Abstract
Discrete-time random walks and their extensions are common tools for analyzing animal movement data. In these analyses, resolution of temporal discretization is a critical feature. Ideally, a model both mirrors the relevant temporal scale of the biological process of interest and matches the data sampling rate. Challenges arise when resolution of data is too coarse due to technological constraints, or when we wish to extrapolate results or compare results obtained from data with different resolutions. Drawing loosely on the concept of robustness in statistics, we propose a rigorous mathematical framework for studying movement models’ robustness against changes in temporal resolution. In this framework, we define varying levels of robustness as formal model properties, focusing on random walk models with spatially-explicit component. With the new framework, we can investigate whether models can validly be applied to data across varying temporal resolutions and how we can account for these different resolutions in statistical inference results. We apply the new framework to movement-based resource selection models, demonstrating both analytical and numerical calculations, as well as a Monte Carlo simulation approach. While exact robustness is rare, the concept of approximate robustness provides a promising new direction for analyzing movement models.
References
Aarts G, Fieberg J, Matthiopoulos J (2011) Comparative interpretation of count, presence–absence and point methods for species distribution models. Methods Ecol Evol 3(1):177–187
Benhamou S (2013) Of scales and stationarity in animal movements. Ecol Lett 17(3):261–272
Benhamou S, Sudre J, Bourjea J, Ciccione S, De Santis A, Luschi P (2011) The role of geomagnetic cues in green turtle open sea navigation. PLoS One 6(10):e26,672
Borger L, Dalziel BD, Fryxell JM (2008) Are there general mechanisms of animal home range behaviour? A review and prospects for future research. Ecol Lett 11(6):637–650
Boyce M, Vernier P, Nielsen S, Schmiegelow F (2002) Evaluating resource selection functions. Ecol Model 157(2):281–300
Breed GA, Costa DP, Goebel ME, Robinson PW (2011) Electronic tracking tag programming is critical to data collection for behavioral time-series analysis. Ecosphere 2(1):art10
Codling EA, Hill NA (2005) Sampling rate effects on measurements of correlated and biased random walks. J Theor Biol 233(4):573
Colchero F, Conde DA, Manterola C, Chávez C, Rivera A, Ceballos G (2010) Jaguars on the move: modeling movement to mitigate fragmentation from road expansion in the Mayan Forest. Anim Conserv 14(2):158–166
Côrtes MC, Uriarte M (2013) Integrating frugivory and animal movement: a review of the evidence and implications for scaling seed dispersal. Biol Rev 88(2):255–272
Costa DP, Breed GA, Robinson PW (2012) New insights into pelagic migrations: implications for ecology and conservation. Annu Rev Ecol Evol Syst 43(1):73–96
Courbin N, Fortin D, Dussault C, Fargeot V, Courtois R (2013) Multi-trophic resource selection function enlightens the behavioural game between wolves and their prey. J Anim Ecol 82(5):1062–1071
Fleming CH, Calabrese JM, Mueller T, Olson KA, Leimgruber P, Fagan WF (2014) From fine-scale foraging to home ranges: a semivariance approach to identifying movement modes across spatiotemporal scales. Am Nat 183(5):E154–E167
Forester JD, Im H, Rathouz P (2009) Accounting for animal movement in estimation of resource selection functions: sampling and data analysis. Ecology 90(12):3554–3565
Fortin D, Beyer HL, Boyce M, Smith D, Duchesne T, Mao J (2005) Wolves influence elk movements: behavior shapes a trophic cascade in Yellowstone National Park. Ecology 86(5):1320–1330
Frair JL, Fieberg J, Hebblewhite M, Cagnacci F, DeCesare NJ, Pedrotti LA (2010) Resolving issues of imprecise and habitat-biased locations in ecological analyses using GPS telemetry data. Philos Trans R Soc B 365(1550):2187–2200
Giuggioli L, Kenkre VM (2014) Consequences of animal interactions on their dynamics: emergence of home ranges and territoriality. Mov Ecol 2:2–20
Hampel FR (1971) A general qualitative definition of robustness. Ann Math Stat 42(6):1887–1896
Hampel FR (1986) Robust statistics: the approach based on influence functions. Wiley, New York
Haran M (2011) Gaussian random field models for spatial data. In: Brooks S, Gelman A, Jones GL, Meng XL (eds) Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC, London, pp 449–478
Hebblewhite M, Merrill E (2008) Modelling wildlife–human relationships for social species with mixed-effects resource selection models. J Appl Ecol 45(3):834–844
Huber PJ, Ronchetti EM (2009) Robust statistics, 2nd edn., Wiley series in probability and statisticsWiley, Hoboken, NJ
Ito TY, Lhagvasuren B, Tsunekawa A, Shinoda M, Takatsuki S, Buuveibaatar B, Chimeddorj B (2013) Fragmentation of the habitat of wild ungulates by anthropogenic barriers in Mongolia. PLoS One 8(2):e56,995
James A, Plank MJ, Edwards A (2011) Assessing Lévy walks as models of animal foraging. J R Soc Interface 8(62):1233–1247
Jerde CL, Visscher DR (2005) GPS measurement error influences on movement model parameterization. Ecol Appl 15(3):806–810
Johnson CJ, Parker KL, Heard DC, Gillingham MP (2002) Movement parameters of ungulates and scale-specific responses to the environment. J Anim Ecol 71(2):225–235
Langrock R, King R, Matthiopoulos J, Thomas L, Fortin D, Morales JM (2013) Flexible and practical modeling of animal telemetry data: hidden Markov models and extensions. Ecology 93(11):2336–2342
Latham ADM, Latham MC, Boyce M, Boutin S (2011) Movement responses by wolves to industrial linear features and their effect on woodland caribou in northeastern Alberta. Ecol Appl 21(8):2854–2865
Lele SR, Keim JL (2006) Weighted distributions and estimation of resource selection probability functions. Ecology 87(12):3021–3028
Lele SR, Merrill EH, Keim J, Boyce MS (2013) Selection, use, choice and occupancy: clarifying concepts in resource selection studies. J Anim Ecol 82(6):1183–1191
Manly BF, McDonald LL, Thomas DL, McDonald TL, Erickson WP (2002) Resource selection by animals: statical design and analysis for field studies, 2nd edn. Kluwer Academic Publishers, Dordrecht
Masden EA, Reeve R, Desholm M, Fox AD, Furness RW, Haydon DT (2012) Assessing the impact of marine wind farms on birds through movement modelling. J R Soc Interface 9(74):2120–2130
McClintock BT, Johnson DS, Hooten MB, Ver Hoef JM, Morales JM (2014) When to be discrete: the importance of time formulation in understanding animal movement. Mov Ecol 2(1):334
Mills KJ, Patterson BR, Murray DL (2006) Effects of variable sampling frequencies on GPS transmitter efficiency and estimated wolf home range size and movement distance. Wildl Soc Bull 34(5):1463–1469
Moorcroft PR, Barnett A (2008) Mechanistic home range models and resource selection analysis: a reconciliation and unification. Ecology 89(4):1112–1119
Mueller T, Lenz J, Caprano T, Fiedler W, Böhning-Gaese K (2014) Large frugivorous birds facilitate functional connectivity of fragmented landscapes. J Appl Ecol 51(3):684–692
Pépin D, Adrados C, Mann C, Janeau G (2004) Assessing real daily distance traveled by ungulates using differential GPS locations. J Mammal 85(4):774–780
Postlethwaite CM, Dennis TE (2013) Effects of temporal resolution on an inferential model of animal movement. PLoS One 8(5):e57,640
Potts JR, Lewis MA (2014) How do animal territories form and change? Lessons from 20 years of mechanistic modelling. Proc R Soc B 281(1784):20140,231
Potts JR, Bastille-Rousseau G, Murray DL, Schaefer JA, Lewis MA (2014) Predicting local and non-local effects of resources on animal space use using a mechanistic step selection model. Method Ecol Evol 5(3):253–262
Pyke GH (2015) Understanding movements of organisms: it’s time to abandon the Lévy foraging hypothesis. Method Ecol Evol 6(1):1–16
Rhodes JR, McAlpine CA, Lunney D, Possingham HP (2005) A spatially explicit habitat selection model incorporating home range behavior. Ecology 86(5):1199–1205
Robert CP, Casella G (2000) Monte Carlo statistical methods. Springer texts in statistics, 2nd edn. Springer, New York
Robinson WD, Bowlin MS, Bisson I, Shamoun-Baranes J, Thorup K, Diehl RH, Kunz TH, Mabey S, Winkler DW (2009) Integrating concepts and technologies to advance the study of bird migration. Front Ecol Environ 8(7):354–361
Rosser G, Fletcher AG, Maini PK, Baker RE (2013) The effect of sampling rate on observed statistics in a correlated random walk. J R Soc Interface 10(85):20130,273
Rowcliffe MJ, Carbone C, Kays R, Kranstauber B, Jansen PA (2012) Bias in estimating animal travel distance: the effect of sampling frequency. Method Ecol Evol 3(4):653–662
Ryan PG, Petersen SL, Peters G, Gremillet D (2004) GPS tracking a marine predator: the effects of precision, resolution and sampling rate on foraging tracks of African Penguins. Mar Biol 145(2):215–223
Sawyer H, Kauffman M, Nielson R, Horne J (2009) Identifying and prioritizing ungulate migration routes for landscape-level conservation. Ecol Appl 19(8):2016–2025
Schick RS, Loarie SR, Colchero F, Best BD, Boustany A, Conde DA, Halpin PN, Joppa LN, McClellan CM, Clark JS (2008) Understanding movement data and movement processes: current and emerging directions. Ecol Lett 11(12):1338–1350
Schlägel UE, Lewis MA (2016) A framework for analyzing the robustness of movement models to variable step discretization. J Math Biol. doi:10.1007/s00285-016-0969-5
Schlather M, Malinowski A, Oesting M, Boecker D, Strokorb K, Engelke S, Martini J, Menck P, Gross G, Burmeister K, Manitz J, Singleton S, Pfaff B, R Core Team (2014) RandomFields: simulation and analysis of random fields. R package version 3.0.10. http://CRAN.R-project.org/package=RandomFields. Accessed 7 May 2014
Smouse PE, Focardi S, Moorcroft PR, Kie JG, Forester JD, Morales JM (2010) Stochastic modelling of animal movement. Philos Trans R Soc B 365(1550):2201–2211
Squires JR, DeCesare NJ, Olson LE, Kolbe JA, Hebblewhite M, Parks SA (2013) Combining resource selection and movement behavior to predict corridors for Canada lynx at their southern range periphery. Biol Conserv 157:187–195
Tanferna A, López-Jiménez L, Blas J, Hiraldo F, Sergio F (2012) Different location sampling frequencies by satellite tags yield different estimates of migration performance: pooling data requires a common protocol. PLoS One 7(11):e49,659
Tsoar A, Nathan R, Bartan Y, Vyssotski A, Dell’Omo G, Ulanovsky N (2011) Large-scale navigational map in a mammal. Proc Natl Acad Sci USA 108(37):E718–E724
Turchin P (1998) Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants. Sinauer Associates, Sunderland, MA
Wiens DP (2000) Bias constrained minimax robust designs for misspecified regression models. In: Balakrishnan N, Melas VB, Ermakov S (eds) Statistics for industry and technology. Birkhäuser, Boston, MA, pp 117–133
Wiens DP, Zhou J (1996) Minimax regression designs for approximately linear models with autocorrelated errors. J Stat Plan Inference 55(1):95–106
Wilcox R (2012) Introduction to robust estimation and hypothesis testing, 3rd edn. Academic Press, Boston
Acknowledgments
We thank members of the Lewis research group for fruitful discussions and an anonymous reviewer for helpful comments on the manuscript. UES was supported by a scholarship from iCORE, now part of Alberta Innovates-Technology Futures and funding from the University of Alberta. MAL gratefully acknowledges Natural Sciences and Engineering Research Council Discovery and Accelerator grants, a Canada Research Chair and a Killam Research Fellowship.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Appendices
Appendix 1: Proofs of results about exact robustness
Proof
(Theorem 1) First, note that for any standard deviation of the kernel, \(\sigma \), the integral \(\int _{\mathbb {R}} k_{\sigma }(y;x)w(y)\,\mathrm {d}y\) reduces to the weighting function evaluated at the kernel’s mean,
because \(k_{\sigma }(\cdot | y)\) is a Gaussian density integrating to 1 and with vanishing first central moment. If we consider w as a linear transformation of a Normally distributed random variable with mean x, then Eq. (24) reflects a special case of Jensen’s inequality, in which equality holds.
We now show robustness of degree n with parameter transformation \(g_n(\sigma ,a,b) = (\sqrt{n}\sigma ,a,b)\) by induction. For \(n=1\), we have the trivial transformation \(g_1(\sigma , a, b) = (\sigma , a, b)\), and there is nothing to show for robustness of degree 1.
We assume that robustness or degree n holds, that is we have the relationship
for all \(x_n, x_0 \in \mathbb {R}\). For \(n+1\), we use the Chapman–Kolmogorov equation and Markov property and obtain
where the last step follows by induction. We can now insert the model’s step probabilities and use Eq. (24) to further calculate,
Note that we have assumed that all movement steps are within the domain \(\mathscr {I}\), where the weighting function is positive. Since \(k_{\sigma }(x_{n+1};x_n) = k_{\sigma }(x_{n+1}-x_n;0)\), the integral in the last expression is the convolution of two Gaussian densities with variances \(\sigma ^2\) and \(n\sigma ^2\) and with means 0 and \(x_0\), respectively. Because of the linearity of Gaussian random variables, this is again a Gaussian density with mean \(x_0\) and variance \((n+1)\sigma ^2\). Because Eq. (24) holds for the kernel with any standard deviation, we can rewrite the denominator as \(w(x_0) = \int _{\mathbb {R}} k_{\sqrt{n+1}\,\sigma }(y;x_0) w(y) \, dy\). Thus,
\(\square \)
Proof
(Theorem 2) We proceed analogously to the previous proof. The integral of weighting function and kernel with arbitrary standard deviation \(\sigma \) and mean x is here given by
By completing the square and using substitution \(u=\frac{1}{\sqrt{2} \sigma } (y-x-a\sigma ^2)\) we obtain
The final integral reduces to \(\sqrt{2\pi }\sigma \), and therefore,
Again, we prove robustness of degree n by induction, using parameter transformation \(g_n(\sigma , C, a, b) = (\sqrt{n}\sigma , C,a,b)\). In the induction step, we obtain, with help of Eq. (29),
\(\square \)
Appendix 2: Proof of result about asymptotic robustness
To highlight the main steps necessary to prove Theorem 3, we establish a series of intermediate results. As a first step, we show that the 2-step transition density can be broken up into a product of the form (5) in Definition 2.
Proposition 1
The 2-step transition density of model with transitions (14) can be written as
where the function v is given by
Note that v depends on \(\tau \) through \(\sigma \). For later convenience, we define
Proof
The proposition can be shown with a straightforward calculation. The 2-step transition density is given by
The product of the two Gaussian densities in the integrand can be transformed as follows
The two-step density therefore becomes
The numerator of the first factor is the desired one-step density up to appropriate normalization. If we extend by the required normalization constant \(\int _{\mathbb {R}} k_{\sqrt{2}\sigma }(y;x_{t-2\tau }) w_{\varvec{\theta }}(y)\, \mathrm {d}y\), we obtain Eqs. (31) and (32). \(\square \)
We are now left to show that the function \(v-1\) is in the order of \(\tau \) on its entire domain \(\mathbb {R}^2\times \mathbb {R}^+\). In particular, this means that for any fixed \(\tau ^{*}\), the function \(v(x_1, x_2; \tau ^{*})-1\) is bounded on \(\mathbb {R}^2\) via \(c\tau ^{*}\) for a constant c. It turns out to be helpful to analyze v separately on \(\mathbb {R}^2\times (0,\tau _0)\) and \(\mathbb {R}^2\times [\tau _0,\infty )\) for some \(\tau _0\). Because the proof is simpler for large \(\tau \), we present this result first.
Lemma 1
Let w be continuous and bounded away from zero, that is there exist L and U such that \(0<L\le w_{\varvec{\theta }}(x) \le U\) for all \(x\in \mathbb {R}\). Let w further be twice differentiable on \(\mathbb {R}\) with \(|w''(x)|<M\) for some M and all \(x\in \mathbb {R}\). For any \(\tau _0 > 0\), we have \(v(x_1, x_2,;\tau ) -1 = \mathscr {O}(\tau )\) on \(\mathbb {R}^2\times [\tau _0,\infty )\).
Proof
Let \(\tau _0\) be a number away from zero and fixed. Our goal is to establish bounds on the functions Q and I, as defined in (33) and (34), and to use these to place a bound on \(v-1\). Because w is twice differentiable we can apply Taylor’s theorem to obtain a linear approximation for w using any point \(x\in \mathbb {R}\),
where R(y) is the remainder term. This leads to
where the first term on the RHS becomes \(w_{\varvec{\theta }}(x)\), because the kernel integrates to 1, and the integral in the second term is the first central moment of the kernel, hence vanishes. The remainder R(y), using the Lagrange form, is given by \(R(y) = \frac{w''(\xi )}{2}(y-x)^2\), for some \(\xi \) between \(x_2\) and y. Since the second derivative of w is assumed to be globally bounded, we have \(|R(y)|\le \frac{M}{2}(y-x)^2\). We use this to place bounds on the third term, recognizing that the remaining integral \(\int _{\mathbb {R}} k_{\sigma }(y;x)\,(y-x)^2\,\mathrm {d}y\) is the second central moment of the Gaussian kernel \(k_{\sigma }\), which is given by its variance \(\sigma ^2 = \omega ^2\tau \). Therefore,
In general, the lower bound can be arbitrarily close to zero, therefore we cannot simply invert this inequality to obtain an estimate on the inverse of the integral. Instead, we use the bounds on w and again the fact \(\int _{\mathbb {R}} k_{\sigma }(y;x)\,\mathrm {d}y=1\) for any \(\sigma \) and any \(x\in \mathbb {R}\) to establish
which can be inverted. Since inequalities (42) and (43) hold for any \(\sigma \) and any \(x \in \mathbb {R}\), they allow us to place bounds on both Q and I. For Q, we obtain
for all \(x\in \mathbb {R}, \tau \in \mathbb {R}^+\). We can avoid the dependency of the bounds on x by again invoking the bounds on w,
For the function I, we only make use of the bounds on w and inequality (43) and get
for all \(x_1, x_2 \in \mathbb {R}, \tau \in \mathbb {R}^+\). We can now continue to calculate \(v-1\). An upper bound is immediately given by
With only few more additional steps, we obtain a lower bound by simply drawing upon \(L \le U\), its squared version and its inverse,
Define \(C:= \frac{U^2-L^2}{L^2\tau _0} + \frac{MU}{L^2}\omega ^2\) for the \(\tau _0\) chosen up front. Then,
The product on the RHS is non-positive for \(\tau \ge \tau _0\), and hence \(|v(x_1, x_2;\tau ) -1 |\le C\tau \) for all \(\mathbb {R}^2\times [\tau _0, \infty )\). \(\square \)
The bounds on Q and I, and thus \(v-1\), established in the preceding proof are not sufficient to conclude the result as \(\tau \rightarrow 0\), unless \(L=U\), which is the trivial case of a constant weighting function. More suitable bounds, however, can be found if inequality (42) can be inverted. This can be achieved by assuming \(\tau \) to be small enough.
Lemma 2
Let w be continuous and bounded away from zero, that is there exist L and U such that \(0<L\le w_{\varvec{\theta }}(x) \le U\) for all \(x\in \mathbb {R}\). Let w further be twice differentiable on \(\mathbb {R}\) with \(|w''(x)|<M\) for some M and all \(x\in \mathbb {R}\). Let \(\tau _0 = \frac{2L}{M\omega ^2}\). Then \(v(x_1, x_2,;\tau ) -1 = \mathscr {O}(\tau )\) on \(\mathbb {R}^2\times (0,\tau _0)\).
Proof
Here we develop bounds on Q and I such that both \(Q-1\) and \(I-1\) are in the order of \(\tau \). Let \(\tau \le \tau _{0}\) for \(\tau _0\) as defined in the lemma. Then the lower bound of Eq. (42) is bounded away from zero,
Hence we can invert the inequality (42) and obtain
Note that the values in the numerators and denominators differ slightly because the variances of the kernel k in the numerator and denominator of Q differ by a factor of 2.
Since \(2w_{\varvec{\theta }}(x)-M\omega ^2\tau \ge 2L-M\omega ^2\tau _0>0\), we can conclude
for all \(x\in \mathbb {R}\) and \(\tau <\tau _0\). Using \(2w_{\varvec{\theta }}(x)+M\omega ^2\tau \ge 2w_{\varvec{\theta }}(x)\ge 2L\), we similarly obtain,
for all \(x\in \mathbb {R}\) and \(\tau <\tau _0\). If we set \(C_1 := \max \left( \frac{M\omega ^2}{2L-2\omega ^2\tau _0},\frac{3M\omega ^2}{2L} \right) \), it follows that \(\vert Q(x;\tau ) - 1 \vert \le C_1\tau \) on \(\mathbb {R}^2\times (0,\tau _0)\).
Using analogous arguments as before, we can fine an find an upper bound on I,
A lower bound is given by
Setting \(C_2 := \frac{M\omega ^2\tau }{2L - M\omega ^2\tau _0}\), we obtain \(|I(x_1,x_2;\tau )-1| \le C_2\tau \) on \(\mathbb {R}^2\times (0,\tau _0)\).
We can now estimate \(v-1\) as follows,
for all \(x_1, x_2\in \mathbb {R}\) and all \(\tau <\tau _0\). \(\square \)
Lemmata 1 and 2, together with proposition 1 prove Theorem 3.
Rights and permissions
About this article
Cite this article
Schlägel, U.E., Lewis, M.A. Robustness of movement models: can models bridge the gap between temporal scales of data sets and behavioural processes?. J. Math. Biol. 73, 1691–1726 (2016). https://doi.org/10.1007/s00285-016-1005-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-016-1005-5