Labour and residential accessibility: a Bayesian analysis based on Poisson gravity models with spatial effects

Abstract

In this study, we measure jointly the labour and the residential accessibility of a basic spatial unit using a Bayesian Poisson gravity model with spatial effects. The accessibility measures are broken down into two components: the attractiveness component, which is related to its socio-economic and demographic characteristics, and the impedance component, which reflects the ease of communication within and between basic spatial units. For illustration purposes, the methodology is applied to a data set containing information about commuters from the Spanish region of Aragón. We identify the areas with better labour and residential accessibility, and we also analyse the attractiveness and the impedance components of a set of chosen localities which allows us to better understand their mobility patterns.

Appendix

This “Appendix” describes the algorithm used to calculate the posterior distribution of the parameters of model given by Eqs. (1)–(7) when f(t _ij ; γ) = γg(t _ij ) where, for instance, g(t _ij ) = −t _ij for the exponential model or g(t _ij ) = − log(t _ij  + 1) for the power model. This algorithm is a modification of that proposed by LeSage et al. (2007) using the recent results of Frühwirth-Schnatter et al. (2009) that increases the efficiency of the estimation procedure. Details are as follows:

For each pair (i, j) $\in$ {1, …, K} × {1, …, K}, we introduce the following latent variables:

$$ \begin{aligned} \tau_{ij,1} & = \frac{{\xi_{ij,1} }}{{\lambda_{ij} }}\quad {\text{with}}\,\xi_{ij,1} \sim \exp (1) \\ \tau_{ij,2} & = \frac{{\xi_{ij,2} }}{{\lambda_{ij} }}\quad {\text{with}}\,\xi_{ij,2} \sim {\text{Gamma}}(n_{ij} ,1)\quad {\text{if}}\,n_{ij} > 0 \\ \end{aligned} $$

As Frühwirth-Schnatter et al. (2009) showed, we have that:

$$\begin{gathered} - \log (\tau_{ij,k} ) = \log (\lambda_{ij} ) + \varepsilon_{ij,k} \quad {\text{where}}\,\varepsilon_{ij,k} \approx \sum\limits_{r = 1}^{{R\left( {\nu_{ij,k} } \right)}} {w_{r} \left( {\nu_{ij,k} } \right)N\left( {m_{r} \left( {\nu_{ij,k} } \right),s_{r}^{2} \left( {\nu_{ij,k} } \right)} \right)} \hfill\\ {\text{where}}\,v_{ij,k} = \left\{ {\begin{array}{cc} 1 & {{\text{if}}\,k = 1} \\ {n_{ij} } & {{\text{if}}\,k = 2} \\ \end{array} } \right. \hfill \\ \end{gathered} $$

To handle the above mixture, we introduce the indicators r _ij,k  $\in$ {1, … ,R(ν _ij , _k )} in such a way that:

$$ \begin{gathered} \log (\tau_{ij,k} )|\lambda_{ij} ,r_{ij,k} \sim {\mathsf{\textit{N}}}\left( {\log (\lambda_{ij} ) + m_{{r_{ij,k} }} (\nu_{ij,k} ),s_{{r_{ij,k} }}^{2} (\nu_{ij,k} )} \right) ;\quad k = 1, \ldots ,m_{ij} \hfill \\ {\text{where}}\,\nu_{ij,k} = \left\{ {\begin{array}{cc} 1 & {{\text{if}}\,n_{ij} = 0} \\ 2 & {{\text{if}}\,n_{ij} > 0} \\ \end{array} } \right. \hfill \\ \end{gathered} $$

Based on these latent variables, the algorithm for calculating a sample of the posterior distribution of model parameters is as follows:

• Step 0 Calculate the values of {R(ν _ij,k ), w _r (ν _ij,k ), m _r (ν _ij, _k ), $s_{r}^{2} (\nu_{ij,k} )$; r = 1, …, R(ν _ij,k )}; k = 1, …, m _ij ; i, j = 1, …, K.⁶

• Step 1 Obtain an initial sample of values of {τ _ij,k, r _ij,k ; k = 1, …, m _ij ; i, j = 1, …, K} taking $\lambda_{ij} = \left\{ {\begin{array}{ll} {0.1} \hfill & {{\text{if}}\,n_{ij} = 0} \hfill \\ {n_{ij} } \hfill & {{\text{if}}\,n_{ij} > 0} \hfill \\ \end{array} } \right.$ and using the full-conditional distributions described below.

• Step 2 Implement Gibbs sampling using the full-conditional distributions given in LeSage et al. (2007) with the following modifications:

  • β|rest of the parameters; data

Let y _ij,k  = −log(τ _ij,k ) − $m_{{r_{ij,k} }} (\nu_{ij,k} )$; k = 1, …, m _ij and y _ij  = (y _ij,k , k = 1, … ,m _ij )′; i, j = 1, …, K.

We have then:

$$ y_{ij} = \alpha {\mathbf{1}}_{{m_{ij} }} + \left( {\varvec{x}_{oi}^{{\prime }}\varvec{\beta}_{o} + \varvec{x}_{dj}^{{\prime }}\varvec{\beta}_{d} + f\left( {t_{ij} ,\gamma } \right) + \theta_{i} + \phi_{j} } \right){\mathbf{1}}_{{m_{ij} }} + \varepsilon_{ij} ;\quad i,j = 1, \ldots ,K $$

with ε _ij  ~ ${\mathsf{\textit{N}}}_{{m_{ij} }} ({\mathbf{0}},{\text{diag(}}s_{{r_{ij,k} }}^{2} (\nu_{ij,k} ) ) ,k = 1, \ldots ,m_{ij} )$ and ${\mathbf{1}}_{{m_{ij} }}$ (m _ij  × 1) vector of ones.

We define:

$$ \begin{gathered} \varvec{y}_{i} = \left( {\begin{array}{cc} {{y}_{i1} } \\ \vdots \\ {{y}_{iK} } \\ \end{array} } \right),\varvec{X}_{oi} = {\mathbf{1}}_{{m_{i} }} ,x_{oi}^{{\prime }} ,\varvec{X}_{di} = \left( {\begin{array}{cc} {{\mathbf{1}}_{{m_{i1} }} {\mathbf{x}}_{d1}^{{\prime }} } \\ \vdots \\ {{\mathbf{1}}_{{m_{iK} }} {\mathbf{x}}_{dK}^{{\prime }} } \\ \end{array} } \right),\varvec{g}_{i} = \left( {\begin{array}{cc} {g\text{(}t_{i1} \text{)}{\mathbf{1}}_{{m_{i1} }} } \\ \vdots \\ {g\text{(}t_{iK} \text{)}{\mathbf{1}}_{{m_{iK} }} } \\ \end{array} } \right) \hfill \\\varvec{\phi}_{i} = \left( {\begin{array}{cc} {\phi_{1} {\mathbf{1}}_{{m_{i1} }} } \\ \vdots \\ {\phi_{K} {\mathbf{1}}_{{m_{iK} }} } \\ \end{array} } \right) = \varvec{A}_{i}\varvec{\phi}\,\quad{\text{with}}\,\quad\varvec{A}_{i} = {\text{diag}}\left( {{\mathbf{1}}_{{m_{ij} }} ,j = 1, \ldots ,K} \right),\,\varvec{\varepsilon} = \left( {\begin{array}{cc} {\varepsilon_{i1}^{{}} } \\ \vdots \\ {\varepsilon_{iK} } \\ \end{array} } \right) \hfill \\ \end{gathered}, $$

and we have then:

$$ \varvec{y}_{i} = \alpha {\mathbf{1}}_{{m_{i} }} + \varvec{X}_{oi}\varvec{\beta}_{o} + \varvec{X}_{di}\varvec{\beta}_{d} + \gamma \varvec{g}_{i} + \theta_{i} {\mathbf{1}}_{{m_{i} }} + \varvec{A}_{i}\varvec{\phi}+\varvec{\varepsilon}_{i} \quad {\text{for}}\,i = 1, \ldots ,K $$

with ε _i  ~ ${\mathsf{\textit{N}}}_{{m_{i} }} (0,\varvec{S}_{{r_{i} }} )$ where m _i  = m _i1 + $\cdots$ + m _iK , $S_{{r_{i} }} = {\text{diag(}}s_{{r_{ij,k} }}^{2} (\nu_{ij,k} );k = 1, \ldots ,m_{ij} ;j = 1, \ldots ,K )$.

Hence, we obtain:

$$ \varvec{y} = \alpha {\mathbf{1}}_{m} + \varvec{X}_{0}\varvec{\beta}_{0} + \varvec{X}_{d}\varvec{\beta}_{d} + \gamma \varvec{g} + \varvec{B\theta } + \varvec{A\phi } +\varvec{\varepsilon} $$

where

$$ \begin{gathered} \varvec{y} = \left( {\begin{array}{cc} {\varvec{y}_{1} } \\ \vdots \\ {\varvec{y}_{K} } \\ \end{array} } \right),\varvec{X}_{o} = \left( {\begin{array}{cc} {\varvec{X}_{o1} } \\ \vdots \\ {\varvec{X}_{oK} } \\ \end{array} } \right),\varvec{X}_{d} = \left( {\begin{array}{cc} {\varvec{X}_{d1} } \\ \vdots \\ {\varvec{X}_{dK} } \\ \end{array} } \right),\varvec{g} = \left( {\begin{array}{cc} {\varvec{g}_{1} } \\ \vdots \\ {\varvec{g}_{K} } \\ \end{array} } \right) \hfill \\ \varvec{B} = {\text{diag}}({\mathbf{1}}_{{m_{i} }} ;i = 1, \ldots ,K),\varvec{A} = \left( {\begin{array}{cc} {\varvec{A}_{1} } \\ \vdots \\ {\varvec{A}_{K} } \\ \end{array} } \right),\varvec{\varepsilon}= \left( {\begin{array}{cc} {\varvec{\varepsilon}_{1} } \\ \vdots \\ {\varvec{\varepsilon}_{K} } \\ \end{array} } \right)\sim {\mathsf{\textit{N}}}_{m} (\varvec{0},\varvec{S}_{r} ) \hfill \\ \end{gathered} $$

with m = m ₁ + $\cdots$ + m _K and S _r  = ${\text{diag(}}\varvec{S}_{{r_{1} }} , \ldots ,\varvec{S}_{{r_{K} }} )$.

Therefore, it follows that:

$$ \varvec{y} = \varvec{X\beta } + \varvec{B\theta } + \varvec{A}\varvec{\phi} +\varvec{\varepsilon}. $$

where X = (1 _m , X _o , X _d , g) and β = $(\alpha ,\varvec{\beta}_{o}^{\prime } ,\varvec{\beta}_{d}^{\prime } ,\gamma )^{\prime }$

Standard calculations show that:

$$ \begin{aligned} & \varvec{\beta} |{\text{rest}}\,{\text{of}}\,{\text{the}}\,{\text{parameters}},\,{\text{data}}\sim {\mathsf{\textit{N}}}_{p + q + 2} \left( {\varvec{\mu}_{\beta } ,\varvec{\varSigma}_{\beta } } \right)\,{\text{with}} \\ &\varvec{\mu}_{\beta } = \left( {\varvec{X}^{\prime} \varvec{S}_{r}^{ - 1} \varvec{X} + \varvec{S}_{\beta }^{ - 1} } \right)^{ - 1} \left( {\varvec{X}^{\prime } \varvec{S}_{r}^{ - 1} \left( {\varvec{y} - \varvec{B\theta - {\rm A}\phi }} \right)} \right),\varvec{\varSigma}_{\beta } = \left( {\varvec{X}^{\prime } \varvec{S}_{r}^{ - 1} \varvec{X} + \varvec{S}_{\beta }^{ - 1} } \right)^{ - 1} \\ \end{aligned} $$

• $(\tau_{ij,k} ;k = 1, \ldots ,m)|{\text{rest}}\,{\text{of}}\,{\text{the}}\,{\text{parameters}},\,{\text{data}}$

  • $$ \begin{aligned} & {\text{If}}\,n_{ij} = 0\quad {\text{take}}\,\tau_{ij,1} = 1 + \xi_{ij} \,{\text{with}}\,\xi_{ij} \sim \text{Exp} (\lambda_{ij} ) \\ & {\text{If}}\,n_{ij} > 0\quad {\text{draw}}\,\xi_{ij} \sim \text{Exp} (\lambda_{ij} )\,{\text{and}}\,\tau_{ij,2} \sim {\text{Beta(}}n_{ij,1} )\,{\text{and}}\,{\text{take}}\,\tau_{ij,1} = 1 - \tau_{ij,2} + \xi_{ij} \\ \end{aligned} $$

• $r_{ij,1} ,r_{ij,2} |{\text{rest}}\,{\text{of}}\,{\text{the}}\,{\text{parameters}},\,{\text{data}}$

Draw r _ij,1 from the discrete distribution with support {1, …, R(ν _ij,1)} and probability function:

$$ P(r_{ij,1} = u) \propto w_{u} (1)\varphi ( - \log \tau_{ij,1} {-}\log \lambda_{{ij}} ; \, m_{u} (1),s_{u} (1)) $$

where φ(x; m, s) denotes the value in x of the density function of a ${\mathsf{\textit{N}}}(m,s)$.

If also n _ij  > 0, draw r _ij,2 from the discrete distribution with support {1, …, R(ν _ij,2)} and probability function:

$$ P(r_{ij,2} = u) \propto w_{u} (n_{ij} )\varphi ( - \log \tau_{ij,1} {-}\log \lambda_{{ij}} ; \, m_{u} (n_{ij} ),s_{u} (n_{ij} )) $$