \tilde{\sigma }_{{cyy}} - 2\left( {x - \tilde{\mu }_{{cx}} } \right)\left( {y - \tilde{\mu }_{{cy}} } \right)\tilde{\sigma }_{{cxy}} + \left( {y - \tilde{\mu }_{{cy}} } \right)^{2} \tilde{\sigma }_{{cxx}} } \right)} \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{2\pi \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\exp \left\{ { - {\frac{{\left( {x - \tilde{\mu }_{{cx}} } \right)^{2} \tilde{\sigma }_{{cyy}} }}{{2\left| {\tilde{\Upsigma }_{c} } \right|}}}} \right\}\int\limits_{R} {y\exp \left\{ { - {\frac{1}{{2\left| {\tilde{\Upsigma }_{c} } \right|}}}\left( {\left( {x - \tilde{\mu }_{{cy}} } \right)^{2} \tilde{\sigma }_{{cxx}} - 2\left( {x - \tilde{\mu }_{{cx}} } \right)\left( {y - \tilde{\mu }_{{cy}} } \right)\tilde{\sigma }_{{cxy}} } \right)} \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{2\pi \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\exp \left\{ { - {\frac{{\left( {x - \tilde{\mu }_{{cx}} } \right)^{2} \tilde{\sigma }_{{cyy}} }}{{2\left| {\tilde{\Upsigma }_{c} } \right|}}}} \right\} \\ & \quad \times \exp \left\{ {{\frac{{\left( {x - \tilde{\mu }_{{cx}} } \right)^{2} \tilde{\sigma }_{{cxy}}^{2} }}{{2\left| {\tilde{\Upsigma }_{c} } \right|\tilde{\sigma }_{{cxx}} }}}} \right\} \times \int\limits_{R} {y\exp \left\{ { - {\frac{1}{{2\left| {\tilde{\Upsigma }_{c} } \right|}}}\left( {\left( {y - \tilde{\mu }_{{cy}} } \right)\sqrt {\tilde{\sigma }_{{cxx}} } - \left( {x - \tilde{\mu }_{{cx}} } \right){\frac{{\tilde{\sigma }_{{cxy}} }}{{\sqrt {\tilde{\sigma }_{{cxx}} } }}}} \right)^{2} } \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{2\pi \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\exp \left\{ { - {\frac{{\left( {x - \tilde{\mu }_{{cx}} } \right)^{2} }}{{2\left| {\tilde{\Upsigma }_{c} } \right|}}}\left( {\tilde{\sigma }_{{cyy}} - {\frac{{\tilde{\sigma }_{{cxy}}^{2} }}{{\tilde{\sigma }_{{cxx}} }}}} \right)} \right\} \times \int\limits_{R} {y\exp \left\{ { - {\frac{1}{{\left( {2\left| {\tilde{\Upsigma }_{c} } \right|} \right)/\tilde{\sigma }_{{cxx}} }}}\left( {y - \left( {\tilde{\mu }_{{cy}} + \left( {x - \tilde{\mu }_{{cx}} } \right){\frac{{\tilde{\sigma }_{{cxy}} }}{{\tilde{\sigma }_{{cxx}} }}}} \right)} \right)^{2} } \right\}{\text{d}}y} \\ & = p_{c} {\frac{{\sqrt {2\pi {\frac{{\left| {\tilde{\Upsigma }_{c} } \right|}}{{\tilde{\sigma }_{{cxx}} }}}} }}{{2\pi \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\exp \left( { - {\frac{{\left( {x - \tilde{\mu }_{{cx}} } \right)^{2} }}{{2{\frac{{\left| {\tilde{\Upsigma }_{c} } \right|}}{{\left( {\tilde{\sigma }_{{cyy}} - {\frac{{\tilde{\sigma }_{{cxy}}^{2} }}{{\tilde{\sigma }_{{cxx}} }}}} \right)}}}}}}} \right)\int\limits_{R} {yN^{1} \left( {y;\tilde{\mu }_{{cy}} + \left( {x - \tilde{\mu }_{{cx}} } \right){\frac{{\tilde{\sigma }_{{cxy}} }}{{\tilde{\sigma }_{{cxx}} }}},{\frac{{\left| {\tilde{\Upsigma }_{c} } \right|}}{{\tilde{\sigma }_{{cxx}} }}}} \right)dy} \\ & = {\frac{{p_{c} }}{{\sqrt {2\pi \tilde{\sigma }_{{cxx}} } }}}\exp \left( { - {\frac{{\left( {x - \tilde{\mu }_{{cx}} } \right)^{2} }}{{2\tilde{\sigma }_{{cxx}} }}}} \right)\left( {\tilde{\mu }_{{cy}} + \left( {x - \tilde{\mu }_{{cx}} } \right){\frac{{\tilde{\sigma }_{{cxy}} }}{{\sigma _{{cxx}} }}}} \right) \\ & = p_{c} N^{1} \left( {x;\tilde{\mu }_{{cx}} ,\tilde{\sigma }_{{cxx}} } \right)\left( {\tilde{\mu }_{{cy}} + \left( {x - \tilde{\mu }_{{cx}} } \right){\frac{{\tilde{\sigma }_{{cxy}} }}{{\tilde{\sigma }_{{cxx}} }}}} \right) \\ \end{aligned} $$

Appendix 2

Equation 24 can be derived using the following derivations.

Let

$$ \begin{array}{*{20}c} {\tilde{\Upsigma }_{c} = \left( {\begin{array}{*{20}c} {\left\{ {\tilde{\sigma }_{{cij}} } \right\}_{{J \times J}} } & {\left\{ {\tilde{\sigma }_{{cj(J + 1)}} } \right\}_{{J \times 1}} } \\ {\left\{ {\tilde{\sigma }_{{c(J + 1)j}} } \right\}_{{1 \times J}} } & {\tilde{\sigma }_{{c(J + 1)(J + 1)}} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\tilde{\sigma }_{{c\vec{x}\vec{x}}} } & {\tilde{\sigma }_{{c\vec{x}y}} } \\ {\tilde{\sigma }_{{cy\vec{x}}} } & {\tilde{\sigma }_{{cyy}} } \\ \end{array} } \right),} \hfill \\ {\tilde{\Upsigma }_{c}^{{ - 1}} = \left( {\begin{array}{*{20}c} {\left\{ {\tilde{\sigma }^{{cij}} } \right\}_{{J \times J}} } & {\left\{ {\tilde{\sigma }^{{cj(J + 1)}} } \right\}_{{J \times 1}} } \\ {\left\{ {\tilde{\sigma }^{{c(J + 1)j}} } \right\}_{{1 \times J}} } & {\tilde{\sigma }^{{c(J + 1)(J + 1)}} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\tilde{\sigma }^{{c\vec{x}\vec{x}}} } & {\tilde{\sigma }^{{c\vec{x}y}} } \\ {\tilde{\sigma }^{{cy\vec{x}}} } & {\tilde{\sigma }^{{cyy}} } \\ \end{array} } \right).} \hfill \\ \end{array} $$

Then,

$$ \begin{aligned} p_{c} \int\limits_{R} {yN^{{J + 1}} \left( {\left( {\begin{array}{*{20}c} {\vec{x}} \\ y \\ \end{array} } \right);\left( {\begin{array}{*{20}c} {\tilde{\vec{\mu }}_{{c\vec{x}}} } \\ {\tilde{\mu }_{{cy}} } \\ \end{array} } \right),\tilde{\Upsigma }_{c} } \right){\text{d}}y} & = \int\limits_{R} {y{\frac{{p_{c} }}{{\left( {2\pi } \right)^{{{\frac{{J + 1}}{2}}}} \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\exp \left\{ { - {\frac{1}{2}}\left[ {\vec{x} - \tilde{\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\mu } }_{{c\vec{x}}} ,y - \tilde{\mu }_{{cy}} } \right] \times \left[ {\begin{array}{*{20}c} {\tilde{\sigma }_{{c\vec{x}\vec{x}}} } & {\tilde{\sigma }_{{c\vec{x}y}} } \\ {\tilde{\sigma }_{{cy\vec{x}}} } & {\tilde{\sigma }_{{cyy}} } \\ \end{array} } \right]^{{ - 1}} \times \left[ {\begin{array}{*{20}c} {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \\ {y - \tilde{\mu }_{{cy}} } \\ \end{array} } \right]} \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{\left( {2\pi } \right)^{{(J + 1)/2}} \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\int\limits_{R} {y\exp \left\{ { - {\frac{1}{2}}\left[ {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} ,y - \tilde{\mu }_{{cy}} } \right] \times \left[ {\begin{array}{*{20}c} {\tilde{\sigma }^{{c\vec{x}\vec{x}}} } & {\tilde{\sigma }^{{c\vec{x}y}} } \\ {\tilde{\sigma }^{{cy\vec{x}}} } & {\tilde{\sigma }^{{cyy}} } \\ \end{array} } \right] \times \left[ {\begin{array}{*{20}c} {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \\ {y - \tilde{\mu }_{{cy}} } \\ \end{array} } \right]} \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{\left( {2\pi } \right)^{{(J + 1)/2}} \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\int\limits_{R} {y\exp \left\{ { - {\frac{1}{2}}\left( {\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}\vec{x}}} \left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right) + \left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} \left( {y - \tilde{\mu }_{{cy}} } \right) + \left( {y - \tilde{\mu }_{{cy}} } \right)\tilde{\sigma }^{{cy\vec{x}}} \left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right) + \left( {y - \tilde{\mu }_{{cy}} } \right)\tilde{\sigma }^{{cyy}} \left( {y - \tilde{\mu }_{{cy}} } \right)} \right)} \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{\left( {2\pi } \right)^{{(J + 1)/2}} \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}}\int\limits_{R} {y\exp \left\{ { - {\frac{1}{2}}\left( {\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}\vec{x}}} \left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right) + 2\left( {y - \tilde{\mu }_{{cy}} } \right)\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} + \left( {y - \tilde{\mu }_{{cy}} } \right)^{2} \tilde{\sigma }^{{cyy}} } \right)} \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{\left( {2\pi } \right)^{{(J + 1)/2}} \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}} \cdot \exp \left( { - {\frac{1}{2}}\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{{\text{T}}} \tilde{\sigma }^{{c\vec{x}\vec{x}}} \left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)} \right) \cdot \int\limits_{R} {y\exp \left\{ { - {\frac{1}{2}}\left( {2\left( {y - \tilde{\mu }_{{cy}} } \right)\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} + \left( {y - \tilde{\mu }_{{cy}} } \right)^{2} \tilde{\sigma }^{{cyy}} } \right)} \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{\left( {2\pi } \right)^{{(J + 1)/2}} \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}} \cdot \exp \left( { - {\frac{1}{2}}\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}\vec{x}}} \left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)} \right) \cdot \exp \left( {{\frac{1}{2}}\left( {{\frac{{(\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} )^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} }}{{\sqrt {\tilde{\sigma }^{{cyy}} } }}}} \right)^{2} } \right) \cdot \int\limits_{R} {y\exp \left\{ { - {\frac{1}{2}}\left( {\left( {y - \tilde{\mu }_{{cy}} } \right)\sqrt {\tilde{\sigma }^{{cyy}} } + {\frac{{\left( {\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} }}{{\sqrt {\tilde{\sigma }^{{cyy}} } }}}} \right)^{2} } \right\}{\text{d}}y} \\ & = {\frac{{p_{c} }}{{\left( {2\pi } \right)^{{(J + 1)/2}} \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}} \cdot \exp \left( { - {\frac{1}{2}}\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}\vec{x}}} \left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)} \right) \cdot \exp \left( {{\frac{1}{2}}\left( {{\frac{{(\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} )^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} }}{{\sqrt {\tilde{\sigma }^{{cyy}} } }}}} \right)^{2} } \right) \cdot \int\limits_{R} {y\exp \left\{ { - {\frac{1}{{2(1/\tilde{\sigma }^{{cyy}} )}}}\left( {y - \left( {\tilde{\mu }_{{cy}} - {\frac{{\left( {\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} }}{{\tilde{\sigma }^{{cyy}} }}}} \right)} \right)^{2} } \right\}{\text{d}}y} \\ & = {\frac{{p_{c} \sqrt {2\pi \left( {1/\tilde{\sigma }^{{cyy}} } \right)} }}{{\left( {2\pi } \right)^{{(J + 1)/2}} \sqrt {\left| {\tilde{\Upsigma }_{c} } \right|} }}} \cdot \exp \left( { - {\frac{1}{2}}\left( {\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}\vec{x}}} \left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right) - {\frac{{((\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} )^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} )^{2} }}{{\tilde{\sigma }^{{cyy}} }}}} \right)} \right) \cdot \int\limits_{R} {yN^{1} \left( {y;\tilde{\mu }_{{cy}} - {\frac{{\left( {\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} }}{{\tilde{\sigma }^{{cyy}} }}},{\frac{1}{{\tilde{\sigma }^{{cyy}} }}}} \right){\text{d}}y} \\ & = {\frac{{p_{c} }}{{\left( {2\pi } \right)^{{J/2}} \sqrt {\tilde{\sigma }^{{cyy}} \left| {\tilde{\Upsigma }_{c} } \right|} }}} \cdot \exp \left( { - {\frac{1}{2}}\left( {\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}\vec{x}}} \left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right) - {\frac{{\left( {\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} } \right)^{2} }}{{\tilde{\sigma }^{{cyy}} }}}} \right)} \right) \cdot \left( {\tilde{\mu }_{{cy}} - {\frac{{\left( {\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} }}{{\tilde{\sigma }^{{cyy}} }}}} \right) \\ & = {\frac{{p_{c} }}{{\left( {2\pi } \right)^{{J/2}} \sqrt {\tilde{\sigma }^{{cyy}} \left| {\tilde{\Upsigma }_{c} } \right|} }}} \cdot \left( {\tilde{\mu }_{{cy}} - {\frac{{\left( {\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{\rm T} \tilde{\sigma }^{{c\vec{x}y}} }}{{\tilde{\sigma }^{{cyy}} }}}} \right) \cdot \exp \left( { - {\frac{1}{2}}\left( {(\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} )^{\rm T} \left( {\tilde{\sigma }^{{c\vec{x}\vec{x}}} - \tilde{\sigma }^{{c\vec{x}y}} \left( {\tilde{\sigma }^{{cyy}} } \right)^{{ - 1}} \tilde{\sigma }^{{cy\vec{x}}} } \right)\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)} \right)} \right) \\ & = {\frac{{p_{c} }}{{\left( {2\pi } \right)^{{J/2}} \sqrt {\tilde{\sigma }^{{cyy}} \left| {\tilde{\Upsigma }_{c} } \right|} }}} \cdot \left( {\tilde{\mu }_{{cy}} - {\frac{{\left( {\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{{\text{T}}} \tilde{\sigma }^{{c\vec{x}y}} }}{{\tilde{\sigma }^{{cyy}} }}}} \right) \cdot \exp \left( { - {\frac{1}{2}}\left( {\left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)^{{\text{T}}} \left( {\tilde{\sigma }^{{c\vec{x}\vec{x}}} } \right)^{{ - 1}} \left( {\vec{x} - \tilde{\vec{\mu }}_{{c\vec{x}}} } \right)} \right)} \right) \\ & = p_{c} N^{J} \left( {\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} ;\tilde{\vec{\mu }}_{{c\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} }} ,\left\{ {\tilde{\sigma }_{{cij}} } \right\}_{{J \times J}} } \right) \cdot \left( {\tilde{\mu }_{{cy}} - {\frac{{\left( {\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} - \tilde{\vec{\mu }}_{{c\overset{\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {x} }} } \right)^{\rm T} \left\{ {\tilde{\sigma }^{{cj(J + 1)}} } \right\}_{{J \times 1}} }}{{\tilde{\sigma }^{{c(J + 1)(J + 1)}} }}}} \right) \\ \end{aligned} $$