## Abstract

A discrete-time Markov process for directions in the plane is introduced. The random direction at any time is influenced both by the direction at the preceding time and by a target direction that depends on the current time. The deviations of the directions from their targets have a limiting distribution as time tends to infinity, and asymptotic approximations to the limiting probability density function are given both for the case of weak dependence and for the case of strong dependence. Tests of independence and of independence of increments are presented. Various types of behaviour of the process are illustrated by simulations. Some physical applications, notably concerning the commensurate versus incommensurate nature of solid materials, are discussed. The application of the process is illustrated also by the analysis of some data on semi-diurnal wind directions.

## 1. Introduction

Observations comprising directions in the plane occur in various contexts. Such data can be represented as unit vectors or points on the unit circle. Examples include directions of migrating birds [1] and orientations of palaeocurrents [2]. Many other examples are considered in Fisher [3], Mardia & Jupp [4] and Jammalamadaka & SenGupta [5]. Some circular observations occur as time series, e.g. hourly measurements of wind direction [6]. Other circular ‘time series’ occur in which the observed directions are ordered not by time but by distance along a line. Examples include directions of face-cleat along a tunnel in a mine (§7.1 of [3]) and rotation angles around bonds in a polypeptide [7].

Various stochastic processes have been proposed as models for circular time series. The main proposals can be put in the following groups.

— Combining a regression function that is a constant rotation of the circle with an arbitrary error distribution gives a Markov process on the circle. Transforming this process by continuous one-to-one transformations of the circle gives the large class of Markov processes considered by Wehrly & Johnson [8].

— Various transformations can be used to convert familiar (possibly multivariate) stochastic processes on the line to processes on the circle:

(i) wrapping the real line to the circle by reduction modulo 2

*π*converts autoregressive (AR) processes to the wrapped autoregressive processes of Breckling [6];(ii) link functions that identify the line with the circle minus a point transform autoregressive moving average (ARMA) processes to the linked ARMA processes of Fisher & Lee [9];

(iii) radial projection from the plane to the circle transforms bivariate Gaussian processes into processes on the circle [9].

— Suitable mappings from the circle into the natural parameter space of von Mises distributions yield the processes of Accardi

*et al.*[10], the von Mises processes of Breckling [6] and the circular AR processes of Fisher & Lee [9].— Letting the parameters of circular distributions follow a Markov chain gives the hidden Markov processes of Holzmann

*et al.*[11].— Combining a regression function that is a Möbius transformation of the circle with a von Mises error distribution gives the Möbius processes of Downs & Mardia [12]; similarly, using a wrapped Cauchy error distribution gives the Möbius processes of Kato [13] and Kato & Jones [14].

Overviews of time-series models for circular data are given in ch. 7 of Fisher [3], §11.5.2 of Mardia & Jupp [4] and §12.8 of Jammalamadaka & SenGupta [5].

The purpose of this paper is to expand the repertoire of methods for the analysis of circular time series by introducing and exploring a simple Markov process which has constant drift, in the sense that the ‘target directions’ move at constant speed round the circle. We therefore call it a *drifting Markov process on the circle*.

## 2. Overview

The aim of this paper is to provide a simple tool for modelling sequences of directions, i.e. sequences of angles. We do so by constructing a stochastic process on the circle, i.e. a sequence of random angles *θ*_{1},*θ*_{2},… in [0,2*π*], where 2*π* is identified with 0. In general, *θ*_{1},*θ*_{2},… are dependent. The stochastic process introduced in §3*b* is Markov, i.e. the probability distribution of *θ*_{j} depends on *θ*_{1},…,*θ*_{j−1} only through *θ*_{j−1}, for *j*=2,3,…. The conditional probability density of *θ*_{j} given *θ*_{j−1} is given explicitly in (3.3).

The conditional distribution of *θ*_{j} given *θ*_{j−1} can be considered as consisting of (i) a deterministic part, given by a ‘target’ direction *ν*_{j} and (ii) a random part describing how the distribution of *θ*_{j} is related to *ν*_{j} and *θ*_{j−1}. The target directions *ν*_{1},*ν*_{2},… satisfy
2.1where *ν*_{0} is the initial angle and the angle Δ indicates the rate at which the target directions move round the circle. The random part is the conditional probability density *f*(*θ*_{j}|*θ*_{j−1};*κ*,*λ*,*ν*_{0},Δ,*m*) of *θ*_{j} given *θ*_{j−1}, which is given in (3.3) and depends on the parameters *κ*,*λ*, *ν*_{0}, Δ and *m*. The parameter *κ* determines the strength of the dependence of *θ*_{j} on *θ*_{j−1}. For large positive *κ*, *θ*_{j} tends to be near *θ*_{j−1}; if *κ* is very negative then *θ*_{j} tends to be near *θ*_{j−1}+*π*; if *κ*=0 then *θ*_{j} and *θ*_{j−1} are independent. The parameter *λ*, which may be assumed to satisfy *λ*≥0, determines how close *θ*_{j} tends to be to *ν*_{j}. If *λ*=0 then the distribution of *θ*_{j} does not depend on *ν*_{j}; if *λ* is large then *θ*_{j} tends to be near *ν*_{j}. The parameter *m* is a positive integer (usually known) and the part of the conditional probability density *f*(*θ*_{j}|*θ*_{j−1};*κ*,*λ*,*ν*_{0},Δ,*m*) of *θ*_{j} that involves *ν*_{j} has *m*-fold circular symmetry, i.e. it is invariant under change of *θ*_{j} to *θ*_{j}+2*kπ*/*m* (for *k*=1,…,*m*).

As time *j* tends to infinity, the distribution of the deviations *θ*_{j}−*ν*_{j} of the directions *θ*_{j} from their targets *ν*_{j} tends to a limiting distribution. Although explicit descriptions of this limiting distribution do not seem to be available, approximations to it are given in §4, both for the case of weak dependence (*κ*≃0) and for the case of strong dependence ( or ).

For *n*=1,2,…, the angles *θ*_{1},…,*θ*_{n} are a random set of *n* points on the circle. As , an appropriate ‘time average’ of these *n* points tends to a probability distribution on the circle. If Δ/(2*π*) is irrational then this limiting distribution is the uniform distribution on the circle, whereas if Δ/(2*π*) is rational then the limiting distribution is non-uniform with symmetry depending on Δ. Thus, as discussed in more detail in §7, there is a connection with commensurate versus incommensurate behaviour in the structural properties of solids.

The random angles *θ*_{1},*θ*_{2},… are independent if and only if *κ*=0, whereas the increments *θ*_{2}−*θ*_{1},*θ*_{3}−*θ*_{2},… are independent if and only if *λ*=0. Tests of independence and of independent increments are presented in §5.

Various types of behaviour of the drifting Markov process on the circle are illustrated by simulations given in §6. Some physical applications (notably to structural chemistry and materials science) are indicated in §7. The process is used to analyse some data on semi-diurnal wind directions.

## 3. The Markov process

### (a) Circular statistics

The most basic probability distribution on the circle is the *uniform distribution*, in which the probability that a random angle lies in an arc is proportional to the length of the arc. Thus, the density is the constant function *f*(*θ*)=(2*π*)^{−1} for 0≤*θ*≤2*π*.

Possibly, the most useful distributions on the circle are the *von Mises distributions* *M*(*μ*,*κ*), which have densities
3.1with *κ*≥0 and 0≤*μ*≤2*π*. The function *I*_{p} denotes the modified Bessel function of the first kind and order *p*, for *p*=0,1,…. The parameter *κ* is the *concentration* and *μ* is the *mean direction*. The case *κ*=0 gives the uniform distribution. For *κ*>0, the *M*(*μ*,*κ*) distribution is symmetrical about its mode, *μ*. As *κ* increases, the distribution becomes more concentrated and tends to a point distribution at *μ* as . For any circular distribution, the *mean resultant length*, *ρ*, and *mean direction*, *μ*, are defined by and with *ρ*≥0. For the *M*(*μ*,*κ*) distribution, *ρ*=*A*(*κ*), where
3.2

Detailed expositions of circular statistics are given in Fisher [3], Jammalamadaka & SenGupta [5] and ch. 1–8 of Mardia & Jupp [4].

### (b) The Markov process

Our aim is to construct a simple probabilistic model for random directions (angles) *θ*_{1},*θ*_{2},… for which each *θ*_{j} is subject to two (possibly conflicting) tendencies: (i) to be near ‘target’ directions *ν*_{j}+2*kπ*/*m* for *k*=0,1,…,*m*−1 (or possibly near *ν*_{j}+*π*+2*kπ*/*m* for *k*=0,1,…,*m*−1) and (ii) to be near *θ*_{j−1} (or possibly near *θ*_{j−1}+*π*). Tendency (i) can be seen in physical terms as a ‘forcing’ effect. It is convenient to assume that (2.1) holds. The parameter Δ measures the rate at which the ‘target directions’ move round the circle. The form (3.1) of the von Mises densities suggests that these tendencies may be formalized in the model for *θ*_{1},*θ*_{2},… in which the conditional density of *θ*_{j} given *θ*_{j−1} is
3.3for *j*=2,3,…, where *m* is a positive integer. Since the density (3.3) is unchanged under simultaneous change of *λ* to −*λ* and *ν*_{0} to *ν*_{0}+*π*/*m*, it may be assumed that *λ*≥0. On the other hand, in contrast to the situation in (3.1), it cannot be assumed that *κ*≥0; when *λ*=0, the increment *θ*_{j}−*θ*_{j−1} has a von Mises distribution with mean direction 0 or *π*, depending on the sign of *κ*. The normalizing constant *c*_{m}(*κ*,*λ*,*μ*) in (3.3) is given by
3.4
3.5

The term in the exponent on the right-hand side of (3.3) can be regarded as a ‘forcing’, ‘drift’ or ‘trend’ term, representing the effect of an ‘external’ influence that tends to make *θ*_{j} cluster near *ν*_{j}+2*kπ*/*m* for *k*=0,1,…,*m*−1. The term in the exponent can be regarded as representing the effect of an ‘internal’ influence that tends to make *θ*_{j} cluster near *θ*_{j−1}. The distribution of *θ*_{j} given *θ*_{j−1} is influenced by both *θ*_{j−1} and *ν*_{j}, the strengths of these influences being given by *κ* and *λ*. The parameter Δ represents the rate of constant drift round the circle. Equation (2.1) can be considered as the deterministic part of the process.

The distribution with density (3.3) is in the class of generalized von Mises GvM_{m} distributions considered by Gatto & Jammalamadaka [15] in the general case and by Yfantis & Borgman [16] in the case *m*=2. The process specified by (3.3) is Markov but is stationary only if *λ*=0 or Δ=0.

Three special cases are worth noting.

(a) If

*κ*=0 then*θ*_{1},*θ*_{2},… are independent and*θ*_{j}has the*m*-fold von Mises distribution with density cf. §3.6.1 of Mardia & Jupp [4].(b) For

*λ*=0, the process is one of the models considered by Accardi*et al.*[10] and Wehrly & Johnson [8]. This process is a stationary Markov process and is a special case of the von Mises process of Breckling [6]. As , the distribution of*θ*_{j}tends to the uniform distribution.(c) In the case

*m*=1, the exponent in (3.3) can be written as , and so the conditional distribution of*θ*_{j}given*θ*_{j−1}is von Mises with mean direction*μ*_{j}and concentration*κ*_{j}, where If also Δ=0 then the process is the cosine process considered by Hughes*et al.*[7].

The normalizing constant *c*_{m}(*κ*,*λ*,*μ*) has the following asymptotic approximations. Use of *I*_{r}(*κ*)=*O*(*κ*^{r}) in (3.5) gives
3.6whereas Laplace approximation to the integral in (3.4) gives
3.7

Given *θ*_{1},…,*θ*_{n}, the maximum-likelihood estimates and of *κ*,*λ*,*ν*_{0} and Δ are the values of these parameters that maximize the log-likelihood (conditional on *θ*_{1})
3.8For given values of *ν*_{0} and Δ, the family with densities (3.3) is a full 2-parameter exponential model, and so (3.8) is a concave function of (*κ*,*λ*).

## 4. The limiting distribution

Although the process is not (in general) stationary, subtraction of the ‘target directions’ produces a stationary process. For *j*=1,2,…, the angle *ϕ*_{j} is defined by *ϕ*_{j}=*θ*_{j}−*ν*_{j}. Then the process is stationary with transition densities
4.1where *c*_{m}(*κ*,*λ*,*ϕ*_{j−1}−Δ) is given by (3.5).

Because the transition densities *g*(*ϕ*_{j}|*ϕ*_{j−1}) of (4.1) are positive and continuous functions on the circle, the process is irreducible and the circle is petite in the sense of Meyn & Tweedie [17]. It follows from theorem 2.1(ii) of Meyn & Tweedie [17] that is positive Harris recurrent. Thus, the process has a unique stationary distribution, to which the distribution of *ϕ*_{j} tends as .

Let *h* denote the density of the stationary distribution. From (4.1),
4.2Then, substituting (3.6) or (3.7) into (4.2), using for *κ* near 0 or approximating the exponent in the integrand in (4.2) by a quadratic for |*κ*| large, and then integrating yields the following asymptotic results:

(i) for

*κ*≃0, so that the limiting mean direction and mean resultant length of*mϕ*_{j}are*O*(*κ*^{m}) (if*λ*>0) and*A*(*λ*)+*O*(*κ*^{m});(ii) for , so that both the limiting mean direction (if defined) and the mean resultant length of

*mϕ*_{j}are*O*(|*κ*|^{−1}).

The limiting behaviour of the process yields the limiting behaviour of realizations *θ*_{1},…,*θ*_{n} from the process . As , the empirical distribution (which assigns weight *n*^{−1} to each of *θ*_{1},…,*θ*_{n}) tends to a limiting distribution on the circle. If Δ/(2*π*) is irrational then the limiting distribution is the uniform distribution, whereas if Δ/(2*π*)=*p*/*q* with *p* and *q* coprime integers then the limiting distribution is (in general) non-uniform with density .

The dichotomy of behaviour of the limiting distribution between cases in which Δ/(2*π*) is rational and those in which it is irrational has significant consequences (discussed in §7) for certain physical applications of the model.

## 5. Tests of independence and of independence of increments

In model (3.3), the angles *θ*_{1},*θ*_{2},… are independent if and only if *κ*=0. In this case, *m*(*θ*_{2}−*ν*_{0}−2Δ),*m*(*θ*_{3}−*ν*_{0}−3Δ),… have the von Mises *M*(0,*λ*) distribution. Then a large-sample test of the hypothesis that *κ*=0 proceeds by fitting an *M*(0,*λ*) distribution to , (where and are the maximum-likelihood estimates of *ν*_{0} and Δ), transforming them by the probability integral transform of the fitted distribution, and then applying a test of uniformity such as Watson's *U*^{2} test (see §§6.4 and 6.3.3 of [4]). Use of Watson's *U*^{2} test of uniformity ensures that the test of independence is consistent, i.e. if *θ*_{1},…,*θ*_{n} are observations on dependent random angles then, as , the probability that the test rejects the null hypothesis of independence tends to 1.

The increments *θ*_{2}−*θ*_{1},*θ*_{3}−*θ*_{2},… of the process with transition probability densities (3.3) are independent if and only if *λ*=0. In this case, the increments have the von Mises *M*(0,*κ*) distribution. Thus, the hypothesis that *λ*=0 can be tested by fitting an *M*(0,*κ*) distribution to *θ*_{2}−*θ*_{1},*θ*_{3}−*θ*_{2},…,*θ*_{n}−*θ*_{n−1}, transforming them by the probability integral transform of the fitted distribution, and then applying a test of uniformity. Again, use of Watson's *U*^{2} test of uniformity ensures that the test of independence of increments is consistent.

## 6. Simulations

Figure 1 displays circular time-series plots of simulations of the Markov process with transition densities (3.3) for selected values of the parameters *κ*,*λ*,*m* and Δ. They illustrate some possible behaviours of the process. For the process simulated in figure 1*a*, *θ*_{1},*θ*_{2},… are independent with the von Mises distribution *M*(0,1). Figure 1*b* is an example in which *κ*/*λ* is large, so that *θ*_{j} tends to be close to *θ*_{j−1}+Δ. Figure 1*c* illustrates that when *λ*/*κ* is large and *m*=6, *θ*_{j} tends to be close to one of *ν*_{j}+2*kπ*/6 with *k*=0,1,…,5. Figure 1*d* is an example in which *κ* is negative and |*κ*|/*λ* is large, so that *θ*_{j} tends to be close to *θ*_{j−1}+Δ+*π*.

Figure 2 displays circular time-series plots of *ϕ*_{1},…,*ϕ*_{n} corresponding to those of *θ*_{1},…,*θ*_{n} in figure 1*b*,*d*. (Because Δ=0 for figure 1*a*,*c*, the corresponding plots of *ϕ*_{1},…,*ϕ*_{n} are the same as those of *θ*_{1},…,*θ*_{n}.) Subtraction of the target direction *ν*_{j} has made figure 2*a*,*b* more regular than figure 1*b*,*d*, respectively.

Because there is no known explicit expression for the density (4.2) of *ϕ* in the stationary distribution, we give in figure 3 density estimates of the stationary distributions arising from the distributions used to generate the data in figure 1. Since *κ*=0 in the distribution leading to figure 1*a*, the corresponding stationary distribution of *ϕ* is the von Mises distribution *M*(0,1), and the estimated density shown in figure 3*a* is very close to this distribution. Figure 3*b*,*d* shows that in these cases, the stationary distribution is almost uniform. Figure 3*c* indicates clear sixfold symmetry, in agreement with the general *m*-fold circular symmetry of the stationary distribution that can be deduced from (4.2).

## 7. Physical applications

We now consider various potential applications of the drifting Markov process on the circle, within the physical, biological and environmental sciences. Section 7*a* focuses on a problem relating to the structural properties of solid materials with a view to obtaining specific insights that are not readily obtained from experimental studies. In §7*b*, we use the process to analyse semi-diurnal observations of wind directions at a coastal site. Section 7*c* considers briefly a range of other potential applications.

### (a) Solid inclusion compounds

A physical situation that can be identified immediately with the models described above concerns the orientational distribution of ‘guest’ molecules arranged in a periodic manner (with period *c*_{g}) along the one-dimensional tunnels in a ‘host’ solid (figure 4). The host tunnel is also periodic, with period *c*_{h}. An actual realization of this type of solid structure is the urea inclusion compounds [18–22] in which guest molecules (typically based on *n*-alkane chains) are located within a honeycomb arrangement of one-dimensional tunnels in a urea host structure (figure 5). The guest molecules are densely packed along the host tunnels (diameter *ca* 5.5 Å), which are constructed from an extensive hydrogen-bonded arrangement of urea molecules.

Focusing now on an individual host tunnel, the guest molecules (labelled *j*=0,1,2,…) are arranged in a periodic manner along the tunnel axis (*z*-axis). If the position of the first (*j*=0) guest molecule along the host tunnel is denoted by *z*_{0}, then the position of each subsequent guest molecule (*j*=1,2,…) is
7.1In general terms, such materials can be subdivided into two types, depending on whether the ratio of the guest and host periodicities, *α*=*c*_{g}/*c*_{h}, is rational or irrational. In the case of rational *α*, the material is classified as commensurate, whereas in the case of irrational *α*, the material is classified as incommensurate [23–26]. For urea inclusion compounds, the relationship between *c*_{g} and *c*_{h} is usually incommensurate, although for certain types of guest molecule, the structure is observed to be commensurate [27–29].

We are interested here in the distribution of orientations *ω*_{j} of the guest molecules, where the rotation angle *ω* refers to rotation about the axis of the host tunnel (figure 4). It is reasonable to assume that the guest molecule does not have any other orientational degrees of freedom as a consequence of the directional constraints imposed by the host tunnel (figure 4). Given that the positions of the guest molecules with respect to the host tunnel are fixed according to (7.1) (except insofar as *z*_{0} may be chosen arbitrarily), the only spatial degree of freedom for each guest molecule is the rotation angle *ω*_{j}.

In certain circumstances, it is reasonable to consider that the host–guest material is constructed by adding each guest molecule sequentially to a pre-formed host tunnel. Assuming that the host tunnel is rigid, the orientational properties of a given guest molecule *j* depend only on (i) its interaction with the host tunnel structure and (ii) its interaction with the adjacent guest molecule *j*−1 already inside the tunnel.

The host–guest interaction energy in (i) is described by the potential energy function *V* _{hg}(*z*,*ω*), which depends on the position and orientation of the guest molecule relative to the host tunnel. Given the periodicity of the host tunnel, it follows that the function *V* _{hg} is periodic with
The guest–guest interaction energy in (ii) depends only on the relative orientations of guest molecule *j* and the guest molecule *j*−1 added to the tunnel in the preceding step, and is described by the potential energy function *V* _{gg}(*ω*_{j}−*ω*_{j−1}). We recall that the relative positions of guest molecules *j* and *j*−1 are fixed by the periodicity of the guest substructure, with *z*_{j}−*z*_{j−1}=*c*_{g}. This physical system translates directly into the drifting Markov process on the circle described above, with the parameters *κ* and *λ* in (3.3) corresponding to and , where *k* is the Boltzmann constant and *T* is the temperature.

Each ‘target direction’ *ν*_{j} of the model may be considered as the orientation of guest molecule *j* ‘forced’ by its interaction with the host structure; thus, *ν*_{j} is the value of *ω* corresponding to the minimum of *V* _{hg}(*z*_{j},*ω*). Clearly, the actual value *ω*_{j} of the orientation of guest molecule *j* may differ from *ν*_{j} because of the (potentially) competing influence of guest–guest interaction.

For a host structure with *m*-fold screw symmetry, the potential energy function *V* _{hg} satisfies
The urea inclusion compounds are an example of *m*-fold screw symmetry with *m*=6 [18–21]. There are also several examples of solid host tunnel structures for which *m*=3.

For a helical host structure *m*-fold screw symmetry (and with the pitch of the helix equal to the host period *c*_{h}) one possible form of *V* _{hg} is
where *V* _{0} is a constant and *γ*_{0} is a constant that depends only on *z*_{0}. In this case, the parameter Δ is
7.2

In §4, we found that there is a dichotomy in the behaviour of the limiting distributions for the drifting Markov process depending on whether Δ/2*π* is rational or irrational. In the context of the one-dimensional host–guest materials discussed here, the limiting distribution refers to the overall orientational behaviour of the guest molecules along the host tunnels. From (7.2), it is immediately apparent that this dichotomy of behaviour coincides exactly with the commensurate versus incommensurate nature of the host–guest material, recalling that *c*_{g}/*c*_{h} is rational for commensurate materials and irrational for incommensurate materials. On this basis, we may surmise that the limiting orientational distributions of the guest molecules in one-dimensional host tunnel structures will exhibit contrasting behaviour depending on whether the material is commensurate or incommensurate.

Several structural properties of the type of solid inclusion compound discussed here (i.e. based on one-dimensional host tunnel structures) can be established from experimental studies. Thus, periodic structural properties may be determined by diffraction-based techniques and different types of local structural property may be established by a variety of spectroscopic techniques. For example, using diffraction techniques, the question of whether the relationship between the host and guest periodicities along the tunnel direction is commensurate or incommensurate can be elucidated. However, knowledge of the specific sequence of guest molecule orientations on moving along the tunnel is significantly more difficult to establish from experimental studies, except in certain special cases (for example, if all guest molecules have the same orientation as a result of a dominant orienting influence imposed by their interaction with the host structure). In cases for which the orienting influence from guest–guest interaction (between adjacent guest molecules) and the orienting influence from host–guest interaction are comparable to each other, more complex orientational distributions of the guest molecules may be expected. It is for such cases that the model developed here is able to provide particularly valuable insights into the sequence and distribution of guest molecule orientations, which in general are not readily accessible from experimental studies. For this reason, the role of the model in this specific physical application is primarily to provide insights beyond the level of knowledge accessible by experiment, rather than to provide a means of analysing experimental data.

### (b) Wind directions

Figure 6 is a circular time series plot of 122 semi-diurnal observations on wind direction at Koeberg, South Africa, from 1 May to 30 June 1986. The observations, nominally at noon and midnight on each day, are the directions of the mean wind vector over each preceding hour and form part of a much larger dataset which is analysed in §12.3 of Zucchini & MacDonald [30] and is available from http://wsopuppenkiste.wiso.uni-goettingen.de/hmm-with-r/data/wind2.txt. Because Koeberg is on the coast, it is reasonable to expect that reversals of these wind directions will be observed as the result of the alternation of sea and land breezes. Figure 6 provides some evidence of such reversals. We fit the drifting Markov model to this dataset. Since there is no reason to anticipate circular *m*-fold symmetry (with *m*>1), we take *m*=1.

Application to this dataset of the test (using Watson's *U*^{2} test of uniformity) of *λ*=0 described in §5 gives a *p*-value less than 0.025, so we reject the hypothesis that the increments are independent with the same *M*(0,*κ*) distribution. The corresponding test of *κ*=0 gives a *p*-value greater than 0.05, so we accept the hypothesis that the observations are independent with *θ*_{j}∼*M*(*ν*_{0}+*jΔ*,*λ*) for *j*=2,…,*n*. In this 3-parameter sub-model with *κ*=0, the maximum-likelihood estimates of *λ*, *ν*_{0} and Δ are , and .

The possible effect of land and sea breezes leads to the hypothesis that Δ=*π*. In the 2-parameter sub-model with *κ*=0 and Δ=*π*, the estimates of *λ* and *ν* are and . From (3.8), the deviance between the two sub-models is . Comparing this deviance with the limiting large-sample distribution gives a *p*-value of less than 10^{−3}, and so we reject the hypothesis that Δ=*π* and conclude that the underlying meteorological process is more complicated than simple alternation of land and sea breezes.

Because figure 6 is reminiscent of figure 1*c*, we also fit the drifting Markov model with *m*=6, although we have no convincing plausibility argument for *m*=6 arising in this particular physical situation. The test of *κ*=0 gives a *p*-value less than 0.05; so, we reject the hypothesis of independence and we fit the model in which *κ*,*λ*,*ν*_{0} and Δ are all unknown. The maximum-likelihood estimates are and . In the 3-parameter sub-model with Δ=*π*, the estimates of *κ*, *λ* and *ν*_{0} are , and , respectively. From (3.8), the deviance between the two sub-models is . Comparing this deviance with the limiting large-sample distribution gives a *p*-value of 0.08, and so we accept the hypothesis that Δ=*π*. The maximized log-likelihoods under the 3-parameter models with (i) *κ*=0,*m*=1 and (ii) Δ=*π*,*m*=6 are 0.09 and 17.16, respectively, so Akaike's Information Criterion indicates that the latter model gives a better fit to the data. We conclude that the wind direction changes between six equally spaced major directions and that there is a significant alternation of land and sea breezes.

### (c) Further potential applications

Further potential applications of the drifting Markov process on the circle include the following.

(i) The process with

*m*=1 and Δ=0 could be used to model the directions in successive segments of the‘outward’ or ‘homeward’ paths of wildlife, such as those considered for bison by Langrock*et al.*[31] and for groups of baboons or individual chimpanzees by Byrne*et al.*[32].(ii) The process with

*m*=1 and Δ=*π*could be used to model circular processes with periodic reversal, e.g. semi-diurnal effects on the tidal stream at a given point, directions of travel of wildlife between sleeping and feeding sites, and directions of the horizontal component of the solar magnetic field in sunspots [33].(iii) Some predators, such as sharks and wolves, turn towards their prey while they attack from the side. The changing directions in which such predators move during an attack could be modelled by the process with

*m*=1 and 0<Δ<*π*.

## Acknowledgements

We thank Dr Roland Langrock and Prof. Walter Zucchini for pointing us towards the data on wind directions. We are grateful to Dr Colan Hughes for preparation of figure 5.

- Received February 11, 2013.
- Accepted May 10, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.