Introduction

Whereas the previous chapters have been concerned with statistical inference for a single sample of data (typically, relating to the reference direction or dispersion) and with comparison of such quantities across several samples, this chapter and the next are concerned with relationships between random variables, or the dependence of a circular random variable on another variable. Examples of the sorts of problems to be studied in this chapter are:

Introduction

We shall be concerned with three types of data in this chapter:

1. (I) Circular observations in relation to a time sequence t1,t2,…

2. (II) Circular observations in relation to a one-dimensional spatial sequence x1,x2,…, in other words a sequence of points along a line.

3. (III) Circular observations in relation to a corresponding set of two-dimensional spatial locations, in other words positions in the plane.

Time-dependent sequences of data of type I arise commonly in Meteorology and Oceanography as measurements of wind or current direction at a particular location; often, but not always, each direction has an associated measurement of strength, so that the data constitute velocities. When the measurements have been recorded at equal intervals (e.g. hourly or daily measurements of wind direction), the sequence is called a time series. Typically, the questions of interest relate to finding models which capture a sufficient amount of the dependence structure to allow future behaviour of the series to be predicted.

Data sequences gathered along a line (type II) can occur in Structural Geology, for example, when mapping along a mining tunnel. Here, interest centres on extracting an overall trend, or detecting sudden changes in the general trend of the sequence. Data of this type differ from those of type I in that they can be looked at in reverse order – the order depended simply on the direction x1,x2,… or xn,xn-1,… used for sampling – whereas time-ordered sequences usually have only one sensible direction.

Introduction

In the Preface, the various types of circular data – vectors/axes, uniform/unimodal/multimodal – were described together with the usual ways of recording them. This chapter is concerned with the most basic aspects of statistical analysis of a single sample of circular measurements θ1,…,θn: methods for displaying the sample, and simple summary quantities which can be calculated from the sample.

Data display

Why do we display data? There is no shortage of reasons, but among the most important are:

• to gain an initial idea of the important characteristics of the sample; for example, does the sample appear to be from a uniform (or isotropic) distribution; or from a unimodal distribution; or from a multimodal distribution?

• to emphasise such characteristics

• to suggest models for the data, such as a von Mises model for a sample which appears to be drawn from a symmetric unimodal distribution

• to try to avoid doing something stupid: note the advice once given to students graduating from a weather-forecasting course – before issuing a forecast, look out of the window.

In view of the first of these points, the nature of any display, apart from a simple plot of the raw data, will depend on the number of modal groups apparent in the sample. Unless the sample is clearly unimodal, constructing a useful display is not a trivial exercise.

Introduction

Probability models are a very important aspect of statistical analysis. If we can fit a probability model to our data, by suitable estimation of parameters in the model, then the data set can be summarised efficiently using the particular form of probability model specified by the parameter estimates. It is perhaps surprising to find that probability models have not found much application to circular data.

To understand the reason for this last comment, we consider three types of data (linear, circular, spherical) and correspondingly, three models for single groups of data (Normal distribution, von Mises distribution, Fisher distribution); each represents the most commonly used model for its data type. These models have two sorts of parameters, one defining the location or reference direction of the distribution and the other the dispersion about that location. For the Normal distribution, dispersion is quantified by the variance σ2, with σ2 near 0 corresponding to a highly concentrated distribution, and with the distribution spreading out more and more over the whole real line as σ2 increases. For the von Mises and Fisher distributions, the dispersion is quantified by a concentration parameter κ, with κ = 0 corresponding to uniformity and increasing κ to increasing concentration about the reference direction.

For linear data, the normal distribution is often found to be a satisfactory model, irrespective of the dispersion in the data, and formal statistical analysis can proceed regardless of the value of σ2.

Circular data analysis is a curious byway of Statistics, sitting as it does somewhere between the analysis of linear data and the analysis of spherical data.

For one thing, various aspects of the subject seem trivial in comparison. A sample of linear data can be so disposed as a collection of clusters, long tails, inliers and outliers over an indefinitely large interval of the real line that no single display shows all features. A sample of spherical data can be so dispersed over the surface of the unit sphere as modal groups, transitional modal-girdle patterns and outliers that it can really only be viewed satisfactorily using a rotating three-dimensional display. By contrast, circular data cannot escape very far from each other (the notion of an outlier is of little consequence in most practical situations) and certainly cannot hide from view. So, many of the basic problems addressed in the exploratory phase of linear or spherical data analysis seem benign or non-existent for circular data.

For another, the problem of extracting practical information from circular data would appear to be tantalisingly close to the same problem for linear data, especially for concentrated data sets (i.e. data sets contained in a small arc of the circle). Approximate linearity of a small arc would seem to justify application of linear methods and so make special treatment of circular data largely unnecessary.

The purpose of the book

Data measured in the form of angles or two-dimensional orientations are to be found almost everywhere throughout Science. They arise commonly in Biology, Geography, Geology, Geophysics, Medicine, Meteorology and Oceanography, and in many other areas. Typical examples include departure directions of birds or animals from points of release, orientations of fracture planes and linear geographical features, directional movement of animals in response to stimuli, wind and ocean current directions, circadian and other biorhythms, times of day of accident occurrences, and so on.

The last 20 years, and more particularly the last 10 years, have seen a vigorous development of statistical methods for analysing such data, with emphasis on problems of data display, correlation, regression, and analysis of data with temporal or spatial structure. In addition, some of the exciting modern developments in general statistical methodology, particularly nonparametric smoothing methods and bootstrap-based methods, have contributed significantly to the data analyst's ability to make progress with problems which have been relatively intractable. The subject has now reached a point of development at which it seems appropriate to provide a unified and up-to-date account of this material for practical use. In this respect, the present book is a companion volume to Statistical Analysis of Spherical Data (Fisher, Lewis & Embleton 1987) which was concerned with three-dimensional unit vectors or orientations, although developments in nonparametric smoothing and bootstrap methods over the last four years have meant that rather more effective use of them has been possible in this book.

Introduction

This chapter contains descriptions of two general classes of statistical methods used extensively in the earlier chapters of the book, namely bootstrap methods and randomisation methods.

The phrase ‘bootstrap methods’ refers to a class of computer-intensive statistical procedures which can often be helpful for carrying out a statistical test or for assessing the variability of a point estimate in situations where more usual statistical procedures are not valid and/or not available (e.g. the sampling distribution of a statistic is not known). Typically, but not always, these situations arise when only small amounts of data are available for analysis. The bootstrap methods not only provide satisfactory results in small samples, but compare favourably with other large-sample methods in larger samples for which the latter may be applicable. Accordingly, if one is in some doubt about the precise sample size at which to switch from a small-sample method to a large-sample method, both approaches can be used and the results compared. Generally, in estimation problems the large sample methods provide a more convenient summary of the data, and are far simpler to implement. See Efron (1982) and Efron & Gong (1983) for accessible introductions to the subject. Hall (1992) has given a theoretical account of many aspects of the bootstrap.

In Section 8.2, the basic elements of the bootstrap method are described, in the context of a simple example for linear data.

Introduction

In many situations, the data set under consideration arises not as a single sample of measurements of a single phenomenon, but in the form of samples measured under a variety of experimental conditions or collected from a variety of localities. For such data, interest usually centres on two issues: first, whether there appear to be real differences between the various responses (usually, the mean directions of the samples), if so which are the responses which differ and whether there is sound evidence that the differences really exist (in other words that they are statistically significant); second, for those responses which are regarded as comparable, how to combine them to get a pooled estimate of their common mean direction. For the most part, it will be assumed that the individual samples have been drawn from unimodal distributions. The sorts of applications we have in mind are the following:

Introduction

The methods in this chapter are concerned with the analysis of a sample of independent observations θ1,…, θn from some common population of vectors or axes. A number of data sets were introduced in Chapter 2, in the context of methods for displaying data. Here, we begin with some other examples to illustrate the range of problems which occur with a single sample of data.

For this edition I have added one new topic, made a number of miscellaneous changes intended to improve the presentation and clarify some of the arguments, and adjusted a few emphases to attune with the changing times. (Microwave ovens are now commonplace, while ballistic galvanometers have been consigned to the laboratory basement.) To keep the length reasonable I have relegated to the problems a few peripheral topics, such as Lorentzians.

The teaching of vibrational physics at undergraduate level cannot ignore for ever the revolutionary ideas grouped under the title ‘chaos’. I believe that a straightforward description of the physical phenomena will provide the student with a platform from which to approach the more detailed, but also more abstract, consideration of these matters in terms of trajectories and attractors in phase space. I have therefore expanded chapter 7 to include a simple discussion of large-amplitude forced vibrations, introducing chaotic vibrations and related non-linear effects.

Our examination of various waves in the previous three chapters has highlighted a fact of great practical significance: most wave systems are dispersive. The chief characteristic of a dispersive system is that it distorts any travelling waves that are not sinusoidal (section 12.1). Sinusoidal waves are not very useful, since they cannot carry any information other than their frequency and amplitude: one is usually interested in sending signals as some kind of modulated wave, and in particular as a train of pulses. Such an enterprise might appear doomed to failure if the chosen wave system is a dispersive one.

The remarkable fact is, however, that dispersive systems can after all support the undegraded propagation of stable pulses of a special kind, known as solitary waves. The effective cancellation of the dispersion has a surprising source in another property of most real wave systems, namely non-linearity.

To see how this comes about, we must first learn how non-linearity on its own affects a wave system. To simplify the discussion, we shall consider in this chapter only travelling plane waves moving in the positive z-direction, and we shall also neglect all non-conservative processes like friction and viscosity, which dissipate energy.

Non-linear wave systems

In chapter 7 we examined free and forced vibrations of non-linear systems.

The wave equation we discussed in chapter 9 has the special property that it allows a disturbance of arbitrary form to be propagated indefinitely as a travelling wave, without having its shape changed. We met several examples of such non-dispersive waves in chapter 10.

Non-dispersive waves are exceptional. In this chapter we examine possible sources of dispersion in a stretched string.

Stiff strings

The stretched string of chapter 9 was assumed to be perfectly flexible, so that there were no transverse return forces other than those due to the tension. Real strings, such as violin and piano strings, are ‘stiff’ and tend to straighten out even when unstretched. The extra return forces due to this lateral stiffness make the string dispersive.

These return forces come from the stresses within curved parts of the string. The stress forces at any cross-section of the string will have components acting along the string direction, and components acting in the plane perpendicular to the string. Each set of components can be replaced by a single force and a single torque: for the components parallel to the string we have the force of tension, and a bending moment, while the perpendicular components give a shear force tending to break the string across, and a twisting moment. Since we already know the return force due to the tension, and since the string is presumably not twisted, we need consider only the other two.