A Bayesian hierarchical spatio-temporal model for significant wave height in the North Atlantic

Abstract

Bad weather and rough seas continue to be a major cause for ship losses and is thus a significant contributor to the risk to maritime transportation. This stresses the importance of taking severe sea state conditions adequately into account in ship design and operation. Hence, there is a need for appropriate stochastic models describing the variability of sea states, taking into account long-term trends related to climate change. Various stochastic models of significant wave height are reported in the literature, but most are based on point measurements without considering spatial variations. As far as the authors are aware, no model of significant wave height to date exploits the flexible framework of Bayesian hierarchical space-time models. This framework allows modelling of complex dependence structures in space and time and incorporation of physical features and prior knowledge, yet at the same time remains intuitive and easily interpreted. This paper presents a Bayesian hierarchical space-time model for significant wave height. The model has been fitted by significant wave height data for an area in the North Atlantic ocean. The different components of the model will be outlined, and the results from applying the model to monthly and daily data will be discussed. Different model alternatives have been tried and long-term trends in the data have been identified for all model alternatives. Overall, these trends are in reasonable agreement and also agree fairly well with previous studies. Furthermore, a discussion of possible extensions to the model, e.g. incorporating regression terms with relevant meteorological data will be presented.

Appendix

1.1 A Derivation of the full conditional distributions

In the following, the derivation of the full conditional distributions, needed for the Gibbs sampler, will be outlined. Vectors will be written in bold, and the notation $P(V | \cdot )$ will be used to represent the conditional distribution of V conditioned on all other random quantities, i.e. the full conditional distribution. The values of the hyperparameters in Table 1 applies for all the specified prior distributions in the following and will not be referred in each case.

Spatial data will be treated as a vector of size X = 153 × 1, so that for example ${\user2{Z}}_{t}, {\varvec{\theta}}_{t}, {\varvec{\mu}}$ are all column vectors of size X. For the temporal components without spatial description, e.g. M _t and T _t , a vector of corresponding size may be obtained by multiplication by a vector of ones, $\user2{1},$ of size X. In the following, e.g. M _t is assumed to be interpreted as $\mathbf{1} M_{t}$ in the instances where the $\user2{1}$ is not explicitly written. The transpose of a matrix or vector $\user2{V}$ will be denoted ${\user2{V}}^{\prime}.$

The derivation of the full conditionals is similar to the derivation in Natvig and Tvete (2007), but whereas Natvig and Tvete (2007) sampled most parameters individually, here some sets of parameters are sampled jointly where this is natural. Presumably, this may speed up convergence of the Gibbs sampler. It is noted that for model alternatives 4 and 5, with only one temporal noise term, all relevant parameters c, d, γ and η were sampled jointly. The derivation of these joint full conditionals are not outlined here, but follow the same general approach.

1.2 A.1 Conditional distribution for ${\user2{H}}(t)$

It is noted that since there are no random noise term in Eq. 2, the full conditional distribution of ${\user2{H}}_{t}$ reduces to just a single, fixed value, at each location, x and time, t:

$$ P({\user2{H}}_{t} | \cdot) = {\varvec{\mu}} + {\varvec{\theta}}_{t} + M_{t} + \hbox{T}_{t} $$

Hence, determining ${\varvec{\mu}}, {\varvec{\theta}}_{t}, \hbox{T}_{t} \hbox { and } M_{t}$ determines ${\user2{H}}_{t} = H(x, t).$

1.3 A.2 Conditional distribution for $\varvec{\mu}$

From the model specification, the following conditional distributions are obtained directly for all the parameters with dependence with $\varvec{\mu}$

$$ {\varvec{\mu}} | {\varvec{\mu}}_{0}, a_{\phi}, a_{\lambda}, \sigma_{\mu}^{2} \sim MVN\left({\varvec{\mu}}_{0}, \sigma_{\mu}^2 {\user2{A}}_\mu^{-1}\right) {\user2{Z}}_{t} | {\varvec{\mu}}, {\varvec{\theta}}_{t}, M_{t}, \hbox{T}_{t}, \sigma_{Z}^{2} \sim MVN\left({\varvec{\mu}} + {\varvec{\theta}}_{t} + {\user2{1}} (M_{t} + \hbox{T}_{t}) , {\user2{I}} \sigma_{Z}^2\right) $$

where the X × X precision matrix ${\user2{A}}_{\mu}$ has elements (see e.g. Smith 2001)

$$ a_{ij} = \left\{ \begin{array}{ll} 1 & \hbox { if } i = j \\ -a_{\phi} & \hbox { for } i, j \hbox { lateral neighbours } \\ -a_{\lambda} & \hbox { for } i, j \hbox { longitudinal neighbours } \\ 0 & \hbox { otherwise } \end{array} \right. $$

$\user2{I}$ is the identity matrix and $\user2{1}$ is a X × 1 vector of ones. It is noted that ${\user2{A}}_{\mu}$ must be positive definite, and sufficient conditions are that both a _ϕ and a _λ are positive and that $a_{\phi} + a_{\lambda}\le \frac{1}{2}$ (see e.g. Besag and Kooperberg 1995, p. 734). Similar constraints were used in Natvig and Tvete (2007).

Now, conditioned on the underlying processes, the observations are independent, and the full conditional distribution for $\varvec{\mu}$ is proportional to a product of the known conditional distributions above

$$ P({\varvec{\mu}} | \cdot)\propto P({\varvec{\mu}}| {\varvec{\mu}}_{0}, a_{\phi}, a_{\lambda}, \sigma_{\mu}^2) \displaystyle\prod_{t=1}^T P({\user2{Z}}_{t} | {\varvec{\mu}}, {\varvec{\theta}}_{t}, M_{t}, \hbox{T}_{t}, \sigma_{Z}^2) \propto e^{-\frac{1}{2\sigma_{\mu}^2}\left( ({\varvec{\mu}} - {\varvec{\mu}}_{0})^{\prime} {\user2{A}}_{\mu} ({\varvec{\mu}} - {\varvec{\mu}}_{0}) \right)} \times e^{-\frac{1}{2\sigma_{Z}^2}\left(\sum_{t} ({\user2{Z}}_{t} - {\varvec{\mu}} - {\varvec{\theta}}_{t} - {\user2{1}}(M_{t} + \hbox{T}_{t}) )^{\prime}\left({\user2{Z}}_t - {\varvec{\mu}} - {\varvec{\theta}}_t - {\user2{1}}(M_t + \hbox{T}_{t}) \right) \right) } \propto e^{-\frac{1}{2}\left[{\varvec{\mu}}^{\prime}\left( \frac{1}{\sigma^2_\mu}{\user2{A}}_\mu + \frac{T}{\sigma_{Z}^2}{\user2{I}} \right){\varvec{\mu}} - 2{\varvec{\mu}}^{\prime} \left( \frac{1}{\sigma_{\mu}^2} {\user2{A}}_\mu {\varvec{\mu}}_0 + \frac{1}{\sigma_{Z}^2}\sum_t({\user2{Z}}_t - {\varvec{\theta}}_t - {\user2{1}}(M_t + \hbox{T}_t) ) \right) \right] },, $$

which gives

$$ {\varvec{\mu}} | \cdot \sim MVN \left[ \bar{\varvec{\mu}}, {\varvec{\Upsigma}}_\mu \right] \quad \hbox {with}\, {\varvec{\Upsigma}}_\mu = \left( \frac{1}{\sigma_{\mu}^2}{\user2 {A}}_\mu + \frac{T}{\sigma_Z^2}{\user2{I}} \right)^{-1} \bar{\varvec{\mu}} = {\varvec{\Upsigma}}_\mu \left( \frac{1}{\sigma_\mu^2} {\user2{A}}_\mu {\varvec{\mu}}_0 + \frac{1}{\sigma_Z^2}\sum_{t}({\user2{Z}}_t - {\varvec{\theta}}_t - {\user2{1}}( M_t + \hbox{T}_t) ) \right) $$

1.4 A.3 Conditional distribution for $\varvec{\theta}(t)$

$P(\varvec{\theta}_t | \cdot )$ for $t = 1, \ldots, T-1:$ To derive the conditional distribution of $\varvec{\theta}_t$ for $t = 1, \ldots, T-1,$ it is noted that the hierarchical model gives

$$ {\varvec{\theta}}_t | b_0, {\varvec{\theta}}_{t-1}, b_N, b_E, b_S, b_W, \sigma_{\theta}^2\sim MVN\left({\user2{B}} {\varvec{\theta}}_{t-1} , {\user2{I}} \sigma_\theta^2 \right) {\varvec{\theta}}_{t+1} | b_0, {\varvec{\theta}}_{t}, b_N, b_E, b_S, b_W, \sigma_{\theta}^{2} \sim MVN\left({\user2{B}} {\varvec{\theta}}_{t} , {\user2{I}} \sigma_\theta^2 \right) {\user2{Z}}_t | {\varvec{\mu}}, {\varvec{\theta}}_t, M_t, \hbox{T}_t, \sigma_Z^2 \sim MVN\left({\varvec{\mu}} + {\varvec{\theta}}_t + {\user2{1}}(M_t + \hbox{T}_t) , {\user2{I}} \sigma_Z^2\right) $$

where the matrix $\user2{B}$ is an X × X matrix with elements

$$ b_{ij} = \left\{ \begin{array}{ll} b_0 &\hbox { if } i = j\\ b_N & \hbox { for } j = \hbox { neighbor to the North of } i \\ b_E & \hbox { for } j = \hbox { neighbor to the East of } i \\ b_S & \hbox { for } j = \hbox { neighbor to the South of } i\\ b_W & \hbox { for } j = \hbox { neighbor to the West of } i \\ 0 & \hbox { otherwise } \end{array}\right. $$

Now, the full conditional is obtained in a similar way as in A.2 above, arriving at the following multinormal distribution, for $t = 1, \ldots, T-1$:

$$ {\varvec{\theta}}_t | \cdot \sim MVN \left[ \bar{\varvec{\theta}}_t, {\varvec{\Upsigma}}_{\theta_t} \right]\quad \hbox {with}\, $$

$$ {\varvec{\Upsigma}}_{\theta_t} = \left( \frac{1}{\sigma_\theta^2}({\user2{I}} + {\user2{B}}^{\prime} {\user2 {B}}) + \frac{1}{\sigma_Z^2}{\user2{I}}\right)^{-1} \bar{\varvec{\theta}}_t = {\varvec{\Upsigma}}_{\theta_t} \left( \frac{1}{\sigma_\theta^2}({\user2{B}} {\varvec{\theta}}_{t-1} + {\user2{B}}^{\prime}{\varvec{\theta}}_{t+1} ) + \frac{1}{\sigma_Z^2}({\user2{Z}}_t - {\varvec{\mu}} - {\user2{1}}(M_t + \hbox{T}_t) )\right) $$

$P(\varvec{\theta}_T | \cdot ):$ From the model specification:

$$ {\varvec{\theta}}_T | b_0, {\varvec{\theta}}_{T-1}, b_N, b_E, b_S, b_W, \sigma_\theta^2 \sim MVN\left({\user2{B}} {\varvec{\theta}}_{T-1} , {\user2{I}}\sigma_\theta^2 \right) {\user2{Z}}_T | {\varvec{\mu}}, {\varvec{\theta}}_T, M_T, \hbox{T}_T, \sigma_Z^2 \sim MVN\left({\varvec{\mu}} + {\varvec{\theta}}_T + {\user2{1}}(M_T + \hbox{T}_T) , {\user2 {I}}\sigma_Z^2\right) $$

and parallel to the above, the full conditional for $\varvec{\theta}_T$ becomes

$$ {\varvec{\theta}}_T | \cdot \sim MVN \left[ \bar{\varvec{\theta}}_T, {\varvec{\Upsigma}}_{\theta_T} \right] \quad \hbox {with}\, {\varvec{\Upsigma}}_{\theta_T} = \left( \frac{1}{\sigma_\theta^2}{\user2{I}} + \frac{1}{\sigma_Z^2}{\user2{I}}\right)^{-1} \bar{\varvec{\theta}}_T = {\varvec{\Upsigma}}_{\theta_T} \left( \frac{1}{\sigma_\theta^2}{\user2{B}}{\varvec{\theta}}_{T-1} + \frac{1}{\sigma_Z^2}({\user2{Z}}_T - {\varvec{\mu}} - {\user2{1}} (M_T + \hbox{T}_T) )\right) $$

$P(\varvec{\theta}_0 | \cdot ){:}$ The model specification and the specification of priors for θ(x, 0), for $x = 1, \ldots, X$ give

$$ {\varvec{\theta}}_0 | \xi_{\theta_0}, \sigma_{\theta_0}^2 \sim MVN\left({\user2{1}}\xi_{\theta_0} , {\user2{I}}\sigma_{\theta_0}^2 \right) {\varvec{\theta}}_{1} | b_0, {\varvec{\theta}}_{0}, b_N, b_E, b_S, b_W, \sigma^2_\theta \sim MVN\left({\user2{B}} {\varvec{\theta}}_{0} , {\user2{I}}\sigma_\theta^2 \right) $$

which gives the full conditional for ${\varvec{\theta}}_0,$

$$ {\varvec{\theta}}_0 | \cdot \sim MVN \left[ \bar{\varvec{\theta}}_0, {\varvec{\Upsigma}}_{\theta_0} \right] \quad \hbox {with}\, {\varvec{\Upsigma}}_{\theta_0} = \left( \frac{1}{\varvec{\sigma}_\theta^2}{\user2{B}}^{\prime} {\user2{B}} + \frac{1}{\sigma_{\theta_0}^2}{\user2{I}}\right)^{-1} \bar{\varvec{\theta}}_0 = {\varvec{\Upsigma}}_{\theta_{0}} \left(\frac{1}{\sigma_\theta^2}{\user2{B}}^{\prime} {\varvec{\theta}}_1 + \frac{1}{\sigma_{\theta_{0}}^2}{\user2{1}}\xi_{\theta_0} \right) $$

1.5 A.4 Conditional distribution for ${\varvec{\mu}}_0$

In order to derive the conditional distribution for ${\varvec{\mu}}_0,$ first define the vector ${\varvec{\mu}}_{0L} = \left(\mu_{0,1}, \mu_{0,2}, \mu_{0,3}, \mu_{0,4}, \mu_{0,5}, \mu_{0,6} \right)^{\prime}.$ It is noted that since there are no noise terms in Eq. 4, ${\varvec{\mu}}_0$ is uniquely determined by ${\varvec{\mu}}_{0L},$ so sampling ${\varvec{\mu}}_{0}$ is equivalent to sampling ${\varvec{\mu}}_{0L}.$ The deterministic relation ${\varvec{\mu}}_0 = {\user2{P}}{\varvec{\mu}}_{0L},$ with $\user2{P}$ the X × 6 trend design matrix with rows $\left(1, m(x), n(x), m(x)^2, n(x)^2, m(x)n(x) \right)$ for $x = 1, \ldots , X = 153,$ holds (analog to Natvig and Tvete 2007; Wikle et al. 1998) and the model and prior specifications give the conditional distributions

$$ {\varvec{\mu}} | {\varvec{\mu}}_{0L}, a_\phi, a_\lambda, \sigma_\mu^2 \sim MVN\left({\user2{P}}{\varvec{\mu}}_{0L}, \sigma_\mu^2 {\user2{A}}_\mu^{-1}\right) {\varvec{\mu}}_{0L} | {\varvec{\xi}}_{\mu_{0}}, \sigma_{\mu_{0}}^2 \sim MVN\left( {\varvec{\xi}}_{\mu_{0}}, {\user2{I}}\sigma_{\mu_0}^2 \right) $$

Hence, the full conditional for ${\varvec{\mu}}_0$ becomes

$$ {\varvec{\mu}}_{0L} | \cdot \sim MVN \left[ \bar{\varvec{\mu}}_{0L}, {\varvec{\Upsigma}}_{\mu_{0L}} \right] \quad \hbox {with}\, {\varvec{\Upsigma}}_{\mu_{0L}} = \left(\frac{1}{\sigma_\mu^2}{\user2{P}}^{\prime}{\user2{A}}_\mu {\user2{P}}+ \frac{1}{\sigma_{\mu_0}^2} {\user2{I}}\right)^{-1} \bar{\varvec{\mu}}_{0L} = {\varvec{\Upsigma}}_{\mu_{0L}} \left( \frac{1}{\sigma_\mu^2}{\user2{P}}^{\prime}{\user2{A}}_\mu {\varvec{\mu}} + \frac{1}{\sigma^2_{\mu_0}} {\varvec{\xi}}_{\mu_0} \right) $$

1.6 A.5 Conditional distribution for b ₀, b _N , b _E , b _S and b _W

The specification of the model and prior distributions give directly, by defining the vector ${\user2{b}}= (b_0, b_N, b_E, b_S, b_W)^{\prime},$

$$ {\varvec{\theta}}_t | b_0, {\varvec{\theta}}_{t-1}, b_N, b_E, b_S, b_W, \sigma^2_\theta \sim MVN\left({\user2{B}} {\varvec{\theta}}_{t-1} , {\user2{I}} \sigma_\theta^2 \right) {\user2{b}}| {\varvec{\xi}}_b, \sigma_b^2 \sim MVN\left({\varvec{\xi}}_b, {\user2{I}} \sigma_b^2\right) $$

Even though it might be easier, implementationwise, to sample the b-parameters individually, it is noted that sampling from the joint conditional distribution for all b-parameters may speed up convergence of the Gibbs sampler. Hence, these parameters will be sampled from the joint full conditional of $\mathbf{b}{:}$

$$ P({\user2{b}} | \cdot ) \propto e^{-\frac{1}{2\sigma_b^2}\left( \left({\user2{b}} - {\varvec{\xi}}_b \right)^{\prime} \left( {\user2{b}} - {\varvec{\xi}}_b\right) \right)} e^{-\frac{1}{2\sigma_\theta^2} \left(\sum_t \left( {\varvec{\theta}}_t - {\user2 {B}}{\varvec{\theta}}_{t-1} \right)^{\prime} \left( {\varvec{\theta}}_t - {\user2{B}}{\varvec{\theta}}_{t-1}\right) \right)} $$

In order to rewrite this as an expression of $\mathbf{b},$ we need to find an X × 5-matrix $\mathbf{J}_{\theta_{t-1}}$ so that

$$ {\mathbf{B}}{\varvec{\theta}}_{t-1} = {\mathbf{J}}_{\theta_{t-1}} {\mathbf{b}} $$

Since, ${\user2{B}} {\varvec{\theta}}_{t-1} = b_0{\varvec{\theta}}_{t-1} + b_N{\varvec{\theta}}_{t-1}^N + b_E{\varvec{\theta}}_{t-1}^E + b_S{\varvec{\theta}}_{t-1}^S + b_W{\varvec{\theta}}_{t-i}^W,$ it is easily seen that, if ${\user2{J}}_{\theta_{t-1}}$ is the matrix with columns ${\varvec{\theta}}_{t-1}, {\varvec\theta}_{t-1}^N,$ ${\varvec{\theta}}_{t-1}^E, {\varvec\theta}_{t-1}^S$ and ${\varvec{\theta}}_{t-1}^W$ respectively, the equation above holds, and by defining the ${\user2{J}}_D$-matrices, for D = N, E, S, W, to be the X × X matrices with ones along the diagonal corresponding to the b _D coefficients in ${\user2{B}}$ and zeros elsewhere then ${\varvec{\theta}}_{t-1}^{D} = {\user2 {J}}_{D}{\varvec{\theta}}_{t-1}$ and this can be written as

$$ {\user2{J}}_{\theta_{t-1}} = \left( {\varvec{\theta}_{t-1}, {\user2{J}}_N{\varvec{\theta}}_{t-1}, {\user2{J}}_E{\varvec{\theta}}_{t-1}, {\user2{J}}_S{\varvec{\theta}}_{t-1}, {\user2{J}}_W{\varvec{\theta}}_{t-1} }\right) $$

where now each of the elements in the vector above is actually an X-dimensional standing vector, defining the rows in the X × 5 dimensional matrix. Hence, completely parallel to the derivations above, the full joint conditional for $\user2{b}$ becomes

$$ {\user2{b}} | \cdot \sim MVN \left[ \bar{\user2{b}}, {\varvec{\Upsigma}}_{\user2{b}} \right] \quad \hbox {with} \,{\varvec{\Upsigma}}_{\user2{b}} = \left(\frac{1}{\sigma_b^2}{\user2{I}} + \frac{1}{\sigma_\theta^2} \sum_t {\user2{J}}_{\theta_{t-1}}^{\prime} {\user2{J}}_{\theta_{t-1}} \right)^{-1} \bar{\user2{b}} = {\varvec{\Upsigma}}_{\user2{b}} \left(\frac{1}{\sigma_b^2}{\varvec{\xi}}_b + \frac{1}{\sigma_\theta^2}\sum_t {\user2{J}}_{\theta_{t-1}}^{\prime}{\varvec{\theta}}_{t} \right) $$

1.7 A.6 Conditional distribution for a _ϕ and a _λ

The following is given by the model specification

$$ {\varvec{\mu}} | {\varvec{\mu}}_0, a_\phi, a_\lambda, \sigma_\mu^2 \sim MVN\left({\varvec{\mu}}_0, \sigma_\mu^2 {\user2 {A}}_\mu^{-1}\right) (a_\phi, a_\lambda) | \xi_a, \sigma_a^2 \sim BVN\left( {\user2{1}}\xi_a, {\user2{I}} \sigma_a^2 \right) $$

which gives the full conditional joint distribution of a _ϕ and a _λ

$$ P(a_\phi, a_\lambda | \cdot ) \propto e^{-\frac{1}{2\sigma_a^2}\left[\left(a_\phi - \xi_a \right)^2 + (a_\lambda - \xi_a)^2 \right]} \frac{1}{|{\user2 {A}}_\mu|^{-1/2}}e^{-\frac{1}{2\sigma_\mu^2} \left( {\varvec{\mu}} - {\varvec{\mu}}_0 \right)^{\prime} {\user2 {A}}_\mu \left( {\varvec{\mu}} - {\varvec{\mu}}_0 \right)} $$

Now, due to the appearance of the determinant $|{\user2 {A}}_\mu | ,$ which contains a _ϕ and a _λ, this distribution is difficult to sample from. Hence, parallel to what was done in Wikle et al. (2001) and Natvig and Tvete (2007), a Metropolis-Hastings step is introduced to sample from this distribution. As proposal distribution for the Metropolis-Hastings step, the pseudo conditional distribution for $\varvec{\mu}$ is introduced:

$$ P^{pseudo}({\varvec{\mu}} | {\varvec{\mu}}_0, a_\phi, a_\lambda, \sigma_\mu^2) \propto \prod_{i = 1}^X p(\mu(x_i) | \mu(x_j), j \neq i; {\varvec{\mu}}_0, a_\phi, a_\lambda, \sigma_\mu^2) $$

where now the μ(x _i )’s, conditional on all other μ(x _j ) are univariate normally distributed as specified by the model. Now, as above, one can obtain the following pseudo distribution $Q(a_\phi, a_\lambda | \cdot),$ where we denote $\Updelta\mu(x) = \mu(x)-\mu_0(x) {:}$

$$ Q(a_\phi, a_\lambda | \cdot) \propto e^{-\frac{1}{2} \left( \frac{1}{\sigma_a^2} + \frac{1}{\sigma_\mu^2}\sum_x \left( \Updelta\mu(x^N) + \Updelta\mu(x^S)\right)^2 \right) a_\phi^2 } \times e^{-\frac{1}{2} \left( \frac{1}{\sigma_a^2} + \frac{1}{\sigma_\mu^2}\sum_x \left( \Updelta\mu(x^E) + \Updelta\mu(x^W)\right)^2 \right) a_\lambda^2 } \times e^{a_\phi \left( \frac{1}{\sigma_a^2}\xi_a + \frac{1}{\sigma_\mu^2} \sum_x \left( \Updelta\mu(x^N) + \Updelta\mu(x^S)\right)\left(\Updelta\mu(x) - a_\lambda\left( \Updelta\mu(x^E) + \Updelta\mu(x^W)\right) \right) \right) } \times e^{a_\lambda \left( \frac{1}{\sigma_a^2}\xi_a + \frac{1}{\sigma_\mu^2} \sum_x \left( \Updelta\mu(x^E) + \Updelta\mu(x^W)\right)\left(\Updelta\mu(x) - a_\phi\left( \Updelta\mu(x^N) + \Updelta\mu(x^S)\right) \right) \right)} $$

When sampling a _ϕ and a _λ, the constraints on these to ensure positive definiteness of ${\user2 {A}}_\mu$ should be kept in mind, and a somewhat different procedure than in Natvig and Tvete (2007) will be adopted, i.e. samples are drawn jointly from a bivariate pseudo proposal distribution and if the conditions are fulfilled (both are positive and the sum not greater than 0.5) they are used in a Metropolis-Hastings step. It is observed that such proposals often have expectations outside the required area but since any proposal is valid, the pseudo-distribution may be adjusted by a shift that brings the expectation within the required area. Since samples are drawn simultaneously, no asymmetry is introduced by this, and this is one advantage by using the bivariate distribution instead of sampling the parameters sequentially as in Natvig and Tvete (2007).

A bivariate normal distribution can be written on the form, with ${\user2 {x}} = (x, y)$

$$ \begin{aligned} f(x, y) &\propto e^{-\frac{1}{2}\left[(x-\mu_x, y - \mu_y)\left( \begin{array}{ll} q_x & q_{xy} \\ q_{yx} & q_y \end{array} \right) \left( \begin{array}{l} x - \mu_x \\ y - \mu_y \end{array}\right) \right] } \\ & \propto e^{-\frac{1}{2} \left[x^2 q_x + y^2 q_y + 2xyq_{xy} - 2x(\mu_x q_x + \mu_y q_{xy}) - 2y(\mu_y q_y + \mu_x q_{xy}) \right] } \end{aligned} $$

where symmetry requires that q _xy  = q _yx . Now, comparing this expression with the expression for the bivariate pseudo-distribution for a _ϕ and a _λ above, it is seen that this has the following elements in its precision matrix ${\user2 {Q}}_{\phi, \lambda}:$

$$ \tilde{q}_\phi = \left( \frac{1}{\sigma_a^2} + \frac{1}{\sigma_\mu^2}\sum_x \left( \Updelta\mu(x^N) + \Updelta\mu(x^S)\right)^2 \right) \tilde{q}_\lambda =\left( \frac{1}{\sigma_a^2} + \frac{1}{\sigma_\mu^2}\sum_x \left( \Updelta\mu(x^E) + \Updelta\mu(x^W)\right)^2 \right) \tilde{q}_{\phi\lambda} =\frac{1}{\sigma_\mu^2} \sum_x \left( \Updelta\mu(x^N) + \Updelta\mu(x^S)\right)\left( \Updelta\mu(x^E) + \Updelta\mu(x^W)\right) $$

The variances and the covariance is found directly by taking the inverse of this matrix, and the variances and correlation becomes.

$$ \tilde{\sigma}^2_\phi = \frac{\tilde{q}_\lambda}{\tilde{q}_\lambda\tilde{q}_\phi - \tilde{q}_{\phi\lambda}^2}, \quad \tilde{\sigma}^2_\lambda = \frac{\tilde{q}_\phi}{\tilde{q}_\lambda\tilde{q}_\phi - \tilde{q}_{\phi\lambda}^2}, \quad \rho = \frac{-\tilde{q}_{\phi\lambda}}{\sqrt{\tilde{q}_\phi \tilde{q}_\lambda}} $$

The expectations may also be found giving the following expectations of the proposal distribution

$$ \tilde{\mu}_\phi = \frac{\tilde{q}_\lambda}{\tilde{q}_\phi \tilde{q}_\lambda - \tilde{q}_{\phi\lambda}^2} \left( \frac{1}{\sigma_a^2} \xi_a + \frac{1}{\sigma_\mu^2} \sum_x \left( \Updelta\mu(x^N) + \Updelta\mu(x^S)\right)\left(\Updelta\mu(x) \right) \right) -\frac{ \tilde{q}_{\phi\lambda}}{\tilde{q}_\phi \tilde{q}_\lambda - \tilde{q}_{\phi\lambda}^2} \left(\frac{1}{\sigma_a^2} \xi_a + \frac{1}{\sigma_\mu^2} \sum_x \left( \Updelta\mu(x^E) + \Updelta\mu(x^W)\right)\left(\Updelta\mu(x) \right)\right) \tilde\mu_\lambda = \frac{\tilde{q}_\phi}{\tilde{q}_\phi \tilde{q}_\lambda - \tilde{q}_{\phi\lambda}^2}\left(\frac{1}{\sigma_a^2} \xi_a + \frac{1}{\sigma_\mu^2} \sum_x \left( \Updelta\mu(x^E) + \Updelta\mu(x^W)\right)\left(\Updelta\mu(x) \right)\right) - \frac{ \tilde{q}_{\phi\lambda}}{\tilde{q}_\phi \tilde{q}_\lambda - \tilde{q}_{\phi\lambda}^2} \left( \frac{1}{\sigma_a^2} \xi_a + \frac{1}{\sigma_\mu^2} \sum_x \left( \Updelta\mu(x^N) + \Updelta\mu(x^S)\right)\left(\Updelta\mu(x) \right) \right) $$

This bivariate normal distribution can now be used as proposal in the Metropolis-Hastings step, which may be adjusted if the expectation of a _ϕ and a _λ is outside the area determined by the positive definiteness requirements.

1.8 A.7 Conditional distribution for M _t

From the model specification, the following distributions are given for each $t = 1, \ldots, T$

$$ {\user2 {Z}}_t | {\varvec{\mu}}, {\varvec{\theta}}_t, M_t, \hbox{T}_t, \sigma_Z^2 \sim MVN\left({\varvec{\mu}} + {\varvec{\theta}}_t + {\user2{1}}(M_t + \hbox{T}_t) , {\user2 {I}} \sigma_Z^2\right) M_t | c, d, \sigma_m^2 \sim N\left( c \cos(\omega t) + d \sin(\omega t), \sigma_m^2 \right) $$

and the full conditional becomes, for all t, the following univariate normal distribution

$$ M_t \sim N\left[ \bar{M}_t, \sigma_{M_t} \right] \quad \hbox {with}\, \sigma_{M_t} = \left( \frac{1}{\sigma_m^2} + \frac{X}{\sigma_Z^2} \right)^{-1} \bar{M}_t = \sigma_{M_t} \left( \frac{1}{\sigma_m^2} \left(c \cos(\omega t) + d \sin(\omega t) \right) + \frac{1}{\sigma_Z^2} {\user2{1}}^{\prime}\left({\user2{Z}}_t - {\varvec{\mu}} - {\varvec{\theta}}_t - {\user2{1}}\hbox{T}_t\right) \right) $$

1.9 A.8 Joint conditional distribution for c and d

Defining ${\varvec{\xi}}_{cd} = \left[ \begin{array}{l} \xi_c \\ \xi_d \end{array} \right], {\varvec{\Upsigma}}_{cd} = \left[ \begin{array}{ll} \sigma_c^2 & 0 \\ 0 & \sigma_d^2 \end{array} \right],$ ${\varvec{\beta}}_{cd} = \left[ \begin{array}{l} c \\ d \end{array} \right]$ and ${\varvec{\omega}}_{t} = \left[ \begin{array}{l} \cos(\omega t) \\ \sin(\omega t) \end{array} \right]$ the model specification gives, for $t = 1, \ldots , T,$

$$ (c, d) | \xi_c, \xi_d, \sigma_c^2, \sigma_d^2 \sim BVN \left( {\varvec{\xi}}_{cd} , {\varvec{\Upsigma}}_{cd} \right) M_t | c, d, \sigma_m^2 \sim N\left( {\varvec{\omega}}_{t}^{\prime} {\varvec{\beta}}_{cd}, \sigma_m^2 \right) $$

and it is straightforward to derive the full conditional for (c, d) as the bivariate normal distribution

$$ (c, d) | \cdot \sim BVN\left[{\user2{Q}}_M^{-1} \left[ \begin{array}{l} \frac{\xi_c}{\sigma_c^2} + \frac{1}{\sigma_m^2}\sum_tM_t \cos(\omega t) \\ \frac{\xi_d}{\sigma_d^2} + \frac{1}{\sigma_m^2}\sum_t M_t \sin(\omega t) \end{array} \right], {\user2{Q}}_M^{-1} \right] $$

with ${\user2{Q}}_M = \left[ \begin{array}{ll} \frac{1}{\sigma_c^2} + \frac{1}{\sigma_m^2}\sum_t(\cos( \omega t))^2 & \frac{1}{\sigma_m^2}\sum_t\cos( \omega t) \sin( \omega t) \\ \frac{1}{\sigma_m^2}\sum_t \cos( \omega t) \sin( \omega t) & \frac{1}{\sigma_d^2} + \frac{1}{\sigma_m^2} \sum_t (\sin (\omega t) )^2 \end{array} \right]$

1.10 A.9 Conditional distribution for T _t

From the model specification, the following distributions are given for each $t = 1, \ldots, T$

$${\user2{Z}}_t | {\varvec{\mu}}, {\varvec{\theta}}_t, M_t, \hbox{T}_t, \sigma_Z^2 \sim MVN\left({\varvec{\mu}} + {\varvec{\theta}}_t + {\user2{1}}(M_t + \hbox{T}_t ) , {\user2{I}} \sigma_Z^2\right)\hbox{T}_t | \gamma, \eta, \sigma_{\rm{T}}^2 \sim N\left( \gamma t + \eta t^2, \sigma^2_{\rm{T}} \right) $$

and the full conditional for T becomes, for all t, 

$$ \hbox{T}_t \sim N \left[ \bar{\hbox{T}}_t, \sigma_{\rm{T}_t} \right] \quad\hbox {with } \sigma_{\rm{{T}_t}} = \left(\frac{1}{\sigma_{\rm{T}}^2} + \frac{X}{\sigma_{Z}^2} \right)^{-1} \bar{\hbox{T}}_t = \sigma_{\rm{{T}_t}} \left( \frac{1}{\sigma_{\rm{T}}^2}(\gamma t + \eta t^2) + \frac{1}{\sigma_Z^2}{\user2{1}}^{\prime}({\user2{Z}}_t - {\varvec{\mu}} - {\varvec{\theta}}_t - {\user2{1}}M_t ) \right) $$

1.11 A.10 Joint conditional distribution for γ and η

Defining ${\varvec{\xi}}_{\gamma \eta} = \left[ \begin{array}{l} \xi_\gamma \\ \xi_\eta \end{array} \right], {\varvec{\Upsigma}}_{\gamma\eta} = \left[ \begin{array}{ll} \sigma_\gamma^2 & 0 \\ 0 & \sigma_\eta^2 \end{array} \right], {\varvec{\beta}}_{\gamma\eta} = \left[ \begin{array}{l} \gamma \\ \eta \end{array} \right]$ and ${\varvec{\tau}}_{t} = \left[ \begin{array}{c} t \\ t^2 \end{array} \right]$ the model specification gives

$$ (\gamma, \eta) | \xi_\gamma, \xi_\eta, \sigma_\gamma^2, \sigma_\eta^2 \sim BVN \left( {\varvec{\xi}}_{\gamma\eta} , {\varvec{\Upsigma}}_{\gamma\eta} \right) \hbox{T}_t | \gamma, \eta, \sigma_{\rm{T}}^2 \sim N\left( \gamma t + \eta t^2, \sigma^2_{\rm{T}}\right) = N\left( {\varvec{\tau}}_{t}^{\prime} {\varvec{\beta}}_{\gamma\eta}, \sigma_{\rm{T}}^2 \right) , \quad \hbox {for} \,t = 1, \ldots , T $$

and the full conditional for (γ, η) becomes,

$$ (\gamma, \eta) | \cdot \sim BVN\left[ {\user2{Q}}_{\rm{T}}^{-1} \left[ \begin{array}{l} \frac{\xi_\gamma}{\sigma_\gamma^2} + \frac{1}{\sigma_{\rm{T}}^2}\sum_t \hbox{T}_t t \\ \frac{\xi_\eta}{\sigma_\eta^2} + \frac{1}{\sigma_{\rm{T}}^2}\sum_t \hbox{T}_t t^2 \end{array} \right] , {\user2{Q}}_{\rm{T}}^{-1} \right] $$

with precision matrix ${\user2{Q}}_{\rm{T}} = \left[ \begin{array}{ll} \frac{1}{\sigma_\gamma^2} + \frac{1}{\sigma_{\rm{T}}^2}\sum_t t^2 & \frac{1}{\sigma_{\rm{T}}^2}\sum_t t^3 \\ \frac{1}{\sigma_{\rm{T}}^2}\sum_t t^3 & \frac{1}{\sigma_\eta^2} + \frac{1}{\sigma_{\rm{T}}^2} \sum_t t^4 \end{array} \right] .$

1.12 A.11 Conditional distribution for σ ² _Z

From the model and prior specification, the following are given

$$ {\user2{Z}}_t | {\varvec{\mu}}, {\varvec{\theta}}_t, M_t, \gamma, \eta, \sigma_Z^2 \sim MVN\left({\varvec{\mu}} + {\varvec{\theta}}_t + {\user2{1}}(M_t + \hbox{T}_t ), {\user2 {I}} \sigma_Z^2\right) \sigma_Z^2 | \alpha_{Z}, \beta_{Z} \sim IG(\alpha_{Z}, \beta_{Z}) $$

and parallel to the above, the full conditional distribution becomes another inverse gamma distribution with updated parameters. Denoting ${\varvec{\Updelta}} = \left({\user2{Z}}_t - {\varvec{\mu}} -{\varvec{\theta}}_t - {\user2{1}}(M_t + \hbox{T}_t)\right),$ gives

$$ \sigma_Z^2 | \cdot \sim IG\left( \alpha_Z + \frac{X T}{2} , \beta_Z + \frac{1}{2}\sum_t {\varvec{\Updelta^{\prime}\Updelta}} \right) $$

1.13 A.12 Conditional distribution for σ ²_μ

Similarly, the model and prior specification gives

$$ {\varvec{\mu}} | {\varvec{\mu}}_0, a_\phi, a_\lambda, \sigma_\mu^2 \sim MVN\left({\varvec{\mu}}_0, \sigma_\mu^2 {\user2 {A}}_\mu^{-1}\right)\sigma_\mu^2 | \alpha_\mu, \beta_\mu \sim IG \left( \alpha_\mu, \beta_\mu \right) $$

Hence,

$$ \sigma_\mu^2 | \cdot \sim IG\left( \alpha_\mu + \frac{X}{2}, \beta_\mu + \frac{1}{2}({\varvec{\mu}} - {\varvec{\mu}}_0)^{\prime}{\user2 {A}}_\mu ({\varvec{\mu}} - {\varvec{\mu}}_0) \right) $$

1.14 A.13 Conditional distribution for σ ²_θ

From the model and prior specification, the following is given for $t = 1, \ldots, T$

$$ {\varvec{\theta}}_t | b_0, {\varvec{\theta}}_{t-1}, b_N, b_E, b_S, b_W, \sigma^2_\theta \sim MVN\left({\user2 {B}} {\varvec{\theta}}_{t-1} , {\user2 {I}} \sigma_\theta^2 \right) \sigma_\theta^2 | \alpha_\theta, \beta_\theta \sim IG(\alpha_\theta, \beta_\theta) $$

Hence,

$$ \sigma_\theta^2 | \cdot \sim IG \left( \alpha_\theta + \frac{X T}{2}, \beta_\theta + \frac{1}{2}\sum_t \left({\varvec{\theta}}_t - {\user2 {B}} {\varvec{\theta}}_{t-1} \right)^{\prime} \left( {\varvec{\theta}}_t - {\user2 {B}} {\varvec{\theta}}_{t-1}\right) \right) $$

1.15 A.14 Conditional distribution for σ ² _m

The following is specified by the model, for all $t = 1, \ldots , T,$

$$ M_{t} | c, d, \sigma_m^2 \sim N\left( c \cos(\omega t) + d \sin(\omega t), \sigma_m^2 \right) \sigma_m^2 | \alpha_m, \beta_m \sim IG(\alpha_m, \beta_m) $$

and thus,

$$ \sigma_m^2 | \cdot \sim IG\left( \alpha_m + \frac{T}{2}, \beta_m + \frac{1}{2}\sum_t\left(M_t - c \cos(\omega t) - d \sin(\omega t) \right)^2 \right) $$

1.16 A.15 Conditional distribution for σ ²_T

Finally, the model and prior specification, gives the following for $t = 1, \ldots, T$

$$ \hbox{T}_t | \gamma, \eta, \sigma_{\rm{T}}^2 \sim N\left( \gamma t + \eta t^2, \sigma_{\rm{T}}^2 \right) \sigma_{\rm{T}}^2 | \alpha_t, \beta_t \sim IG(\alpha_t, \beta_t) $$

Hence, finally

$$ \sigma_{\rm{T}}^2 | \cdot \sim IG\left( \alpha_t + \frac{T}{2}, \beta_t + \frac{1}{2}\sum_t\left(\hbox{T}_t - \gamma t - \eta t^2 \right)^2 \right) $$