# Circular analysis, one sample

The module plots a rose diagram (polar histogram) of directions. For plotting current-oriented specimens, orientations of trackways, fault lines, etc. Also appropriate for time-of day data (0-24 hours).

One column of directional (0-360) or orientational (0-180) data in degrees is expected. Directional or periodic data in other forms (radians, hours, etc.) must be converted to degrees using e.g. the Evaluate Expression module (Transform menu).

By default, the 'mathematical' angle convention of anticlockwise from east is chosen. If you use the 'geographical' convention of clockwise from north, tick the box.

You can also choose whether to have the abundances proportional to radius in the rose diagram, or proportional to area (equal area).

The "Kernel density" option plots a circular kernel density estimate.

#### Descriptive statistics

(For mathematical details see the Past manual).

The mean angle takes circularity into account.

The 95 percent confidence interval on the mean is estimated according to Fisher (1983). It assumes circular normal distribution, and is not accurate for very large variances (confidence interval larger than 45 degrees) or small sample sizes. The bootstrapped 95% confidence interval on the mean uses 5000 bootstrap replicates. The graphic uses the bootstrapped confidence interval.

The concentration parameter κ is estimated by iterative approximation.

#### Rayleigh’s test for uniform distribution

The R value (mean resultant length) is tested against a random distribution using Rayleigh's test for directional data (Davis 1986). Note that this procedure assumes evenly or unimodally (von Mises) distributed data - the test is not appropriate for e.g. bimodal data. The p values are approximated according to Mardia (1972).

#### Rao’s spacing test for uniform distribution

The Rao's spacing test (Batschelet 1981) for uniform distribution is nonparametric, and does not assume e.g. von Mises distribution. The p value is estimated by linear interpolation from the probability tables published by Russell & Levitin (1995).

A Chi-square test for uniform distribution is also available, with 4 bins.

#### The Watson’s U2 goodness-of-fit test for von Mises distribution

The test statistic U is estimated by numerical integration. Critical values for the test statistic are obtained by linear interpolation into Table 1 of Lockhart & Stevens (1985). They are acceptably accurate for n>=20.

#### Axial data

The 'Orientations' option allows analysis of linear (axial) orientations (0-180 degrees). The Rayleigh and Watson tests are then carried out on doubled angles (this trick is described by Davis 1986); the Chi-square uses four bins from 0-180 degrees; the rose diagram mirrors the histogram around the origin.

#### References

Batschelet, E. 1981. Circular statistics in biology. Academic Press.

Davis, J.C. 1986. Statistics and Data Analysis in Geology. John Wiley & Sons.

Fisher, N.I. 1983. Comment on "A Method for Estimating the Standard Deviation of Wind Directions". Journal of Applied Meteorology 22:1971.

Lockhart, R.A. & M.A. Stephens 1985. Tests of fit for the von Mises distribution. Biometrika 72:647-652.

Mardia, K.V. 1972. Statistics of directional data. Academic Press, London.

Russell, G. S. & D.J. Levitin 1995. An expanded table of probability values for Rao's spacing test. Communications in Statistics: Simulation and Computation 24:879-888.

Published Aug. 31, 2020 3:23 PM - Last modified Aug. 31, 2020 3:23 PM