VECTOR STATISTICS AND TESTS

Theory

In the last topic, we dealt with simple scalar distributions and the primary statistics which describe them. In this section we go on to consider 2-D distributions, vectors (directed lines) and axes (undirected lines). Since we more often encounter vectors in geology, we will only discuss them; but remember that the same principles apply to axes. Also, in most cases the vectors will only indicate directions. That is they will not have variable intensity measurements associated with them. Therefore, we will treat these vectors as 'unit vectors', vectors of length, 1, whose tips will all fall on a sphere of diameter, 1, centered on the origin.

We can describe the distribution of a group of N unit vectors by considering the distribution that their vector tips make on the surface of the unit sphere. Let us assume we have a unit sphere where any point on the surface can be described by an inclination (I) and a declination (D). The inclination defines the vertical angle which each vector makes with the horizontal plane, the declination defines the horizontal angle which each angle makes with 0° N. For a distribution of N unit vectors, each vector has direction cosines

The resultant vector (average direction or vector mean) is then

The resultant (average) vector direction is then

The two most commonly described vector distributions are exact analogs of the scalar uniform and normal distributions. On the unit sphere, a uniform distribution can be viewed as a completely random shotgun pattern over the entire sphere surface. Any single vector direction has an equal likelihood of occurrance. The spherical analog of the normal distribution is called Fisher's distribution. Fisherian distributions have a mean value which can be calculated from the equations noted above and a variance that is usually described by the a95 statistic. The a95 is equivalent to 2s, where s is the scalar standard deviation, and describes the half-angle that encloses 95% of the vector end points around the mean value. The a95 is defined as

An inverse measure of the vector dispersion described by the a95 is the precision parameter, k. The precision parameter is estimated from

The remaining moments of vector distributions are not used for description. Instead, specialized shape statistics are used to characterize the distribution of unit vectors about the mean direction. The Fisherian distribution assumes that the vectors are equally spaced in azimuth around the mean and decrease in abundance away from the mean as is defined by the Fisherian probability distribution. It is common, however, for unit vectors to be elliptically distributed around the mean value. This is a different vector distribution and the shape is defined on the basis of the degree of ellipticity. The ellipticity is usually defined on the basis of a 2-D projection of the data into a plane perpendicular to the mean vector direction where the x-axis is equal to 0° declination. The 2-D distribution of the data points (xi, yi; i=1, N) can then be calculated by moment-of-inertia. The shape statistics that can be defined on the basis of moment-of-inertia calculations are described by Engebretson and Beck (1978).

Algorithm Development/Canned Programs

The problem sets associated with this topic require you to calculate simple vector statistics for two data sets. Unfortunately, most subroutine packages do not provide subroutines to do elementary vector statistics. I will place in your accounts four Fortran source programs to do vector statistical analysis. ALPHA95 is a simple program to calculate alpha95s. VECTOR is a more 'sophisticated' program to calculate simple vector statistics including the alpha95. SHAPE (and its subroutine ROTATE) calculates the shape statistics associated with a 2-D unit vector distribution.

Another practical solution is to use the Macintosh program called STEREONET. This program is on the server and will let you input a vector data set, get simple vector statistics, etc.

References

(1) N. I. Fisher, T. Lewis, and B. Embleton, 1987, Statistical Analysis of Spherical Data, Cambridge University Press.

(2) D. C. Engebretson and M. E. Beck Jr., 1978, On the shape of directional data sets, J. Geophys. Res., v. 83, p. 5979-5982.