2 - Examples

Summary

This chapter collects together, and describes in a simple manner, a number of applications of the theory presented in this book. The examples are based on real data, and, where possible, results of parametric inference in the literature are cited for comparison with the new nonparametric inference theory.

Data example on S¹: wind and ozone

The wind direction and ozone concentration were observed at a weather station for 19 days. Table 2.1 shows the wind directions in degrees. The data are taken from Johnson and Wehrly (1977). The data viewed on the unit circle S¹ are plotted in Figure 3.1. We compute the sample extrinsic and intrinsic mean directions, which come out to be 16.71 and 5.68 degrees, respectively. They are displayed in the figure. We use angular coordinates for the data in degrees lying between [0°, 360°) as in Table 2.1. An asymptotic 95% confidence region for the intrinsic mean as obtained in Section 3.7, Chapter 3, turns out to be

{(cos θ, sin θ) : - 0.434 ≤ θ ≤ 0.6324}.

The corresponding end points of this arc are also displayed in Figure 3.1.

Johnson and Wehrly (1977) computed the so-called angular–linear correlation ρAL = max_α{ρ(cos(θ - α), X)}, where X is the ozone concentration when the direction of wind is θ. Here ρ denotes the true coefficient of correlation. Based on the sample counterpart r_AL, the 95% confidence interval for ρAL was found to be (0.32, 1.00).