A General SEM Framework for Integrating Moderation and Mediation: The Constrained Approach

Abstract

Modeling the combination of latent moderating and mediating effects is a significant issue in the social and behavioral sciences. Chen and Cheng (Structural Equation Modeling: A Multidisciplinary Journal 21: 94–101, 2014) generalized Jöreskog and Yang’s (Advanced structural equation modeling: Issues and techniques (pp. 57–88). Mahwah, NJ: Lawrence Erlbaum, 1996) constrained approach to allow for the concurrent modeling of moderation and mediation within the context of SEM. Unfortunately, due to restrictions related to Chen and Cheng’s partitioning scheme, their framework cannot completely conceptualize and interpret moderation of indirect effects in a mediated model. In the current study, the Chen and Cheng (abbreviated below as C & C) framework is extended to accommodate situations in which any two pathways that constitute a particular indirect effect in a mediated model can be differentially or collectively moderated by the moderator variable(s). By preserving the inherent advantage of the C & C framework, i.e., the matrix partitioning technique, while at the same time further generalizing its applicability, it is expected that the current framework enhances the potential usefulness of the constrained approach as well as the entire class of the product indicator approaches.

Appendices

Appendix A: Expansions of ${\boldsymbol{\upeta}}_{{\mathbf{F}}^{\ast }}$, ${\boldsymbol{\upeta}}_{{\mathbf{S}}^{\ast }}$, ${\mathbf{y}}_{{\mathbf{F}}^{\ast }},$ and ${\mathbf{y}}_{{\mathbf{S}}^{\ast }}$

Before ${\boldsymbol{\upeta}}_{{\mathbf{F}}^{\ast }}$, ${\boldsymbol{\upeta}}_{{\mathbf{S}}^{\ast }}$, ${\mathbf{y}}_{{\mathbf{F}}^{\ast }},$ and ${\mathbf{y}}_{{\mathbf{S}}^{\ast }}$ are discussed in the subsequent paragraph, it is necessary to gain familiarity with the notation of four basic types of matrices. The $n \times n$ identity matrix will be denoted as Ι _n and the $mn \times mn$ commutation matrix will be indicated as K _mn for $m\ \ne\ n$ and K _n for $m = n$ (see definition 3.1 of Magnus and Neudecker 1979). The $n\left(n + 1\right)/2 \times {n}^2$ elimination matrix and ${n}^2 \times n\left(n + 1\right)/2$ duplication matrix will be denoted as L _n and D _n , respectively (see definitions 3.1a and 3.2a of Magnus and Neudecker 1980).

The expansions of ${\boldsymbol{\upeta}}_{{\mathbf{F}}^{\ast }}$, ${\boldsymbol{\upeta}}_{{\mathbf{S}}^{\ast }}$, ${\mathbf{y}}_{{\mathbf{F}}^{\ast }},$ and ${\mathbf{y}}_{{\mathbf{S}}^{\ast }}$ can be obtained with the aid of several theorems and properties of the Kronecker product, vech and vec operators shown in Magnus and Neudecker (1980, 1988). The resulting forms of these expansions are expressed as below.

$$ {\boldsymbol{\upeta}}_{{\mathbf{F}}^{\ast }} = {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}\mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\upalpha}}_{\mathbf{F}}{\boldsymbol{\upalpha}}_{\mathbf{F}}^{\mathrm{T}}\right) + {\boldsymbol{\upzeta}}_{{\mathbf{F}}^{\ast }}, $$

$$ {\boldsymbol{\upeta}}_{{\mathbf{S}}^{\ast }} = {\mathbf{W}}_{\mathbf{2}}{\mathbf{S}}_{\mathbf{1}}\mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\upalpha}}_{\mathbf{S}}{\boldsymbol{\upalpha}}_{\mathbf{F}}^{\mathrm{T}}\right) + {\mathbf{W}}_{\mathbf{2}}{\mathbf{S}}_{\mathbf{1}}{\mathbf{S}}_{\mathbf{2}}{\boldsymbol{\upeta}}_{\mathbf{F}} + {\mathbf{W}}_{\mathbf{2}}{\mathbf{S}}_{\mathbf{1}}{\mathbf{S}}_{\mathbf{3}}{\boldsymbol{\upeta}}_{\mathbf{F}}^{\ast } + {\boldsymbol{\upzeta}}_{{\mathbf{S}}^{\ast }}, $$

$$ {\mathbf{y}}_{{\mathbf{F}}^{\ast }} = {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p\mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\upnu}}_{\mathbf{F}}{\boldsymbol{\upnu}}_{\mathbf{F}}^{\mathrm{T}}\right) + {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p{\mathbf{E}}_{\mathbf{1}}{\boldsymbol{\upeta}}_{\mathbf{F}} + {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p{\mathbf{E}}_{\mathbf{2}}{\mathbf{D}}_f{\mathbf{W}}_{\mathbf{1}}^{\mathrm{T}}{\boldsymbol{\upeta}}_{{\mathbf{F}}^{\ast }} + {\boldsymbol{\upvarepsilon}}_{{\mathbf{F}}^{\ast }}, $$

$$ {\mathbf{y}}_{{\mathbf{S}}^{\ast }} = {\mathbf{W}}_{\mathbf{4}}\mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\upnu}}_{\mathbf{S}}{\boldsymbol{\upnu}}_{\mathbf{F}}^{\mathrm{T}}\right) + {\mathbf{W}}_{\mathbf{4}}{\mathbf{A}}_{\mathbf{1}}{\boldsymbol{\upeta}}_{\mathbf{F}} + {\mathbf{W}}_{\mathbf{4}}{\mathbf{A}}_{\mathbf{2}}{\boldsymbol{\upeta}}_{\mathbf{S}} + {\mathbf{W}}_{\mathbf{4}}{\mathbf{A}}_{\mathbf{3}}{\mathbf{W}}_{\mathbf{2}}^{\mathrm{T}}{\boldsymbol{\upeta}}_{{\mathbf{S}}^{\ast }} + {\boldsymbol{\upvarepsilon}}_{{\mathbf{S}}^{\ast }}, $$

where ${\boldsymbol{\upzeta}}_{{\mathbf{F}}^{\ast }}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}\left({\mathbf{F}}_{\mathbf{2}}{\boldsymbol{\upzeta}}_{\mathbf{F}} + {\boldsymbol{\upzeta}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{F}}\right)$,

$$ {\boldsymbol{\upzeta}}_{{\mathbf{S}}^{\ast }}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{2}}{\mathbf{S}}_{\mathbf{1}}\left[{\mathbf{S}}_{\mathbf{4}}{\boldsymbol{\upzeta}}_{\mathbf{F}} + {\mathbf{S}}_{\mathbf{5}}{\boldsymbol{\upzeta}}_{\mathbf{S}} + {\mathbf{S}}_{\mathbf{6}}\left({\boldsymbol{\upzeta}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{F}}\right) + {\mathbf{S}}_{\mathbf{7}}\left({\boldsymbol{\upzeta}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{F}}\right) + {\boldsymbol{\upzeta}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{S}}\right], $$

$$ {\boldsymbol{\upvarepsilon}}_{{\mathbf{F}}^{\ast }}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p\left[{\mathbf{E}}_{\mathbf{3}}{\boldsymbol{\upvarepsilon}}_{\mathbf{F}} + {\mathbf{E}}_{\mathbf{4}}\left({\boldsymbol{\upvarepsilon}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{F}}\right) + {\mathbf{E}}_{\mathbf{5}}\left({\boldsymbol{\upzeta}}_{\mathbf{F}} \otimes {\boldsymbol{\upvarepsilon}}_{\mathbf{F}}\right) + {\boldsymbol{\upvarepsilon}}_{\mathbf{F}} \otimes {\boldsymbol{\upvarepsilon}}_{\mathbf{F}}\right], $$

$$ \begin{aligned}{\boldsymbol{\upvarepsilon}}_{{\mathbf{S}}^{\ast }}\ &\underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{4}}\big[{\mathbf{A}}_{\mathbf{4}}{\boldsymbol{\upvarepsilon}}_{\mathbf{F}} + {\mathbf{A}}_{\mathbf{5}}{\boldsymbol{\upvarepsilon}}_{\mathbf{S}} + {\mathbf{A}}_{\mathbf{6}}\left({\boldsymbol{\upvarepsilon}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{F}}\right) + {\mathbf{A}}_{\mathbf{7}}\left({\boldsymbol{\upzeta}}_{\mathbf{F}} \otimes {\boldsymbol{\upvarepsilon}}_{\mathbf{S}}\right) + {\mathbf{A}}_{\mathbf{8}}\left({\boldsymbol{\upvarepsilon}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{S}}\right)\\ &\quad+ {\mathbf{A}}_{\mathbf{9}}\left({\boldsymbol{\upvarepsilon}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{F}}\right) + {\boldsymbol{\upvarepsilon}}_{\mathbf{F}} \otimes {\boldsymbol{\upvarepsilon}}_{\mathbf{S}}\big]\end{aligned} $$

(here “$\underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}$” is the symbol for “defined as”) in which ${\mathbf{F}}_{\mathbf{1}} = [({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}}) \otimes ({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}}{)]}^{\hbox{-} 1}$, ${\mathbf{F}}_{\mathbf{2}} = {\mathbf{I}}_f \otimes {\boldsymbol{\upalpha}}_{\mathbf{F}} + {\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\mathbf{I}}_f$, ${\mathbf{S}}_{\mathbf{1}} = [({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{FF}}) \otimes ({\mathbf{I}}_s - {\mathbf{B}}_{\mathbf{S}\mathbf{S}}{)]}^{\hbox{-} 1}$, ${\mathbf{S}}_{\mathbf{2}} = {\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\mathbf{B}}_{\mathbf{S}\mathbf{F}}$, ${\mathbf{S}}_{\mathbf{3}} = {\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}$, ${\mathbf{S}}_{\mathbf{4}} = {\mathbf{I}}_f \otimes [{\boldsymbol{\upalpha}}_{\mathbf{S}} + {\mathbf{B}}_{\mathbf{S}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1}{\boldsymbol{\upalpha}}_{\mathbf{F}} + {\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}({\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\boldsymbol{\upalpha}}_{\mathbf{F}})]$, ${\mathbf{S}}_{\mathbf{5}} = {\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\mathbf{I}}_s$, ${\mathbf{S}}_{\mathbf{6}} = {\mathbf{I}}_f \otimes [{\mathbf{B}}_{\mathbf{S}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1} + {\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}{\mathbf{F}}_{\mathbf{2}}]$, ${\mathbf{S}}_{\mathbf{7}} = {\mathbf{I}}_f \otimes ({\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}})$, ${\mathbf{E}}_{\mathbf{1}} = {\boldsymbol{\Lambda}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\upnu}}_{\mathbf{F}} + {\boldsymbol{\upnu}}_{\mathbf{F}} \otimes {\boldsymbol{\Lambda}}_{\mathbf{F}\mathbf{F}}$, ${\mathbf{E}}_{\mathbf{2}} = {\boldsymbol{\Lambda}}_{\mathbf{FF}} \otimes {\boldsymbol{\Lambda}}_{\mathbf{FF}}$, ${\mathbf{E}}_{\mathbf{3}} = {\mathbf{I}}_p \otimes [{\boldsymbol{\upnu}}_{\mathbf{F}} + {\boldsymbol{\Lambda}}_{\mathbf{F}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1}{\boldsymbol{\upalpha}}_{\mathbf{F}}] + [{\boldsymbol{\upnu}}_{\mathbf{F}} + {\boldsymbol{\Lambda}}_{\mathbf{F}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1}{\boldsymbol{\upalpha}}_{\mathbf{F}}] \otimes {\mathbf{I}}_p$, ${\mathbf{E}}_{\mathbf{4}} = {\mathbf{I}}_p \otimes ({\boldsymbol{\Lambda}}_{\mathbf{FF}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{FF}})}^{\hbox{-} 1})$, ${\mathbf{E}}_{\mathbf{5}} = ({\boldsymbol{\Lambda}}_{\mathbf{FF}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{FF}})}^{\hbox{-} 1}) \otimes {\mathbf{I}}_p$, ${\mathbf{A}}_{\mathbf{1}} = {\boldsymbol{\Lambda}}_{\mathbf{FF}} \otimes {\boldsymbol{\upnu}}_{\mathbf{S}}$, ${\mathbf{A}}_{\mathbf{2}} = {\boldsymbol{\upnu}}_{\mathbf{F}} \otimes {\boldsymbol{\Lambda}}_{\mathbf{SS}}$, ${\mathbf{A}}_{\mathbf{3}} = {\boldsymbol{\Lambda}}_{\mathbf{FF}} \otimes {\boldsymbol{\Lambda}}_{\mathbf{SS}}$, ${\mathbf{A}}_{\mathbf{4}} = {\mathbf{I}}_p \otimes [{\boldsymbol{\upnu}}_{\mathbf{S}} + {\boldsymbol{\Lambda}}_{\mathbf{S}\mathbf{S}}{({\mathbf{I}}_s - {\mathbf{B}}_{\mathbf{S}\mathbf{S}})}^{\hbox{-} 1}({\boldsymbol{\upalpha}}_{\mathbf{S}} + {\mathbf{B}}_{\mathbf{S}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1}{\boldsymbol{\upalpha}}_{\mathbf{F}} + {\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}({\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\boldsymbol{\upalpha}}_{\mathbf{F}}))]$, ${\mathbf{A}}_{\mathbf{5}} = [{\boldsymbol{\upnu}}_{\mathbf{F}} + {\boldsymbol{\Lambda}}_{\mathbf{F}\mathbf{F}}{({\mathbf{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1}{\boldsymbol{\upalpha}}_{\mathbf{F}}] \otimes {\mathbf{I}}_q$, ${\mathbf{A}}_{\mathbf{6}} = {\mathbf{I}}_p \otimes [{\boldsymbol{\Lambda}}_{\mathbf{S}\mathbf{S}}{({\mathbf{I}}_s - {\mathbf{B}}_{\mathbf{S}\mathbf{S}})}^{\hbox{-} 1}({\mathbf{B}}_{\mathbf{S}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1} + {\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}{\mathbf{F}}_{\mathbf{2}})]$, ${\mathbf{A}}_{\mathbf{7}} = ({\boldsymbol{\Lambda}}_{\mathbf{FF}}{\left({\mathbf{I}}_f - {\mathbf{B}}_{\mathbf{FF}}\right)}^{\hbox{-} 1}) \otimes {\mathbf{I}}_q$, ${\mathbf{A}}_{\mathbf{8}} = {\mathbf{I}}_p \otimes ({\boldsymbol{\Lambda}}_{\mathbf{SS}}{({\mathbf{I}}_s - {\mathbf{B}}_{\mathbf{SS}})}^{\hbox{-} 1})$ and ${\mathbf{A}}_{\mathbf{9}} = {\mathbf{I}}_p \otimes ({\boldsymbol{\Lambda}}_{\mathbf{S}\mathbf{S}}{({\mathbf{I}}_s - {\mathbf{B}}_{\mathbf{S}\mathbf{S}})}^{\hbox{-} 1}{\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}})$.

Here, ${\boldsymbol{\upzeta}}_{{\mathbf{F}}^{\ast }}$ and ${\boldsymbol{\upzeta}}_{{\mathbf{S}}^{\ast }}$ are vectors of disturbance terms of ${\boldsymbol{\upeta}}_{{\mathbf{F}}^{\ast }}$ and ${\boldsymbol{\upeta}}_{{\mathbf{S}}^{\ast }},$ while ${\boldsymbol{\upvarepsilon}}_{{\mathbf{F}}^{\ast }}$ and ${\boldsymbol{\upvarepsilon}}_{{\mathbf{S}}^{\ast }}$ are vectors of measurement errors of ${\mathbf{y}}_{{\mathbf{F}}^{\ast }}$ and ${\mathbf{y}}_{{\mathbf{S}}^{\ast }}$. Meanwhile, F ₁ , F ₂ , S ₁ to S ₇ , E ₁ to E ₅ , and A ₁ to A ₉ are all constant matrices.

Appendix B: Partitioned Matrices Ψ and Θ

The disturbance covariance matrix Ψ is partitioned into a $5 \times 5$ array of submatrices as expressed below:

$$ \boldsymbol{\Psi} = \left[\begin{array}{ccccc} {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}} & & & & \\ {} {\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}} & {\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{S}} & & & \\ {} {\boldsymbol{\Psi}}_{\mathbf{TF}} & {\boldsymbol{\Psi}}_{\mathbf{TS}} & {\boldsymbol{\Psi}}_{\mathbf{TT}} & & \\ {} {\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast}\mathbf{F}} & {\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast}\mathbf{S}} & {\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast}\mathbf{T}} & {\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast }{\mathbf{F}}^{\ast }} & \\ {} {\boldsymbol{\Psi}}_{{\mathbf{S}}^{\ast}\mathbf{F}} & {\boldsymbol{\Psi}}_{{\mathbf{S}}^{\ast}\mathbf{S}} & {\boldsymbol{\Psi}}_{{\mathbf{S}}^{\ast}\mathbf{T}} & {\boldsymbol{\Psi}}_{{\mathbf{S}}^{\ast }{\mathbf{F}}^{\ast }} & {\boldsymbol{\Psi}}_{{\mathbf{S}}^{\ast }{\mathbf{S}}^{\ast }} \end{array}\right], $$

where ${{ {\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast}\mathbf{F}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}{\mathbf{F}}_{\mathbf{2}}{\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}}}$, ${\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast}\mathbf{S}}\underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}{\mathbf{F}}_{\mathbf{2}}{\left({\boldsymbol{\Psi}}_{\mathbf{SF}}\right)}^{\mathrm{T}}$, ${\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast}\mathbf{T}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}{\mathbf{F}}_{\mathbf{2}} {\left({\boldsymbol{\Psi}}_{\mathbf{TF}}\right)}^{\mathrm{T}}$, ${\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast }{\mathbf{F}}^{\ast }}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}\left[{\mathbf{F}}_{\mathbf{2}}{\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}{\mathbf{F}}_{\mathbf{2}}^{\mathrm{T}} + \left({\mathbf{I}}_{f^2} + {\mathbf{K}}_{ff}\right)\left({\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right)\right]{\mathbf{F}}_{\mathbf{1}}^{\mathrm{T}}{\mathbf{L}}_f^{\mathrm{T}}{\mathbf{W}}_{\mathbf{1}}^{\mathrm{T}}$,

$$ \begin{array}{l}{\boldsymbol{\Psi}}_{{\mathbf{S}}^{\ast}\mathbf{F}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{2}}{\mathbf{S}}_{\mathbf{1}}\big[{\mathbf{S}}_{\mathbf{4}}{\boldsymbol{\Psi}}_{\mathbf{FF}} + {\mathbf{S}}_{\mathbf{5}}{\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\\ {} + {\mathbf{S}}_{\mathbf{7}}{\Delta}_{f^3 \times f}\left[{\mathbf{K}}_{f{f}^3}\left(\mathrm{v}\mathrm{e}\mathrm{c}\left(\left({\mathbf{I}}_{f^2} + {\mathbf{K}}_{ff}\right)\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right.\right.\\ {} + \left.\mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right) \otimes \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\big]\big],\end{array} $$

$$ \begin{array}{l}{\boldsymbol{\Psi}}_{{\mathbf{S}}^{\ast}\mathbf{S}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{2}}{\mathbf{S}}_{\mathbf{1}}\big[{\mathbf{S}}_{\mathbf{4}}{\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}} + {\mathbf{S}}_{\mathbf{5}}{\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{S}}\\ {} + {\mathbf{S}}_{\mathbf{7}}{\varDelta}_{f^3 \times s}\left[{\mathbf{K}}_{s{f}^3}\left(\mathrm{v}\mathrm{e}\mathrm{c}\left(\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right) + {\mathbf{K}}_{fs}\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right.\right.\\ {} \left.+ \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right) \otimes \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)\right)\big]\big],\end{array} $$

$$ \begin{array}{l}{\boldsymbol{\Psi}}_{{\mathbf{S}}^{\ast}\mathbf{T}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{2}}{\mathbf{S}}_{\mathbf{1}}\big[{\mathbf{S}}_{\mathbf{4}}{\left({\boldsymbol{\Psi}}_{\mathbf{TF}}\right)}^{\mathrm{T}} + {\mathbf{S}}_{\mathbf{5}}{\left({\boldsymbol{\Psi}}_{\mathbf{TS}}\right)}^{\mathrm{T}}\\ {} + {\mathbf{S}}_{\mathbf{7}}{\varDelta}_{f^3 \times t}\left[{\mathbf{K}}_{t{f}^3}\left(\mathrm{v}\mathrm{e}\mathrm{c}\left(\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{TF}}\right) + {\mathbf{K}}_{ft}\left({\boldsymbol{\Psi}}_{\mathbf{TF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right.\right.\\ {} \left.+ \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right) \otimes \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{TF}}\right)\right)\big]\big],\end{array} $$

$${\fontsize{9}{11}\selectfont{ \begin{array}{l}{\boldsymbol{\Psi}}_{{\mathbf{S}}^{\ast }{\mathbf{F}}^{\ast }}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{2}}{\mathbf{S}}_{\mathbf{1}}\big[{\mathbf{S}}_{\mathbf{4}}{\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}{\mathbf{F}}_{\mathbf{2}}^{\mathrm{T}} + {\mathbf{S}}_{\mathbf{5}}{\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}{\mathbf{F}}_{\mathbf{2}}^{\mathrm{T}}\\ {} + {\mathbf{S}}_{\mathbf{7}}{\Delta}_{f^3 \times f}\left[{\mathbf{K}}_{f{f}^3}\left(\mathrm{v}\mathrm{e}\mathrm{c}\left(\left({\mathbf{I}}_{f^2} + {\mathbf{K}}_{ff}\right)\left({\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right)\right) + \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right) \otimes \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right)\right)\right]{\mathbf{F}}_{\mathbf{2}}^{\mathrm{T}}\\ {} + {\mathbf{S}}_{\mathbf{6}}\left({\mathbf{I}}_{f^2} + {\mathbf{K}}_{ff}\right)\left({\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right) + {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}} + {\mathbf{K}}_{fs}\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right)\big]{\mathbf{F}}_{\mathbf{1}}^{\mathrm{T}}{\mathbf{L}}_f^{\mathrm{T}}{\mathbf{W}}_{\mathbf{1}}^{\mathrm{T}},\end{array} }} $$

$$ {\fontsize{8.5}{10.5}\selectfont{ \begin{array}{l}{\boldsymbol{\Psi}}_{{\mathbf{S}}^{\ast }{\mathbf{S}}^{\ast }}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{2}}{\mathbf{S}}_{\mathbf{1}}\big[{\mathbf{S}}_{\mathbf{4}}{\boldsymbol{\Psi}}_{\mathbf{FF}}{\mathbf{S}}_{\mathbf{4}}^{\mathrm{T}} + {\mathbf{S}}_{\mathbf{5}}{\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{S}}{\mathbf{S}}_{\mathbf{5}}^{\mathrm{T}} + {\mathbf{S}}_{\mathbf{6}}\left({\mathbf{I}}_{f^2} + {\mathbf{K}}_{ff}\right)\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right){\mathbf{S}}_{\mathbf{6}}^{\mathrm{T}}\\[10pt] {} + {\mathbf{S}}_{\mathbf{7}}\big[\left({\mathbf{I}}_{f^3} + {\mathbf{K}}_{f{f}^2} + {\mathbf{K}}_{f^2f} + {\mathbf{I}}_f \otimes {\mathbf{K}}_{ff} + {\mathbf{K}}_{ff} \otimes {\mathbf{I}}_f + \left({\mathbf{I}}_f \otimes {\mathbf{K}}_{ff}\right){\mathbf{K}}_{f{f}^2}\right)\\[10pt] {} \cdot \left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right) + \left({\mathbf{I}}_{f^3} + {\mathbf{K}}_{f{f}^2} + {\mathbf{K}}_{f^2f}\right)\left(\left(\mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\mathrm{v}\mathrm{e}\mathrm{c}{\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right)}^{\mathrm{T}}\right) \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\\[10pt] {} \cdot \left({\mathbf{I}}_{f^3} + {\mathbf{K}}_{f{f}^2} + {\mathbf{K}}_{f^2f}\right)\big]{\mathbf{S}}_{\mathbf{7}}^{\mathrm{T}} + \left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{S}}\right) + {\mathbf{K}}_{fs}\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}} \otimes {\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}\right)\\[10pt] {} + {\mathbf{S}}_{\mathbf{5}}{\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}{\mathbf{S}}_{\mathbf{4}}^{\mathrm{T}} + {\mathbf{S}}_{\mathbf{4}}{\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}{\mathbf{S}}_{\mathbf{5}}^{\mathrm{T}}\\[10pt] {} + {\mathbf{S}}_{\mathbf{7}}{\Delta}_{f^3 \times f}\left[{\mathbf{K}}_{f{f}^3}\left(\mathrm{v}\mathrm{e}\mathrm{c}\left(\left({\mathbf{I}}_{f^2} + {\mathbf{K}}_{ff}\right)\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right) + \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right) \otimes \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right]{\mathbf{S}}_{\mathbf{4}}^{\mathrm{T}}\\[10pt] {} + {\mathbf{S}}_{\mathbf{4}}{\Delta}_{f \times {f}^3}\left[{\mathbf{K}}_{f^3f}\left(\mathrm{v}\mathrm{e}\mathrm{c}\left(\left({\mathbf{I}}_{f^2} + {\mathbf{K}}_{ff}\right)\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right) + \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right) \otimes \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right]{\mathbf{S}}_{\mathbf{7}}^{\mathrm{T}}\\[10pt] {} + {\mathbf{S}}_{\mathbf{7}}{\Delta}_{f^3 \times s}\left[{\mathbf{K}}_{s{f}^3}\left(\mathrm{v}\mathrm{e}\mathrm{c}\left(\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right) + {\mathbf{K}}_{fs}\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right) + \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right) \otimes \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)\right)\right]{\mathbf{S}}_{\mathbf{5}}^{\mathrm{T}}\\[10pt] {} + {\mathbf{S}}_{\mathbf{5}}{\Delta}_{s \times {f}^3}\left[{\mathbf{K}}_{f^3s}\left(\mathrm{v}\mathrm{e}\mathrm{c}\left[\left({\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right) + {\mathbf{K}}_{ff}\left({\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right]\right.\right.\\[10pt] {} + \mathrm{v}\mathrm{e}\mathrm{c}\left({\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}\right) \otimes \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\big)\big]{\mathbf{S}}_{\mathbf{7}}^{\mathrm{T}}\\[10pt] {} + \left[\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right) + {\mathbf{K}}_{fs}\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right]{\mathbf{S}}_{\mathbf{6}}^{\mathrm{T}}\\[10pt] {} + {\mathbf{S}}_{\mathbf{6}}\left[\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}\right) + {\mathbf{K}}_{ff}\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}\right)\right]\big]{\mathbf{S}}_{\mathbf{1}}^{\mathrm{T}}{\mathbf{W}}_{\mathbf{2}}^{\mathrm{T}}\end{array} }} $$

(here the symbol “${\Delta}_{n \times m}$” is used to transform a $nm \times 1$ vector into an $n \times m$ matrix).

The measurement error covariance matrix Θ is partitioned into a $5 \times 5$ array of submatrices as expressed below:

$$ \boldsymbol{\Theta} = \left[\begin{array}{ccccc} {\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}} & & & & \\ {} {\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}} & {\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{S}} & & & \\ {} {\boldsymbol{\Theta}}_{\mathbf{TF}} & {\boldsymbol{\Theta}}_{\mathbf{TS}} & {\boldsymbol{\Theta}}_{\mathbf{TT}} & & \\ {} {\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast}\mathbf{F}} & {\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast}\mathbf{S}} & {\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast}\mathbf{T}} & {\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast }{\mathbf{F}}^{\ast }} & \\ {} {\boldsymbol{\Theta}}_{{\mathbf{S}}^{\ast}\mathbf{F}} & {\boldsymbol{\Theta}}_{{\mathbf{S}}^{\ast}\mathbf{S}} & {\boldsymbol{\Theta}}_{{\mathbf{S}}^{\ast}\mathbf{T}} & {\boldsymbol{\Theta}}_{{\mathbf{S}}^{\ast }{\mathbf{F}}^{\ast }} & {\boldsymbol{\Theta}}_{{\mathbf{S}}^{\ast }{\mathbf{S}}^{\ast }} \end{array}\right], $$

where ${\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast}\mathbf{F}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p{\mathbf{E}}_{\mathbf{3}}{\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}}$, ${\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast}\mathbf{S}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p{\mathbf{E}}_{\mathbf{3}}{\left({\boldsymbol{\Theta}}_{\mathbf{SF}}\right)}^{\mathrm{T}}$, ${\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast}\mathbf{T}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p{\mathbf{E}}_{\mathbf{3}}{\left({\boldsymbol{\Theta}}_{\mathbf{TF}}\right)}^{\mathrm{T}}$,

$$ \begin{aligned}{\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast }{\mathbf{F}}^{\ast }}\ &\underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p\big[{\mathbf{E}}_{\mathbf{3}}{\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}}{\mathbf{E}}_{\mathbf{3}}^{\mathrm{T}} + {\mathbf{E}}_{\mathbf{4}}\left({\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right){\mathbf{E}}_{\mathbf{4}}^{\mathrm{T}} + {\mathbf{E}}_{\mathbf{5}}\left({\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}}\right){\mathbf{E}}_{\mathbf{5}}^{\mathrm{T}}\\ &+ \left({\mathbf{I}}_{p^2} + {\mathbf{K}}_{pp}\right)\left({\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}}\right) + {\mathbf{E}}_{\mathbf{5}}{\mathbf{K}}_{fp}\left({\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right){\mathbf{E}}_{\mathbf{4}}^{\mathrm{T}} \\ &+ {\mathbf{E}}_{\mathbf{4}}{\mathbf{K}}_{pf}\left({\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}}\right){\mathbf{E}}_{\mathbf{5}}^{\mathrm{T}}\big]{\mathbf{L}}_p^{\mathrm{T}}{\mathbf{W}}_{\mathbf{3}}^{\mathrm{T}},\end{aligned} $$

$$ {\fontsize{10}{12}\selectfont{{\boldsymbol{\Theta}}_{{\mathbf{S}}^{\ast}\mathbf{F}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{4}}\left[{\mathbf{A}}_{\mathbf{4}}{\boldsymbol{\Theta}}_{\mathbf{FF}} + {\mathbf{A}}_{\mathbf{5}}{\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}} + {\mathbf{A}}_{\mathbf{9}}{\Delta}_{\left(p{f}^2\right) \times p}\left[{\mathbf{K}}_{p\left(p{f}^2\right)}\mathrm{v}\mathrm{e}\mathrm{c}\left({\mathbf{K}}_{fp}\left({\boldsymbol{\Theta}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right]\right],}} $$

$$ {\fontsize{9}{11}\selectfont{{\boldsymbol{\Theta}}_{{\mathbf{S}}^{\ast}\mathbf{S}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{4}}\left[{\mathbf{A}}_{\mathbf{4}}{\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{5}}{\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{S}} + {\mathbf{A}}_{\mathbf{9}}{\Delta}_{\left(p{f}^2\right) \times q}\left[{\mathbf{K}}_{q\left(p{f}^2\right)}\mathrm{v}\mathrm{e}\mathrm{c}\left({\mathbf{K}}_{fq}\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right]\right],}} $$

$$ {\fontsize{9}{11}\selectfont{{\boldsymbol{\Theta}}_{{\mathbf{S}}^{\ast}\mathbf{T}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{4}}\left[{\mathbf{A}}_{\mathbf{4}}{\left({\boldsymbol{\Theta}}_{\mathbf{TF}}\right)}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{5}}{\left({\boldsymbol{\Theta}}_{\mathbf{TS}}\right)}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{9}}{\Delta}_{\left(p{f}^2\right) \times r}\left[{\mathbf{K}}_{r\left(p{f}^2\right)}\mathrm{v}\mathrm{e}\mathrm{c}\left({\mathbf{K}}_{fr}\left({\boldsymbol{\Theta}}_{\mathbf{TF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right]\right],}} $$

$$ {\fontsize{10}{12}\selectfont{\begin{aligned}&{\boldsymbol{\Theta}}_{{\mathbf{S}}^{\ast }{\mathbf{F}}^{\ast }}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{4}}\big[{\mathbf{A}}_{\mathbf{4}}{\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}}{\mathbf{E}}_{\mathbf{3}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{5}}{\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}}{\mathbf{E}}_{\mathbf{3}}^{\mathrm{T}} \\[-10pt] &\quad+ {\mathbf{A}}_{\mathbf{9}}{\Delta}_{\left(p{f}^2\right) \times p}\left[{\mathbf{K}}_{p\left(p{f}^2\right)}\mathrm{v}\mathrm{e}\mathrm{c}\left({\mathbf{K}}_{fp}\left({\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right)\right)\right]{\mathbf{E}}_{\mathbf{3}}^{\mathrm{T}}\\ &\quad+ {\mathbf{A}}_{\mathbf{6}}\left({\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right){\mathbf{E}}_{\mathbf{4}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{7}}{\mathbf{K}}_{fq}\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right){\mathbf{E}}_{\mathbf{4}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{8}}\left({\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right){\mathbf{E}}_{\mathbf{4}}^{\mathrm{T}}\\ &\quad+ {\mathbf{A}}_{\mathbf{6}}{\mathbf{K}}_{pf}\left({\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}}\right){\mathbf{E}}_{\mathbf{5}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{7}}\left({\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}}\right){\mathbf{E}}_{\mathbf{5}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{8}}{\mathbf{K}}_{ps}\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}} \otimes {\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}}\right){\mathbf{E}}_{\mathbf{5}}^{\mathrm{T}}\\ &\quad+ \left({\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}}\right) + {\mathbf{K}}_{pq}\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}} \otimes {\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}}\right)\big]{\mathbf{L}}_p^{\mathrm{T}}{\mathbf{W}}_{\mathbf{3}}^{\mathrm{T}},\end{aligned}}} $$

$$ \begin{aligned}{\boldsymbol{\Theta}}_{{\mathbf{S}}^{\ast }{\mathbf{S}}^{\ast }} &= {\mathbf{W}}_{\mathbf{4}}\big[{\mathbf{A}}_{\mathbf{4}}{\boldsymbol{\Theta}}_{\mathbf{FF}}{\mathbf{A}}_{\mathbf{4}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{5}}{\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{S}}{\mathbf{A}}_{\mathbf{5}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{6}}\left({\boldsymbol{\Theta}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right){\mathbf{A}}_{\mathbf{6}}^{\mathrm{T}} \\ &\quad+ {\mathbf{A}}_{\mathbf{7}}\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{S}}\right){\mathbf{A}}_{\mathbf{7}}^{\mathrm{T}}+ {\mathbf{A}}_{\mathbf{8}}\left({\boldsymbol{\Theta}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{S}}\right){\mathbf{A}}_{\mathbf{8}}^{\mathrm{T}} \\ &\quad+ {\mathbf{A}}_{\mathbf{9}}\big[\left({\mathbf{I}}_{p{f}^2} + {\mathbf{I}}_p \otimes {\mathbf{K}}_{ff}\right)\left({\boldsymbol{\Theta}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\\ &\quad+ {\mathbf{K}}_{p\left({f}^2\right)}\left(\left(\mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\mathrm{v}\mathrm{e}\mathrm{c}{\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right)}^{\mathrm{T}}\right) \otimes {\boldsymbol{\Theta}}_{\mathbf{FF}}\right){\mathbf{K}}_{\left({f}^2\right)p}\big]{\mathbf{A}}_{\mathbf{9}}^{\mathrm{T}}\\ &\quad+ \left({\boldsymbol{\Theta}}_{\mathbf{FF}} \otimes {\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{S}}\right) + {\mathbf{K}}_{pq}\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}} \otimes {\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}\right)\\ &\quad+ {\mathbf{A}}_{\mathbf{5}}{\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}}{\mathbf{A}}_{\mathbf{4}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{9}}{\Delta}_{\left(p{f}^2\right) \times p}\left[{\mathbf{K}}_{p\left(p{f}^2\right)}\mathrm{v}\mathrm{e}\mathrm{c}\left({\mathbf{K}}_{fp}\left({\boldsymbol{\Theta}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right]{\mathbf{A}}_{\mathbf{4}}^{\mathrm{T}}\\ &\quad+ {\mathbf{A}}_{\mathbf{4}}{\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}{\mathbf{A}}_{\mathbf{5}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{9}}{\Delta}_{\left(p{f}^2\right) \times q}\left[{\mathbf{K}}_{q\left(p{f}^2\right)}\mathrm{v}\mathrm{e}\mathrm{c}\left({\mathbf{K}}_{fq}\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right]{\mathbf{A}}_{\mathbf{5}}^{\mathrm{T}}\\ &\quad+ {\mathbf{A}}_{\mathbf{7}}{\mathbf{K}}_{fq}\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{FF}}\right){\mathbf{A}}_{\mathbf{6}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{8}}\left({\boldsymbol{\Theta}}_{\mathbf{FF}} \otimes {\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right){\mathbf{A}}_{\mathbf{6}}^{\mathrm{T}}\\ &\quad+ {\mathbf{A}}_{\mathbf{6}}{\mathbf{K}}_{pf}\left({\boldsymbol{\Psi}}_{\mathbf{FF}} \otimes {\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}\right){\mathbf{A}}_{\mathbf{7}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{8}}{\mathbf{K}}_{ps}\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}} \otimes {\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}\right){\mathbf{A}}_{\mathbf{7}}^{\mathrm{T}}\\ &\quad+ {\mathbf{A}}_{\mathbf{6}}\left({\boldsymbol{\Theta}}_{\mathbf{FF}} \otimes {\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}\right){\mathbf{A}}_{\mathbf{8}}^{\mathrm{T}} + {\mathbf{A}}_{\mathbf{7}}{\mathbf{K}}_{fq}\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}} \otimes {\left({\boldsymbol{\Psi}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}\right){\mathbf{A}}_{\mathbf{8}}^{\mathrm{T}}\\ &\quad+ {\mathbf{A}}_{\mathbf{4}}{\varDelta}_{p \times \left(p{f}^2\right)}\left[{\mathbf{K}}_{\left(p{f}^2\right)p}\left(\mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Theta}}_{\mathbf{FF}}\right) \otimes \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right]{\mathbf{A}}_{\mathbf{9}}^{\mathrm{T}}\\ &\quad+ {\mathbf{A}}_{\mathbf{5}}{\Delta}_{q \times \left(p{f}^2\right)}\left[{\mathbf{K}}_{\left(p{f}^2\right)q}\left(\mathrm{v}\mathrm{e}\mathrm{c}\left({\left({\boldsymbol{\Theta}}_{\mathbf{S}\mathbf{F}}\right)}^{\mathrm{T}}\right) \otimes \mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{\Psi}}_{\mathbf{FF}}\right)\right)\right]{\mathbf{A}}_{\mathbf{9}}^{\mathrm{T}}\big]{\mathbf{W}}_{\mathbf{4}}^{\mathrm{T}}.\end{aligned} $$