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Table 1 Simulation findings comparing the azimuthal telemetry model (ATM) with the average of the component-wise azimuth intersections (simple) and Lenth (1981) maximum likelihood estimator (Lenth, sigloc) and M-estimators (Andrews, Huber)

From: Accounting for location uncertainty in azimuthal telemetry data improves ecological inference

   κ=100 κ=25
   Simple sigloc Lenth Huber Andrews ATM Simple sigloc Lenth Huber Andrews ATM
Encircle design comparison:
nθ=3 \(n_{\hat {\boldsymbol {\mu }}}\) 600 597 600 600 600 600 600 581 598 600 600 600
  d0.5 (m) 22.4 20.8 20.5 20.4 20.4 21.0 45.3 43.6 42.7 42.8 43.5 42.8
  Coverage 0.430 0.432 0.432 0.433 0.888 0.422 0.425 0.423 0.435 0.858
nθ=4 \(n_{\hat {\boldsymbol {\mu }}}\) 600 539 600 600 600 600 600 470 592 595 599 600
  d0.5 (m) 9.9 9.3 8.7 8.7 8.8 8.7 19.2 19.6 17.5 17.6 17.4 17.5
  Coverage 0.575 0.592 0.585 0.592 0.923 0.553 0.542 0.538 0.541 0.917
Random design comparison:
nθ=3 \(n_{\hat {\boldsymbol {\mu }}}\) 600 533 595 593 593 600 600 439 564 561 566 600
  d0.5 (m) 32.6 32.2 25.1 25.1 25.3 25.0 62.9 75.1 54.2 53.4 53.7 55.6
  Coverage 0.403 0.418 0.417 0.417 0.883 0.328 0.348 0.348 0.352 0.850
nθ=4 \(n_{\hat {\boldsymbol {\mu }}}\) 600 454 594 594 598 600 600 367 573 573 579 600
  d0.5 (m) 14.2 13.0 9.9 10.0 9.9 10.0 25.3 34.0 19.5 19.6 20.0 20.3
  Coverage 0.559 0.581 0.567 0.572 0.920 0.526 0.560 0.550 0.556 0.912
Road design comparison:
nθ=3 \(n_{\hat {\boldsymbol {\mu }}}\) 600 499 593 593 597 600 600 409 573 571 576 600
  d0.5 (m) 56.7 44.4 39.0 39.0 38.5 40.5 95.5 110.4 85.1 84.6 83.9 86.4
  Coverage 0.397 0.418 0.418 0.412 0.877 0.296 0.316 0.310 0.312 0.822
nθ=4 \(n_{\hat {\boldsymbol {\mu }}}\) 600 443 600 600 600 600 600 316 592 593 595 600
  d0.5 (m) 53.8 33.4 26.9 27.7 28.1 26.5 90.3 83.5 54.6 54.8 55.2 55.8
  Coverage 0.580 0.618 0.595 0.588 0.923 0.487 0.561 0.543 0.545 0.883
  1. ‘sigloc’ implements Lenth’s maximum likelihood estimator via an alternative algorithm than suggested by Lenth (1981)
  2. Notes: Random, encircle, and road indicate different telemetry study designs. κ is the concentration parameter of the von Mises distribution and controls the amount of azimuthal uncertainty; larger values indicate higher precision. We use κ=100 as moderate uncertainty and a κ=25 for high uncertainty. nθ indicates the number of observer locations used for each spatial animal location. \(n_{\boldsymbol {\hat {\mu }}}\) is the number of animal spatial location estimates that were appropriately estimated; we simulated a total of 600 locations per scenario. d0.5 is the median of the Euclidean distance between the estimated animal location and true location (\(d(\hat {\boldsymbol {\mu }},\boldsymbol {\mu })\)). Coverage is defined as the number of 95% isopleths that contained the true μ out of the total \(n_{\hat {\boldsymbol {\mu }}}\)