Mapping species distributions in one dimension by non-homogeneous hidden Markov models: the case of freshwater pearl mussels in the River Dee

Abstract

The investigation of species distributions in rivers involves data which are inherently sequential and unlikely to be fully independent. To take these characteristics into account, we develop a Bayesian hierarchical model for mapping the distribution of freshwater pearl mussels in the River Dee (Scotland). At the top of the hierarchy the likelihood is used to describe the sequence of sites in which mussels were observed or not. Given that false observations can occur, and that “not observed” means both that the species was not present and that it was not observed, a Markov prior is introduced at the second level of the hierarchy to represent the sequence of sites in which mussels are estimated to occur. The Markov prior allows modelling the spatial dependency between neighbouring sites. A third level in the hierarchy is given by the representation of the transition probabilities of the Markov chain in terms of site-specific explanatory variables, through a logistic regression. The selection of the explanatory variables which influence the Markov process is performed by means of a simulation-based procedure, in the complex case of association between covariates. Four features were found to be associated with reduced chance of finding a local mussel population: tributaries, bridges, dredging, and waste water treatment works. These results complement the results of a previous study, providing new evidence for the causes of the deterioration of a highly threatened species.

Appendix

The basic MCMC sampler used for variable selection is here described. Given the draws $\theta ^{(h-1)}$, $\alpha ^{(h-1)}$, $\eta ^{(h-1)}$, $x^{T(h-1)}$, $y^{*(h-1)}$, generated at the $(h-1)$-th iteration under the constraint $\theta _{0}^{(h-1)}<\theta _{1}^{(h-1)}$, the steps of the generic $h$-th iteration are the following.

[Step 1] The sequence of the hidden states $x^{T(h)}$ is generated by the forward filtering-backward sampling (ff-bs) algorithm (Chib 1996). It is so called because firstly the filtered probabilities of the hidden states are computed going forwards; then the conditional probabilities of the hidden states are computed going backwards, sampling the states from the full conditional:

$$\begin{aligned} p\left( x^{T}\mid y^{T}\right) =p\left( x_{T}\mid y^{T}\right) \overset{T-1}{ \underset{t=1}{\prod }}p\left( x_{t}\mid x_{t+1},y^{t}\right) , \end{aligned}$$

suppressing the conditioning on $\theta , \alpha , \eta$, and $z$.

Let $\xi _{t+1\mid t}$ be the bidimensional column vector whose generic entry is $p\left( x_{t}=i\mid y^{t}\right)$; $\xi _{t\mid t}$ be the bidimensional column vector whose generic entry is $p\left( x_{t}=i\mid y^{t}\right)$; $\xi _{t}$ be the bidimensional column vector whose generic entry is $p\left( x_{t}=i\mid x_{t+1},y^{t}\right)$, for any $i\in S_{X}$. The iterative scheme of the ff-bs algorithm is the following.

• (1.1) Place

  $$\begin{aligned} \xi _{1\mid 0}^{(h)}=\delta ^{(h)\prime }, \end{aligned}$$

  where $\delta ^{(h)}$ is the initial distribution of the Markov chain.

• (1.2) Compute

  $$\begin{aligned} \xi _{t\mid t}^{(h)}=\frac{\xi _{t\mid t-1}^{(h)}\odot F_{t}^{(h-1)}}{ 1_{(2)}^{\prime }\left( \xi _{t\mid t-1}^{(h)}\odot F_{t}^{(h-1)}\right) } \qquad \hbox {and}\qquad \xi _{t+1\mid t}^{(h)}=\left[ \Gamma ^{t(h-1)}\right] ^{\prime }\ \xi _{t\mid t}^{(h)}, \end{aligned}$$

  for any $t=1,\ldots ,T-1,$ where $F_{t}^{(h-1)}=\left( p\left( y_{t}\left| x_{t}=0,\theta ^{(h-1)}\right. \right) ;p\left( y_{t}\left| x_{t}=1,\right. \right. \right.$ $\left. \left. \left. \theta ^{(h-1)}\right. \right) \right) ^{\prime }$, $1_{(2)}$ is the bidimensional column vector of ones, and $\odot$ denotes the Hadamard product.

• 1.3) Compute

  $$\begin{aligned} \xi _{T\mid T}^{(h)}=\frac{\xi _{T\mid T-1}^{(h)}\odot F_{T}^{(h-1)}}{ 1_{(2)}^{\prime }\left( \xi _{T\mid T-1}^{(h)}\odot F_{T}^{(h-1)}\right) }. \end{aligned}$$

• 1.4) Generate $x_{T}^{(h)}$ from $\xi _{T\mid T}^{(h)}$.

• 1.5) Compute

  $$\begin{aligned} \xi _{t}^{(h)}=\frac{\xi _{t\mid t}^{(h)}\odot \Gamma _{\bullet x_{t+1}^{(h)}}^{t(h-1)}}{1_{(2)}^{\prime }\left( \xi _{t\mid t}^{(h)}\odot \Gamma _{\bullet x_{t+1}^{(h)}}^{t(h-1)}\right) }, \end{aligned}$$

  and generate $x_{t}^{(h)}$ from $\xi _{t}^{(h)}$, for any $t=T-1,\ldots ,1$. Vector $\Gamma _{\bullet x_{t+1}^{(h)}}^{t(h-1)}$ represents the column of $\Gamma ^{t(h-1)}$ corresponding to the state generated previously.

[Step 2] The logarithmic transformations of the parameters $\theta _{0}^{(h)}$ and $\theta _{1}^{(h)}$ are independently generated from the random walk logit$\left( \theta _{i}^{(h)}\right) =$ logit $\left( \theta _{i}^{(h-1)}\right) +U_{\Theta }$ ($i\in S_{X}$), where $U_{\Theta }$ is a Gaussian noise with zero mean and constant variance $\sigma _{\Theta }^{2}$. Then the pair of parameters $\left( \theta _{0}^{(h)};\theta _{1}^{(h)}\right)$ is permuted if $\theta _{0}^{(h)}>\theta _{1}^{(h)}$, in order to respect the prior constraint $\theta _{0}<\theta _{1}$ (Green and Richardson 2002).

The vector $\theta ^{(h)}$ is accepted with probability

$$\begin{aligned} A\left( \theta ^{(h-1)};\theta ^{(h)}\right) =\min \left\{ 1;\frac{p\left( y^{T}|x^{T(h)},\theta ^{(h)}\right) }{p\left( y^{T}|x^{T(h)},\theta ^{(h-1)}\right) }\frac{q\left( \theta ^{(h-1)}\mid \theta ^{(h)}\right) }{ q\left( \theta ^{(h)}\mid \theta ^{(h-1)}\right) }\right\} , \end{aligned}$$

where the proposal ratio $q\left( \theta ^{(h-1)}\mid \theta ^{(h)}\right) / q\left( \theta ^{(h)}\mid \theta ^{(h-1)}\right)$ reduces to the ratio of the Jacobians of the logit transformation of $\theta _{0}$ and $\theta _{1}$, i.e. $J_{\theta ^{(h-1)}}/J_{\theta ^{(h)}}$, because the proposal densities cancel out, due to their symmetry:

$$\begin{aligned} \frac{q\left( \theta ^{(h-1)}\mid \theta ^{(h)}\right) }{q\left( \theta ^{(h)}\mid \theta ^{(h-1)}\right) }=\frac{J_{\theta ^{(h-1)}}}{J_{\theta ^{(h)}}}=\frac{\theta _{0}^{(h)}\left( 1-\theta _{0}^{(h)}\right) \theta _{1}^{(h)}\left( 1-\theta _{1}^{(h)}\right) }{\theta _{0}^{(h-1)}\left( 1-\theta _{0}^{(h-1)}\right) \theta _{1}^{(h-1)}\left( 1-\theta _{1}^{(h-1)}\right) }. \end{aligned}$$

[Step 3] Any coefficient $\alpha _{i,j,k}^{(h)}$ ($i,j\in S_{X}, i\ne j, k=0,1,\ldots ,K$) is independently generated from the random walk $\alpha _{i,j,k}^{(h)}=\alpha _{i,j,k}^{(h-1)}+U_{A}$, where $U_{A}$ is a Gaussian noise of zero mean and constant variance $\sigma _{A}^{2}$. Then, each parameter $\eta _{k(i)}^{(h)}$ ($i\in S_{X}$, $k=1,\ldots ,K$) is independently generated from a Bernoulli distribution of parameter $a_{k(i)}^{(h)}/ \left( a_{k(i)}^{(h)}+b_{k(i)}^{(h)}\right) ,$ where $a_{k(i)}^{(h)}$ is the product of the transition probabilities $\gamma _{x_{t-1},x_{t}}^{(h)}$, for any $t=2,\ldots ,T$, when $x_{t-1}^{(h)}=i$ and $x_{t}^{(h)}=j$ ($i\ne j$), replacing $\eta _{k(i)}$ in any $\gamma _{x_{t-1},x_{t}}^{(h)}$ by 1, whereas $b_{k(i)}^{(h)}$ is the product of the transition probabilities $\gamma _{x_{t-1},x_{t}}^{(h)}$, for any $t=2,\ldots ,T$, when $x_{t-1}^{(h)}=i$ and $x_{t}^{(h)}=j$ ($i\ne j$), replacing $\eta _{k(i)}$ in any $\gamma _{x_{t-1},x_{t}}^{(h)}$ by 0:

$$\begin{aligned} a_{k(i)}^{(h)}&\propto \underset{\left\{ t\ge 2:x_{t-1}^{(h)}=i,x_{t}^{(h)}=j,i\ne j\right\} }{\prod }\frac{\exp \left( z_{t}H_{i}^{[k;1]}\alpha _{i,j}^{\prime (h)}\right) }{1+\exp \left( z_{t}H_{i}^{[k;1]}\alpha _{i,j}^{\prime (h)}\right) }\lambda , \\ b_{k(i)}^{(h)}&\propto \underset{\left\{ t\ge 2:x_{t-1}^{(h)}=i,x_{t}^{(h)}=j,i\ne j\right\} }{\prod }\frac{\exp \left( z_{t}H_{i}^{[k;0]}\alpha _{i,j}^{\prime (h)}\right) }{1+\exp \left( z_{t}H_{i}^{[k;0]}\alpha _{i,j}^{\prime (h)}\right) }\left( 1-\lambda \right) , \end{aligned}$$

where $H_{i}^{[k;1]}$ is the diagonal matrix $H_{i}$ in which $\eta _{k(i)}^{(h)}$ is replaced by 1, i.e. $H_{i}^{[k;1]}=$ diag$\left( 1,\eta _{1(i)}^{(h)},\ldots ,\right.$ $\left. \eta _{k-1(i)}^{(h)},1,\eta _{k+1(i)}^{(h-1)},\ldots ,\eta _{K(i)}^{(h-1)}\right)$, and $H_{i}^{[k;0]}$ is the diagonal matrix $H_{i}$ in which $\eta _{k(i)}^{(h)}$ is replaced by 0, i.e. $H_{i}^{[k;0]}=$ diag$\left( 1,\eta _{1(i)}^{(h)},\ldots ,\eta _{k-1(i)}^{(h)},0,\eta _{k+1(i)}^{(h-1)},\right.$ $\left. \ldots ,\eta _{K(i)}^{(h-1)}\right)$.

Finally each pair of vectors $\left( \alpha _{i,j}^{(h)};\eta _{i}^{(h)}\right) , i,j\in S_{X}$ ($i\ne j$), is accepted in block with probability

$$\begin{aligned} \begin{array}{l} A\left[ \left( \alpha _{i,j}^{(h-1)};\eta _{i}^{(h-1)}\right) ;\left( \alpha _{i,j}^{(h)};\eta _{i}^{(h)}\right) \right] \\ \qquad =\min \left\{ 1;\frac{p\left( \alpha _{i,j}^{(h)};\eta _{i}^{(h)}\left| x^{T(h)};z;\delta \right. \right) q\left( \alpha _{i,j}^{(h-1)};\eta _{i}^{(h-1)}\left| \alpha _{i,j}^{(h)};\eta _{i}^{(h)};z;\delta \right. \right) }{p\left( \alpha _{i,j}^{(h-1)};\eta _{i}^{(h-1)}\left| x^{T(h)};z;\delta \right. \right) q\left( \alpha _{i,j}^{(h)};\eta _{i}^{(h)}\left| \alpha _{i,j}^{(h-1)};\eta _{i}^{(h-1)};z;\delta \right. \right) }\right\} \\ \\ \qquad =\min \left\{ 1;\frac{p\left( x^{T(h)}\left| \alpha _{i,j}^{(h)};\eta _{i}^{(h)};z;\delta \right. \right) p\left( \eta _{i}^{(h)}\right) p\left( \alpha _{i,j}^{(h)}\right) q\left( \eta _{i}^{(h-1)}\left| \alpha _{i,j}^{(h-1)};\alpha _{i,j}^{(h)};\eta _{i}^{(h)};z;\delta \right. \right) }{p\left( x^{T(h)}\left| \alpha _{i,j}^{(h-1)};\eta _{i}^{(h-1)};z;\delta \right. \right) p\left( \eta _{i}^{(h-1)}\right) p\left( \alpha _{i,j}^{(h-1)}\right) q\left( \eta _{i}^{(h)}\left| \alpha _{i,j}^{(h)};\alpha _{i,j}^{(h-1)};\eta _{i}^{(h-1)};z;\delta \right. \right) }\right\} \end{array} \end{aligned}$$

by the factorization of the proposal density, by the independence of $\alpha _{i,j}^{(h)}$ on $\eta _{i}^{(h-1)}$ and cancelling the ratio $q\left( \alpha _{i,j}^{(h-1)}\left| \alpha _{i,j}^{(h)};z;\delta \right. \right) / q\left( \alpha _{i,j}^{(h)}\left| \alpha _{i,j}^{(h-1)};z;\delta \right. \right)$ for the symmetry of the proposal distribution.

[Step 4] Every missing observation $y_{t}^{*}$, given the hidden state $x_{t}^{(h)}=i$ ($i\in S_{X}$), is generated from the Bernoulli distribution $\mathcal B e\left( \theta _{i}^{(h)}\right)$. This concludes the $h$-th iteration.