Scale-insensitive estimation of speed and distance traveled from animal tracking data

Background Speed and distance traveled provide quantifiable links between behavior and energetics, and are among the metrics most routinely estimated from animal tracking data. Researchers typically sum over the straight-line displacements (SLDs) between sampled locations to quantify distance traveled, while speed is estimated by dividing these displacements by time. Problematically, this approach is highly sensitive to the measurement scale, with biases subject to the sampling frequency, the tortuosity of the animal’s movement, and the amount of measurement error. Compounding the issue of scale-sensitivity, SLD estimates do not come equipped with confidence intervals to quantify their uncertainty. Methods To overcome the limitations of SLD estimation, we outline a continuous-time speed and distance (CTSD) estimation method. An inherent property of working in continuous-time is the ability to separate the underlying continuous-time movement process from the discrete-time sampling process, making these models less sensitive to the sampling schedule when estimating parameters. The first step of CTSD is to estimate the device’s error parameters to calibrate the measurement error. Once the errors have been calibrated, model selection techniques are employed to identify the best fit continuous-time movement model for the data. A simulation-based approach is then employed to sample from the distribution of trajectories conditional on the data, from which the mean speed estimate and its confidence intervals can be extracted. Results Using simulated data, we demonstrate how CTSD provides accurate, scale-insensitive estimates with reliable confidence intervals. When applied to empirical GPS data, we found that SLD estimates varied substantially with sampling frequency, whereas CTSD provided relatively consistent estimates, with often dramatic improvements over SLD. Conclusions The methods described in this study allow for the computationally efficient, scale-insensitive estimation of speed and distance traveled, without biases due to the sampling frequency, the tortuosity of the animal’s movement, or the amount of measurement error. In addition to being robust to the sampling schedule, the point estimates come equipped with confidence intervals, permitting formal statistical inference. All the methods developed in this study are now freely available in the ctmmR package or the ctmmweb point-and-click web based graphical user interface.


RMS speed
The RMS speed is easily related to the velocity variance where for a stationary process, v RMS = √ tr σ, or v RMS (t) = µ(t) 2 + tr σ(t) instantaneously.

Mean speed
The mean speed is derived from the mean absolute deviation |v| , which is difficult to calculate in general. First we will derive the mean speed under the assumption of µ = 0, which is sufficient for the time-average of a stationary process. Next we will derive the mean speed for a symmetric covariance matrix σ. Finally, we will combine these exact results into an approximate formula for the general case.

Mean speed: zero mean
If µ = 0, then in two dimensions the mean speed is determined by the integral where u(θ) = (cos θ, sin θ) is the unit vector. This relation can be simplified by rotating our polar coordinate system so that θ = 0 occurs when u(θ) is parallel with one of the eigen-vectors of the velocity covariance. Let σ ± represent the two eigen-values of σ, where σ + ≥ σ − . The mean speed integral then reduces to where E(k) is the complete elliptic integral of the second kind. In the isotropic case where σ ± = σ 0 and E(0) = π/2, the mean speed reduces to π/2 σ 0 .

Mean speed: zero eccentricity
If σ = σ 0 I, then the two-dimensional mean speed is given by the integral where I m (z) the m th order modified Bessel function of the first kind. The remaining integral resolves to Mean speed: combined approximation To construct a generally applicable approximation, we compare the combined limits of zero mean and eccentricity, which results in a mean speed of π/2 σ 0 . Relation (11) generalizes this result to non-zero eccentricity, while relation (15) generalizes this result to non-zero mean. In (15), one can identify the mean-zero limit as a factor of the first term, while the second term contains the large mean limit. Therefore we combine the two results directly to obtain the approximate relation 2σ 0 . (16) where σ 0 = (σ + + σ − )/2. This result is exact in three limits: µ 2 → 0, µ → ∞, and σ + → σ − . χ 2 and χ confidence intervals In this appendix, we detail how we translate point estimates and standard errors into non-standard confidence intervals, which can be more appropriate than normal confidence intervals if the sampling distribution more resembles another. All of these confidence intervals obey the central limit theorem and share the same first two moments or cumulants.
If a statistic X is proportionally χ 2 k , then its mean and variance obey the relation and so the degrees of freedom are given by We use CIs derived from this distribution on square speed estimates, because they are exact in some cases then.

χ confidence intervals
If a statistic X is proportionally χ 1 k , then its mean and variance obey the relation where B denotes the beta function. We use CIs derived from this distribution on mean speed estimates, because they are exact in some cases. To solve k = f −1 (R), we expand about a numerical estimate k i and update to get where ψ(z) = ψ 0 (z) is the digamma function. For the initial numerical estimate, we use the asymptotic expansion of the ratio of the variance to the square mean which is analogous to the χ 2 relation and well behaved.