Matrix-variate Smooth Transition Models for Temporal Networks

Abstract

In many fields, network analysis is used to investigate complex relationships. The increased availability of temporal network data opens the way to the statistical analysis of the network topology and its dynamics. In addition network data are subject to measurement errors and random fluctuations. This calls for realistic time series models which account for relevant features of the data. In this chapter, we propose a new modeling and inference framework for studying matrix-valued panel data characterized by nonlinear dynamics and heavy tails. We assume a smooth transition model for the dynamics and a matrix-variate t distribution for the error term and show how the model can be used in temporal network analysis. Some properties of the model including the close-form expression for the predictor are given. We adopt a Bayesian approach to inference and design an efficient Markov chain Monte Carlo algorithm for approximating the posterior distribution. We apply the proposed model to a volatility network among European firms and an international oil production network and show its ability to account for structural changes. Our framework is motivated by temporal network data, nevertheless, it is general and can be of interest to all researchers interested in the analysis of matrix-variate time series.

Appendix—Posterior approximation

The Gibbs sampler iterates the following steps:

• 1. Draw $(\nu ,\textbf{W})$ from the joint posterior distribution $P(\nu ,\textbf{W} | \textbf{Y},\textbf{x},B_1,B_2,\Sigma _1,\Sigma _2,\alpha ,\delta )$ with a collapsed-Gibbs step that first samples $\nu \!\sim \! P(\nu |\textbf{Y},B_1,B_2,\Sigma _1,\Sigma _2,\alpha ,\delta )$ and then $\textbf{W} \!\sim \! P(\textbf{W} | \textbf{Y},\textbf{x},B_1,B_2,\Sigma _1,\Sigma _2,\nu ,\alpha ,\delta )$.

• 2. Draw ${\text {vec}}(B_j)$ from the multivariate normal distribution $P(\textrm{vec}\big (B_j\big ) | \textbf{Y},\textbf{x},\textbf{W},\Sigma _1,\Sigma _2,\alpha ,\delta )$.

• 3. Draw $\Sigma _1$ from the Wishart distribution $P(\Sigma _1 | \textbf{W},\nu )$.

• 4. Draw $\Sigma _2$ from the inverse Wishart distribution $P(\Sigma _2 | \textbf{Y},\textbf{x},B_1,B_2,\textbf{W},\alpha ,\delta )$.

• 5. Draw $\alpha$ and $\delta$ from their joint full conditional distribution $P(\alpha ,\delta |\textbf{Y},\textbf{x},B_1,B_2,\textbf{W},\Sigma _2)$ using an adaptive Metropolis–Hastings algorithm.

In the following, we provide the derivation of the full conditional distributions used in the Gibbs sampler and discuss the sampling methods.

Sampling $B_1$. Let $\tilde{Y}_t = Y_t - (1-\phi (x_t; \alpha ,\delta ))B_2$. The coefficient matrix $B_1$ is drawn from the posterior full conditional distribution

$$\begin{aligned}&P(B_1 | \textbf{Y},\textbf{W},\textbf{x},B_2,\Sigma _1,\Sigma _2) \propto P(B_1) P(\textbf{Y},\textbf{W}|\textbf{x},B_1,B_2,\Sigma _1,\Sigma _2,\alpha ,\delta ) \\&\propto \exp \Big ( \!\!-\!\frac{1}{2} {\text {tr}}\big ( \textrm{vec}\big (B_1\big )' (\underline{\Omega }_2 \otimes \underline{\Omega }_1)^{-1} \textrm{vec}\big (B_1\big ) \big ) \\&\quad -\frac{1}{2} \sum _{t=1}^T {\text {tr}}\big ( \textrm{vec}\big (\tilde{Y\big )_t-\phi (x_t; \alpha ,\delta )B_1}' (\Sigma _2 \otimes W_t)^{-1} \textrm{vec}\big (\tilde{Y\big )_t-\phi (x_t; \alpha ,\delta )B_1} \big ) \Big ) \\&\propto \exp \Big ( \!\!-\!\frac{1}{2} {\text {tr}}\big ( \textrm{vec}\big (B_1\big )' \big [ (\underline{\Omega }_2 \otimes \underline{\Omega }_1)^{-1} + \!\sum _{t=1}^T (\Sigma _2 \otimes W_t)^{-1} \phi (x_t; \alpha ,\delta )^2 \big ] \textrm{vec}\big (B_1\big ) \\&\quad -2 \!\sum _{t=1}^T \textrm{vec}\big (\tilde{Y\big )_t}' (\Sigma _2 \otimes W_t)^{-1} \phi (x_t; \alpha ,\delta ) \textrm{vec}\big (B_1\big ) \big ) \Big ) \\&\propto \exp \Big ( \!\!-\!\frac{1}{2} {\text {tr}}\big ( \textrm{vec}\big (B_1\big )' \overline{\Omega }_1^{-1} \textrm{vec}\big (B_1\big ) -2\textrm{vec}\big (\overline{M\big )_1}' \overline{\Omega }_1^{-1} \textrm{vec}\big (B_1\big ) \big ) \Big ), \end{aligned}$$

where

$$\begin{aligned} \overline{\Omega }_1&= \Big [ (\underline{\Omega }_2 \otimes \underline{\Omega }_1)^{-1} + \sum _{t=1}^T (\Sigma _2 \otimes W_t)^{-1} \phi (x_t; \alpha ,\delta )^2 \Big ]^{-1},\\ \textrm{vec}\big (\overline{M\big )_1}&= \overline{\Omega }_1 \sum _{t=1}^T (\Sigma _2 \otimes W_t)^{-1} \phi (x_t; \alpha ,\delta ) \textrm{vec}\big (\tilde{Y\big )_t} \end{aligned}$$

meaning that $\textrm{vec}\big (B_1\big ) \sim \mathcal {N}_{n^2}(\textrm{vec}\big (\overline{M\big )_1},\overline{\Omega }_1)$.

Sampling $B_2$. Let $\bar{Y}_t = Y_t - \phi (x_t; \alpha ,\delta )B_1$. The coefficient matrix $B_2$ is drawn from the posterior full conditional distribution

$$\begin{aligned}&P(B_2 | \textbf{Y},\textbf{W},\textbf{x},B_1,\Sigma _1,\Sigma _2,\alpha ,\delta ) \propto P(B_2) P(\textbf{Y},\textbf{W}|\textbf{x},B_1,B_2,\Sigma _1,\Sigma _2,\alpha ,\delta ) \\&\propto \exp \Big ( \!\!-\!\frac{1}{2} {\text {tr}}\big ( \textrm{vec}\big (B_2\big )' (\underline{\Omega }_2 \otimes \underline{\Omega }_1)^{-1} \textrm{vec}\big (B_2\big ) \big ) \\&\quad -\frac{1}{2} \!\sum _{t=1}^T {\text {tr}}\big ( \textrm{vec}\big (\bar{Y\big )_t-(1-\phi (x_t; \alpha ,\delta ))B_2}' (\Sigma _2 \otimes W_t)^{-1} \textrm{vec}\big (\bar{Y\big )_t-(1-\phi (x_t; \alpha ,\delta ))B_2} \big ) \Big ) \\&\propto \exp \Big ( \!\!-\!\frac{1}{2} {\text {tr}}\big ( \textrm{vec}\big (B_2\big )' \big [ (\underline{\Omega }_2 \otimes \underline{\Omega }_1)^{-1} + \!\sum _{t=1}^T (\Sigma _2 \otimes W_t)^{-1} (1-\phi (x_t; \alpha ,\delta ))^2 \big ] \textrm{vec}\big (B_2\big ) \\&\quad - 2 \!\sum _{t=1}^T \textrm{vec}\big (\bar{Y\big )_t}' (\Sigma _2 \otimes W_t)^{-1} (1-\phi (x_t; \alpha ,\delta )) \textrm{vec}\big (B_2\big ) \big ) \Big ) \\&\propto \exp \Big ( \!\!-\!\frac{1}{2} {\text {tr}}\big ( \textrm{vec}\big (B_2\big )' \overline{\Omega }_2^{-1} \textrm{vec}\big (B_2\big ) -2\textrm{vec}\big (\overline{M\big )_2}' \overline{\Omega }_2^{-1} \textrm{vec}\big (B_2\big ) \big ) \Big ), \end{aligned}$$

where

$$\begin{aligned} \overline{\Omega }_2&= \Big [ (\underline{\Omega }_2 \otimes \underline{\Omega }_1)^{-1} + \sum _{t=1}^T (\Sigma _2 \otimes W_t)^{-1} (1-\phi (x_t; \alpha ,\delta ))^2 \Big ]^{-1},\\ \textrm{vec}\big (\overline{M\big )_2}&= \overline{\Omega }_2 \sum _{t=1}^T (\Sigma _2 \otimes W_t)^{-1} (1-\phi (x_t; \alpha ,\delta )) \textrm{vec}\big (\bar{Y\big )_t} \end{aligned}$$

meaning that $\textrm{vec}\big (B_2\big ) \sim \mathcal {N}_{n^2}(\textrm{vec}\big (\overline{M\big )_2},\overline{\Omega }_2)$.

Sampling $W_t$. The auxiliary covariance matrices $W_t$, $t=1,\ldots ,T$, are drawn from the posterior full conditional distribution

$$\begin{aligned} P(W_t&| \textbf{Y},\textbf{x},B_1,B_2, \Sigma _1, \Sigma _2,\nu ,\alpha ,\delta ) \propto P(\textbf{Y},\textbf{W}|\textbf{x},B_1,B_2,\Sigma _1,\Sigma _2,\nu ,\alpha ,\delta ) \\&\propto {\big |W_t\big |}^{-\frac{n+\nu +n+1 +n-1}{2}} \exp \Big ( \!\!-\!\frac{1}{2} {{\,\textrm{tr}\,}}\big ( (Y_t-M_t) \Sigma _2^{-1} (Y_t-M_t)' W_t^{-1} \big ) -\frac{1}{2} {{\,\textrm{tr}\,}}\big ( \Sigma _1 W_t^{-1}) \big ) \Big ) \\&\propto {\big |W_t\big |}^{-\frac{(\nu +2n-1)+n+1}{2}} \exp \Big ( \!\!-\!\frac{1}{2} {{\,\textrm{tr}\,}}\big ( \Sigma _1 + (Y_t-M_t) \Sigma _2^{-1} (Y_t-M_t)' W_t^{-1} \big ) \Big ) \\&\propto \mathcal{I}\mathcal{W}_n( \overline{W}, \overline{\nu } ) \end{aligned}$$

where

$$\begin{aligned} \overline{W} = \Sigma _1 + (Y_t-M_t) \Sigma _2^{-1} (Y_t-M_t)', \qquad \overline{\nu } = \nu +2n-1. \end{aligned}$$

Sampling $\Sigma _1$. Given a Wishart prior, the posterior full conditional distribution for $\Sigma _1$ is conjugate and obtained as follows:

$$\begin{aligned} P(\Sigma _1 | \textbf{W},\nu )&\propto P(\Sigma _1|\gamma ) P(\textbf{W}|\Sigma _1,\nu ) \\&\propto {\big |\Sigma _1\big |}^{\frac{\underline{\kappa }_1-n-1}{2}} \exp \Big ( \!\!-\!\frac{1}{2} {{\,\textrm{tr}\,}}\big (\underline{\Psi }_1 \Sigma _1\big ) \Big ) \prod _{t=1}^T {\big |\Sigma _1\big |}^{\frac{\nu +n-1}{2}} \exp \Big ( \!\!-\!\frac{1}{2} {{\,\textrm{tr}\,}}\big (\Sigma _1 W_t^{-1}\big ) \Big ) \\&\propto {\big |\Sigma _1\big |}^{\frac{(\underline{\kappa }_1+T(\nu +n-1))-n-1}{2}} \exp \Big ( \!\!-\!\frac{1}{2} {{\,\textrm{tr}\,}}\big (\underline{\Psi }_1 \Sigma _1 + \sum _{t=1}^T [ W_t^{-1} ] \Sigma _1\big ) \Big ) \\&\propto \mathcal {W}_n( \overline{\Psi }_1, \overline{\kappa }_1 ) \end{aligned}$$

where

$$\begin{aligned} \overline{\Psi }_1 = \big ( \underline{\Psi }_1 + \sum _{t=1}^T W_t^{-1} \big )^{-1}, \qquad \overline{\kappa }_1 = \underline{\kappa }_1 +T(\nu +n-1). \end{aligned}$$

Sampling $\Sigma _2$. Given an inverse Wishart prior, the posterior full conditional distribution for $\Sigma _2$ is conjugate. Using the properties of the Kronecker product and of the vectorization and trace operators, we obtain

$$\begin{aligned} P(\Sigma _2&| \textbf{Y},\textbf{x},B_1,B_2, \textbf{W},\alpha ,\delta ) \propto P(\Sigma _2) P(\textbf{Y},\textbf{W}|\textbf{x},B_1,B_2,\Sigma _2,\alpha ,\delta ) \\&\propto {\big |\Sigma _2\big |}^{-\frac{\underline{\kappa }_2+Tn +n+1}{2}} \exp \Big ( \!\!-\!\frac{1}{2} \big ( {{\,\textrm{tr}\,}}\big (\underline{\Psi }_2 \Sigma _2^{-1}\big ) + {{\,\textrm{tr}\,}}\big (\sum _{t=1}^T (Y_t-M_t)' W_t^{-1} (Y_t-M_t) \Sigma _2^{-1}\big ) \big ) \Big ) \\&\propto {\big |\Sigma _2\big |}^{-\frac{\underline{\kappa }_2+Tn +n+1}{2}} \exp \Big ( \!\!-\!\frac{1}{2} {{\,\textrm{tr}\,}}\big (\underline{\Psi }_2 \Sigma _2^{-1} + S_2 \Sigma _2^{-1}\big ) \Big ) \\&\propto \mathcal{I}\mathcal{W}_n(\overline{\Psi }_2, \overline{\kappa }_2) \end{aligned}$$

where we defined $S_2 = \sum _{t=1}^T (Y_t-M_t)' W_t^{-1} (Y_t-M_t)$ and

$$\begin{aligned} \overline{\Psi }_2 = \underline{\Psi }_2 + S_2, \qquad \overline{\kappa }_2 = \underline{\kappa }_2 + T n. \end{aligned}$$

Sampling $\nu$. Combining the prior distribution and the likelihood in Eqs. (19)–(21), one gets

$$\begin{aligned}&P(\nu | \textbf{Y},\textbf{x},B_1,B_2,\Sigma _1,\Sigma _2,\alpha ,\delta ) \propto P(\nu ) P(\textbf{Y} | \textbf{x},B_1,B_2,\Sigma _1,\Sigma _2,\nu ,\alpha ,\delta ) \\&\propto \nu ^{\underline{a}_\nu -1} e^{-\nu /\underline{b}_\nu } \prod _{t=1}^T \frac{\Gamma _n\big ( \frac{\nu +2n-1}{2} \big )}{\Gamma _n\big ( \frac{\nu +n-1}{2} \big )} \big | I_n + \Sigma _1^{-1}(Y_t-M_t)\Sigma _2^{-1}(Y_t-M_t)' \big |^{-\frac{\nu +2n-1}{2}} \mathbb {I}_{(2,+\infty )}(\nu ) \\&\propto \nu ^{\underline{a}_\nu -1} e^{-\nu /\underline{b}_\nu } \bigg [ \frac{\Gamma _n\big ( \frac{\nu +2n-1}{2} \big )}{\Gamma _n\big ( \frac{\nu +n-1}{2} \big )} \bigg ]^T \bigg [ \prod _{t=1}^T \big | I_n + \Sigma _1^{-1}(Y_t-M_t)\Sigma _2^{-1}(Y_t-M_t)' \big | \bigg ]^{-\frac{\nu +2n-1}{2}} \!\! \mathbb {I}_{(2,+\infty )}(\nu ) \\&\propto \mathcal{T}\mathcal{G}a(\overline{a}_\nu , \overline{b}_\nu ; 2,+\infty ) \bigg [ \frac{\Gamma _n\big ( \frac{\nu +2n-1}{2} \big )}{\Gamma _n\big ( \frac{\nu +n-1}{2} \big )} \bigg ]^T \end{aligned}$$

where

$$\begin{aligned} \overline{a}_\nu = \underline{a}_\nu \qquad \overline{b}_\nu = \Big [ \frac{1}{\underline{b}_\nu } +\frac{1}{2}\sum _{t=1}^T \log \Big ( \big | I_n + \Sigma _1^{-1}(Y_t-M_t)\Sigma _2^{-1}(Y_t-M_t)' \big | \Big ) \Big ]^{-1}. \end{aligned}$$

We sample from this distribution using an adaptive RWMH step with truncated lognormal proposal distribution [5, 6].

Sampling $\alpha$ and $\delta$. Combining the prior distribution and the likelihood in Eqs. (20)–(21) one gets

$$\begin{aligned} P(\alpha ,\delta&| \textbf{Y},\textbf{x},B_1,B_2,\textbf{W},\Sigma _2) \propto P(\alpha )P(\delta ) P(\textbf{Y} | \textbf{x},B_1,B_2,\textbf{W},\Sigma _2,\delta ,\alpha ) \\&\propto \alpha ^{\underline{a}_\alpha -1} \exp (-\alpha /\underline{b}_\alpha )\exp (-(\delta -\underline{\mu }_\delta )^2(2\underline{\sigma }_\delta ^2)^{-1}) \\&\quad \cdot \prod _{t=1}^T \exp \Big ( \!\!-\!\frac{1}{2} {\text {tr}}\big ( \Sigma _2^{-1} (Y_t-M_t)' W_t^{-1} (Y_t-M_t) \big ) \Big ) \\&\propto \alpha ^{\underline{a}_\alpha -1} \exp (-\alpha /\underline{b}_\alpha ) \exp (-(\delta -\underline{\mu }_\delta )^2(2\underline{\sigma }_\delta ^2)^{-1})\\&\quad \cdot \exp \Big ( \!\!-\!\frac{1}{2} \sum _{t=1}^T {\text {tr}}\big ( \Sigma _2^{-1} \big ( Y_t-B_1\phi (x_t; \alpha ,\delta )-B_2(1-\phi (x_t; \alpha ,\delta )) \big )'W_t^{-1} \\&\quad \cdot \big ( Y_t-B_1\phi (x_t; \alpha ,\delta )-B_2(1-\phi (x_t; \alpha ,\delta )) \big ) \big ) \Big ) \end{aligned}$$

We sample from this distribution using an adaptive RWMH step [5, 6, see] with lognormal proposal for the parameter $\alpha$. Let $\boldsymbol{\eta }=(\log \alpha , \delta )'$ the value of the MCMC chain the iteration $(j-1)$, the candidate is generated as follows:

$$\begin{aligned} \boldsymbol{\eta }^{*}=\boldsymbol{\eta }^{(j-1)}+\lambda ^{(j)}\hbox {Chol}(S^{(j-1)})\boldsymbol{\varepsilon }^{(j)}, \qquad \boldsymbol{\varepsilon }^{(j)}\sim \mathcal {N}(\textbf{0},I_2) \end{aligned}$$

(25)

and $\boldsymbol{\eta }^{(j)}=\boldsymbol{\eta }^{*}$ with probability $\rho ^{(j)}=\min (1,r^{(j)})$, where $r^{(j)}=$ and $\boldsymbol{\eta }^{(j)}=\boldsymbol{\eta }^{(j-1)}$ with probability $1-\rho ^{(j)}$. The updating scheme for the adaptation parameters is as follows

$$\begin{aligned} \log \lambda ^{(j)}= & {} \log \lambda ^{(j-1)}+\gamma ^{(j)}(\rho ^{(j)}-\rho ^{*})\end{aligned}$$

(26)

$$\begin{aligned} \boldsymbol{\mu }^{(j)}= & {} \boldsymbol{\mu }^{(j-1)}+\gamma ^{(j)}(\boldsymbol{\eta }-\boldsymbol{\mu }^{(j)})\end{aligned}$$

(27)

$$\begin{aligned} \Upsilon ^{(j)}= & {} \Upsilon ^{(j-1)}+\gamma ^{(j)}((\boldsymbol{\eta }-\boldsymbol{\mu }^{(j)})(\boldsymbol{\eta }-\boldsymbol{\mu }^{(j)})'-\Upsilon ^{(j-1)}) \end{aligned}$$

(28)

where $\gamma ^{(j)}=C/j^\kappa$ with $\kappa =0.65$, and the target acceptance rate is $\rho ^* = 0.30$.