# Identifying latent behavioural states in animal movement with M4, a nonparametric Bayesian method

**Handling Editor: ** Robert Freckleton

## Abstract

- Understanding animal movement often relies upon telemetry and biologging devices. These data are frequently used to estimate latent behavioural states to help understand why animals move across the landscape. While there are a variety of methods that make behavioural inferences from biotelemetry data, some features of these methods (e.g. analysis of a single data stream, use of parametric distributions) may limit their generality to reliably discriminate among behavioural states.
- To address some of the limitations of existing behavioural state estimation models, we introduce a nonparametric Bayesian framework called the mixed-membership method for movement (M4), which is available within the open-source bayesmove R package. This framework can analyse multiple data streams (e.g. step length, turning angle, acceleration) without relying on parametric distributions, which may capture complex behaviours more successfully than current methods. We tested our Bayesian framework using simulated trajectories and compared model performance against two segmentation methods (behavioural change point analysis (BCPA) and segclust2d), one machine learning method [expectation-maximization binary clustering (EMbC)] and one type of state-space model [hidden Markov model (HMM)]. We also illustrated this Bayesian framework using movements of juvenile snail kites
*Rostrhamus sociabilis*in Florida, USA. - The Bayesian framework estimated breakpoints more accurately than the other segmentation methods for tracks of different lengths. Likewise, the Bayesian framework provided more accurate estimates of behaviour than the other state estimation methods when simulations were generated from less frequently considered distributions (e.g. truncated normal, beta, uniform). Three behavioural states were estimated from snail kite movements, which were labelled as ‘encamped’, ‘area-restricted search’ and ‘transit’. Changes in these behaviours over time were associated with known dispersal events from the nest site, as well as movements to and from possible breeding locations.
- Our nonparametric Bayesian framework estimated behavioural states with comparable or superior accuracy compared to the other methods when step lengths and turning angles of simulations were generated from less frequently considered distributions. Since the most appropriate parametric distributions may not be obvious a priori, methods (such as M4) that are agnostic to the underlying distributions can provide powerful alternatives to address questions in movement ecology.

## 1 INTRODUCTION

Our understanding of animal movement has advanced considerably in recent decades with the emergence of the field of movement ecology (Fraser et al., 2018; Joo, Picardi, et al., 2020), which focuses on understanding where animals go, what they are doing and how they are influenced by their surrounding environment (Nathan et al., 2008). As telemetry and biologging devices continue to increase in their battery life, data resolution and affordability (Hussey et al., 2015; Kays et al., 2015), statistical methods that can efficiently analyse these large datasets will become ever more important (Patterson et al., 2017; Potts et al., 2018). To fully understand animal movement, it is necessary to account for behaviour since space and resource use are directly linked to an animal's internal state (Gurarie et al., 2016; Nathan et al., 2008).

Since the direct observation of animal behaviour can be challenging in many situations, recorded tracks from biologging devices are increasingly used to infer potential behaviour by estimating latent states. These latent states can be estimated from a variety of data streams (i.e. time series of variables) such as step lengths, turning angles, ambient temperature and acceleration, among others (Edelhoff et al., 2016). State estimation is often performed using segmentation and clustering methods, as well as state-space models (SSMs). Segmentation methods partition tracks into segments by detecting shifts in the data stream(s), whereas clustering methods classify these segments (or the observations directly) into discrete states. Alternatively, SSMs estimate latent states per observation based on the transition probabilities among a given number of states (Edelhoff et al., 2016; Gurarie et al., 2016). While existing state estimation methods provide fast or powerful predictive capacity (Edelhoff et al., 2016; Patterson et al., 2017), they possess a number of limitations that can impact the inference made on behavioural states.

For instance, segmentation methods commonly infer behaviour using only a single data stream such as persistence velocity or speed (Edelhoff et al., 2016; but see Patin et al., 2020). This can be problematic when underlying behaviours are complex and not well-represented by a single metric alone. Additionally, many segmentation methods, clustering methods and SSMs typically estimate behavioural states by fitting the data streams to parametric probability distributions (e.g. Edelhoff et al., 2016; Joo, Boone, et al., 2020; Patterson et al., 2017), such as Gaussian, gamma or wrapped Cauchy distributions. When the structure in the data streams is not well-captured by parametric distributions, this can often result in overestimation of the true number of states when information criteria are used due to model misspecification (Gurarie et al., 2016; Pohle et al., 2017). Furthermore, running SSMs and some clustering methods can be computationally costly: model runtime can take minutes to days depending on the type of model, sample size, number of estimated states and computer hardware. This is further exacerbated when model selection (e.g. determining the likely number of groups by fitting models with different numbers of groups) and multi-model inference are performed.

Given the limitations posed by existing state estimation methods, there is a need to develop a framework that is based on as few parametric assumptions as possible while also being fast and flexible. Here, we introduce a new two-stage modelling framework called the mixed-membership method for movement (M4) that implements nonparametric Bayesian methods to: (a) jointly segment multiple data streams into relatively homogeneous units of behaviours; and (b) subsequently determine the likely number of behavioural states using a mixed-membership method where segments can be comprised of more than one behavioural state. Latent behavioural states are estimated for entire track segments (as opposed to individual observations) since this reflects our understanding that behaviour is inherently autocorrelated, especially when observations are sampled at short time intervals (Pohle et al., 2017; Potts et al., 2018). Additionally, track segments are expected to be characterized by multiple states (Patin et al., 2020; Pohle et al., 2017). This M4 model framework is available within the open-source R package bayesmove available on CRAN (Cullen & Valle, 2021). In this article, we describe the model structure and the Bayesian sampling methods used to estimate from the posterior distribution. We then demonstrate that M4 can successfully recover breakpoints and behavioural states based on simulated trajectories and compare our model's performance against two common segmentation methods (behavioural change point analysis (BCPA), Gurarie et al., 2009; segclust2d, Patin et al., 2020), one machine learning method (expectation-maximization binary clustering (EMbC), Garriga et al., 2016) and one type of SSM, a hidden Markov model (HMM). Finally, we illustrate our novel approach on the movements of an endangered raptor species, the Everglade snail kite *Rostrhamus sociabilis*, and interpret the results within the context of natal and breeding dispersal events.

## 2 MATERIALS AND METHODS

### 2.1 Model structure

Most existing segmentation methods (e.g. BCPA, segclust2d, behavioural movement segmentation), some machine learning methods (e.g. EMbC) and most SSMs (e.g. HMMs, multistate random walks) experience one or more common limitations to behavioural state estimation. These limitations include the reliance on parametric distributions, analysis of only a single data stream, as well as reliance on information criteria to determine the most likely number of states.

#### 2.1.1 Discretization of data streams

We address the problem of parametric distributions by providing an approach that relaxes parametric assumptions through the discretization of data streams (Figure 1a–c). Although data streams (e.g. step lengths and turning angles) are not typically discretized into bins, we expect that this may lead to more robust estimates in the face of parametric distribution uncertainty. This is because bins are estimated independently of one another and extreme values lose their influence when added to the first or last bins with the rest of the data. Therefore, the discretization of data streams is expected to increase model flexibility (John & Langley, 1995; Kitagawa, 1987).

Selecting the number of bins and the binning method is relatively subjective and therefore it is important that prior biological reasoning be used to inform these decisions. For example, discretization methods may include the use of equal bin widths or quantiles. However, the number of bins should be sufficient to characterize the shape of the density distribution. These assumptions during the discretization process are not unlike assumptions made for HMMs when selecting probability distributions to fit data streams, but require practitioners to make more decisions up front. Based on a sensitivity analysis of binning methods used on a right-skewed data stream (i.e. step lengths), the use of quantiles resulted in greater discrimination of behavioural states than bins of equal widths (Appendix S1). However, data streams with circular distributions (e.g. turning angles) will likely be more interpretable when using bins of equal widths. At a minimum, discretized values for each data stream and the associated track IDs are required to begin analysing the data.

#### 2.1.2 Segmentation

*P*breakpoints , where

*k*is the model number. Each breakpoint is restricted to being an integer between 2 and across all observations, where is the total number of observations for individual

*i*. Given a particular model , its breakpoints define track segments. We assume that for any given track segment

*c*:

*i*at time

*t*for data stream

*j*and is a vector of probabilities that sum to 1. The vector indicates the probability that observations within segment

*c*are assigned to one of the

*L*bins. Overall, the model is seeking breakpoints that define relatively homogeneous track segments. The use of a categorical distribution to characterize track segments (as opposed to continuous distributions) is what makes this framework nonparametric. Our prior is given by:

Since the posterior distribution of models can vary greatly in the number and position of breakpoints, but only a single set of breakpoints can be used to define track segments, we select the *maximum a posteriori* (MAP) estimate (i.e. the breakpoints of the model with the greatest log marginal likelihood) (Figure 1d). Although the MAP estimate does not account for uncertainty in breakpoint number and position, it appears to be in good agreement with estimates from the entire posterior distribution as described in Appendix S4. These MAP breakpoints are then used to define segments per individual track, which are subsequently clustered into latent behavioural states by a mixed-membership model.

#### 2.1.3 Mixed-membership clustering

Although most existing state estimation methods assign a single discrete state to observations or track segments (e.g. Garriga et al., 2016; McClintock & Michelot, 2018; Patin et al., 2020; but see Jonsen et al., 2019), animal movement may not be entirely comprised of a single behaviour over a given sampling interval (Patin et al., 2020; Pohle et al., 2017). Latent Dirichlet allocation (LDA), a mixed-membership clustering method, can be used to classify each track segment as a mixture of multiple states (Hudon et al., 2021; Valle et al., 2014). For example, a proportion of observations within a given track segment might belong to state 1 while another proportion might belong to state 2 and so on.

*t*of track segment

*c*from data stream

*j*for individual

*i*. Additionally, is the latent behavioural state membership associated with and is a vector of probabilities for each behaviour and data stream. Notice that influences the distribution for by determining the subscript

*k*for the vector . We assume that the latent state membership is given by:

*K*(i.e. the number of clusters or states) that sum to 1 and indicates the likelihood of assigning an observation at time

*t*of track segment

*c*for individual

*i*to each behavioural state

*k*. This formulation assumes that each observation within a particular track segment must belong to a single behavioural state, but that track segments are comprised of multiple states. For our priors, we assume that:

*k*to track segment

*c*(i.e. ) to approximately zero as

*k*approaches

*K*. As a result, fewer and fewer observations will be assigned to states with large values of

*k*, enabling the model to identify the most likely number of behavioural states (Valle et al., in press; Valle et al., 2017). This is an improvement on existing state estimation methods in the sense that our model only needs to be run once, whereas several other common methods (e.g. HMMs, segclust2d and other clustering methods) are typically run multiple times with varying numbers of behavioural states to then determine the best model via information criteria (e.g. AIC or BIC).

This LDA model is fitted using a Gibbs sampler and a complete description of the full conditional distribution can be found in Appendix S5. Similar to the segmentation model, convergence was assessed by inspecting trace plots of the log-likelihood. The posterior mean for all was then used to identify the most likely number of behaviours. The estimated state-dependent distributions for each data stream (from ) were evaluated and used to corroborate the findings based on the posterior means from all ’s by determining whether the distributions were biologically relevant (Figure 1e–g). This combination of results provides a straightforward approach to selecting the most likely number of behavioural states. A list of the primary functions to analyse data using the M4 framework within the bayesmove R package is included in Appendix S6.

### 2.2 Simulation study

We assessed the performance of M4 compared to other methods via simulations. We first evaluated the ability of our track segmentation method to detect true breakpoints and compared its results to those obtained by two segmentation methods (i.e. BCPA and segclust2d). We then evaluated the ability of our clustering method to estimate the true number of behavioural states and to properly assign behaviour proportions to track segments. For this component, we compared the results of our model to those obtained by a HMM (McClintock & Michelot, 2018) and two additional clustering methods (i.e. segclust2d and EMbC).

#### 2.2.1 Generating simulated trajectories

We generated multiple three-state trajectories from a correlated random walk at regular time intervals, where five tracks were simulated at each of four durations (1,000, 5,000, 10,000 and 50,000 observations), resulting in a total of 20 tracks. Each track was comprised of 10, 50, 100 or 500 segments that each included 100 observations. Each of these segments included a dominant behavioural state (80% of observations), which was randomly assigned to each segment. The three behavioural states were parameterized to represent (a) little to no movement (‘encamped’), (b) slow and tortuous movement (‘area-restricted search’ or ARS), as well as (c) fast and directed movement (‘transit’). To assess model performance on tracks generated from different types of distributions, we generated two sets of tracks with 20 simulations in each. In the first set of simulations, step lengths for each behaviour were drawn from a truncated normal distribution and turning angles were drawn from either a beta, uniform or truncated normal distribution (hereafter referred to as ‘uncommon distributions’) (Figure 2a). For the second set of simulations, step lengths for each behavioural state were generated from a gamma distribution and turning angles were drawn from a wrapped Cauchy distribution (hereafter referred to as ‘common distributions’) (Figure 2b). Both sets of simulations were designed to generally resemble each other in their step length and turning angle distributions. Additional comparisons of simulated tracks generated from a HMM can be found in Appendix S11.

#### 2.2.2 Implementation of Bayesian M4 framework

Step lengths and turning angles were the data streams used to make inference on latent behavioural states. Step lengths were separated into five bins using the 25th, 50th, 75th, 90th and 100th quantiles as upper limits. Quantiles were used to discretize highly right-skewed step lengths as suggested by our sensitivity analysis (Appendix S1). Turning angles were discretized into eight bins from to using equal widths since the distribution of this variable was relatively balanced and bounded by lower and upper limits.

Each simulated track was analysed by the M4 segmentation model using a vague prior where the hyperparameter was set to 1. Trace plots of the log marginal likelihood indicated that the model reached convergence for each simulation, where 40,000 MCMC iterations were used for all but the longest tracks, which used 60,000 iterations (Appendix S7). We then assessed how well our model identified the true breakpoints for each simulation, where a threshold of ±10 observations was used to distinguish an accurate from an inaccurate estimated breakpoint. If no breakpoints were estimated within ±30 observations of a true breakpoint, then the model was considered to have ‘missed’ that breakpoint. Other thresholds were tested and all resulted in the same relative pattern of accuracy (Appendix S1).

Estimated track segments were used as input for the LDA model of M4, which was run using 1,000 MCMC iterations, a burn-in of 500 iterations and vague priors where hyperparameters were set to and . We set the maximum number of behavioural states to seven since this was expected to include the true number of states, estimated via the truncated stick-breaking prior of the LDA. Trace plots of the log-likelihood indicated that the model reached convergence for each simulation (Appendix S7). The true number of states was estimated by calculating the arithmetic mean of behaviour proportions across all track segments and selecting the set of states that together represented ≥90% of all observations on average. Additionally, state-dependent distributions of step lengths and turning angles were inspected so that we only selected states that were also biologically interpretable. Since the LDA treats track segments as a combination of behavioural states, proportions of each state were estimated per track segment. The accuracy of state estimates was evaluated by two methods: (a) we calculated the percentage of observations where the dominant behaviour of each track segment was accurately classified, and (b) we calculated the root mean square error (RMSE) of the estimated behaviour proportions compared to the true behaviour proportions over all states and track segments.

#### 2.2.3 Method comparison

We compared the performance of M4 on the simulated trajectories against BCPA, EMbC, HMM and segclust2d (see Appendix S8 for details regarding model properties and assumptions). All models were run using a 2.6 GHz i7 CPU with 16 GB RAM.

#### Segmentation models

The BCPA model performed segmentation based on persistence velocity (PV), which is a combination of velocity (*V*) and turning angle () [i.e. ], using the R package bcpa v1.1 (Gurarie, 2014). Parameters for BCPA were tuned to provide a close approximation of the true number and location of simulated breakpoints with window size set to 80, sensitivity set to 2 and clusterwidth set to 30. Breakpoint accuracy was evaluated using the same method as for M4.

The segclust2d model performed segmentation on step lengths and the absolute value of turning angles using the R package segclust2d v0.2.0 (Patin et al., 2019). This method models each data stream using a Gaussian distribution, so the absolute value of turning angles was used to accommodate this unimodal assumption. Tuning parameters were chosen within the bounds of the simulated tracks, such that the maximum number of segments was set to 1.5 × *s*, where *s* is the true number of segments, the minimum observations per segment was set to 50 and the number of potential clusters (i.e. states) ranged from 2 to 4. Since the model was still analysing the longest simulations (50,000 observations) after 2 days, these tracks were omitted from the reported results for segclust2d. Breakpoint accuracy was assessed in the same manner as for M4.

#### Clustering models

The EMbC model was fitted to step lengths and the absolute value of turning angles using the R package EMbC v2.0.3 (Garriga et al., 2019). The absolute value of turning angles was used to achieve better discrimination among states given the use of a unimodal distribution like the Gaussian distribution. This model uses binary clustering to partition each of *n* data streams into a ‘low’ and ‘high’ class, resulting in a total of 2* ^{n}* possible states. For our analysis, this resulted in four states estimated from a bivariate Gaussian distribution. To make these results comparable to the other models, both states with ‘high’ step lengths (and ‘low’ or ‘high’ turning angles) were merged into a single state to produce three states overall. State classification accuracy was assessed at the segment level so that results were directly comparable with the Bayesian M4 model. This was achieved by using true breakpoints to segment the time series of states estimated by the EMbC model and then calculating the proportion of these behaviours within each track segment. Additionally, the resulting state-dependent distributions of step lengths and turning angles were discretized using the bin limits defined for M4 to compare the accuracy of distribution shapes. Accuracy was measured by RMSE across bins of all states and data streams per simulation.

A discrete-time HMM was also fitted to each of the simulated trajectories using the R package momentuHMM v1.5 (McClintock & Michelot, 2018). Step lengths were modelled using a gamma distribution, and turning angles were assumed to arise from a wrapped Cauchy distribution. The HMMs for each simulation were run using a range of two to four possible behavioural states () and each -state model was run 30 times using different starting values to increase the chance of finding the global (as opposed to local) maximum of the likelihood. The selection of ‘good’ starting values is critical since it can affect computation time and the ability of the model to identify the global maximum of the likelihood (McClintock & Michelot, 2018; Michelot et al., 2016). The most likely number of states was selected using a combination of AIC and BIC, where the model with the lowest value was considered to be most likely. However, if the difference in AIC or BIC (ΔAIC/BIC) of the next best model was <10 (Burnham & Anderson, 2002), the more parsimonious model was chosen. Behaviour classification accuracy was assessed in the same manner as for EMbC.

The segclust2d model clustered segments previously estimated by this method into *K* states. The number of likely states was estimated using BIC in the same manner as for HMM. The likely number of states (and associated breakpoints) were used to assign behavioural states to track segments, which were then compared to the other methods using the proportion of each state per estimated segment (which were all either 0 or 1). Additionally, the accuracy of the state-dependent distributions was evaluated in the same manner as for EMbC and HMM.

### 2.3 Snail kite case study

As part of a larger investigation on the effects of wetland management on wildlife, solar-powered GPS-GSM transmitters (Ecotone Telemetry) were attached to juvenile snail kites (*n* = 26) prior to fledging at Lakes Tohopekaliga, East Tohopekaliga and Kissimmee in central Florida during 2018 and 2019. Tagging of snail kites was conducted under US Geological Survey BBL Permit #23906. Subsequent movement of each individual resulted in a total of 40,720 observations (Figure 3). Locations were collected once per hour only during daylight at an accuracy of ±30 m. As a result of the programmed duty cycle and time periods where GPS tags failed to transmit data, track time intervals were irregular. To ensure comparable step lengths and turning angles, we filtered our data to the most common time interval (i.e. 1 hr). We chose to omit all other observations since imputation procedures for long time gaps would increase the number of artificial data and the use of linear interpolation would artificially inflate the number of turning angles at zero radians.

Step lengths and turning angles were used to estimate latent behavioural states. As was performed on the simulated tracks, step lengths for the empirical data were discretized into five bins using the 25th, 50th, 75th, 90th and 100th quantiles as upper limits. This resulted in bin limits at 0.00, 0.03, 0.09, 0.32, 1.63 and 72.56 km. Turning angles were discretized into eight bins from to using equal bin widths, resulting in bin limits at .

Step lengths and turning angles for each of the 26 snail kites were analysed by the M4 segmentation model using 80,000 iterations, a burn-in of 40,000 iterations and hyperparameter . The MAP estimates of breakpoints for each snail kite were used to define track segments per individual. Subsequently, track segments were analysed across all individuals via LDA to estimate the most likely number of states, to define state-dependent distributions and to estimate the proportion of each state that characterized each track segment. This was performed using 1,000 MCMC iterations with a burn-in of 500 iterations and vague priors were used with hyperparameters set at and for a maximum possible number of seven states. Trace plots of the segmentation and LDA models from M4 indicated that all had reached convergence (Appendix S7). The proportions of behavioural states were evaluated over time in relation to emigration from natal sites and peak breeding season of snail kites in Florida (1 March–30 June; Reichert et al., 2020) to discern any patterns associated with these events.

## 3 RESULTS

### 3.1 Segmentation model comparison

The M4 segmentation model successfully recovered breakpoints from the simulations and outperformed both BCPA and segclust2d. Among the three methods, the segclust2d model took much longer to run (0.46 to 418 min) compared to M4 (0.24 to 11 min) and BCPA models (0.25 to 21 min), particularly for longer tracks (Figures 4a and 5a). While all three models exhibited similar accuracy on the shortest simulations, M4 was much more accurate on all larger simulations. For these large simulations, the accuracy of the M4 segmentation model was >80% on average when simulations were generated from uncommon distributions and >90% on average when generated from common distributions (Figures 4b, 5b and 6a). Additionally, M4 missed the lowest proportion of true breakpoints (uncommon: 21%; common: 0.3%) compared to BCPA (uncommon: 67%; common: 66%) and segclust2d (uncommon: 26%; common: 30%) across simulations of all analysed track lengths.

### 3.2 Clustering model comparison

When estimating the true number of states on both sets of simulations, M4 correctly determined the number of true states more frequently than the other methods and exhibited greater computational efficiency over all other clustering methods besides EMbC. The Bayesian LDA model took 2–23 s to run, highlighting the computational efficiency of this particular model. When added to the duration of the segmentation model, the proposed method ran much faster than the HMM (20–36×) and segclust2d (2–178×) at all track lengths despite these models being fitted with only two to four states, whereas our method allows for up to seven states (Figures 4a and 5a). However, the time to run each EMbC model increased very little with increases in track length, but also automatically assumed four states were present. The LDA model from M4 correctly suggested three states as most likely for 18 of the 20 simulations generated from uncommon distributions and 19 of 20 simulations generated from common distributions (Figure 6b, Appendix S9). By comparison, the HMM suggested (via AIC and BIC) that three states were most likely in 17 and 16 of the 20 analysed simulations generated by uncommon and common distributions, respectively. The segclust2d model suggested that three states were most likely in only six and five of the 15 analysed simulations generated from uncommon and common distributions, respectively, based on BIC.

To enable direct comparisons among all four models that estimated behavioural states, we assumed three states were most likely for all 20 simulations when calculating model accuracy. Using this assumption, we find that M4 demonstrated high accuracy in behavioural state estimation for both sets of simulations, often equivalent or superior to the other clustering methods. When analysing simulations generated from uncommon distributions, mean accuracy of M4 to classify the dominant state within each segment was greater than that of the HMM and segclust2d models at all track lengths (Figure 4c). However, mean accuracy of the EMbC model was slightly greater than M4 on this set of simulations at a track length of 5,000 observations. When analysing simulations generated from common distributions, mean accuracy of M4 was slightly below that of the HMM, but greater than the mean accuracy of the EMbC and segclust2d models at all track lengths (Figure 5c). Additionally, accuracy measures displayed little variability in M4 across tracks of different lengths and on each set of simulations, highlighting the increased stability of this framework. Similar to the pattern found for estimates of dominant behavioural states, the accuracy of behavioural state proportions was higher in M4 for all but the HMM on simulations generated from common distributions, as denoted by low RMSE (Figures 4d, 5d and 6c).

The accuracy of the estimated step lengths and turning angles distributions was relatively consistent across each set of simulations. For tracks generated from uncommon distributions, M4 was slightly more accurate than the HMM, but much more accurate than EMbC and segclust2d across all track lengths (Appendix S9). However, HMM estimates were slightly more accurate than the Bayesian model on tracks of all lengths when generated from common distributions (Appendix S9). When viewed as continuous distributions, it is clear that the HMM, EMbC and segclust2d models had difficulty estimating the true distributions of step lengths and turning angles regardless of track length for the simulation with uncommon distributions (Appendix S10). On the other hand, the HMM was able to perfectly estimate the state-dependent distributions of the simulations generated from common distributions (Appendix S10).

### 3.3 Snail kite analysis

The segmentation of 26 snail kite trajectories using M4 took 4 min to run and estimated 1 to 64 breakpoints for these individuals. Breakpoints were then used to define 444 track segments from all individuals (Figure 7a). These segments were clustered into states using M4, which took approximately 27 s to run. It appeared that there were likely three behavioural states, which comprised 91.6% of all state assignments on average (Figure 7b). To ensure that these three states were biologically interpretable, distributions of step lengths and turning angles were also evaluated (Figure 7c). The distributions showed: (a) a slow and tortuous behaviour; (b) a tortuous behaviour with intermediate speed; and (c) a fast and directed behaviour. For this reason, these behaviours were labelled ‘encamped, ‘ARS’ and ‘transit’ respectively.

Some individuals were only tracked for a short period of time and did not leave the natal area. However, 17 birds did emigrate from their natal site. Dispersal events were typically denoted by a brief period of ARS or transit behaviour (Figure 8a,b; Appendix S9). The three longest tracks, which belonged to snail kites tracked for more than a year (SNIK 12, SNIK 14 and SNIK 15), displayed relatively synchronous behaviour before, during and after their first breeding season. Two brief periods of high activity behaviour that occurred during and immediately following peak breeding season in 2019 may potentially represent pre- and post-breeding dispersal events (Figure 8a,c,d).

## 4 DISCUSSION

We demonstrated that our Bayesian M4 framework (available within the bayesmove R package) can accurately identify changes in behavioural states, reliably estimate the most likely number of behavioural states and properly characterize the state-dependent distributions of data streams. This two-stage model treats track segments as the unit of interest (as opposed to observations) and relies on the discretization of data streams to avoid the need to specify parametric probability distributions. Importantly, the proposed method is computationally efficient, a key characteristic given the ever-increasing storage capacities of modern sensors and their ability to measure a growing number of intrinsic and environmental variables (Whitford & Klimley, 2019; Williams et al., 2020). A comparison of model performance in addition to the analysis of an empirical dataset highlight the utility of the M4 framework.

### 4.1 Method comparison

Although BCPA displayed comparable speed to M4 during track segmentation, the accuracy of the estimated breakpoints was much higher in the latter. Additionally, M4 was much faster and exhibited greater accuracy of breakpoint estimates than the segclust2d method, which was not able to successfully analyse the simulated tracks of 50,000 observations. Since the accuracy of the segclust2d method was not much greater than the BCPA for either set of simulations (Figures 4b and 5b), it appears that BCPA’s reliance on a single derived variable (i.e. persistence velocity) instead of separate data streams was not as limiting as was initially expected.

While HMMs are powerful methods that can incorporate individual-level random effects and account for cyclical patterns (McClintock & Michelot, 2018; Patterson et al., 2017), they can also be restrictive in some of their assumptions. Standard forms of HMMs require the use of parametric distributions, which may not fit the data streams well (Appendix S9; Langrock et al., 2018). While HMMs displayed greater accuracy than M4 when the selected parametric distributions matched the true underlying distributions (Figure 5c), we find that the proposed methodology performed better than HMMs when the selected parametric distributions did not match the true underlying distribution. By comparison, the segclust2d and EMbC methods are straightforward to apply when estimating latent behavioural states from a set of tracks, but appear limited by their assumption of Gaussian distributions when partitioning observations into segments or into states, respectively. Since the most common data streams (i.e. step lengths and turning angles) are not typically modelled with a Gaussian distribution (McClintock et al., 2020), this likely contributes to the lower accuracy of these models.

The determination of the most likely number of states is another issue when fitting clustering models and HMMs since this is typically unknown a priori and is directly impacted by how well the selected parametric distributions characterize the states (Pohle et al., 2017). Unfortunately, HMMs often require multiple models to be fit and compared using information theoretic approaches, which tend to favour a greater number of states than are truly present and come at a high computational cost (Li & Bolker, 2017; Pohle et al., 2017). Importantly, while M4 allows for up to seven behavioural states, we only attempted to fit HMMs with two to four behavioural states. Even in this limited context, fitting HMMs was already much slower than fitting M4. Had we attempted to fit HMMs with two to seven behavioural states, the amount of time required for this would be substantially larger than what we report in Figures 4 and 5. A similar issue is present in segclust2d, where models are fit with every possible number of track segments and states before comparing via BIC. A different problem is posed by the EMbC model, which imposes four states by default when analysing step lengths and turning angles. These issues are directly addressed by our framework since we use a mixed-membership model (LDA) with a penalizing prior to cluster track segments, enabling the estimation of the most likely number of states and the state-dependent distributions in a single step. While existing methods can provide useful behavioural inference depending upon the ecological question and dataset, the M4 framework provides a powerful alternative when behaviours are complex, multiple data streams are available and these data are not well-characterized by parametric distributions and/or when datasets are large.

### 4.2 Empirical applications

Three behavioural states were clearly estimated for the snail kite dataset, which was supported by biologically relevant distributions of step lengths and turning angles. The ‘encamped’ state likely represents fine-scale behaviours that include resting, feeding and time spent at the nest (as a fledgling or reproductive adult). On the other hand, the ‘ARS’ state likely includes exploration for nearby suitable habitat as well as foraging bouts (Martin et al., 2006; Pias et al., 2016). Finally, the ‘transit’ state includes fast, directed movements associated with dispersal of snail kites in addition to departure from wetlands experiencing low water levels (Robertson et al., 2017).

The time series of snail kite behaviour proportions showed variability in the timing of emigration from natal sites among individuals, but changes in behaviour were generally synchronous in the three birds that reached maturity. This variability in the timing of emigration from natal sites could be due to a variety of factors, such as hatching date, body condition and local environmental conditions (Cattau et al., 2016; Fletcher et al., 2015; Rodgers & Schwikert, 2003). The shifts in behaviour proportions appeared to show multiple phases of high and low activity, some of which seem to match the phenology of natal dispersal (summer), pre-breeding dispersal (early spring) and post-breeding dispersal (late summer) (Bennetts & Kitchens, 2000). While the continued monitoring of these tagged birds should provide greater evidence for the characterization of activity budgets over ontogeny, future research could also explore the primary drivers of snail kite movement and habitat use within each behavioural state through the inclusion of environmental covariates.

### 4.3 Caveats and extensions

In addition to the M4 method proposed by this study, other nonparametric state estimation methods have been previously developed (Langrock et al., 2018; Nams, 2014; Sur et al., 2014). In one such example, the behavioural movement segmentation (BMS) model proposed by Nams (2014) uses a combination of direct search optimization, iterative sampling and k-means clustering to estimate latent states from track segments. BMS is similar to our proposed M4 framework in that both methods are nonparametric, partition multiple data streams into segments and cluster segments into latent states (Nams, 2014). However, M4 differs both technically and conceptually from BMS in that M4 proposes breakpoints using RJMCMC, the number of likely states are estimated within a single model run (instead of using multi-model selection) and track segments are expected to be comprised of multiple states rather than just one. We believe that practitioners should carefully evaluate the properties and assumptions of different methods to determine the best method to properly analyse their data and address their objectives.

Although M4 effectively classified behavioural states from both simulated and empirical tracks, there are some limitations to this approach. The selection of the number and width of bins when discretizing data streams is a subjective choice that impacts the results from the segmentation model and ultimately the estimation of behavioural states. Therefore, practitioners may need to test different binning methods if the segmentation model does not produce reliable breakpoints that match up with plots of the data stream(s). Additionally, our model implicitly assumes that location error is negligible or requires that it be accounted for via another method. Although our model can analyse data streams from regular or irregular time intervals, this will also depend on the inherent properties of the data streams themselves. Since step lengths and turning angles are calculated from multiple successive observations, these values will not be comparable once the data are not close to a regular time interval. However, variables such as net squared displacement (the squared distance from the starting location to all other relocations) can be analysed over irregular time intervals.

M4 can be extended to analyse other types of data streams and can include prior knowledge on the timing of behavioural shifts. Although only step lengths and turning angles were analysed for the simulated and empirical tracks, additional ancillary data coming from the sensor (e.g. elevation, salinity, temperature or accelerometer data) could be used to make behavioural inference. These data streams could come from all types of distributions (i.e. continuous, discrete, bounded between 0 and 1). It is also relatively straightforward to deal with zero-inflated data by including all zeroes in a single bin. Additionally, our segmentation model can be implemented in a semi-supervised fashion, by which practitioners pre-specify breakpoints for the time series based on a priori knowledge and these breakpoints will be considered by the RJMCMC algorithm. This may be particularly useful if daily activity patterns are expected or if only one of several possible states can be clearly identified.

## ACKNOWLEDGEMENTS

This work was supported in part by the University of Florida Biodiversity Institute (UFBI), the US Department of Agriculture's National Institute of Food and Agriculture (USDA-NIFA) McIntire–Stennis project 1005163, the US National Science Foundation (NSF) award 1458034 and 2040819, the Florida Fish and Wildlife Conservation Commission (FWC) award 13416 and by the US Army Corp of Engineers (USACE). The tagging of snail kites was approved and conducted under UF IACUC no. 202008334. J.A.C. was supported by a postdoctoral fellowship from the University of Florida Informatics Institute (UFII) Fellowship Program. The authors thank Tyler Beck for his assistance with fieldwork and Brenda Betancourt for her feedback during model and package development. Additionally, they are grateful to Rocío Joo and Mathieu Basille for their valuable comments on an earlier draft of this paper.

## CONFLICT OF INTEREST

The authors declare that there is no conflict of interest.

## AUTHORS' CONTRIBUTIONS

J.A.C. and D.V. conceived the ideas and designed the methodology; C.L.P. and R.J.F. collected the empirical data; J.A.C., C.L.P. and D.V. analysed the data; J.A.C. and D.V. led the writing of the manuscript. All authors contributed critically to the drafts and gave final approval for publication.

## Open Research

# PEER REVIEW

The peer review history for this article is available at https://publons.com/publon/10.1111/2041-210X.13745.

# DATA AVAILABILITY STATEMENT

All code mentioned here for the Bayesian framework are available within the bayesmove package for R hosted on CRAN at https://CRAN.R-project.org/package=bayesmove. The development version of the package is available on GitHub at https://github.com/joshcullen/bayesmove and a full set of vignettes can be found at https://joshcullen.github.io/bayesmove. The code to generate the simulations and perform method comparison are available on Zenodo at https://doi.org/10.5281/zenodo.4245254 (Cullen, 2021). The Everglade snail kite telemetry data have not been made available since this is a federally listed endangered species and the location data are sensitive.