A. Derivation of full conditionals of the improved BMTME model under the matrix normal distribution

Full conditional distribution for vec(β)

$$\begin{aligned} &\text{\(vec}(\mathbf{b}\_{1})\text{)} \text{LLS} \\ &\propto \exp\left\{-\frac{1}{2} \left[\text{vec}(\mathbf{Y}) - \text{vec}(\mathbf{X}\boldsymbol{\mathfrak{\mathcal{B}}}) - (\mathbf{I}\_{L} \otimes \mathbf{Z}\_{1})\text{vec}(\mathbf{b}\_{1}) \\ &- \text{vec}(\mathbf{Z}\_{2}\mathbf{b}\_{2})\right]^{T} \left[\mathbf{R}\_{\varepsilon}^{-1} \otimes \mathbf{I}\_{n}\right] \left[\text{vec}(\mathbf{Y}) - \text{vec}(\mathbf{X}\boldsymbol{\mathfrak{\mathcal{B}}}) - (\mathbf{I}\_{L} \otimes \mathbf{Z}\_{1})\text{vec}(\mathbf{b}\_{1}) - \text{vec}(\mathbf{Z}\_{2}\mathbf{b}\_{2})\right] \\ &- \frac{1}{2} \left[\text{vec}(\mathbf{b}\_{1})^{\top} \left[\mathbf{E}\_{t}^{-1} \otimes \mathbf{G}\_{\mathbf{g}}^{-1}\right]\right] \text{vec}(\mathbf{b}\_{1}) \right\} \\ &\propto \exp\left\{-\frac{1}{2} \left[\text{vec}(\mathbf{b}\_{1}) - \text{vec}(\tilde{\mathbf{b}}\_{1})\right]^{T} \tilde{\mathbf{E}}\_{\mathbf{b}\_{1}}^{-1} \left[\text{vec}(\mathbf{b}\_{1}) - \text{vec}(\tilde{\mathbf{b}}\_{1})\right] \right\} \\ &\propto N \{\text{vec}(\tilde{\mathbf{b}}\_{1}), \tilde{\mathbf{E}}\_{\mathbf{b}\_{1}}\} \end{aligned} \tag{A2}$$

$$\begin{split} \text{where } \tilde{\boldsymbol{\Sigma}}\_{\mathsf{b}\_{1}} &= \left(\boldsymbol{\Sigma}\_{\mathsf{t}}^{-1} \otimes \boldsymbol{\mathsf{G}}\_{\mathcal{S}}^{-1} + \boldsymbol{\mathsf{R}}\_{\boldsymbol{\varepsilon}}^{-1} \otimes \boldsymbol{\mathsf{Z}}\_{\mathbf{1}}^{T} \boldsymbol{\mathsf{Z}}\_{1}\right)^{-1} \text{ and } \boldsymbol{\mathsf{vec}}\big(\tilde{\boldsymbol{\theta}}\_{1}\big) &= \tilde{\boldsymbol{\Sigma}}\_{\mathsf{b}\_{1}} \big(\boldsymbol{\mathsf{R}}\_{\boldsymbol{\varepsilon}}^{-1} \otimes \boldsymbol{\mathsf{Z}}\_{1}^{T}\big) \big[\mathsf{vec}(\boldsymbol{\mathsf{Y}}) - \mathsf{vec}(\boldsymbol{\mathsf{X}}\boldsymbol{\mathsf{\theta}}) - \boldsymbol{\mathsf{Z}}\_{\boldsymbol{\mathsf{Y}}} \\ \boldsymbol{\mathsf{vec}}(\boldsymbol{\mathsf{Z}}\_{2}\boldsymbol{\mathsf{b}}\_{2})\big]. \end{split}$$

## Full conditional for Σ<sup>E</sup>

$$\begin{split} P(\Sigma\_{\mathbb{E}} | ELS) &\propto P(b\_{2} | \Sigma\_{\mathbb{E}}) P(\Sigma\_{\mathbb{E}}) \\ &\propto |\Sigma\_{\mathbb{E}}|^{-\frac{\mu}{2}} |\Sigma\_{\mathbb{E}} \otimes \mathcal{G}\_{\mathcal{S}}|^{-\frac{\mathsf{c}}{2}} \exp\left\{-\frac{1}{2} tr \left[b\_{2}^{\*T} \left(\Sigma\_{t}^{-1} \otimes \mathcal{G}\_{\mathcal{S}}^{-1}\right) b\_{2}^{\*} \Sigma\_{\mathbb{E}}^{-1}\right] \right\} |\mathcal{S}\_{\mathbb{E}}|^{\frac{\mathsf{c}\tau + l - 1}{2}} \\ &\times |\Sigma\_{\mathbb{E}}|^{-\frac{\mathsf{c}\tau + l}{2}} \exp\left\{-\frac{1}{2} tr \left(\mathcal{S}\_{\mathbb{E}} \Sigma\_{\mathbb{E}}^{-1}\right)\right\} \\ &\propto |\Sigma\_{\mathbb{E}}|^{-\frac{\mathsf{c}\tau + l + |L|}{2}} \exp\left\{-\frac{1}{2} tr \left[\left(b\_{2}^{\*T} \left(\Sigma\_{t}^{-1} \otimes \mathcal{G}\_{\mathcal{S}}^{-1}\right) b\_{2}^{\*} + \mathcal{S}\_{\mathbb{E}}\right)\right] \Sigma\_{\mathbb{E}}^{-1}\right\} |\mathcal{S}\_{\mathbb{E}}|^{\frac{\mathsf{c}\tau + l - 1}{2}} \\ &\propto IV(\upsilon\_{\mathbb{E}} + \left|L + I - 1, b\_{2}^{\*T} \left(\Sigma\_{t}^{-1} \otimes \mathcal{G}\_{\mathcal{S}}^{-1}\right) b\_{2}^{\*} + \mathcal{S}\_{\mathbb{E}}\right) \end{split} \tag{A5}$$

References

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Unmixing. Boca Raton, FL: Chapman & Hall/CRC; 2003

Version 1.0.4 http://CRAN.R-project.org/package=BGLR. 2014

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### Full conditional for R<sup>e</sup>

P Re ð Þ jELSE ∝ P Yjβ, b1, b2, Re � �Pð Þ Re <sup>∝</sup>j j Re �<sup>n</sup> <sup>2</sup> exp � <sup>1</sup> 2 tr Y � Xβ � Z1b<sup>1</sup> � Z2b<sup>2</sup> � �<sup>T</sup> I<sup>n</sup> Y � Xβ � Z1b<sup>1</sup> � Z2b<sup>2</sup> � �R�<sup>1</sup> e � � h i � <sup>S</sup><sup>e</sup> j j<sup>υ</sup>eþL�<sup>1</sup> <sup>2</sup> j j Re �υeþ<sup>L</sup> <sup>2</sup> exp � <sup>1</sup> 2 tr SeR�<sup>1</sup> e � � � � <sup>∝</sup>j j Re �υeþnþL�1þ<sup>1</sup> <sup>2</sup> exp � <sup>1</sup> 2 tr Y � Xβ � Z1b<sup>1</sup> � Z2b<sup>2</sup> � �<sup>T</sup> <sup>Y</sup> � <sup>X</sup><sup>β</sup> � <sup>Z</sup>1b<sup>1</sup> � <sup>Z</sup>2b<sup>2</sup> � � <sup>þ</sup> <sup>S</sup><sup>e</sup> h iR�<sup>1</sup> e � � ∝IW υ<sup>e</sup> þ n þ L � 1; Y � Xβ � Z1b<sup>1</sup> � Z2b<sup>2</sup> �� � � � � þ S<sup>e</sup> � � (A6)

where k(Y � Xβ � Z1b<sup>1</sup> � Z2b2)k = (Y � Xβ � Z1b<sup>1</sup> � Z2b2) T (Y � Xβ � Z1b<sup>1</sup> � Z2b2) .

## Author details

Osval A. Montesinos-López<sup>1</sup> \*, Abelardo Montesinos-López<sup>2</sup> , José Cricelio Montesinos-López<sup>3</sup> , José Crossa4 , Francisco Javier Luna-Vázquez<sup>1</sup> and Josafhat Salinas-Ruiz<sup>5</sup>

\*Address all correspondence to: oamontes1@ucol.mx

1 Facultad de Telemática, Universidad de Colima, Colima, México

2 Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e Ingenierías (CUCEI), Universidad de Guadalajara, Guadalajara, Jalisco, México

3 Centro de Investigación en Matemáticas (CIMAT), Guanajuato, México

