7.2. Models

The Wiener process (or Brownian motion) is the continuous analog to the standard random walk (Ross, 1985). The Wiener process with drift can be derived from a biased random walk, a random walk in which the probabilities of moving to the right and to the left are not equal (but are constant). The process X(t) is said to be the Wiener drift process if it has the following properties (Ross, 1985):

• 1) X(0) = 0;
• 2) for t > 0, X(t) is normally distributed with mean rt and variance ²t;
• 3) each disjoint segment of an individual path is independent.

telegrapher's equation

A natural extension of this model that incorporates correlation among movements is based on a correlated random walk. The correlated random walk is presented as follows. Let X_t be a discrete time, discrete space process with and t = 0, 1, 2, ... . The transition probabilities of X are defined as follows:

p = Pr (particle moves one unit in the same direction as the previous movement)

q = Pr (particle moves one unit in the opposite direction of the last step)

p + q = 1. (7.1)

The standard initial conditions are that x₀ = 0, and for the first step, the probability of moving to the right = the probability of moving to the left = 1/2.

Following the approach of Goldstein (1951), it is possible to derive a limiting continuous distribution based on this process called the telegrapher's equation:

. (7.2)

The same result can be obtained by beginning with the continuous (in time and space) analog to the correlated random walk. In this process, a particle moves in one direction with a constant speed until it reverses direction and then moves in the opposite direction with the same speed. The direction reversing process is governed by a Poisson process with parameter .

The first two moments of the displacement process defined by the telegraph equation are easily obtained and are quite tractable:

E(X) = 0 (7.3)

. (7.4)

For small t

(7.5)

which is characteristic of wave equations. Also when t is large

(7.6)

which is similar to that of the Wiener process with diffusion coefficient D = ²/2.

Equation (7.2) can be solved for p(x,t) with initial conditions p(x,0) = 0 and

(7.7)

, (7.8)

where I_o and I₁ are modified Bessel functions and is the Dirac distribution. Unfortunately, the pdf derived from this equation is rather complex and is probably not practical as a model of animal movement at the level of the individual, although it has been applied to population patterns (Holmes, 1993).

O-U based model

An alternative model of correlated movement is based on the Ornstein-Uhlenbeck (O-U) process (Uhlenbeck and Ornstein, 1930). The O-U process was first presented as an alternative model for Brownian motion and was developed to describe the velocities of particles. The model operates under the assumption that as a particle travels with greater velocity, it is increasingly likely to contact another particle and meet resistance. Thus, there is a tendency for particles to be brought back to zero velocity, and with the O-U process, the strength of this tendency is linearly related to the magnitude of the velocity. There is correlation between movements occurring over short periods of time and a tendency to return to zero velocity. This type of process resembles, in some cases, the movement patterns of animals on a short time scale. From this velocity based model, the distribution of displacements can be obtained, which is compatible with individual movement data.

To begin, let X(t) be the position of a particle at time t. Define V(t) as the velocity at time t. Since the O-U process applies to particles with zero mean velocity, the mean must be subtracted off. Denote

. (7.9)

If U(t) follows an Ornstein-Uhlenbeck process, then:

. (7.10)

The parameter characterizes the spread of the particles, and the parameter characterizes the propensity of the particle to return to its mean velocity. The conditional distribution p(u, t | u₀, s), t > s, is a Gaussian distribution with

(7.11)

. (7.12)

In contrast to the Wiener process, the variance of the O-U process stabilizes as t gets large. The displacements predicted by the process can be obtained by integrating:

. (7.13)

Here T_i is defined as the time interval t_i - t_i-1. This integration is considered a stochastic integration because U(s) is a stochastic process (Cox and Miller, 1965). As reported by Doob (1942) Y has a Gaussian distribution with

(7.14)

. (7.15)

Interestingly, the mean and variance are the same as those of the telegrapher's equation. Also,

. (7.16)

Thus the joint distribution of Y_i and Y_i+1 is a bivariate normal with mean and variance given in equations (7.14) and (7.15) and with correlation coefficient

. (7.17)

An important feature of this equation is that as the time scale gets larger, the correlation decays. Also, the correlation coefficient depends only on and not on , and Figure 7.1 shows that there is an inverse relationship between and . In addition, as (with / a constant), the variance approaches a linear relationship with time, and the covariance goes to 0. The process then becomes indistinguishable from the Wiener process.

If the time increments are equal, the data can be analyzed with standard time series analysis. If the data have unequal time increments, as with many radio-tracking studies, the analysis is not as simple. The equations describing the Wiener drift process and the O-U displacement process do not require equal time increments, though, and thus can form the basis of the analysis of unequal time increment data.