Circular Statistical Methods Research Articles

Automated conformational analysis from crystallographic data. 4. Statistical descriptors for a distribution of torsion angles

The methods of linear and circular statistics are used to derive summary descriptors for a unimodal distribution of torsion angles (r;). The arithmetic methods of normal linear statistics are complicated by the phase restriction - 180 < 7 i ~--- 180 °. Phase adjustments must be made to generate a correct arithmetic mean (-L,) which spans _180 ° in a Newman projection of the re. A general single-pass algorithm is described for the calculation of this mean, its standard error ~r(-L,) and the sample standard deviation o'(r,,). The single-pass technique is restricted to distributions in the range r_< 180°; a preliminary pass is required for broader distributions. The r; are, however, properly represented as a circular distribution and the established formalisms of circular statistics should be applied. Here a circular mean or preferred direction (-L.) may be derived in a single pass for a distribution of any range. The variance of the distribution may be assessed in terms of the concentration, R, of data points around the mean -~c. A circular standard error o-(~c) and a circular sample standard deviation ~r(rc) may then be derived. It is shown that the arithmetic and circular descriptors are numerically similar, except for broad distributions. The circular method has computational advantages in minimizing phase-shift operations and the results are more realistic and reliable when used in further statistical tests.

Multivariate analysis of initial orientation in homing pigeons: Is there a “preferred compass direction”?

The mean vanishing direction of homing pigeons of a given release often deviates from the home direction considerably. Any kind of geographical systematic in these deviations might give us a hint on the distribution of the information involved in the homing process. Analysis of releases arranged symmetrically around the home loft with methods of circular statistics could not resolve the pending question of site-specificity vs. directional effects in the initial orientation of homing pigeons unambiguously. In part, this ambiguity comes about by the facts that 1. any release site is correlated with a specific cardinal direction and any cardinal direction is composed of several release sites; 2. most methods of higher order circular statistics combine two different characteristics of a sample, namely a measurement of direction (mean vanishing direction) and a measurement of directedness (length of mean vector.).