1980, Geophysical Journal Royal Astronomical Society, v. 61, p. 101 - 113.

Analysis of the modes of directional data with particular reference to paleomagnetism

D. R. Van Alstine, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA

Abstract. Advantages of using the mode in analysis of paleomagnetic vectors are discussed, and a computer technique is described for contouring and precisely locating the modes of vector distributions that may be highly skewed. In contrast to conventional determinations of the mode, unit vectors from a given data set are treated not as discrete points, but as identical Fisherian probability density functions defined (at an angle θ from the unit vector) by p=exp(sk(cosθ−1)), where k is the estimate of the Fisherian concentration parameter, and s is an arbitrarily assigned "smoothing parameter." A grid, representing the cumulative probability distribution of the total sample of vectors, is contoured to provide a graphical display of the distribution around the most probable value, the mode. By repeatedly contouring the same sample of vectors with successively larger values of s, and by treating the mode as a vector with length given by the total probability value at the mode, "progressive modal diagrams" can be constructed, to aid in determining the stable position of the mode of skewed distributions. In addition, a new statistic, "β₉₅" is suggested as an error estimator for the mode. The statistic β₉₅ is derived from the largest subset of the total sample that has a mean identical with the mode of the total sample; this statistic is defined as the Fisherian half-angle of the cone of 95% confidence for the mean of this subset.