Statistics

Currently Fisher statistics (from equations in Fisher et al., 1987) are available in the Least Squares and Locality Equal Area plots. Four sets of statistics are calculated: geographic and stratigraphic coordinates, and "true" and "hemisphere" Fisher statistics. The "true" statistics are just that: traditional Fisher statistics. Because individual localities often combine normal and reversed polarities, a second statistic, the "hemisphere" statistic was devised. Starting from the original "true" mean direction, those directions in the opposite hemisphere from the starting mean are inverted through the origin before being added to the total direction vector. The resulting mean is then used in place of the "true" mean in the next iteration, and so on, until five iterations have occurred or the mean direction at the end is nearly the same as the beginning. The estimated kappa and alpha-95 for this statistic reflect this collection of directions that all lie in the same hemisphere as the final mean. This should have a greater kappa and smaller alpha-95 than the true Fisher statistics (or will be equal if all the data lies in one hemisphere) and in principle should be the mean direction desired for a locality mean when two polarities are present.

For those plots where a confidence cone is plotted, the statistic type used in computing that mean direction and confidence cone are boldfaced.

Because of some problems in implementing the correct approximations for computing alpha-95 for collections of directions with small kappa (kappa < 5) and a small number of directions (less than 30), the program presently averages the approximation in Fisher et al., 1987 for kappa>5 with that for n>30. You will hear a beep whenever this is done, for a maximum of four (Least Squares) or six (Locality) beeps. The confidence cones in question can be identified by the low kappa and n.

A slight variation of Fisher statistics was suggested by McFadden and McElhinny (1988). This allows the combination of line and plane or circle fits in obtaining a Fisher mean; of greater impact (and the reason that this was in fact added) is the ability to restrict the plane or circle fits to those arcs that are reasonable given the paleomagnetic observations. In this code, this arc is taken to extend from the "farthest" point along the plane or circle (generally the last point used) to the antipode of the "nearest" point on the plane or circle (generally the first point used; see the figure in the description of the .lsq file format, below). Any such combination of line and plane fits must be approached with caution, as noted by Bailey (1990); it seems clear that some work remains to separate the best estimate of the mean and its uncertainty (McFadden and McElhinny's formulation) from the best estimate of the uncertainty of the mean that includes non-Fisherian biases inherent in the data or its reduction (Bailey's objection and the Bingham and non-parametric statistics below). The algorithm of McFadden and McElhinny has been coded from their paper and is not their original code or a modification; it was tested against the example of McFadden and McElhinny and found to match their estimated means and uncertainties within about 0.1-0.2°ree;. Note that McFadden and McElhinny noted that the uncertainties, while probably good approximations, are not rigorously derived.

Bingham statistics (Onstott, 1980) are available whereever Fisher statistics are. Although similar to Fisher stats in use, certain points should be noted. The code does not perform the complete integration to get exact concentration parameters; instead it uses a lookup table from Mardia and Zemroch (1977) with extensions by Gillett (1986). This should be adequate for most purposes. However, we have found that the approximations that are commonly used in computing Bingham statistics (and are used at present in the code) are poor for calculating alpha-95's for poles to planes or circles (the ||kappa1| and |kappa2| > > 0 condition is violated). This can be seen by using the "Combine lines and planes" option in the Least-Squares plot for a collection of best-fit lines and comparing the result with that obtained using the "Treat as lines" option.

Because of some of the problems with the Bingham distribution, non-parametric bipolar/girdle statistics were added to the Least Squares Equal Area Plot's elliptical statistics option and, for convenience (and because Watson has discussed these uncertainty calculations), have been dubbed "Watson" statistics despite the fact these are NOT the symmetrical Watson bipolar or girdle distributions (Fisher et al., 1987, p. 155ff.). This statistic appears to yield a more stable error ellipse regardless of the type of data (line data will have the same ellipse when either "Treat as lines" or "Combine lines and planes" are chosen). However, this statistic, being non-parametric, lacks estimated concentration parameters as described in Fisher et al. [1987]. This statistic is approximate for less than 25 directions. For poorly grouped data the 95% confidence cone might exceed 90°ree; in general the statistics on such data are meaningless. But to provide some comparative measure, the alphan cone is displayed, where n is the greatest confidence level producing a cone with semi-major axis of 90°ree;. The text in the upper right corner will indicate n.

Statistics presented in the window include the two kappa's of the plotted points (for the Bingham case), the alpha-95's of the major and minor axis of the error ellipse, the mean direction, and the VGP latitude and longitude. The "Oval Azimuth" is the angle measured clockwise from a line through the mean direction and the stereonet's center to the major axis of the error ellipse. The statistic uses the eigenvalues (e1>e2>e3) and associated eigenvectors (e-sub-1,e-sub-2,e-sub-3) of a orientation or principal component matrix constructed from the selected paleomagnetic directions. "Plot Girdle" to Bingham or Watson error ellipses are simply 2 error ellipses with the same mean and oval azimuth but with axes equal to 90°ree;+alpha and 90°ree;-alpha, where alpha is the either the semi-major or semi-minor axis.

The performance and use of the Bingham option is usually straightforward, but some circumstances require special attention to the values actually calculated (see below). For the elliptical option in the "Locality Equal Area..." plot (which still uses Bingham statistics), the program determines if the selected points seem to describe a single direction (e1 > 2e2) or a girdle (e1 < 2e2). If a single direction, the error ellipse about the mean direction (e-sub-1) is plotted. If a girdle, the error ellipse about the pole (e-sub-3) of the best fitting plane through the points is plotted, as are the bounding ellipses (girdles) of the distribution. One could interpret these girdles as the confidence bounds on the possible positions of two different components, where each of the measured directions reflect a different linear combination of the two components.

For the "Least Squares Equal Area...," the Bingham or Watson statistics are manually controlled in a dialog box that pops up after selecting the directions to plot. Generally, if all the directions selected were derived using line fits, use the "Treat as Lines" option; if all the directions were plane or circle fits, use the "Treat as Planes" option. If you wish to combine planes and lines, use the "Combine lines and planes" option.

There are some instances where you might wish to override the choices outlined above; for instance, if you have a set of plane-fits from data that clearly represent the overlap of two distinct (and unvarying) directions, all of the plane fits will have similar poles. Using "Treat as Planes" will produce an elongated ellipse centered on the data's mean; this might have little meaning in terms of confidence regions of either underlying direction. In this case, using the "Treat as Lines" option with the "Plot Girdle" option will give the direction and uncertainty of the pole to the plane containing both directions; the girdle should enclose the region containing both directions.

For reference, when data are "treated as lines," each direction is added to the orientation (or principal component) matrix as usual, the mean direction and associated uncertainties are for e-sub-1. When "treated as planes," directions are again added as usual, but the mean direction and its uncertainties now are of the e-sub-3 vector of the moment-of-inertia matrix. When "combining lines and planes," plane fits (identified by a "P" or "C" in the id in the least-squares file .LSQ) are added as usual, but line fits are instead added as two mutually orthogonal directions, each orthogonal to the line-fit direction. The plotted mean and uncertainties again are of e-sub-3. This is "Method 2" of Kirschvink (1980). NOTE: There are several errors in Kirschvink (1980) regarding this method of combining lines and planes; that paper implies that this technique is only approximate, changes the mean direction of line data, and is probably inferior to "Method 1." In fact it appears that there were bugs in the codes used by Kirschvink and Method 2 is in fact exact both in producing a mean and in producing error ellipses with the non-parametric statistic (with the one change that each of the two orthogonal directions are added at full weight, not half weight as suggested by Kirschvink (1980)). Using either "Treat as lines" or "Combine lines and planes" will produce identical means if all the selected directions are line-fits. As noted above, the "Watson" statistics were added because they are more consistent than the Bingham statistics when comparing line data analyzed as lines with line data transformed to pairs of planes; the "Watson" and Bingham mean directions are the same. A discussion of these statistics has been prepared and is available from

C. H. Jones (cjones@mantle.colorado.edu)
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