Fig 1.
Directed acyclic graph demonstrating the Bayesian state-space model (SSM) approach developed in this paper.
The unobserved position at time step n is generated following a combination of movement parameters (the process model) of the fish:
(position of the center of the home range), k and radius, and depends in the previous position
at n-1. The observed data (number of detection, ND) at time step n consists in the number of detections over n by each of the omnidirectional receivers (j in R). Note that ND at n is independent of the ND at n-1 and is generated using the probability of detection by receiver j at n time unit (PDj,n) determined by a logit function (with parameters α and β at the n time) of the distance (d at n) between the (unobserved) fish position and the (known) receiver position at the n time (observational model). The parameters of the state-space model (movement parameters) were estimated using a Bayesian approach.
Table 1.
Characteristics of the temporal series of acoustic detections considering four combinations of movement parameters (sim 1, sim 2, sim 3 and sim 4) to test the performance and accuracy of the Bayesian state-space model developed here.
The table shows the number of days of the time series and the total number, mean and s.d. of detections generated considering a time-step period of 15 min. The table also shows the specific values of the movement parameters simulated (exploration rate of the home range k min-1, the radius in meters of the circular home range, and the latitude and longitude in meters of the center of the home range).
Fig 2.
Simulated data for testing the feasibility and accuracy of our approach.
Discrete-time trajectories of the four movement parameters combinations (sim 1, sim 2, sim 3 and sim 4, Table 1) generated for 6 different time-steps periods in min (5, 10, 15, 30, 60 and 90 min) to test the accuracy and the performance of the analytical approach. The resulting numbers of acoustic detections were obtained through the simulation experiment for the 24 movement trajectories, and the movement parameters and positions were estimated using a Bayesian tate-space model proposed here. Results were compared with the (known) real values.
Fig 3.
Simulated and real probability of detection in function of the distance.
Daily (n = 10 days) probability of detections (logit function with parameters α and β) against the distance considered in the simulated exercise (upper panel) and real-data (down panel) implemented in the observational model of the Bayesian state-space model described in Fig 1. Note the low variability observed in our case study. The variability in the logit models can be easily modified and adapted to any other case as the parameters of the model (α and β) can be estimated (using a control tag) and included in the analytical approach for each time step (see Material and Methods).
Table 2.
Characteristics of the time-series of acoustic detections obtained from 6 individual of pearly razorfish, Xyrichtys novacula tracked in the waters of Mallorca Island (NW Mediterranean).
The table shows the characteristics of the individuals tracked, the acoustic tag identification (ID) and the gender and the total length in mm of the individual. The time-steps defined the number of 15 min periods (n), the total detections shows the overall number of detection received by a given individual, and mean and standard deviation (s.d.) of detections shows the average number of detection per time-step generated for each individual tracked.
Fig 4.
Estimated and real movement parameters resulted from the simulation experiment.
Estimated Bayesian Credibility Intervals (BCI, 2.5% and 97.5% as point range in red) using the Bayesian state-space model proposed here and real values (as a horizontal dashed blue line) of the movement parameters (latitude and longitude in meters of the center of the home range, k in min-1 and radius in meters) in each of the combinations of different time-steps considered here (5, 10 15, 30, 60 and 90 min) and four simulations (overall 24 simulated trajectories). In most cases the estimated BCI included the real value suggesting good performance of the model. Only the estimated values for the time-steps periods 30, 60 and 90 min in simulation 3 and 4 were consistently biased, suggesting a poor performance for this particular type of fish movement.
Fig 5.
Estimated and real fish trajectories resulted from the simulation experiment.
First six days of the estimated (in blue) and real (in red) discrete-time trajectories of the 24 simulated trajectories. The estimated trajectory corresponds to the Bayesian mean, and the error is represented in the figure as a density plot of 100 trajectories re-sampled from the posterior distribution generated by the state-space model.
Table 3.
Percentages of agreement (% of replicas where the estimated Bayesian Credibility Interval, BCI, 2.5% and 97.5% included the true-known value) for each movement parameter obtained from the second series of simulation experiments based in 50 replications of sim 1 and sim 4 considering a time-step of 15 and 30 min.
Table 4.
Posterior distributions of the Bayesian state-space model fitted to the temporal series of acoustic detection generated by 6 individuals of pearly razorfish, Xyrichtys novacula, tracked in 2011 to estimate the home range movement parameters (the exploration rate of the home range k, in min-1), the radius of the circular home range (in m) and the latitude and the longitude in UTM).
The table shows the Bayesian mean and the uncertainty associated with the movement parameters through the Bayesian Credibility Interval (BCI, 2.5% and 97.5%).
Fig 6.
Estimated (plus uncertainty) movement parameters in case study applied to pearly reazorfish Xyrichtys novacula, using a Bayesian state-space model.
Estimated Bayesian Credibility Interval (BCI, 2.5% and 97.5% as point range in red) of the movement parameters using a Bayesian state-space model for discrete-time (based in a time-step of 15 min) trajectories of 6 individuals of pearly razorfish, Xyrichtys novacula tracked in 2011 using an array of omnidirectional receivers in the waters of Mallorca Island (NW Mediterranean).
Fig 7.
Estimated trajectories in case study of pearly razorfish, Xyrichtys novacula, using a Bayesian state-space model.
Estimated trajectory using a Bayesian states-space model for discrete-time (based in a time-step of 15 min) trajectories of six individuals of pearly razorfish, Xyrichtys novacula (the picture shows an image of the species) tracked in 2011 using an array of omnidirectional receivers in the waters of Mallorca Island (NW Mediterranean). The plot shows the continuous path and the estimated positions as points in latitude and longitude (UTM). The trajectories have been centered to the same center of the home range to improve visualization.