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

Abstract

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.

Appendices

Appendix 1: Formula Derivation

In Eq. (38), the Fisher probability density on $(\varphi ,\theta )\in \{[\mathrm{0,2}\pi ],[0,\pi /2]\}$ is defined as

$$\begin{aligned}&F\left(\varphi ,\theta ,\overline{\varphi },\overline{\theta },\kappa \right)\\&\quad=\frac{\kappa sin\theta }{2\pi sinh\kappa }\mathrm{cosh}\left[\kappa \left(sin\theta sin\overline{\theta }\mathrm{cos}\left(\varphi -\overline{\varphi }\right)+cos\theta cos\overline{\theta }\right)\right] \end{aligned}$$

(38)

In Eq. (39), L’s lower bound H is defined as

$$H=\sum _{i=1}^{N}\sum _{j=1}^{k}{\gamma }_{ij}log\left[\frac{{\alpha }_{j}}{{\gamma }_{ij}}\frac{{\kappa }_{j}sin{\theta }_{j}}{2\pi sinh{\kappa }_{j}}\mathrm{cosh}({\kappa }_{j}{v}_{ij})\right]$$

(39)

where ${v}_{ij}$ is defined as

$${v}_{ij}=sin{\theta }_{i}sin\overline{{\theta }_{j}}\mathrm{cos}\left({\varphi }_{i}-\overline{{\varphi }_{j}}\right)-cos{\theta }_{i}cos\overline{{\theta }_{j}}$$

The specific formulation of Eq. (40) is

$$G\left({\kappa }_{j}\right)=\frac{\partial H}{\partial {\kappa }_{j}}=\sum _{i=1}^{N}{\gamma }_{ij}\left[\frac{1}{{\kappa }_{j}}-coth{\kappa }_{j}+{v}_{ij}\mathrm{tanh}({\kappa }_{j}{v}_{ij})\right]$$

(40)

The specific formulation of Eq. (41) is

$${G}^{\prime}\left({\kappa }_{j}\right)=\frac{{\partial }^{2}H}{\partial {{\kappa }_{j}}^{2}}=\sum _{i=1}^{N}{\gamma }_{ij}\left[\frac{1}{{{\kappa }_{j}}^{2}}+{csch}^{2}{\kappa }_{j}+{v}_{ij}^{2}{sech}^{2}({\kappa }_{j}{v}_{ij})\right]$$

(41)

In Eq. (42), the rotation matrix $Rot$ is derived as

$$Rot=\left[\begin{array}{ccc}\mathrm{cos}\left(-{\varphi }_{\mathrm{s}}\right)&\quad -\mathrm{sin}\left(-{\varphi }_{\mathrm{s}}\right)&\quad 0\\ \mathrm{sin}\left(-{\varphi }_{\mathrm{s}}\right)&\quad \mathrm{cos}\left(-{\varphi }_{\mathrm{s}}\right)&\quad 0\\ 0&\quad 0&\quad 1\end{array}\right]\cdot \left[\begin{array}{ccc}\mathrm{cos}\left(-{\theta }_{\mathrm{s}}\right)&\quad 0&\quad -\mathrm{sin}\left(-{\theta }_{\mathrm{s}}\right)\\ 0&\quad 1&\quad 0\\ -\mathrm{sin}\left(-{\theta }_{\mathrm{s}}\right)&\quad 0&\quad \mathrm{cos}\left(-{\theta }_{\mathrm{s}}\right)\end{array}\right]$$

(42)

The specific formulation of Eq. (43) is

$$\text{prob}_{sc}={\int }_{0}^{2\pi }d\varphi {\int }_{0}^{T}\frac{{\kappa }_{c}sin{\theta }_{0}}{2\pi sinh{\kappa }_{c}}\mathrm{cosh}[{\kappa }_{c}(sin\theta sin{\theta }_{c}^{\mathrm{^{\prime}}}cos\varphi +cos\theta cos{\theta }_{c}^{\mathrm{^{\prime}}})]d\theta$$

(43)

The specific formulation of Eq. (44) is

$$\text{prob}_{sc}=\frac{2\pi }{2}\cdot \frac{r}{2} \sum _{i=1}^{ng}{A}_{i}\sum _{j=1}^{ng}{B}_{j}\frac{{\kappa }_{c}sin{\widetilde{\theta }}_{j}}{2\pi sinh\kappa }\mathrm{cosh}\left[{\kappa }_{c}\left(-sin{\widetilde{\theta }}_{j}sin{\theta }_{c}^{\mathrm{^{\prime}}}cos{\widetilde{\varphi }}_{i}-cos{\widetilde{\theta }}_{j}cos{\theta }_{c}^{\mathrm{^{\prime}}}\right)\right]$$

(44)

Appendix 2 List of symbols

$p$:

3D point

${M}_{p}$:

Covariance matrix

$\mathrm{knn}$:

Nearest point number for normal vector calculation

${e}_{i}$:

The ${i}\text{th}$ eigenvector of ${M}_{p}$

${\lambda }_{i}$:

The ${i}\text{th}$ eigenvalue of ${M}_{p}$

M:

Sample point number

m:

Elements in $\{-M,-M+1,...,M-1,M\}$

M′:

Number of m

$\Phi$:

Fibonacci sample point threshold

${\mathrm{lon}}_{m}$:

Longtitude value corresponding to m

${\mathrm{lati}}_{m}$:

Latitude value corresponding to m

$x{s}_{m}$:

Coordinate x corresponding to m

$y{s}_{m}$:

Coordinate y corresponding to m

$z{s}_{m}$:

Coordinate z corresponding to m

Vec:

Normal vector set

$N$:

Normal vector number

$\mathrm{vec}$:

The ${i}\text{th}$ normal vector

Sp:

Sample point set

${\mathrm{sp}}_{i}$:

The ${i}\text{th}$ sample point

Neig:

Neighbor normal vector set of sample points

T:

Neighbor angle threshold

${r}_{0}$:

Radius of the circular sampling window on the stereographic projection plane

$\rho$:

Neighbor density set of sample points

${\rho }_{i}$:

Neighbor density of the ${i}\text{th}$ sample point

$\delta$:

Angle set of sample points to the nearest higher density sample points

${\delta }_{i}$:

Angle of the ${i}\text{th}$ sample point to the nearest higher density sample point

${\mathrm{src}}_{i}$:

Priority score for the ${i}\text{th}$ sample point

Cen:

Grouping center set

${\mathrm{cen}}_{i}$:

The ${i}\text{th}$ grouping center

Fcen:

Index set of the sample points selected as grouping centers

${\mathrm{fcen}}_{i}$:

Index set of the sample point selected as the ${i}\text{th}$ grouping center

$\alpha$:

Weight set of Fisher mixed distribution

${\alpha }_{i}$:

Weight of the ${i}\text{th}$ independent Fisher distribution

$\kappa$:

Fisher constant

${\kappa }_{i}$:

Fisher constant of the ${i}\text{th}$ independent distribution

$\varphi$:

Dip direction

$\theta$:

Dip

$\overline{\varphi }$:

Mean dip direction

$\overline{\theta }$:

Mean dip

$f$:

Fisher probability density function

${\gamma }_{ij}$:

Affiliation of the ${i}\text{th}$ normal vector belonging to the ${j}\text{th}$ Fisher distribution

$L$:

Maximum likelihood function

$H$:

Lower bound of L

$G\left({\kappa }_{j}\right)$:

Partial derivative of H to ${\kappa }_{j}$

$lr$:

NR-EM learning rate

$\mathrm{coef}$:

Reduce coefficient of $lr$

$\mathrm{Neig}\_{\mathrm{cen}}_{i}$:

Set of normal vectors in the neighborhood of the ${i}\text{th}$ grouping center

$|\mathrm{Neig}\_{\mathrm{cen}}_{i}|$:

Element number in $\mathrm{Neig}\_{\mathrm{cen}}_{i}$

${R}_{i}$:

Mean vector of $\mathrm{Neig}\_{\mathrm{cen}}_{i}$

${\varphi }_{c}$:

Grouping center dip direction

${\theta }_{c}$:

Grouping center dip

${\varphi }_{s}$:

Sample point dip direction

${\theta }_{c}$:

Sample point dip

${x}_{c}$:

Coordinate x of grouping centers

${y}_{c}$:

Coordinate y of grouping centers

${z}_{c}$:

Coordinate z of grouping centers

$Rot$:

Rotation matrix

${\theta }_{c}^{\mathrm{^{\prime}}}$:

Grouping center dip after rotation

${x}_{c}\mathrm{^{\prime}}$:

Coordinate x of grouping centers after rotation

${y}_{c}\mathrm{^{\prime}}$:

Coordinate y of grouping centers after rotation

${z}_{c}\mathrm{^{\prime}}$:

Coordinate z of grouping centers after rotation

${\mathrm{prob}}_{sc}$:

Neighbor density of the ${c}\text{th}$ Fisher distribution at the ${s}\text{th}$ sample point

$ng$:

Number of Gaussian quadrature points

${\widetilde{\theta }}_{j}$:

Gaussian quadrature value of the ${j}\text{th}$ dip

${\widetilde{\varphi }}_{i}$:

Gaussian quadrature value of the ${i}\text{th}$ dip direction

$k$:

Grouping number

$\mathrm{Sgp}$:

Grouping score set of different group numbers

${\mathrm{Sgp}}_{k}$:

Grouping score set with group number k

${\mathrm{Fit}}_{k}$:

Fitting accuracy of the Fisher mixed distribution with group number k

${\mathrm{Domi}}_{k}$:

Dominance of the Fisher mixed distribution with group number k

${\mathrm{Prob}}_{i}$:

Predicted neighbor density of the ${i}\text{th}$ sample point

${\mathrm{Truth}}_{i}$:

True neighbor density of the ${i}\text{th}$ sample point

${k}_{\mathrm{opt}}$:

Optimal group number

$\mathrm{Gp}$:

Grouping label set of sample points

$g{p}_{si}$:

Grouping label set of the ${i}\text{th}$ sample point

$G{p}_{\mathrm{vec}}$:

Grouping label set of normal vectors

$g{p}_{vi}$:

Grouping label of the ${i}\text{th}$ normal vector

${v}_{\mathrm{par}}$:

Variation ratio of parameter par during two adjacent NR-EM iteration steps defined as ${v}_{par}=\frac{|pa{r}_{t+1}-pa{r}_{t}|}{|pa{r}_{t+1}|}$