On modeling animal movements using Brownian motion with measurement error

Abstract

Modeling animal movements with Brownian motion (or more generally by a Gaussian process) has a long tradition in ecological studies. The recent Brownian bridge movement model (BBMM), which incorporates measurement errors, has been quickly adopted by ecologists because of its simplicity and tractability. We discuss some nontrivial properties of the discrete‐time stochastic process that results from observing a Brownian motion with added normal noise at discrete times. In particular, we demonstrate that the observed sequence of random variables is not Markov. Consequently the expected occupation time between two successively observed locations does not depend on just those two observations; the whole path must be taken into account. Nonetheless, the exact likelihood function of the observed time series remains tractable; it requires only sparse matrix computations. The likelihood‐based estimation procedure is described in detail and compared to the BBMM estimation.

Introduction

Wildlife biologists have long depended on remote monitoring of individual animals to determine movements, behaviors, utilization distributions, and home ranges (Heezen and Tester 1967, Marshall and Whittington 1969, Hutton et al. 1976, Dunn and Gipson 1977). Animals move in continuous trajectories through their environment. Nevertheless, full continuous knowledge of a wild animal's trajectory is, today, almost always a technical impossibility (but see Powell and Mitchell [2012]). Usually the continuous trajectory is only observable at discrete times; that is, only locations at certain time points are available. This leads to a problem of interpolation, or estimation, of an animal's path between two consecutive observations, which provide limits on an animal's location during the intervening time when no positions are known.

Many stochastic models have been used for animal movements; see Smouse et al. (2010) for a recent review. One model that incorporates measurement errors is Brownian motion (BM) that models animal movements coupled with an additive independent normally distributed noise that models the measurement errors. This model's parameters are the BM variance, which models animal mobility, and the measurement‐error variance, which is a property of the biotelemetry device. This is essentially the Brownian bridge movement model (BBMM) of Horne et al. (2007), except that they treated the measurement‐error variance as known. The BBMM quickly gained popularity in the ecological community (e.g., Lonergan et al. 2009, Willems and Hill 2009, Farmer et al. 2010, Takekawa et al. 2010), and has even been implemented in R packages such as BBMM (Nielson et al. 2012) and adehabitat (Calenge 2006). Most recently, a dynamic modification of the BBMM has been proposed to allow the BM variance to be time varying (Kranstauber et al. 2012) with an implementation available in the R package move (Kranstauber and Smolla 2013).

Nevertheless, users of the BBMM might not be fully aware of its problems and limitations. Horne et al. (2007) treated the measurement‐error variance as known, and proposed to estimate the instantaneous variance of the BM via a conditional likelihood of the odd‐numbered locations given the even‐numbered locations. This estimation method is not a standard statistical practice. The conditional likelihood makes the two variance components unidentifiable when the data are equally spaced in time. We will show that the resulting estimator of the BM variance is biased and that the bias does not vanish as the sample size increases. The estimation procedure only works correctly when the measurement‐error variance is negligible, which defeats the purpose of including it in the first place. Even when there is no measurement error, in which case the observed process reduces to a BM, its estimation is inefficient because it uses only half of the data.

The main purpose of this note is to demonstrate that the standard likelihood‐based inference of estimating the multivariate normal distribution is available in this case, and, moreover, it will provide a better, more precise solution. If the variance of the measurement errors is of the same magnitude as the variability that comes from the Brownian motion, the BBMM likelihood produces a significant bias in parameter estimation, which we show using simulations. We also show that the introduced process has properties with nontrivial implications for interesting, derived quantities such as expected occupation time.

The rest of the article is organized as follows. We first present the distributional properties of a BM with measurement error. Then, we derive the expected occupation time based on the model and provide an full likelihood estimation approach. A simulation study demonstrates the bias of the BBMM approach and the efficiency of our full likelihood approach. We conclude with a real data set example and a discussion.

Brownian Motion with Measurement Error

Model

Without loss of generality, we present the BM model with measurement error (BMME) and its properties in the one‐dimensional case. (Generalization to higher‐dimensional cases is straightforward and discussed later.) Let {B_t, t ≥ 0} be the standard one‐dimensional BM with instantaneous variance σ² and B₀ = 0. Let {ξ_i}_0≤i≤n be independent and identically normally distributed random variables with mean 0 and variance δ². Assume that {B_t, t ≥ 0} and {ξ_i}_0≤i≤n are independent. A Brownian motion with measurement error observed at 0 = t₀ < … < t_n = T is modeled by Z_i = + ξ_i, i = 0, 1, … , n. Let X_i = Z_i – Z_i−1 for i = 1, … , n, and X = (X₁, … , X_n)^⊤. Then X is the increment of the observed vector Z. One can think of B_t as the x‐coordinate of the true animal location at time t, ξ_i as the x‐error of that location at time of measurement t_i, Z_i as the observed locations at times t_i, and X_i as increments of the observed process. Note that this is the same model as the BBMM of Horne et al. (2007) except that it is more general because the measurement‐error variance can be either fixed and given or estimated from the data.

Theoretically, using BM is supported by Donsker's theorem type results (e.g., Durrett 2010:386). Donsker's theorem states that, under some general conditions, a random walk with weakly dependent increments converges in distribution to BM. However, some caution should be exercised when animal movement is modeled by BM. For example, Brownian motion is not periodic. So, if an animal demonstrates a certain cyclic routine in its behavior, then BM might not be appropriate. Also BM assumes constant movement so, if an animal stays in the same location for long periods of time, BM cannot be suitable. There are also some technical properties of BM that cause some difficulty interpreting theoretical findings. For example, the trajectory of Brownian motion is differentiable nowhere and is of unbounded variation. Consequently, the instantaneous speed and covered distance are not well defined. Instead one uses surrogates like instantaneous variance and quadratic variation.

Preliminaries on the multivariate normal distribution

Before presenting the properties of the model, we review some basic facts about the multivariate normal distribution that will be used in our derivations. A k‐dimensional random (column) vector has a multivariate normal distribution if any linear combination of its components has univariate normal distribution. Note that from a mathematical point of view it is convenient to treat a constant as a normally distributed random variable with variance zero. A k‐normal vector Y is characterized by its mean vector μ and variance matrix Σ. If the variance matrix Σ has rank k (that is, it is invertible), then the density of Y is given by

It follows from the definition that any linear transformation of a normally distributed random vector has a multivariate normal distribution (e.g., Ravi‐shanker and Dey 2002:153). More specifically, if Y is a k‐normal vector with mean μ and covariance matrix Σ, and A is a m × k matrix, then AY is m‐normal vector with mean Aμ and covariance matrix AΣA^⊤.

Finally, in what follows we need formulas for the conditional distribution of a subvector of a normal random vector. This theoretical result is well known (e.g., Ravishanker and Dey 2002:156). Let Y be a k‐normal vector with mean μ and covariance matrix Σ of full rank k. Partition Y as Y^⊤ = (, ), where Y₁ is q‐dimensional subvector and 0 < q < k. Partition μ and Σ as

The conditional distribution of q‐vector Y₁ given that Y₂ = a is multivariate normal with mean

and covariance matrix

Distribution of observed process

The observed random vector Z = (Z₀, … , Z_n)^⊤ has a multivariate normal distribution because (, , … , , ξ₁, … , ξ_n) is jointly a normal random vector and Z is a linear transformation of it. It is easy to see that

and

Therefore, the multivariate normal distribution of Z has mean vector 0 and an (n + 1) × (n + 1)‐covariance matrix:

Distribution of increments

The vector of increments of the observed process X also has a multivariate normal distribution because it is just another linear transformation of (, , … , , ξ₁, … , ξ_n). One can check that

and

That is, the multivariate normal distribution of X has mean vector 0 and an n × n covariance matrix:

where τ_i = t_i – t_i−1. The sparsity of Σ can be exploited in evaluating the likelihood when parameters are estimated with the likelihood method. Compared to the increments of a Brownian motion, which is multivariate normal with a diagonal covariance matrix, we see that the covariance matrix is no longer diagonal because of the added measurement error. That is, {Z_i}_0≤i≤n does not have independent increments. Moreover, it is not Markov. Using the formulas for the conditional distribution of normal subvector 1 and 2, one can easily verify that

where dx is some infinitesimal x‐region. That is, the future behavior of the process {Z_i}_0≤i≤n after time i depends on the entire path up to time i, not just on the present value at time i. This is an important difference from the BM, which is a continuous Markov process with stationary independent increments.

Expected Occupation Time

Definition of expected occupation time

Following Horne et al. (2007), we define the expected occupation time in x‐region A between two consecutive times t_i and t_i+1 by

where Pr(B_t ∈ A | Z) is the probability of finding the animal in x‐region A at time t ∈ [0,T] given information on all the observed locations. This quantity represents the expected fraction of time that the animal spends in region A. Horne et al. (2007) demonstrated that if

then the corresponding density of the expected occupation time is given by

The main objective of this section is to find h(x).

Occupation time without measurement error

Assume for a moment that there is no measurement error, δ² = 0; that is, we directly observe the BM B_t. If one needs to “interpolate” the animal movement for t ∈ [t_i, t_i+1] between two known locations at times t_i and t_i+1 then, because of the Markovian property of the BM, we need to know only and . More specifically, in this case for 0 = t₀ < … < t_i < t < t_i+1 < t_n = T,

This is because time reversal does not change the distribution of the BM. That is, V_t = B_T – B_T−t is a BM as well and a Markov process with respect to a suitable filtration. As a consequence, the well‐known standard formulas for the Brownian bridge can be employed to do the “interpolation” to generate possible sample paths from their distribution. More specifically, using statements 1 and 2 for t_i ≤ t ≤ t_i+1 we find that B_t given = a and = b has a normal distribution with mean

and variance

Therefore, the density of the expected occupation time in this case is given by

Occupation time with measurement error

The Markov property is no longer true for the observed data Z_i; an equation similar to Eq. 4 is not available. Specifically, for 0 = t₀ < … < t_i < t < t_i+1 < t_n = T,

This, again, follows from formulas for the conditional distribution of normal subvector 1 and 2. Therefore, to calculate the probability of finding the animal in a certain area at a given time t one cannot use Pr(B_t ∈ dx | Z_i, Z_i+1) for t_i < t < t_i+1 if the whole vector Z is observed. Instead, one should use the density

Because random vector (B_t, Z₀, … , Z_n)^⊤ follows a multivariate normal distribution, again by employing the result on conditional distribution of normal subvector, we find that f(x, t) is the density of univariate normal random variable with mean

and variance

where Σ₁₂ = σ²[min(t, t₀), … , min(t, t_n)] and Σ₂₁ = .

Thus, the density of the expected occupation time is given by

Note that, when δ = 0, Eq. 8 reduces to statement 2.

To use Eq. 8 in an application, one needs to first estimate standard deviation σ (and δ, if necessary). This can be done with the likelihood method as given in the next section. The covariance matrix Σ_Z is then constructed. Eqs. 6 and 7 and the density f(t, x) in Eq. 5 can be calculated for every t ∈ [0, T]. The computations are not demanding. The most time consuming procedure, inverting Σ_Z, needs to be done only once after the parameters have been estimated.

Example

Consider three data sets, each containing four observations taken at times 100, 110, 120, and 130 minutes.

• Data set 1: 

  Z(100) = 0, Z(110) = 200, Z(120) = 400, Z(130) = 200.

• Data set 2: 

  Z(100) = 0, Z(110) = 200, Z(120) = 400, Z(130) = 400.

• Data set 3: 

  Z(100) = 0, Z(110) = 200, Z(120) = 400, Z(130) = 600.

In Fig. 1, we plot the densities of the expected occupation time between 110 and 120 minutes for three models:

Figure 1

Densities of the expected occupation time for pure Brownian motion (BM) model (standard deviation σ = 25.35), Brownian bridge movement model (BBMM) of Horne, Garton, Krone, and Lewis (HGKL) (σ = 25.35 and δ = 28.85), and BM model with measurement error (BMME; σ = 25.35 and measurement error δ = 28.85) between 110 and 120 minutes. The densities are based on three data sets, each containing four observations taken at times 100, 110, 120, and 130 minutes: Data set 1, where Z(i) are the observed locations at time x: Z(100) = 0, Z(110) = 200, Z(120) = 400, Z(130) = 200. Data set 2: Z(100) = 0, Z(110) = 200, Z(120) = 400, Z(130) = 400. Data set 3: Z(100) = 0, Z(110) = 200, Z(120) = 400, Z(130) = 600.

1. the pure BM model with σ = 25.35 (and δ = 0),

2. the BBMM of Horne, Garton, Krone, and Lewis (BBMM (HGKL)) with σ = 25.35 and δ = 28.85, and

3. the BM model with measurement error (BMME) with σ = 25.35 and δ = 28.85.

Since the values of Z(110) and Z(120) are the same for all three data sets, the BBMM and pure BM model yield the same curve in all three cases. The key observation here is that Eq. 8 produces three different curves depending on values of Z(130). This is because Z process is not Markov. The density is symmetric about the 300 when Z(100) and Z(130) are also symmetric about 300. They are skewed to the right when Z(130) is closer to 200 than to 400. This is just an illustration with four data points. In real data with n data points, all points affect the density.

Parameter Estimation

Estimating parameters σ and δ is straightforward because the full likelihood of {X_i}_1≤i≤n is available. The joint density of the multivariate normal vector (X₁, … , X_n)^⊤ is given by

where x ∈ ℝⁿ. Because of the banded nature of the covariance matrix Σ_X, the likelihood can be evaluated very efficiently using banded‐matrix inversion, which facilitates the optimization even for large dimension n. (We used the banded‐matrix implementation bandSparse in the R package Matrix (Bates and Maechler 2012). For statistical inferences, a point estimator needs to be accompanied by its variance estimator for uncertainty assessment. From the standard likelihood approach, the covariance matrix of the estimator is estimated by the inverse of the Fisher information matrix evaluated at the parameter estimates.

Extending to a two‐dimensional space is easy. If we assume that the two components of the BM (easting and northing) are independent and the same is true for measurement errors, then the full likelihood is just a product of two joint densities, like Eq. 9. With a little more effort, we can derive similar formulas even if we assume some dependency structure so long as all the random variables are normally distributed. Another straightforward extension is to make the variance of error dependent on the locations. Then the technique described in Kranstauber et al. (2012) can be employed for the estimation. Also, as it was done in Kranstauber et al. (2012), one can introduce time‐dependent σ² to accommodate dynamic animal movement in different “speed.”

Computing tools

All BBMM‐related computations are done with the help of the R package BBMM (Nielson et al. 2012). The full likelihood calculations rely on the banded‐matrix implementation bandSparse in the R package Matrix (Bates and Maechler 2012). Our R code for the full likelihood estimation and simulations is included as a Supplement. An open‐source R package smam for statistical modeling of animal movement is currently under development.

Simulation Study

We conducted a simulation study to validate our findings. We generated animal‐movement locations using a two‐dimensional BM with σ = 25.35 and measurement standard deviation δ = 28.85, which equal those values in Horne et al. (2007). The time intervals between observations (τ = τ_i = t_i – t_i−1) control the relative importance of the measurement error: the longer the intervals, the less important the measurement error. We chose time intervals of 5, 10, 20, and 40 minutes. For each interval, we generated data with four possible sample sizes n (the number of observed locations): 200, 400, 800, and 1600. For each data set, we fixed δ = 28.85 as known and estimated σ with both the BBMM approach as implemented in the R package BBMM and our full likelihood approach. We did 1000 replicates for each simulation configuration. The results about empirical bias, empirical standard error of the estimates (SEE), and empirical root mean square error (RMSE) are reported in Table 1.

Table 1. Comparison of Brownian bridge movement model (BBMM) estimator and the full likelihood estimator with standard deviation σ = 25.35 when measurement error δ = 28.85 is assumed to be known.

We begin the explain of our simulation results by discussing the relationship between the animal mobility characteristic σ² and the measurement variance δ². These two parameters are not directly comparable. The parameter σ² is related to Brownian motion; it tells us how much variance is gained per unit time so σ² is measured in square meters per minute in our case. The parameter δ² is a variance of observations, and its units here are square meters. Comparing these two quantities requires accounting for the time intervals between observations. This leads us to define the ratio Δ = δ²/τσ², which is the ratio of the measurement‐error variance to the animal‐movement variance in a time period of duration τ. Larger values of Δ correspond to a larger impact of measurement errors.

From Table 1, the BBMM estimator is clearly biased, and the bias does not diminish even when n is increased from 200 to 1600. The bias is bigger when τ is higher: the measurement error is relatively important. In contrast, the full likelihood estimator is virtually unbiased. As for standard error, even in the best scenario for the BBMM estimator to work, as Δ approaches zero, the full likelihood estimator is still better because the BBMM method throws away half of the data points. The ratio of the empirical standard errors of the two method in the case of Δ = 3.24% is close to . This is because, in typical statistical inferences, the variance of an estimator is inverse proportional to the sample size and, therefore, the SEE is proportional to the reciprocal of the square root of the sample size. The full likelihood estimator's RMSE is much lower than that of the BBMM estimator because of its bias and its inefficiency. The bias produced by the BBMM is smaller when Δ is small. It is not surprising, because similar results can be obtained by just using the standard BM with no measurement errors.

The BBMM needs a δ² value as input, but the full likelihood method can estimate it along with σ². This might be interesting if a measurement device's error is not always the same under lab and field conditions, which is true for global positioning system (GPS) receivers (Lawrence‐Apfel et al. 2012). The BBMM likelihood function cannot yield the estimation of δ. It is easy to see from the formula of the conditional likelihood in the BBMM approach (Horne et al. 2007:2357) that σ and δ are unidentifiable when the time intervals are equally spaced. In Table 2, we present the bias, standard error of the estimates (SEE), and the average of model‐based standard error (ASE) from the full likelihood estimator based on 1000 replicates under the settings of τ = 5 and τ = 10. As suggested by general likelihood theory, the estimators are asymptotically unbiased, consistent, and normally distributed. The agreement between SEE and ASE is reasonably good, especially as sample size increases, suggesting that the variance estimator from the full likelihood approach estimates the variation of the estimator well. It is worth noting that the estimator for δ has higher SEE. This is because Δ is much less than 100%, and, as a consequence, the random noise that comes from the BM itself is much higher than the variance of measurement error.

Table 2. Estimation results of σ and δ via full likelihood method when true parameters are σ = 25.35 and δ = 28.85.

An Example

Let us consider the data set locations from the R package BBMM. The data set is 25 GPS locations from a female mule deer (Odocoileus hemionus). The locations' coordinates are in meters and were collected 120 minutes apart. If the measurement standard deviation δ is set to be 20, then the BBMM estimate of σ is 56.10. Note that for these values for σ, δ, and the time interval, the ratio Δ is just 0.11%. Therefore, we do not expect changing values of δ to produce any significant impact on σ's estimation. Indeed, running the BBMM R code again but setting δ = 0 produces a BBMM estimate of σ of 56.13. Practically, there is no change.

Now let us run the full likelihood estimation procedure. If we try to estimate both parameters then the estimate for σ is 67.39 and for δ is 0.18. As we explained in the end of previous section, when both Δ and n are small the estimator of δ is not very reliable; however, the measurement error δ is relatively uninfluential. To demonstrate this point, we run the full likelihood estimation for σ alone when δ is set to be 20, and the estimate of σ is again 67.39. That is, for this data set, the measurement error is not important for either our method or the BBMM method.

However, the estimates of σ are noticeably different. So which one is closer to the truth? We argue in favor of ours as follows. When δ = 0 (or when Δ is very small) we are practically dealing with two‐dimensional Brownian motion. Let x₁, … , x₂₄ be increments of the x‐coordinates, and y₁, … , y₂₄ be increments of the y‐coordinates. Because of the Brownian motion assumption, all 48 increments are independent and have a normal distribution with mean zero and variance 120σ². Therefore, according to well‐known results for the normal distribution, the estimate for σ is

This is almost exactly our value and it must be because, when δ = 0, the maximization of the full likelihood Eq. 9 will lead us to the same formula. Now, when δ = 0, the BBMM likelihood function is the correct conditional density of odd locations given the even ones. As a result, the estimation is also asymptotically unbiased. However, since only half of the data points are used, the SEE for the BBMM estimation is times larger. That is, σˆ = 67.39 is produced by a more efficient statistical procedure.

Discussion

One of the key observations here is that adding random noise to a Brownian motion results in a stochastic process that does not satisfy the Markov property. That is, in contrast to BM, the entire sequence of previous positions up to the current time provides more information than just the current position. Consequently, when an “interpolation” between two observed locations is needed it is not sufficient to take into account just those two points. The whole trajectory should be used. However, the dependence is weak; therefore, the impact from a few (but not two!) of neighboring points is the greatest. The strength of the dependence increases as Δ increases.

We believe the BBMM has three major issues. First, its conditional likelihood function produces a bias in the estimation of σ, and the bias is greater when the measurement error δ is large relative to σ. The second issue is the unidentifiability of parameters of the model. Even if we do not need to estimate δ, the fact that σ and δ are fused together in the BBMM likelihood tells us that there is a problem with the model design. These two parameters, the animal mobility characteristic σ and the measurement standard error δ, are very different in their meaning. And, therefore, they have to enter into likelihood function as two independent variables as is the case for the full likelihood function Eq. 9. But when the time intervals are equally spaced, the BBMM likelihood function depends directly on a linear combination of σ² and δ², which makes separating these two parameters impossible. Finally, when Δ is small, the BBMM method is inefficient by using only a half of the data.

Supplemental Material

Supplement

R code used for the full likelihood estimation and simulations (Ecological Archives E095‐021‐S1).