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 1 Noonan et al.
 2 
 3 
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 5 METHODOLOGY
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 7 
 8 
 9 Scale-insensitive estimation of speed and distance
10 
11 traveled from animal tracking data
12 
13 
Michael J Noonan1,2*†, Christen H Fleming1,2†, Thomas S Akre114 , Jonathan Drescher-Lehman
1,3, Eliezer
Gurarie215 , Autumn-Lynn Harrison
4, Roland Kays5,6 and Justin M Calabrese1,2
16 
17 
*
18 Correspondence:
NoonanM@si.edu
19 1Smithsonian Conservation
20 Biology Institute, National
21 Zoological Park, 1500 Remount Abstract
22 Rd, 22630 Front Royal, USA
Full list of author information is
23 Background: Speed and distance traveled provide quantifiable links betweenavailable at the end of the article
24 †Equal contributors behavior and energetics, and are among the metrics most routinely estimated
25 from animal tracking data. Researchers typically sum over the straight-line
26 displacements (SLDs) between sampled locations to quantify distance travelled,
27 while speed is estimated by dividing these displacements by time. Problematically,
28 this approach is highly sensitive to the measurement scale, with biases subject to
29 the sampling frequency, the tortuosity of the animal’s movement, and the amount
30 of measurement error. Compounding the issue of scale-sensitivity, SLD estimates
31 do not come equipped with confidence intervals to quantify their uncertainty.
32 Methods: To overcome the limitations of SLD estimation, we outline a
33 
continuous-time speed and distance (CTSD) estimation method. An inherent
34 
35 property of working in continuous-time is the ability to separate the underlying
36 continuous-time movement process from the discrete-time sampling process,
37 making these models less sensitive to the sampling schedule when estimating
38 parameters. The first step of CTSD is to estimate the device’s error parameters
39 to calibrate the measurement error. Once the errors have been calibrated, model
40 selection techniques are employed to identify the best fit continuous-time
41 movement model for the data. A simulation-based approach is then employed to
42 sample from the distribution of trajectories conditional on the data, from which
43 the mean speed estimate and its confidence intervals can be extracted.
44 Results: Using simulated data, we demonstrate how CTSD provides accurate,
45 scale-insensitive estimates with reliable confidence intervals. When applied to
46 empirical GPS data, we found that SLD estimates varied substantially with
47 sampling frequency, whereas CTSD provided relatively consistent estimates, with
48 
49 often dramatic improvements over SLD.
50 Conclusions: The methods described in this study allow for the computationally
51 efficient, scale-insensitive estimation of speed and distance traveled, without
52 biases due to the sampling frequency, the tortuosity of the animal’s movement, or
53 the amount of measurement error. In addition to being robust to the sampling
54 schedule, the point estimates come equipped with confidence intervals,
55 permitting formal statistical inference. All the methods developed in this study
56 are now freely available in the ctmm R package or the ctmmweb point-and-click
57 web based graphical user interface.
58 
59 Keywords: Continuous-time; correlated velocity; ctmm; GPS; movement models;
60 step length; telemetry; travel distance
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 1 Noonan et al. Page 2 of 21
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 6 Background
 7 Understanding how far animals must travel to meet their nutritional and/or re-
 8 productive requirements, as well as the rate at which these distances are covered,
 9 are fundamental components of ecological research [1, 2]. Collectively, speed- and
10 
distance-related movement metrics provide quantifiable links between behavior and
11 
12 energetics [1, 3, 4, 5, 6], can inform on risk/reward tradeoffs (sensu Charnov [7]),
13 and can be important signals for the extent of anthropogenic disturbance [8, 9].
14 Accurately quantifying variations in an animal’s movement speed over time can
15 also enable explorations into the behavioral mechanisms animals use to navigate
16 
17 their environment [10]. For instance, when individuals exhibit area restricted search
18 (sensu Kareiva [11]), they are expected to slow down and move more tortuously in
19 areas of high resource density, and speed up and move more ballistically in areas of
20 low resource density (see also [12]).
21 
Animal tracking data are becoming an increasingly important resource for ad-
22 
23 dressing these questions [13], with distance traveled typically being quantified by
24 summing the straight-line displacement (SLD) between discretely sampled locations
25 [14, 15, 16, 17]. Similarly, dividing this value by the time elapsed between location
26 observations is used to estimate an animal’s speed (but see the instantaneous-speed
27 
28 estimation method of Johnson et al. [18], and the Gaussian, mean-speed estimation
29 methods of Calabrese et al. [19], and Gurarie et al. [20]). Although straightforward
30 to calculate, approximating a non-linear movement path by a series of linear seg-
31 ments has long been known to underestimate the true distance traveled at coarse
32 
sampling frequencies [21, 22, 14, 12, 15, 16, 17]. All else being equal, the extent of
33 
34 this bias will tend to increase with both the amount of tortuosity in the animal’s
35 movement and the coarseness of the sampling [16]. As a correction to this scale-
36 sensitivity, it is suggested that increasing the sampling frequency will improve the
37 accuracy of SLD estimates, as linear segments of smaller lengths more accurately
38 
39 capture the shape of non-linear paths [16]. Problematically however, animal track-
40 ing data are also subject to measurement error [23, 24]. When paths are sampled at
41 fine temporal scales, measurement error becomes a major source of bias and SLD
42 will tend to over-estimate the true distance traveled [25]. To see this, consider an
43 
individual tracked at a one-minute sampling interval. If, during that interval, it
44 
45 travels an average of 5m, but the measurement error on each location is 10m, the
46 error will be larger than the scale of the movement, and will dominate the esti-
47 mated distance traveled. The suggested approach to correct for error induced bias
48 is to smooth the data by fitting a movement model to the data to jointly estimate
49 
50 measurement and process variances, and then apply SLD on the smoothed data
51 [26, 27]. However, the fundamental limitations with this type of approach are that
52 joint estimation has serious identifiability issues [28] which can lead to under- or
53 over-smoothing, while coarse-scale tortuosity induced bias is still not accounted for.
54 
Compounding the issue of the sensitivity of SLD estimation, these estimates do not
55 
56 come equipped with confidence intervals to quantify their uncertainty. This means
57 that it is not currently possible to determine if a set of SLD-based estimates are
58 statistically different from one another. These issues present serious problems for
59 any comparative analyses because SLD estimates are not only influenced by how far
60 
61 the animal traveled, but also by the sampling frequency [22, 14, 15], the tortuosity
62 of the animal’s movement [16], and the amount of measurement error [25].
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 6 Importantly, the continuous nature of animal movement means that as individuals
 7 navigate through their environment their positions and, crucially in the context of
 8 speed/distance estimation, velocities are necessarily autocorrelated over time [20].
 9 Here, we take advantage of these fundamental properties of motion to overcome
10 
11 the scale-sensitivity of SLD estimation. We outline how to estimate speed, both
12 average and instantaneous, and distance traveled in a scale-insensitive way that
13 builds upon the existing continuous-time movement modeling framework [29, 18, 30,
14 31, 19, 32, 33]. Modeling movement in this framework separates the continuous-time
15 
16 structure of the underlying movement process from the discrete-time structure of
17 the sampling process [34, 29, 35, 36], which allows for inference that is less sensitive
18 to the sampling schedule than discrete-time approaches [37]. Our approach makes
19 use of the error [29, 32], and correlated velocity components of these models [29, 20]
20 
21 to estimate speed and distance traveled as latent variables (i.e., indirectly observed
22 variables that are inferred from directly observed variables). Crucially, not only does
23 this approach allow for scale-insensitive estimation of these movement metrics, but
24 it also provides a means of obtaining confidence intervals. We first use a series of
25 
26 simulations to demonstrate the influence of each source of bias on SLD estimation
27 (i.e., sampling frequency; random data loss; tortuosity; and measurement error). We
28 then use a similar set of simulations to show how the continuous-time approach we
29 detail can correct for these sources of bias and provide accurate estimates. Finally,
30 
31 we demonstrate the utility of our approach, and the sometimes radical improvements
32 it can provide versus both conventional and model-smoothed SLD, on GPS data
33 from a wood turtle (Glyptemys insculpta) tracked in Virginia, USA, and a white-
34 nosed coati (Nasua narica) tracked on Barro Colorado Island, Panama.
35 
36 
37 Methods
38 Universal data limitations for speed/distance estimation
39 A currently unrecognized aspect of speed/distance estimation is that, irrespective
40 
of what estimator is applied to the data, this analysis is not necessarily appropriate
41 
42 for every dataset. We therefore begin by detailing this limitation so as to place the
43 work that follows in its proper context.
44 An animal’s true location in two dimensions, r(t), at time t is defined by the
45 location vector
46 
47 
48 r(t) = (x(t), y(t)) . (1)
49 
50 While an animal’s displacement over a certain timeframe, (t , t ), is the straight line
51 1 2
52 displacement between true locations r(t1) and r(t2), the distance that it traveled,
53 d(t1, t2), is the integral of its speed, v(t), with respect to time
54 
55 ∫ t2
56 d(t1, t2) = v(t) dt , (2)
57 t1
58 
59 where speed is the m√agnitude of the velocity vector, v(t), given by60 
61 v(t) = |v(t)| = v (t)2x + vy(t)2 . (3)
62 
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 6 Finally, for any given time, an animal’s velocity is the derivative of its true position
 7 with respect to time,
 8 
 9 d
10 v(t) = r(t) . (4)dt
11 
12 From these fundamental relationships, we see that estimating speed and/or distance
13 
traveled from location data requires that there be information on velocity in the
14 
15 data. Conversely, if no velocity information exists, then speed/distance estimation
16 is inappropriate, irrespective of what estimator is used.
17 As noted above, the continuous nature of animal movement means that positions
18 
19 and velocities are necessarily autocorrelated over time [20, 38]. Animals with strong
20 directional persistence (e.g., as in a migratory individual), will tend to have long
21 velocity autocorrelation timescales, τv. Animals with more tortuous movement in
22 contrast, will tend to have a much shorter τv. The relationship between τv and the
23 
24 sampling interval, ∆t, is, therefore, critical for determining whether there will be
25 any signature of the animal’s velocity, and hence movement path, in the data. More
26 specifically, because velocity autocorrelation decays exponentially at rate 1/τv, the
27 time required for the proportion of the original velocity autocorrelation to decay
28 
29 to α is τα = τv ln(1/α). Conventionally 5% or less autocorrelation remaining in the
30 data is considered effectively independent, so∼ 3τv is the time it takes for 95% of the
31 velocity autocorrelation to decay. Therefore, if ∆t > 3τv, no statistically significant
32 signature of the animal’s velocity will remain in the location data, leaving insufficient
33 
34 information for accurate speed or distance estimation (Fig. 1). This means that such
35 a dataset is simply too coarsely sampled to support speed/distance estimation, and
36 this limitation applies regardless of which estimator is used. Further mathematical
37 proofs on this universal data limitation are provided in Appendix S1.
38 
39 
40 Bias in straight-line displacement (SLD) estimation
41 Animal tracking data are obtained by discretely sampling an animal’s location, r,
42 at times ti ∈ {t1, . . . , tn}. From these data, distance traveled is typically quantified43 
44 by summing the SLD between locations
45 √
46 d̂ = |∆r| = ∆x2 + ∆y2. (5)
47 
48 
49 Further dividing this estimate by the change in time over which the movement
50 occurred is used to estimate speed
51 
52 d̂
53 v̂ = . (6)
54 ∆t
55 
56 Problematically, measuring the length of a non-linear movement path by summing
57 a series of linear segments between true locations will always underestimate the true
58 distance traveled unless the focal animal actually moved in perfectly straight lines
59 between observations (Fig. 2a). This happens because discretely sampled tracking
60 
61 data represents only a subset of the animal’s full path, and the shortest distance
62 between two points is a straight line. All else being equal, the extent of this bias will
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 6 also be greater for individuals with more tortuous movement (see the blue, dotted
 7 line in Fig. 2c; see also [16]). Increasing the sampling frequency is often suggested
 8 as way of reducing this negative bias [22, 14, 15, 16], since decreasing the time
 9 between successive relocations results in shorter segments that better approximate
10 
11 the non-linear shape of the movement path — effectively functioning as a Riemann
12 sum approximation of the path length [39].
13 Crucially, this approach is only valid if the true positions are known exactly (i.e.,
14 the red, dashed line in Fig. 2c). In reality however, the true positions are not known,
15 
16 as there is generally some extent of measurement error on the observations [23, 24].
17 If these errors are uncorrelated in time, SLD estimates actually diverge to infinity
18 as the samp∣ling frequency i∣ncreases19 
20 ∣ ∣
21 ∣∣ ∆ ∣lim (r + error)∣ =∞. (7)
22 ∆t→0 ∣∣∆t ︸ ︷︷ ︸∣∣
23 observable
24 
25 This happens because the actual distance traveled by the animal goes to 0 in the
26 limit where ∆t→ 0, but the magnitude of uncorrelated measurement error is inde-
27 pendent of ∆t (e.g., Fig. 2b). As a result, at short sampling intervals, the estimate
28 becomes dominated by measurement error (see the gray, dashed line in Fig. 2c; see
29 
30 also [25]). Jointly estimating the movement and error variances, and then smooth-
31 ing the data conditional on these fitted models has been suggested as a means of
32 correcting for error induced bias [26, 27]. However, this type of approach is limited
33 by the serious identifiability issues of joint estimation [28] which can lead to under-
34 
35 or over-smoothing of the data, while the coarse-scale, tortuosity induced bias is still
36 not accounted for.
37 Collectively, this scale-sensitivity means that when animals are tracked at coarse
38 temporal scales SLD will tend to underestimate their speed and distance traveled,
39 
40 yet will tend to overestimate these quantities when tracked at fine temporal scales.
41 While, in principle, it is possible to adjust the sampling frequency such that these
42 sources of bias cancel out, this would require knowing the error magnitude of the
43 deployed tracking device and tortuousity in the animal’s movement a priori. Fur-
44 
45 thermore, tortuousity might vary substantially from one individual to the next [40]
46 even within the same species tracked in the same place, at the same time [16],
47 and measurement error can vary between tracking devices. In practice therefore,
48 it would be extremely difficult to reliably hit this ‘Goldilocks’ sampling frequency,
49 
and missing it would mean biasing the results in one direction or the other. Using
50 
51 the sampling frequency to strike a balance between these sources of bias is thus an
52 unreliable way of accounting for the scale-sensitivity of SLD estimation.
53 
54 Continuous-time estimation of speed/distance traveled
55 
56 To alleviate the scale-sensitivity of SLD estimation, we outline a scale-insensitive,
57 continuous-time speed and distance estimation (CTSD) method that builds upon
58 the existing continuous-time movement modeling framework [29, 18, 30, 31, 19, 33].
59 As described above, an inherent property of working in continuous-time is the abil-
60 
61 ity to separate the underlying continuous-time movement process from the discrete-
62 time sampling process. Consequently, continuous-time models are less sensitive to
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 6 the sampling schedule when estimating parameters. Starting with some tracking
 7 data (Fig. 3a), the first step in our approach is to account for error in the posi-
 8 tion measurements [29, 41, 32]. This is done by using calibration data, where the
 9 tracking device has been left in a fixed location for a period of time (Fig. 3b), to
10 
11 estimate the device’s root mean square (RMS) user equivalent range error (UERE).
12 RMS UERE is the device specific error, in meters, defined by the sum of errors
13 resulting from receiver noise, satellite clocks, and tropospheric/ionospheric effects,
14 given ideal satellite coverage [42]. For GPS data, the device specific RMS UERE is
15 
16 then used as a proportionality constant to translate the unit-less location specific
17 errors, recorded in GPS dilution of precision (DOP) values (both horizontal, HDOP,
18 and vertical VDOP), into standard deviations of mean-zero error (Fig. 3c), where
19 the location error = RMS UERE×HDOP [43]. Assuming functional devices, RMS
20 
21 UERE values should apply to all tags of a given type, while DOP values capture
22 the large location-to-location differences in measurement error. Note, ARGOS data
23 [44], and some brands of GPS tracking devices come pre-calibrated, so the additional
24 step of collecting calibration data to transform the DOP values is not necessary in
25 
26 such cases. To calibrate the errors we used the uere.fit() function from the ctmm
27 package [45]. After data import and error calibration, we recommend that the data
28 be inspected for outlying data points, and all outliers should be removed prior to
29 analysis (for examples of this process see Appendix S2).
30 
31 The next step is to fit a continuous-time, correlated-velocity movement model
32 that appropriately describes the animal movement data. As noted above, speed and
33 distance travelled are properties of an animal’s velocity over time, and the capacity
34 to estimate these quantities is linked to the ability to resolve τv. If the data are
35 
36 too coarsely sampled, relative to the animal’s movement, to be able to fit a corre-
37 lated velocity model [20], it will not be possible to estimate speed/distance, as the
38 data will no longer contain any signature of the path the animal traveled between
39 locations (see also Appendix S1). Here, it is also important to fit the error and
40 
41 movement models separately because, if fit simultaneously, it can be difficult for
42 the models to distinguish between actual movement and error, and parameters can
43 be confounded [28]. This second step, therefore, begins by holding the error model
44 fixed after calibration, and then employing model selection techniques to identify
45 
46 the best continuous-time movement process for the data [36, 38]. Models are fit us-
47 ing perturbative hybrid residual maximum likelihood (pHREML; [46]), and the best
48 movement model for the data selected using small-sample-size corrected Akaike’s
49 Information Criterion (AICc; [19]), using the R package ctmm, applying the workflow
50 
51 described by [19]. Notably, if model selection favors a model without correlated ve-
52 locities, such as OU motion [47], or Brownian Motion [48], this is an indication that
53 the data are too coarsely sampled to support velocity estimation. The selection of
54 a correlated velocity process, such as Integrated Ornstein-Uhlenbeck (IOU) motion
55 
56 [29] or Ornstein-Uhlenbeck Foraging (OUF) motion [30], is necessary to proceed to
57 the next steps of speed and distance estimation (Fig. 3d, e). To fit and select the
58 movement, and error models, we use the R package ctmm, applying the workflow
59 described by [19], which includes all stationary, continuous time-models currently
60 
61 in use in the ecological literature [32]. Although these models return immediate
62 Gaussian estimates of the RMS speed [19, 20] (detailed in Appendix S3), RMS
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 6 speed is not necessarily proportional to the total distance traveled, and the true ve-
 7 locities, v(t), are not necessarily normally distributed. Obtaining a non-parametric
 8 estimate of speed, whose time average is proportional to distance traveled, requires
 9 an additional simulation step that we describe here.
10 
11 Once appropriate error and movement models have been estimated, the final step
12 is to simulate a series of error-free trajectories conditioned on the data, with a sam-
13 pling interval that is much smaller than the velocity autocorrelation timescales (Fig.
14 3f). At scales much shorter than the velocity autocorrelation timescales, the instan-
15 
16 taneous velocities become approximately constant over short time intervals, and
17 the simulated data are therefore more appropriate for straight-line interpolation.
18 When calculating mean speeds and distances, numerical errors from this discretiza-
19 tion are O(∆t3), with shorter intervals (∆t) producing more accurate estimates.
20 
21 The computation time, however, scales inversely with ∆t, where shorter intervals
22 increase the computation time. Consequently, there is a trade-off between accuracy
23 and computation time, and we chose ∆t = τv10 , where τv is the velocity autocorre-
24 lation timescale, which has a corresponding relative error of O(10−3). In terms of
25 
26 the number of simulated trajectories, our approach first simulates 20 trajectories
27 and then continues to batch simulate trajectories until the standard error reaches
28 the target error threshold (here 10−3). For each of these simulated trajectories, we
29 calculate the √instantaneous speeds30 31 
32 v(ti) = vx(t )2 + v (t )2i y i , (8)
33 
34 and use these to estimate total distance traveled (d), and average speed (v̄) using
35 
36 the trapezoidal rule (i.e., the average of the left∑and right Riemann sums; [39])
[1]
37 
38 ∑ i∑(∆ti|v(ti)|)d = (∆ti|v(ti)|) v̄ = . (9)39 j(∆ti j)
40 
41 
42 Repeating this third step over multiple rounds of simulations (Fig. 3g) provides
43 an ensemble of estimates from which the mean speed, 〈v̄〉, and/or distance 〈d〉 can
44 be estimated. Because this method relies on generating an ensemble of estimates
45 that contain process, measurement, and parameter uncertainty, it is also possible to
46 
47 calculate the variance around the point estimate as well as confidence intervals. The
48 estimates range on a scale from 0 to infinity, so as an improvement over normal CIs,
49 which can include negative values, we summarize the uncertainty of this ensemble
50 with χ statistics. These are exact for the mean speed of a stationary Gaussian
51 
52 process with isotropic variance, as its location (and derivatives thereof) are normally
53 distributed with equal variance in every direction (see Appendix S3).
54 The methods we describe here are fully implemented in the R package ctmm (ver-
55 sion 0.5.7 and higher), as well as in the point-and-click web based graphical user
56 
interface at ctmm.shinyapps.io/ctmmweb/ (version 0.2.5; [49]). Average speed or
57 
58 distance travelled can be estimated via the speed() function, whereas instanta-
59 neous speeds can be estimated using the speeds() function. While this workflow
60 
[1]
61 More computationally efficient numerical integrators exist, but they require evenly
62 sampled data.
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 6 involves several steps, the ctmm R package and ctmmweb point-and-click web based
 7 graphical user interface streamline this procedure, and full examples of the workflow
 8 are shown in Appendix S2.
 9 
10 
11 Simulation study
12 We first used simulated data to explore how the bias of SLD estimation, both con-
13 ventional and model-smoothed, as well as CTSD, varied with sampling frequency,
14 movement tortuosity, random data loss, and measurement error. Although CTSD
15 
16 permits estimation of both instantaneous and mean speed, as well as total distance
17 travelled, for conciseness we only evaluated the distance travelled estimates in our
18 simulation study, as these are the most directly related to the conventional SLD
19 estimates. Data were simulated based on an OUF process, which features a home
20 
21 range, correlated positions, and correlated velocities (for full details on this model
22 see [30]). The OUF process is representative of modern GPS tracking data com-
23 monly used in these analyses [50], and tends to apply frequently in practice [40].
24 Data were simulated according to four sets of manipulations:
25 
26 i) Sampling frequency. In our first set of simulations, we tested how variation
27 in sampling frequencies influenced estimates. We set the position and velocity
28 autocorrelation timescales to 1 day, and 1 hour respectively, which are typical
29 timescales for these parameters in many medium-sized, range-resident mam-
30 
31 mals [36, 19, 51]. From this model, we simulated a fine scale trajectory, sam-
32 pled for 10 days at a frequency of 4096 locations/day. This fine-scale, error-free
33 trajectory was used to estimate the true distance travelled — for small time
34 steps the Riemann sum converges to the truth. After determining the truth,
35 
36 mean-zero Gaussian error with a standard deviation of 10m was added to each
37 location. Using the data with error, we estimated the total distance traveled
38 using both conventional SLD and CTSD estimation. Further to conventional
39 SLD, we also estimated model-smoothed SLD sensu [26, 27]. For this latter
40 
41 approach, we applied the standard ctmm workflow [19, 52] to jointly estimated
42 the process and error variances. We then used the estimated movement and
43 error models to smooth the data by predicting the most likely location at each
44 of the sampled times. Finally, we calculated SLD estimates on these smoothed
45 
46 data. We note that because all of the simulated data were generated from sta-
47 tionary, OUF processes, the true model was within the set of candidate models.
48 So this was a best case scenario for how model-smoothed SLD can be expected
49 to perform in practice. We then compared these three estimates to the truth.
50 
51 We then thinned down the fine-scale trajectory by removing every second loca-
52 tion, and repeated the model fitting and estimation process. This thinning and
53 re-estimation was repeated to generate increasingly coarse data with sampling
54 frequencies that ranged from the full resolution of 4096 locations/day, down to
55 
56 8 locations/day in a halving series. Fewer than 8 fixes per day resulted in an
57 OU model being selected for this parameterization (i.e., with a velocity auto-
58 correlation timescale of 1 hour, a 3 hour interval was where ∆t = 3τv and no
59 statistically significant signature of the animal’s velocity remains in the data).
60 
61 ii) Irregular sampling. In our second set of simulations, we tested the perfor-
62 mance of SLD and CTSD on data with irregular sampling, where we mimicked
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 6 the effect of sporadic data loss, which is a common issue with tracking data
 7 [53], and known to present issues to discrete time methods [54, 55]. We set the
 8 position and velocity autocorrelation timescales to 1 day, and 1 hour respec-
 9 tively, and simulated a trajectory sampled for 10 days at a constant frequency
10 
11 of 64 locations/day. Again, after determining the truth, mean-zero Gaussian
12 error with a standard deviation of 10m was added to each location. We then
13 randomly dropped a percentage of the collected locations (ranging from 0% —
14 
15 i.e., no data loss — to 70%, and increasing by 5% increments), where increasing
16 the percentage of data loss resulted in increasingly irregular data. Using the
17 irregularly thinned data with error, we estimated the total distance traveled us-
18 ing both conventional and model-smoothed SLD, as well as CTSD estimation,
19 
20 and compared these estimates to the truth.
21 iii) Movement tortuosity. In our third set of simulations, we tested how varia-
22 tion in the tortuosity of an individual’s movement influenced estimates. Here,
23 we simulated a trajectory sampled for 10 days at a constant frequency of 64
24 
25 locations/day. We set the position autocorrelation timescales to 1 day, but ma-
26 nipulated the velocity autocorrelation timescale (ranging from 11.25 minutes
27 to 1 day in a doubling series), where increasing the duration of velocity auto-
28 
29 correlation generates movement that is decreasingly tortuous (i.e., more linear,
30 [30]). After determining the truth, mean-zero Gaussian error with a standard
31 deviation of 10m was added to each location. The total distance traveled was
32 then estimated using both conventional and model-smoothed SLD and CTSD
33 
34 as described above, and these estimates were compared to the truth.
35 iv) Location error. In our fourth set of simulations, we tested how variation in
36 the amount of measurement error influenced estimates. Here, we simulated 100
37 trajectories, sampled for 10 days at a fixed frequency of 64 locations/day. We
38 
39 set the position and velocity autocorrelation timescales to 1 day, and 1 hour
40 respectively, resulting in ∆t ≈ 13τv. After simulation, we again added mean-zero
41 Gaussian error to each location, but here manipulated the standard deviation
42 
43 (ranging from 0, i.e., no error, to 51.2 meters, in a doubling series of the minimal
44 value of 0.1 m error).
45 The simulations we described above were aimed at determining how CTSD, with
46 a correctly calibrated error model, compared to SLD estimation. However, bias can
47 
48 still be introduced to the CTSD method if the error model is poorly specified. To
49 evaluate the potential severity of this bias, we further compared CTSD distance
50 travelled estimates for three different model fitting approaches; 1) fitting the move-
51 
52 ment model without error; 2) fitting the movement and error models simultaneously
53 sensu [28]; and 3) fitting the movement and error models separately (i.e., the full
54 approach described above). The parameterization of the simulation was identical
55 to the Sampling Frequency simulation described above. The total distance traveled
56 
57 was then estimated using SLD and CTSD with the three error handling approaches,
58 and these estimates were compared to the truth.
59 Each of these simulation studies was repeated 100 times, and we compared the
60 
61 mean performance of each estimator. All simulations were performed in the R envi-
62 ronment (version 3.5.1; [56]) using the methods implemented in the R package ctmm
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 6 (version 0.5.7; [19]), and the computations were conducted on the Smithsonian In-
 7 stitution High Performance Cluster (SI/HPC). The code necessary to reproduce
 8 these simulations is presented in Appendix S4.
 9 
10 
11 Empirical case studies
12 To verify that the estimators would, in practice, perform as they did on the sim-
13 ulated data, we tested both conventional and model-smoothed SLD, and CTSD
14 on GPS relocation data for a wood turtle, and a white-nosed coati [57]. For the
15 
16 wood turtle, locations were sampled every hour over a 42 day period in autumn,
17 2016. Calibration data for this animal’s tracking tag were collected by leaving two
18 devices of the same model in a fixed location for 1 day, and sampling at 10-minute
19 intervals. From these calibration data, the tracking device was found to have a hor-
20 
21 izontal RMS UERE of 10.6 meters, while the tracking data had a median HDOP of
22 1.4 (ranging from 0.8 – 9.9). For the white-nosed coati, which tend to exhibit very
23 tortuous movement [58], locations were sampled every 15 minutes over a 41 day pe-
24 riod in spring 2010, using e-obs collars with a median horizontal accuracy estimate
25 
26 of 15.6 meters (ranging from 2.6 – 78.3 meters). E-obs devices come pre-calibrated,
27 so, for these data, no additional calibration was necessary.
28 We selected these datasets not because CTSD is restricted to terrestrial, GPS
29 tracking data, but to highlight two general cases that are likely to occur in practice:
30 
31 i) the case where the movement and measurement error are on approximately the
32 same scale, resulting in a priori unpredictable biases in SLD estimates (i.e., the
33 white-nosed coati data); and ii) the case where the amount of measurement error
34 is much larger than the amount of movement that occurs between positional fixes,
35 
36 resulting in positively biased SLD (i.e., the wood turtle data). However, in addition
37 to these GPS examples, Appendix S2 provides a worked example of CTSD applied
38 to ARGOS data from a brown pelican (Pelecanus occidentalis), tracked on the
39 eastern coast of the United States.
40 
41 For each of these datasets we first fit the full suite of movement models described
42 above, and performed model selection to identify the most appropriate model for the
43 data. We then estimated the total distance traveled using SLD, both conventional
44 and model-smoothed, and CTSD. To evaluate the scale-sensitivity of these empirical
45 
46 estimates, we subsequently thinned the data by dropping every second location, and
47 repeated the model fitting/selection, and distance estimation steps on these coarser
48 data. This thinning and estimation process was repeated iteratively until the data
49 became too coarse to be able to select a correlated-velocity model (i.e., ∆t > 3τv).
50 
51 To further evaluate how SLD and CTSD estimates might compare in practice, we
52 also estimated the daily distance traveled using SLD and CTSD, which is a routinely
53 estimated metric.
54 
55 Results
56 
57 Simulation results
58 From these simulations, we found SLD estimates to be significantly biased by
59 v√ariation in sampling frequency, with substantial under-estimation at coarse res-60 olutions, over-estimation at fine resolutions, and only a narrow window when61 
VAR[error]
62 VAR]velocity]  ∆t  τv where these contrasting sources of bias cancelled out
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 6 to provide an accurate estimate (Fig. 4a). Model-smoothed SLD did provide some
 7 correction for error induced bias in SLD estimation for finely sampled data, but
 8 still resulted in negatively biased estimated for coarsely sampled data. In contrast,
 9 CTSD provided consistently accurate estimates across the majority of the sampling
10 
11 frequencies we examined, and was the only scale-insensitive estimator of those ex-
12 amined here. We note that when ∆t > τv, CTSD resulted in some positive bias.
13 Despite this positive bias, we found that as the sampling became increasingly coarse,
14 the 95% confidence intervals on the CTSD estimates widened, providing accurate
15 
16 coverage for all but the coarsest sampling regimes (Fig. 5). We also found SLD
17 and model-smoothed SLD estimates to become increasingly negatively biased as
18 the amount of random data loss increased, whereas CTSD was, again, consistently
19 accurate across the data loss regimes we examined (Fig. 4b).
20 
21 Similarly, when the sampling frequency was fixed, SLD estimates varied substan-
22 tial as the underlying movement differed, with, again, only a narrow window where
23 the different sources of bias cancelled out to provided an accurate estimate. Model-
24 smoothed SLD was generally more stable than conventional SLD, but did still suffer
25 
26 from scale-sensitivity, particularly for highly tortuous movement. In contrast, CTSD
27 provided consistently accurate estimates, and was not biased by variation in tortu-
28 osity (Fig. 4c).
29 SLD estimates varied substantial as the underlying movement differed, with,
30 
31 again, only a narrow window where the different sources of bias cancelled out to
32 provided an accurate estimate (Fig. 4c). In contrast, CTSD provided consistently
33 accurate estimates, and was not biased by variation in tortuosity. Finally, as the
34 amount of measurement error increased, the bias in SLD estimates, both conven-
35 
36 tional and model-smoothed, increased exponentially, whereas CTSD was not biased
37 by measurement error (Fig. 4d).
38 Importantly, while we found that CTSD, with a correctly calibrated error model,
39 provided accurate estimates with reliable confidence intervals, CTSD with an in-
40 
41 correct error model resulted in inaccurate estimates (Fig. 6). For instance, when
42 the movement model was fit without error, speed and distance estimates were even
43 more biased that SLD estimates. Simultaneously fitting the movement and error
44 models also resulted in biased estimates, though the extent of the bias was not as
45 
46 extreme as the scale-sensitive bias of conventional SLD estimation.
47 
48 Empirical results
49 Consistent with our simulated findings, SLD estimates of total distance traveled
50 
51 varied substantially with sampling frequency, whereas CTSD provided relatively
52 consistent estimates except at very coarse sampling frequencies, but with appropri-
53 ately wide confidence intervals. For instance, SLD estimation for the wood turtle’s
54 tracking data at the full, 1 hr resolution, suggested this animal traveled 12.8 km over
55 
56 the 42 day sampling period, whereas CTSD estimated the distance traveled as 0.86
57 km (95% CIs: 0.57 – 1.15 km). Coarsening these data resulted in drastic changes to
58 both of the SLD estimate (Fig. 7b), whereas CTSD point estimates and 95% CIs
59 were all consistent. Interestingly, both of the scale-sensitive SLD estimates of daily
60 
61 movement distances varied substantially from day to day, whereas CTSD suggested
62 relatively consistent behavior across the study period (Fig. 7c). The instantaneous
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 6 speed estimates, averaged over each 24 hour cycle, showed how the animal tended to
 7 move more in the early morning, with reduced movement throughout the rest of the
 8 day (Fig. 7d). SLD estimation does not readily allow for estimating instantaneous
 9 speeds from data that are coarse and irregular, precluding any formal comparison.
10 
11 SLD estimation for the coati at the full, 15-min resolution suggested this animal
12 traveled 97.9 km over the 41 day sampling period, whereas CTSD estimated the
13 distance traveled as 79.5 km (95% CIs: 77.2 – 81.8 km). Again, iteratively coarsening
14 these data resulted in more than a two-fold decrease in the SLD estimate (Fig. 8b),
15 whereas CTSD point estimates and 95% CIs were all consistent, albeit with some
16 
17 positive bias and wide confidence intervals at the coarsest sampling frequencies.
18 Similarly, there were significant differences in the daily distance traveled estimates
19 between the two methods, where on only ca. 50% of the days were the SLD estimates
20 within the 95% CIs of the CTSD estimates (Fig. 8c). The instantaneous speed
21 
22 estimates, averaged over each 24 hour cycle, showed how the coati tended to move
23 only during daylight hours, with a number of peak periods of activity, and little
24 to no movement at night (Fig. 8d). This animal’s GPS collar was programmed to
25 turn off at night, however. In this respect, note how the night time instantaneous
26 speed estimates are accompanied by substantially wider confidence intervals than
27 
28 the daytime estimates, which is related to the large time-gap in the location data.
29 
30 Discussion
31 Speed and distance traveled are among the metrics most routinely estimated from
32 GPS tracking data. Problematically however, the commonly used approach of esti-
33 
34 mating these using straight-line displacements is severely scale-sensitive, with biases
35 arising from multiple sources [22, 14, 15, 59, 16, 25, 17]. Even more problematic is the
36 fact that each of these sources of bias operates in a different direction, and can be of
37 variable magnitude. As the combination of sampling irregularities, inter-individual
38 
39 variation in movement, and measurement error are nearly ubiquitous aspects of an-
40 imal tracking data, accurate speed/distance estimation requires statistical methods
41 that can handle these complications, without being subject to artifactual differ-
42 ences due purely to estimator bias, or without having to know the magnitudes of
43 these biases a priori to target the sampling rate accordingly. To date, corrections
44 
45 to these issues have included suggestions to increase the sampling frequency [16], ad
46 hoc quantification of correction factors [17], and model-smoothing [26, 27]. These
47 are unreliable solutions as they do not account for all sources of bias and also fail
48 to provide a means of quantifying uncertainty in the estimates. While Johnson et
49 
50 al. [18] laid out a general approach to estimating trajectory-derived metrics, such
51 as speed and distance travelled, by sampling from the posterior distribution of con-
52 ditional trajectories, they did not implement this in readily accessible tools. The
53 differences between our approach here and a hypothetical application of [18] are
54 
that we rely on a parametric bootstrap rather than treating the likelihood function
55 
56 as a Bayesian prior and we also take careful note from the recent results of [28]
57 to not simultaneously fit movement and error parameters. In our view, it is unfor-
58 tunate that the methods introduced by [18] have not been more widely adopted
59 in movement ecology to date, while scale-sensitive SLD (whether model-smoothed
60 
61 or conventional) is still the estimator of choice for the majority of ecologists and
62 practitioners.
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 6 As a solution to the outlined problems, we have developed CTSD as a new scale-
 7 insensitive method for estimating speed and distance traveled from animal tracking
 8 data that builds upon the existing continuous-time movement modeling framework
 9 [30, 19]. Using a combination of simulated and empirical data, we have demonstrated
10 
11 how CTSD provides accurate, scale-insensitive estimates with reliable confidence
12 intervals, provided ∆t is small enough to estimate τv (i.e., ∆t > 3τv), and teleme-
13 try error is properly calibrated. The net results are speed and distance traveled
14 estimates that can validly be compared across studies, sites, species, and times.
15 
16 For example, because the ∼15m median measurement error of the wood turtle’s
17 tracking data was larger than the scale of the turtle’s movement over the 1-hour
18 sampling intervals (<1m), we found that the SLD estimates were dominated by
19 error-driven bias. Consequently, the estimates varied more than 12-fold across the
20 
21 thinned sampling intervals, and when estimating the daily movement distances for
22 this individual, the scale-sensitivity of the SLD resulted in estimates that varied
23 substantially from one day to the next. The CTSD estimates in contrast, which
24 accounted for the error structure of the telemetry data, suggested relatively consis-
25 
26 tent movement behavior throughout the study period. Had an analysis been based
27 off of the SLD estimates, one would have erroneously concluded that this turtle
28 covered large distances at highly variable rates, as opposed to the slow and steady
29 movement it actually exhibited.
30 
31 In the CTSD formalism, whole-path estimates, such as mean speed and distance
32 traveled, are constructed from instantaneous speed estimates, which are also in-
33 teresting in their own right. Instantaneous speeds averaged over cycles (e.g., 24hr,
34 monthly, or seasonal cycles), such as those depicted in figures 7d and 8d, can serve
35 
36 as the basis of visual diagnostic tools for identifying multiple behavioral states.
37 When different behaviors are associated with clear differences in speed/velocity
38 (e.g., active versus inactive, range-residency versus migration), instantaneous speed
39 estimates can be used as the basis for formally estimating an individual’s behav-
40 
41 ioral state [10, 60]. For example, figure 7d shows how the turtle’s rate of movement
42 changes throughout the day, with consistently more activity in the early morning,
43 versus minimal movement throughout the rest of the day. Patterns in instantaneous
44 speed over time can also allow researchers to identify the times and/or places where
45 
46 changes in movement and behavior occur [10].
47 While CTSD is, by itself, very general, it relies on a fitted movement model that
48 adequately captures the underlying movement behavior in the data. In our expe-
49 rience, the current family of continuous time models covers a very broad array of
50 
51 cases [30, 38, 19], that are useful for a wide range of species [40]. However, in cases
52 where no appropriate model exists, then CTSD estimates may not be representa-
53 tive of the true speed/distance (for further details on how this may affect estimates
54 see Appendix S5). The statistical efficiency of our method follows straightforwardly
55 
56 from related methods in time-series Kriging [61]. For a Gaussian stochastic process
57 with a mean and autocorrelation function that are correctly specified by the move-
58 ment model, the velocity estimates are minimum variance and unbiased (MVU;
59 [62]). For non-Gaussian processes with correctly specified movement model, the ve-
60 
61 locity estimates are best linear unbiased estimates (BLUE; [62]). For asymptotic
62 consistency, the movement model does not have to be correctly specified and only
63 
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 6 ‘compatibility’ (i.e., matching continuity) is required, but the variance of the errors
 7 does need to be correctly estimated [63] (see also Fig. 6). In other words, because
 8 speed and distance travelled are estimated as latent variables of the velocity param-
 9 eter, asymptotic consistency requires a correlated velocity movement model where
10 
11 only the initial curvature of the model autocorrelation function needs to match
12 that of the true autocorrelation function. The BLUE and asymptotic consistency
13 properties of our method stand in contrast to the Gaussian mean-speed parame-
14 ter estimates of [19], and [20], which are only accurate when the process is truly
15 
16 Gaussian. Moreover, the library of continuous-time movement models on which our
17 method can be based is expanding rapidly [29, 64, 32, 65, 66, 60], including multi-
18 state continuous-velocity models [67], so model misspecification should become less
19 problematic going forward.
20 
21 A further caveat to CTSD, and, indeed, any accurate method, is that it can not
22 necessarily be applied to any dataset. If the data are too coarsely sampled, rela-
23 tive to the animal’s movement, to be able to fit a correlated velocity model [20],
24 it will not be possible to estimate speed. This illustrates a fundamental aspect of
25 
26 studying movement through the use of tracking data, that when the sampling is
27 too coarse to contain any signature of the animal’s velocity, this kind of analysis
28 becomes inappropriate. For coarsely sampled data, although it is still mathemati-
29 cally possible to calculate the straight line displacement between any two locations,
30 
31 without a signature of τv these estimates are, ultimately, meaningless as measures
32 of speed or distance travelled. In other words, just because an estimate can be
33 produced when ∆t > 3τv does not mean said estimate is meaningful, as we demon-
34 strate in Appendix S1. In this respect, the model selection step of our approach
35 
36 allows researchers to identify whether or not their data are of sufficient resolution
37 to estimate these metrics in a statistically rigorous way. A corollary of this is that,
38 if estimating speed/distance traveled is a primary goal of a study, we suggest re-
39 searchers tailor their sampling design to ensure data of sufficient resolution to detect
40 
41 τv . As a general rule of thumb, we suggest that the sampling interval should be
42 less than or equal to τv. On the other hand, because the effective sample size for
43 velocity estimation, Nvelocity, corresponds to the equivalent number of statistically
44 independent velocity observations, choosing a sampling interval much smaller than
45 
46 τv will produce marginal benefit. While τv is likely to differ between individuals,
47 species, populations, seasons, etc., it tends to be on the order of minutes to hours for
48 many range-resident species [30, 19, 51, 68]. In practice, sampling resolutions tend
49 to be fine enough to estimate τv for the majority of GPS data for range-resident
50 
51 birds and mammals [40]. Although the empirical examples included in this work
52 involved GPS data from terrestrial species, CTSD can can be applied to any form
53 of tracking data (terrestrial, marine, avian, GPS, ARGOS, VHF, etc...) sampled
54 at a finely enough to resolve τv. Related to this, there will be some positive bias
55 
56 in the CTSD estimates when τv can not be accurately estimated, which happens
57 when 3τv > ∆t > τv. This is the result of small sample size bias, and happens
58 because at coarse sampling frequencies, the ability to estimate τv is reduced and
59 both the point estimate, and lower confidence interval on this parameter approach
60 
61 0. CTSD uses the sampling distribution of τ̂v when parameterizing the simulations,
62 so as more of this sampling distribution’s density becomes concentrated near zero,
63 
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 6 the simulated trajectories become more tortuous, and the estimated speed and/or
 7 distance traveled becomes increasingly large.
 8 Our approach also requires being able to adequately account for measurement er-
 9 ror in the data (i.e., by collecting calibration data, or by using pre-calibrated track-
10 
11 ing devices). Without properly accounting for error, even CTSD with a perfectly
12 specified movement model can result in arbitrarily biased speed/distance estimates.
13 In this respect, while there is no substitute for true calibration data, there are viable
14 alternatives if such data are not available. With GPS data, for instance, a default
15 
16 RMS UERE of 10-15m is often very reasonable — for example the wood turtle’s
17 calibration estimated an RMS UERE of 10.6 meters. Furthermore, ‘opportunistic’
18 calibration data, like dead or sleeping animals can also be used in place of separately
19 collected calibration data. Although these are viable alternatives, we do recommend
20 
that the collection of error calibration data becomes a standard component of future
21 
22 animal tracking studies.
23 
24 Conclusion
25 In conclusion, the methods developed in this study allow for the scale-insensitive
26 
27 estimation of mean speed, instantaneous speeds, and distance traveled from animal
28 tracking data, that can correct for the often massive biases introduced by the sam-
29 pling frequency [22, 14, 15], the tortuosity of the animal’s movement [16], and the
30 amount of measurement error [59, 25], provided ∆t > 3τv and measurement error
31 
32 can be properly accounted for. In addition to being statistically rigorous, CTSD
33 also benefits from being computationally efficient, a property that is well suited to
34 the growing volume of data used in these analyses [13]. All the methods developed
35 in this study are now freely available in the R package ctmm (version 0.5.7; [19])
36 
via the speed() and speeds() functions, or through the point-and-click web based
37 
38 graphical user interface at ctmm.shinyapps.io/ctmmweb/ (version 0.2.5; [49]).
39 
40 List of abbreviations
41 GPS: Global Positioning System
42 RMS: Root Mean Square
UERE: User Equivalent Range Error
43 DOP: Dilution of Precision
44 HDOP: Horizontal Dilution of Precision
45 SLD: Straight Line Displacement
46 CTSD: Continuous-Time Speed and Distance
ctmm: continuous-time movement modelling
47 OU: Ornstein-Uhlenbeck
48 IOU: Integrated Ornstein-Uhlenbeck
49 OUF: Ornstein-Uhlenbeck Foraging
50 MVU: minimum variance and unbiasedBLUE: best linear unbiased estimates
51 
52 
53 Ethics approval and consent to participate
54 All data were collected using IACUC approved protocols.
55 Consent for publication
56 Not Applicable
57 Availability of data and material
58 The white-nosed coati data used in this manuscript are available from the Movebank online repository (DOI:
59 10.5441/001/1.41076dq1), the turtle data are included in the ctmm package, and the source code for the ctmm
60 package is available on CRAN.
61 Competing interests
62 The authors declare that they have no competing interests.
63 
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 6 Funding
This work was supported by the US NSF Advances in Biological Informatics program (ABI-1458748 to JMC). MJN
 7 was supported by a Smithsonian Institution CGPS grant.
 8 
 9 Author’s contributions
MJN and CHF contributed equally to this work. JMC conceptualised the study; MJN conducted the analyses,
10 drafted the manuscript, and contributed to the method’s development; CHF developed the statistical framework and
11 R implementation; EG contributed to the Gaussian speed calculations; RK collected the white nosed coati data; TA
12 and JDL collected the wood turtle data; all authors contributed to the writing of the manuscript.
13 Acknowledgements
14 Not Applicable.
15 Author details
16 1Smithsonian Conservation Biology Institute, National Zoological Park, 1500 Remount Rd, 22630 Front Royal, USA.
17 2Department of Biology, University of Maryland, 20742 College Park, USA. 3Department of Biology, George Mason
18 University, 4400 University Drive, 22030 Fairfax, USA.
4Migratory Bird Center, Smithsonian Conservation Biology
Institute, 20008 Washington, DC, USA. 5North Carolina Museum of Natural Sciences, Biodiversity Lab, 27601
19 Raleigh, USA. 6Department of Forestry & Environmental Resources, North Carolina State University, 4400
20 University Drive, 27695 Raleigh, USA.
21 
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47 
Figures
48 
49 
50 Figure 1 The results of simulations demonstrating the inability to obtain an accurate estimate via
51 straight line displacement (SLD) when the sampling interval, ∆t, is longer the the velocity
52 autocorrelation timescale, τv , and the severe bias when ∆t ≥ 3τv . For details on the simulations,
53 see Appendix S1.
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 7 Figure 2 Examples of the sources of bias in straight line displacement (SLD) estimation for a)
 8 coarsely sampled data that fail to capture the tortuosity of the animal’s movement; and b) finely
 9 sampled data that are subject to measurement error. In both panels the blue line depicts the path
10 the simulated animal actually traveled, the red dots the sampled locations, and the black lines the
straight line distances between locations. Note how SLD using the coarsely sampled data misses
11 movement the animal actually made, whereas SLD using the finely sampled data introduces
12 movement the animal did not make. In panel c), the results of simulations depict the trade-off of
13 these sources of bias across scales. The solid black line depicts the true value to which the
14 estimates should converge (scaled to 1), and both axes are log scaled. Movement paths weresimulated from Ornstein-Uhlenbeck Foraging (OUF) processes. For the simulations depicted by
15 the red and gray curves, the velocity autocorrelation timescale (τv) was set to 1 hour. For the
16 blue curve, τv was set to 1 min, which produced more tortuous movement.
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Figure 3 A walkthrough of the steps involved in our continuous-time speed and distance
22 estimation (CTSD) method using simulated data. Beginning with the tracking data (panel a; here
23 with a 1-hr sampling interval), the first step is to use some calibration data (panel b) to estimate
24 the device’s user equivalent range error (UERE). Once the errors have been calibrated (panel c),
25 model selection techniques are employed to identify the best fit model for the fine-scale (panel d)and coarse-scale (panel e) features of the data — SVF represents the semi-variance function. A
26 trajectory is then simulated, conditional on the data, the fitted movement model, and the
27 calibrated error model (panel f), and the distance/speed of that trajectory is calculated. The
28 simulated animal had a velocity autocorrelation timescale of 25 minutes, so the trajectory in panel
29 f) is simulated at a frequency of 2.5 min. The simulation and estimation step is then repeated overmultiple rounds of simulation (panel g), and the ensemble provides a point estimate and 95% CIs.
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35 Figure 4 Figure depicting the results of simulations quantifying distance traveled via straight line
36 displacement, and the continuous-time estimation method for manipulations of a) sampling
frequency; b) the amount of random, irregular data loss; c) the tortuosity of the underlying
37 movement; and d) the amount of measurement error. For the red line, the shaded area represents
38 the 95% CIs (SLD estimates, both model-smoothed and conventional, do not come with CIs).
39 The arrow in panel a) depicts the point at which the sampling interval, ∆t, is the same as the
40 velocity autocorrelation timescale, τv . In all panels, the dashed line at y = 1 depicts the truevalue to which the estimates should converge and the x-axis is log scaled. Note: the truth has
41 been scaled to 1.
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47 Figure 5 Figure depicting a) the coverage of the 95% confidence intervals, as well as b) theproportion of cases where the coverage of the confidence intervals was higher than, and did not
48 include the true value; and c) lower than, and did not include the true value. In all panels the
49 error bars represent the 95% confidence intervals on the estimated coverage, the dashed line
50 depicts nominal coverage, and the x-axis is log scaled.
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55 Figure 6 The results of simulations quantifying distance traveled via straight line displacement
56 (SLD), and the continuous-time (CTSD) estimates from three different model fitting approaches;
57 i) fitting the movement model without an error model; ii) fitting the movement and error models
simultaneously; and iii) fitting the movement and error models separately via error calibration.
58 The solid lines depict the mean accuracy, and the shaded areas the 95% CIs (SLD estimates, both
59 model-smoothed and conventional, do not come with CIs). The dashed line at y = 1 depicts the
60 true value to which the estimates should converge and the x-axis is log scaled.
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14 Figure 7 Figure depicting: a) GPS data for a wood turtle (Glyptemys insculpta) tracked in
15 Virginia, USA; b) the total distance traveled estimated via conventional straight line displacement(SLD), model-smoothed SLD, and continuous-time speed and distance estimation (CTSD)
16 approach using progressively thinned data; c) the daily distance traveled again using conventional
17 SLD, model-smoothed SLD, and CTSD; and d) CTSD instantaneous speed estimates, ± 95% CIs,
18 averaged over a 24hr cycle. The gray circles in panel a depict the 50% error circles for GPS
location estimates, the trajectory the most likely path between those locations, colored by the
19 instantaneous speed estimates, while the gray shading in panel d depicts night time. Note how the
20 measurement error is larger than the scale of the turtle’s movement (panel a) and, as a result,
21 SLD estimates become dominated by error driven bias as the sampling frequency is increased
22 (panel b), and vary substantially from day to day (panel c). Model-smoothing provided a
reasonable, but insufficient correction to the error induced bias. In contrast, by accounting for the
23 error structure of the telemetry data, the CTSD estimates are consistent across sampling
24 frequencies, and suggest relatively consistent movement behavior throughout the study period.
25 Panel d depicts how the turtle’s tends to move more in the early morning, with minimal
26 movement throughout the rest of the day.
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44 Figure 8 Figure depicting: a) GPS data for a white nosed coati (Nasua narica) tracked on Barro
45 Colorado Island, Panama; b) the total distance traveled estimated via conventional straight line
46 displacement (SLD), model-smoothed SLD, and continuous-time speed and distance estimation
(CTSD) approach using progressively thinned data; c) the daily distance traveled again using
47 conventional SLD, model-smoothed SLD, and CTSD; and d) CTSD instantaneous speed
48 estimates, ± 95% CIs, averaged over a 24hr cycle. The gray circles in panel a depict the 50%
49 error circles for GPS location estimates, the trajectory the most likely path (MLP) between those
50 locations, colored by the instantaneous speed estimates, while the gray shading in panel d depicts
nighttime. Note how the animal’s trajectory does not necessarily move through the center of each
51 location, as measurement error is accounted for when estimating the MLP. In panel d one can see
52 how the coati tends to only move during daylight hours, and becomes stationary at night.
53 However, note the appropriately wide CIs during the night time as the GPS unit was programmed
54 to turn off after sundown.
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Additional file S1 — Proofs of SLD biases
 7 Mathematical proof of the two SLD biases—overestimation at small sampling interval ∆t and underestimation at
 8 large sampling interval, as well as a additional simulation based results demonstrating SLD’s inability to return an
 9 accurate estimate when the sampling interval, ∆t, is longer the the velocity autocorrelation timescale τv .
10 
11 Additional file S2 — Workflow for estimating speed and distance travelled using CTSD in ctmm
12 
13 
14 Additional file S3 — Technical details
Details on estimating the mean speed and root mean square (RMS) speed from either a time-averaged stationary
15 Gaussian stochastic process or from instantaneous Kriged velocity estimates, and how we translate point estimates
16 and standard errors into non-standard confidence intervals.
17 
18 Additional file S4 — R script for reproducing the simulations
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21 Additional file S5 — CTSD and model misspecification
22 Evaluations of the performance of CTSD with misspecified models for scenarios that are likely to occur in real data.
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Figure_1 Click here to access/download;Figure;Figure_1.pdf
33.1
20.1 ∆t < τv
Accuracy
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∆t = τv 0.6
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Sampling frequency (fixes/day)
τv (hours)
Fa)igure_2 Click here to
access/download;Fig
-400 -300 -200 -100 0 100 200 300
x (meters)
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16 No errorNo error, more tortuosity
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y (meters) y (meters) SVF (hm²) y (meters) y (meters)
200 250 200 250 0 20 40 -20 -10 0 10 20 200 250
SVF (km²) y (meters)
0 10 30 150 250 350
Figuare)_4 Click here to
access/download;
2.0 SLD
Model−smoothed SLD
CTSD
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0.1 0.4 1.6 6.4 25.6
Error (m)
Estimated distance travelled Estimated distance travelled Estimated distance travelled Estimated distance travelled
Figuare)_5 Click here to access/downloabd) ;Figure;Figure_5.pdf
1.00
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Coverage of 95% CIs
95% CIs are too low 95% CIs are too high
Fig2u.0re_6 Click here toSLD access/download;Fig
Model−smoothed SLD
CTSD − Unmodelled error
CTSD − Fitted error
CTSD − Calibrated error
1.5
1.0
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16 64 256 1024
Sampling frequency (locations/day)
Estimated distance travelled
Figure_7 Click here to access/download;Figure;Figure_7.png
Figure_8 Click here to access/download;Figure;Figure_8.png
Appendix_S1
  
Click here to access/download
Supplementary Material
Bias_Proofs.pdf
Appendix_S2
  
Click here to access/download
Supplementary Material
Method_Walkthrough.pdf
Appendix_S3
  
Click here to access/download
Supplementary Material
RMS_Speed_Appendix_REVISED.pdf
Appendix_S4
  
Click here to access/download
Supplementary Material
Smoothed_SLD_Appendix.pdf
Appendix_S5
  
Click here to access/download
Supplementary Material
Model_Misspecification.pdf
Response_Letter
  
Click here to access/download
Supplementary Material
Mov_Ecol_Response_Letter.pdf