3 - Models

Summary

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.