OCFMD: An Automatic Optimal Clustering Method of Discontinuity Orientation Based on Fisher Mixed Distribution

Discontinuities largely influence the mechanical properties of rock joints. However, discontinuity orientation clustering methods often rely on the aggregation and separation of orientation data without full consideration of the prior probability structure of orientation data. This paper proposes a method of optimal clustering by Fisher mixed distribution (OCFMD) for automatic grouping of discontinuity orientation. Based on the Fisher prior probability structure of orientation data, OCFMD can identify optimal group centers and group numbers by balancing the fitting accuracy and dominance of Fisher mixed distributions, and optimal grouping results can be generated by membership calculation. A Newton–Raphson expectation maximization (NR-EM) algorithm is derived for the parameter fitting of Fisher mixed distributions. The Fibonacci sequence is used to generate sample points. In addition, the neighbor probability and density of sample points based on Fisher mixed distributions is derived for fitting accuracy calculation. Several cases of rock slopes and rock tunnel excavation faces are adopted for analyzation. Three clustering algorithms combined with four clustering validity indexes of discontinuity grouping are used for comparison. The results show that OCFMD is more accurate and robust than the other automatic grouping methods in optimal grouping result generation. Highlights An automatic optimal clustering method of discontinuity orientation is proposed based on Fisher mixed distributions. The balance between fitting accuracy and dominance of Fisher mixed distributions is derived for the selection of optimal grouping results. The grouping results of several traditional clustering algorithms combined with clustering validity indexes are observed to be inconsistent with manual results. The proposed method is more accurate and robust than the compared traditional methods in optimal grouping result generation. The convergence effectiveness, sensitivity of neighbor angle selection and robustness to normal vector variations are validated.