class: center, middle, inverse, title-slide # Marine mammal exposure to Navy sonar: a continuous-time discrete-space model ##
vISEC2020
### Charlotte M. Jones-Todd
@cmjonestodd
### University of Auckland .typed[] ###
.small[Charlotte Dunn] --- background-image: url("img/tag.jpg") .footnote[
Charlotte Dunn] .bold[ - Blainville’s beaked whales tagged between 2009---2015 (Atlantic Undersea Test and Evaluation Center (AUTEC), Bahamas) - Region regularly used by the US Navy to carry out military exercises with active sonar - Tags attached on or near the dorsal fin - Location estimates of tagged whales were provided by the Argos system - Sonar information from records in the US Navy’s SPORTS database ] --- ### Vote in Slack: hit 🏃 if you use a running App -- .center[ <img src="img/run.png" width="829" height="500" /> ] --- .center[ <img src="img/whale.gif" height="600" /> ] --- ### Continuous-time correlated random walk model (CTCRW) For each coordinate `\(c = 1,2\)` of the observed location of an animal at time `\(t\)` ( `\(t = 1,2,...,n\)` ): <br> .footnote[Fitted using the **R** package **argosTrack** Albertsen, C. M. (2017). **argosTrack**: Fit Movement Models to Argos Data for Marine Animals. R package version 1.1.0.] -- - instantaneous velocity of the animal `\(v_{ct}\)` is described by a Ornstein-Uhlenbeck process; -- <br> - integrating over the velocity process gives the location process `\(\mu_{ct}\)`; -- <br> - measurement equation is given by `\(y_{ct} = \mu_{ct} + \epsilon_{ct}\)`, where `\(y_{ct}\)` is the `\(c\)`th coordinate of the observed location of an animal at time `\(t\)` with measurement error term `\(\epsilon_{ct}\)`. The joint distribution of `\(\epsilon_{1t}\)` and `\(\epsilon_{2t}\)` is a bivariate `\(t\)`-distribution. --- ### Continuous-time correlated random walk model (CTCRW) .center[ <img src="img/track_example.png" width="640" height="500" /> ] .footnote[Fitted using the **R** package **argosTrack** Albertsen, C. M. (2017). **argosTrack**: Fit Movement Models to Argos Data for Marine Animals. R package version 1.1.0.] --- ### Discrete-space continuous-time A continuous-time Markov model describes how an individual transitions between states in continuous time. .center[ <img src="visec2020_files/figure-html/markov-1.png" width="600" /> ] --- ### Discrete-space continuous-time <br> Let `\(q_{rs}(t,z(t))\)` represent the immediate risk of moving from one state `\(r\)` to another state `\(s\)`: <br> `$$q_{rs}(t,z(t)) = \text{lim}_{\delta t \rightarrow 0} \mathbb{P}(S(t + \delta t) = s|S(t) = r)/\delta t.$$` <br> These transition rates form a square matrix `\(\bf{Q}\)` with elements `\(q_{rs}\)`. -- <br> Here `\(q_{rr}=-\Sigma_{s{\neq}r}q_{rs}\)` (i.e., the rows of **Q** sum to zero and `\(q_{rs}\geq0\)` for `\(r{\neq}s\)`). --- ### Discrete-space continuous-time <br> We have `\(r, s = \{1,2\}\)` where state `\(1=\)` off-range (i.e., outside the area used by the Navy for military operations) and state `\(2=\)` on-range: <br> <br> `$$\begin{array}{ccc} \bf{Q} = \left [\begin{array}{cc} q_{1 1} & q_{1 2} \\ q_{2 1} & q_{2 2} \end{array}\right ] & \text{where} \: q_{r r} = -q_{r s}, & \text{for}\: r \neq s. \end{array}$$` --- ### Including exposure information We let <br> `$$\text{log}(q_{k, rs}(\mathbf{z}_k(t))) = (\beta_{0,rs} + u_{k, rs}) + \beta_{1,rs}\text{exp}(- \beta_{2,rs} \mathbf{z}_k(t)),$$` where `$$\mathbf{z}_k(t) \left\{ \begin{array}{rl} = 0 & \text{during exposure} \\ \geq 0 & \text{otherwise}\end{array} \right.$$` is the number of days since an individual was exposed to a sonar event and `\(\beta_{2,rs} \geq 0\)` `\(\forall\: r \neq s\)`. --- ### Including exposure information We let <br> `$$\text{log}(q_{k, rs}(\mathbf{z}_k(t))) = (\color{red}{\beta_{0,rs}} + u_{k, rs}) + \beta_{1,rs}\text{exp}(- \beta_{2,rs} \mathbf{z}_k(t)),$$` where `$$\mathbf{z}_k(t) \left\{ \begin{array}{rl} = 0 & \text{during exposure} \\ \geq 0 & \text{otherwise}\end{array} \right.$$` is the number of days since an individual was exposed to a sonar event and `\(\beta_{2,rs} \geq 0\)` `\(\forall\: r \neq s\)`. <br> .red[some baseline transition rate] --- ### Including exposure information We let <br> `$$\text{log}(q_{k, rs}(\mathbf{z}_k(t))) = (\beta_{0,rs} + u_{k, rs}) + \color{red}{\beta_{1,rs}}\text{exp}(- \beta_{2,rs} \mathbf{z}_k(t)),$$` where `$$\mathbf{z}_k(t) \left\{ \begin{array}{rl} = 0 & \text{during exposure} \\ \geq 0 & \text{otherwise}\end{array} \right.$$` is the number of days since an individual was exposed to a sonar event and `\(\beta_{2,rs} \geq 0\)` `\(\forall\: r \neq s\)`. <br> .red[change in transition rate during exposure] --- ### Including exposure information We let <br> `$$\text{log}(q_{k, rs}(\mathbf{z}_k(t))) = (\beta_{0,rs} + u_{k, rs}) + \beta_{1,rs}\text{exp}(\color{red}{- \beta_{2,rs}} \mathbf{z}_k(t)),$$` where `$$\mathbf{z}_k(t) \left\{ \begin{array}{rl} = 0 & \text{during exposure} \\ \geq 0 & \text{otherwise}\end{array} \right.$$` is the number of days since an individual was exposed to a sonar event and `\(\beta_{2,rs} \geq 0\)` `\(\forall\: r \neq s\)`. <br> .red[exponential decay of transition rates towards some baseline] --- ### Including exposure information We let <br> `$$\text{log}(q_{k, rs}(\mathbf{z}_k(t))) = (\beta_{0,rs} + \color{red}{u_{k, rs}}) + \beta_{1,rs}\text{exp}(- \beta_{2,rs} \mathbf{z}_k(t)),$$` where `$$\mathbf{z}_k(t) \left\{ \begin{array}{rl} = 0 & \text{during exposure} \\ \geq 0 & \text{otherwise}\end{array} \right.$$` is the number of days since an individual was exposed to a sonar event and `\(\beta_{2,rs} \geq 0\)` `\(\forall\: r \neq s\)`. <br> .red[individual level random effect] --- ### Likelihood The transition probability matrix `\(\textbf{P}(t) = \text{Exp}(t\textbf{Q}).\)` <br> The likelihood, `\(L(\bf{Q})\)`, is calculated as the product over all individuals and all transitions, of the probabilities that individual `\(k\)` is in state `\(S(t_{j + 1})\)` at time `\(t_{j+1}\)` given they were in state `\(S(t_j)\)` at time `\(t_j\)`, evaluated at time `\(t_{j + 1} - t_j\)` (for `\(j = 1,...,n_k\)`): <br> `$$L(\boldsymbol{Q}) = \prod_{k, j} L_{k,j} = \prod_{k, j} p_{S(t_j)S(t_{j + 1})}(t_{j + 1} - t_j).$$` .footnote[Parameter estimates are obtained via minimisation of the negative log-likelihood, log(L(**Q**)), using Template Model Builder (TMB).] --- .center[ <img src="img/autec_results.png" width="600" height="600" /> ] --- .panelset[ .panel[.panel-name[93232] .center[ <br> <br> <img src="img/whale1.png" width="80%" /> ] ] .panel[.panel-name[11164] .center[ <br> <br> <img src="img/whale2.png" width="80%" /> ] ] .panel[.panel-name[111670] .center[ <br> <br> <img src="img/whale3.png" width="80%" /> ] ] .panel[.panel-name[129715] .center[ <br> <br> <img src="img/whale4.png" width="80%" /> ] ] .panel[.panel-name[129719] .center[ <br> <br> <img src="img/whale5.png" width="80%" /> ] ] .panel[.panel-name[129720] .center[ <br> <br> <img src="img/whale6.png" width="80%" /> ] ] .panel[.panel-name[129721] .center[ <br> <br> <img src="img/whale7.png" width="80%" /> ] ] ] --- ### Diolch am wrando .pull-left[ .animate__animated.animate__bounceInDown[ ``` ## -------------- ## Any Questions? ## -------------- ## \ ## \ ## \ ## ## __ \ / __ ## / \\ | / \\ ## \\|/ ## _.---v---.,_ ## / \\ /\\__/\\ ## / \\ \\_ _/ ## |__ @ |_/ / ## _/ / ## \\ \\__, / ## ~~~~\\~~~~~~~~~~~~~~`~~~[EOC] ``` ] ] .pull-right[ .center[.large[
@cmjonestodd]] <hr> *Continuous-time discrete-space models of marine mammal exposure to Navy sonar.* (under review) Jones-Todd, C. M., Pirotta, E., Baird, R. W., Durban, J. W., Falcone, E. A., Joyce, T. W., Schorr, G. S., Watwood, S., & Thomas, L. <hr> .center[.card[ ![](img/visec2020.png) [cmjt.github.io/slides/visec2020](https://cmjt.github.io/slides/visec2020) ] ] ]