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Matrix-Variate Hidden Markov Regression Models: Fixed and Random Covariates

  • Special Issue: IFCS 2022
  • Published:
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Abstract

Two families of matrix-variate hidden Markov regression models (MV-HMRMs) are here introduced. The distinction between them relies on the role of the covariates, which can be treated as fixed or random. Parsimony is achieved by using the eigen-decomposition of the components’ covariance matrices. This generates a different number of parsimonious models between the two families: 98 MV-HMRMs with fixed covariates and 9604 MV-HMRMs with random covariates. ECM algorithms are discussed for parameter estimation and, because of the high number of parsimonious models, convenient initialization, and fitting strategies are proposed. Our models are first applied to simulated data, and then to a four-way real dataset concerning the relationship between unemployment and labor force participation in the Italian provinces.

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Code Availability

The code, examples of simulated analyses, and the real dataset used in this manuscript are publicly available at https://github.com/danieletomarchio/Parsimonious-MV-HMRMs-with-fixed-and-random-covariates.

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Acknowledgements

This research also contributes to the PNRR GRInS Project. This research has also been partially supported by MIUR via "The SMILE project: Statistical Modelling and Inference to Live the Environment".

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The authors have no financial interests to report.

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Authors and Affiliations

  1. Department of Economics and Business, University of Catania, Catania, Italy

    Salvatore D. Tomarchio & Antonio Punzo

  2. Department of Law, Economics, Politics, and Modern Languages, LUMSA University, Rome, Italy

    Antonello Maruotti

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  1. Salvatore D. Tomarchio
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  2. Antonio Punzo
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Correspondence to Salvatore D. Tomarchio.

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Appendices

Appendix A

Here, we report the parameters used to generate the simulated datasets of Section 4.1 for the MV-HMRMFCs. We used the following parameters for all three models: \(\pi _1=\pi _2=\pi _3=1/3,\)

$$\begin{aligned} \varvec{\Pi }=\left( \begin{array}{lll} 0.70 &{} 0.20 &{} 0.10 \\ 0.10 &{} 0.80 &{} 0.10 \\ 0.10 &{} 0.30 &{} 0.60 \end{array}\right) , \quad \textbf{B}_1=\left( \begin{array}{llll} 0.00 &{} 1.00 &{} \quad 1.00 &{} -1.00 \\ 1.00 &{} 1.00 &{} -1.00 &{} \quad 1.00 \\ 0.00 &{} 1.00 &{} \quad 1.00 &{} -1.00 \end{array}\right) . \end{aligned}$$

Additionally, for \(\textbf{B}_2\) and \(\textbf{B}_3\), we use the same regression coefficients of \(\textbf{B}_1\), with the exclusion of the first column containing the intercepts, which are replaced by the following vectors \(\varvec{\beta }_{2}=(-7,-8,-7)'\) and \(\varvec{\beta }_{3}=(7,8,7)'\), respectively. Then, for the VVV-VV MV-HMRMFC, we have

$$\begin{aligned} \varvec{\Sigma }_{\mathcal {Y}|1}=\left( \begin{array}{lll} 1.56 &{} \quad 0.22 &{} \quad 0.77 \\ 0.22 &{} \quad 0.41 &{} -0.07 \\ 0.77 &{} -0.07 &{} \quad 1.04 \end{array}\right) , \quad \varvec{\Sigma }_{\mathcal {Y}|2}=\left( \begin{array}{lll} 1.98 &{} \quad 0.39 &{} \quad 0.10 \\ 0.39 &{} \quad 2.07 &{} -0.09 \\ 0.10 &{} -0.09 &{} \quad 2.08 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Sigma }_{\mathcal {Y}|3}=\left( \begin{array}{lll} \quad 3.93 &{} -1.28 &{} 0.10 \\ -1.28 &{} \quad 6.21 &{} 1.29 \\ \quad 0.10 &{} \quad 1.29 &{} 3.09 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {Y}|1}=\left( \begin{array}{lll} \quad 0.78 &{} 0.05 &{} -0.20 \\ \quad 0.05 &{} 1.11 &{} \quad 0.56 \\ -0.20 &{} 0.56 &{} \quad 1.50 \end{array}\right) , \quad \varvec{\Psi }_{\mathcal {Y}|2}=\left( \begin{array}{lll} \quad 2.77 &{} -0.49 &{} \quad 0.10 \\ -0.49 &{} \quad 0.67 &{} -0.03 \\ \quad 0.10 &{} -0.03 &{} \quad 0.72 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {Y}|3}=\left( \begin{array}{lll} \quad 4.01 &{} -1.49 &{} \quad 0.10 \\ -1.49 &{} \quad 0.99 &{} -0.69 \\ \quad 0.10 &{} -0.69 &{} \quad 1.36 \end{array}\right) . \end{aligned}$$

For the VVI-VI MV-HMRMFC, we set \(\varvec{\Sigma }_{\mathcal {Y}|1}=\text {diag}(0.64,2.12,0.25)\), \(\varvec{\Sigma }_{\mathcal {Y}|2}=\text {diag}(1.50,\) 2.16, 2.47), \(\varvec{\Sigma }_{\mathcal {Y}|3}=\text {diag}(7.16,2.51,3.56)\), \(\varvec{\Psi }_{\mathcal {Y}|1}=\text {diag}(0.58,1.92,0.90)\), \(\varvec{\Psi }_{\mathcal {Y}|2}=\text {diag}(0.46,0.73,2.98)\) and \(\varvec{\Psi }_{\mathcal {Y}|3}=\text {diag}(0.49,0.37,5.50)\), where diag\((\cdot )\) identifies a diagonal matrix.

For the EVE-EE MV-HMRMFC, we use

$$\begin{aligned} \varvec{\Sigma }_{\mathcal {Y}|1}=\left( \begin{array}{lll} \quad 1.30 &{} -0.61 &{} -0.10 \\ -0.61 &{} \quad 2.83 &{} -2.24 \\ -0.10 &{} -2.24 &{} \quad 4.48 \end{array}\right) , \quad \varvec{\Sigma }_{\mathcal {Y}|2}=\left( \begin{array}{lll} 1.98 &{} \quad 0.39 &{} \quad 0.29 \\ 0.00 &{} \quad 2.07 &{} -0.09 \\ 0.29 &{} -0.09 &{} \quad 2.08 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Sigma }_{\mathcal {Y}|3}=\left( \begin{array}{lll} \quad 2.68 &{} -0.69 &{} -0.58 \\ -0.69 &{} \quad 2.15 &{} \quad 0.68 \\ -0.58 &{} \quad 0.68 &{} \quad 1.78 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {Y}|1}=\varvec{\Psi }_{\mathcal {Y}|2}=\varvec{\Psi }_{\mathcal {Y}|3}= \left( \begin{array}{lll} \quad 2.77 &{} -0.49 &{} -0.49 \\ -0.49 &{} \quad 0.67 &{} -0.03 \\ -0.49 &{} -0.03 &{} \quad 0.72 \end{array}\right) . \end{aligned}$$

Appendix B

Here, we report the parameters used to generate the simulated datasets of Section 4.1 for the MV-HMRMRCs. We used the following parameters for all three models: \(\pi _1=\pi _2=0.50,\)

$$\begin{aligned} \varvec{\Pi }=\left( \begin{array}{ll} 0.60 &{} 0.40 \\ 0.20 &{} 0.80 \end{array}\right) , \quad \textbf{B}_1=\left( \begin{array}{llll} 2.00 &{} 1.00 &{} \quad 1.00 &{} -1.00 \\ 3.00 &{} 1.00 &{} -1.00 &{} \quad 1.00 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \textbf{M}_1=\left( \begin{array}{lllll} -4 &{} -3 &{} -4 &{} -3 &{} -4 \\ -2 &{} -3 &{} -3 &{} -2 &{} -2 \\ -3 &{} -3 &{} -4 &{} -4 &{} -3 \end{array}\right) . \end{aligned}$$

Similarly to before, for \(\textbf{B}_2\) we use the same regression coefficients of \(\textbf{B}_1\), with the exclusion of the first column containing the intercepts, which is replaced by the following vector \(\varvec{\beta }_{2}=(-12,-13)'\). Additionally, \(\textbf{M}_2\) is obtained by adding a constant \(c=9\) to each element in \(\textbf{M}_1\).

Then, for the VVV-VV-VVV-VV MV-HMRMRC, we have

$$\begin{aligned} \varvec{\Sigma }_{\mathcal {Y}|1}=\left( \begin{array}{ll} 1.70 &{} 0.18 \\ 0.18 &{} 0.61 \end{array}\right) , \quad \varvec{\Sigma }_{\mathcal {Y}|2}=\left( \begin{array}{ll} \quad 8.78 &{} -2.69 \\ -2.69 &{} \quad 2.65 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {Y}|1}=\left( \begin{array}{lllll} \quad 1.08 &{}-0.09 &{}-0.01 &{}-0.08 &{} 0.03 \\ -0.09 &{} \quad 1.00 &{} \quad 0.08 &{}-0.03 &{} 0.20 \\ -0.01 &{} \quad 0.08 &{} \quad 0.82 &{} \quad 0.13 &{} 0.17 \\ -0.08 &{}-0.03 &{} \quad 0.13 &{} \quad 1.17 &{} 0.21 \\ \quad 0.03 &{} \quad 0.20 &{} \quad 0.17 &{} \quad 0.21 &{} 1.12 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {Y}|2}=\left( \begin{array}{lllll} \quad 0.84 &{} \quad 0.11 &{} -0.26 &{} \quad 0.00 &{} -0.44 \\ \quad 0.11 &{} \quad 0.96 &{} \quad 0.20 &{} -0.12 &{} -0.20 \\ -0.26 &{} \quad 0.20 &{} \quad 1.65 &{} -0.21 &{} \quad 0.77 \\ \quad 0.00 &{} -0.12 &{} -0.21 &{} \quad 0.73 &{} -0.05 \\ -0.44 &{} -0.20 &{} \quad 0.77 &{} -0.05 &{} \quad 1.77 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Sigma }_{\mathcal {X}|1}=\left( \begin{array}{lll} \quad 4.30 &{} \quad 0.26 &{} -0.33 \\ \quad 0.26 &{} \quad 4.24 &{} -1.31 \\ -0.33 &{} -1.31 &{} \quad 3.94 \end{array}\right) , \quad \varvec{\Sigma }_{\mathcal {X}|2}=\left( \begin{array}{lll} 1.04 &{} 0.52 &{} 0.18 \\ 0.52 &{} 1.16 &{} 0.52 \\ 0.18 &{} 0.52 &{} 1.31 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {X}|1}=\left( \begin{array}{lllll} \quad 0.76 &{} -0.10 &{} \quad 0.03 &{} \quad 0.02 &{} \quad 0.07 \\ -0.10 &{} \quad 1.16 &{} \quad 0.15 &{} -0.21 &{} \quad 0.15 \\ \quad 0.03 &{} \quad 0.15 &{} \quad 1.09 &{} -0.39 &{} \quad 0.22 \\ \quad 0.02 &{} -0.21 &{} -0.39 &{} \quad 1.19 &{} -0.09 \\ \quad 0.07 &{} \quad 0.15 &{} \quad 0.22 &{} -0.09 &{} \quad 1.10 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {X}|2}=\left( \begin{array}{lllll} \quad 1.59 &{} \quad 0.21 &{} \quad 0.77 &{} -0.25 &{} -0.14 \\ \quad 0.21 &{} \quad 0.65 &{} \quad 0.10 &{} -0.11 &{} -0.23 \\ \quad 0.77 &{} \quad 0.10 &{} \quad 1.30 &{} -0.05 &{} \quad 0.35 \\ -0.25 &{} -0.11 &{} -0.05 &{} \quad 0.77 &{} \quad 0.47 \\ -0.14 &{} -0.23 &{} \quad 0.35 &{} \quad 0.47 &{} \quad 2.01 \end{array}\right) . \end{aligned}$$

For the VVI-VI-VVI-VI MV-HMRMRC, we set \(\varvec{\Sigma }_{\mathcal {Y}|1}=\text {diag}(1.73,0.58)\), \(\varvec{\Sigma }_{\mathcal {Y}|2}=\text {diag}(1.63,9.80)\), \(\varvec{\Psi }_{\mathcal {Y}|1}=\text {diag}(0.74,1.48,0.74,1.11,1.11)\), \(\varvec{\Psi }_{\mathcal {Y}|2}=\text {diag}(2.64,1.32,0.66,\) 0.66, 0.66), \(\varvec{\Sigma }_{\mathcal {X}|1}=\text {diag}(5.55,2.77,4.16)\), \(\varvec{\Sigma }_{\mathcal {X}|2}=\text {diag}(1.00,2.00,0.50)\), \(\varvec{\Psi }_{\mathcal {X}|1}=\text {diag}(1.06,1.77,1.06,0.71,0.71)\) and \(\varvec{\Psi }_{\mathcal {X}|2}=\text {diag}(0.57,0.57,2.30,2.30,0.57)\).

For the EVE-EE-EEV-EV MV-HMRMRC, we use

$$\begin{aligned} \varvec{\Sigma }_{\mathcal {Y}|1}=\left( \begin{array}{ll} 1.70 &{} 0.18 \\ 0.18 &{} 0.61 \end{array}\right) , \quad \varvec{\Sigma }_{\mathcal {Y}|2}=\left( \begin{array}{ll} \quad 0.46 &{} -0.32 \\ -0.32 &{} \quad 2.40 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {Y}|1}=\varvec{\Psi }_{\mathcal {Y}|2}=\left( \begin{array}{lllll} \quad 1.08 &{} -0.09 &{} -0.01 &{} -0.08 &{} 0.03 \\ -0.09 &{} \quad 1.00 &{} \quad 0.08 &{} -0.03 &{} 0.20 \\ -0.01 &{} \quad 0.08 &{} \quad 0.82 &{} \quad 0.13 &{} 0.17 \\ -0.08 &{} -0.03 &{} \quad 0.13 &{} \quad 1.17 &{} 0.21 \\ \quad 0.03 &{} \quad 0.20 &{} \quad 0.17 &{} \quad 0.21 &{} 1.12 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Sigma }_{\mathcal {X}|1}=\left( \begin{array}{lll} \quad 2.21 &{} 0.64 &{} -0.18 \\ \quad 0.64 &{} 5.59 &{} \quad 2.07 \\ -0.18 &{} 2.07 &{} \quad 6.20 \end{array}\right) , \quad \varvec{\Sigma }_{\mathcal {X}|2}=\left( \begin{array}{lll} 4.14 &{} 2.06 &{} 0.71 \\ 2.06 &{} 4.62 &{} 2.07 \\ 0.71 &{} 2.07 &{} 5.24 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {X}|1}=\left( \begin{array}{lllll} \quad 0.76 &{} -0.10 &{} \quad 0.03 &{} \quad 0.02 &{} \quad 0.07 \\ -0.10 &{} \quad 1.16 &{} \quad 0.15 &{} -0.21 &{} \quad 0.15 \\ \quad 0.03 &{} \quad 0.15 &{} \quad 1.09 &{} -0.39 &{} \quad 0.22 \\ \quad 0.02 &{} -0.21 &{} -0.39 &{} \quad 1.19 &{} -0.09 \\ \quad 0.07 &{} \quad 0.15 &{} \quad 0.22 &{} -0.09 &{} \quad 1.10\end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {X}|2}=\left( \begin{array}{lllll} \quad 1.11 &{} \quad 0.21 &{} -0.25 &{} -0.10 &{} \quad 0.02 \\ \quad 0.21 &{} \quad 0.84 &{} -0.22 &{} -0.10 &{} \quad 0.08 \\ -0.25 &{} -0.22 &{} \quad 1.29 &{} \quad 0.12 &{} -0.32 \\ -0.10 &{} -0.10 &{} \quad 0.12 &{} \quad 1.12 &{} -0.13 \\ \quad 0.02 &{} \quad 0.08 &{} -0.32 &{} -0.13 &{} \quad 0.95 \end{array}\right) . \end{aligned}$$

Appendix C

Here, we report the remaining parameters used to generate the simulated datasets of Section 4.3. In particular, we have \(\textbf{M}_2=\textbf{M}_1\) in Scenario A and the following parameters in common between the two scenarios

$$\begin{aligned} \varvec{\Sigma }_{\mathcal {Y}|1}=\varvec{\Sigma }_{\mathcal {Y}|2}=\left( \begin{array}{ll} 2.56 &{} 0.56 \\ 0.56 &{} 1.69 \end{array}\right) , \quad \varvec{\Sigma }_{\mathcal {X}|1}=\varvec{\Sigma }_{\mathcal {X}|2}=\left( \begin{array}{lll} \quad 1.47 &{} -0.30 &{} -0.50 \\ -0.30 &{} \quad 2.23 &{} -0.60 \\ -0.50 &{} -0.60 &{} \quad 2.90 \end{array}\right) , \end{aligned}$$
$$\begin{aligned} \varvec{\Psi }_{\mathcal {Y}|1}=\varvec{\Psi }_{\mathcal {Y}|2}=\varvec{\Psi }_{\mathcal {X}|1}=\varvec{\Psi }_{\mathcal {X}|2}\left( \begin{array}{lllll} \quad 0.75 &{} -0.53 &{} -0.52 &{} 0.03 &{} -0.22 \\ -0.53 &{} \quad 2.90 &{} -0.93 &{} 0.76 &{} \quad 0.39 \\ -0.52 &{} -0.93 &{} \quad 5.06 &{} 0.22 &{} \quad 0.26 \\ \quad 0.03 &{} \quad 0.76 &{} \quad 0.22 &{} 4.72 &{} \quad 1.49 \\ -0.22 &{} \quad 0.39 &{} \quad 0.26 &{} 1.49 &{} \quad 4.17 \end{array}\right) . \end{aligned}$$

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Tomarchio, S.D., Punzo, A. & Maruotti, A. Matrix-Variate Hidden Markov Regression Models: Fixed and Random Covariates. J Classif (2023). https://doi.org/10.1007/s00357-023-09438-y

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