Statistical quantification of the microstructural homogeneity of size and orientation distributions

Abstract

Methodologies to quantify the microstructural homogeneity, or uniformity, have been developed based on the proposed statistical homogeneity theory. Two kinds of homogeneities are considered, for the size and orientation distributions, respectively. In the case of size distribution, the homogeneity is quantified using two parameters, H _0.1 and H _0.2, which are defined as the probabilities falling into the ranges of μ ± 0.1μ and μ ± 0.2μ, respectively, where μ is the mean size. Whereas in the case of orientation distribution, three parameters are used to quantify the homogeneity: H _R, the mean resultant length that is a simple measure of the angular data concentration, and H _0.1 and H _0.2, which are the probabilities in particular angular ranges of the circular or spherical data. These homogeneity quantities are formularized using the common statistical models, and typical examples are demonstrated.

Appendices

Appendix 1: Size distribution

Normal and lognormal distribution

The normal and lognormal distributions were studied previously [23]. For a normal distribution,

$$ H_{0. 1} = 6. 8 8 4 3\times 10^{ - 5} + 7. 9 6 4\times 10^{ - 2} (\mu /\sigma ) + 1.0 4 3\times 10^{ - 4} (\mu /\sigma )^{ 2} - 1. 6 2 8 6\times 10^{ - 4} (\mu /\sigma )^{ 3} + 3. 8 6 3 9\times 10^{ - 6} (\mu /\sigma )^{ 4} \quad(r = 1), $$

(19a)

$$ H_{0. 2} = - 4.0 1 1 7\times 10^{ - 4} + 0. 1 60 5 6(\mu /\sigma ) - 2. 8 1 1 8\times 10^{ - 4} (\mu /\sigma )^{ 2} - 1. 1 8 2 6\times 10^{ - 3} (\mu /\sigma )^{ 3} + 5. 60 8 4\times 10^{ - 5} (\mu /\sigma )^{ 4} \quad(r = 1). $$

(19b)

For a lognormal distribution,

$$ H_{0. 1} = 1. 1 5 3 9\times 10^{ - 2} + 7. 5 9 3 3\times 10^{ - 2} (\mu /\sigma ) + 6. 6 8 3 8\times 10^{ - 4} (\mu /\sigma )^{ 2} - 1. 9 1 6 9\times 10^{ - 4} (\mu /\sigma )^{ 3} + 3. 9 20 1\times 10^{ - 6} (\mu /\sigma )^{ 4} \quad(r = 0. 9 9 9 9 8), $$

(20a)

$$ H_{0. 2} = 2. 2 6 6\times 10^{ - 2} + 0. 1 5 6 2 9(\mu /\sigma ) + 4. 4 4 2\times 10^{ - 4} (\mu /\sigma )^{ 2} - 1. 2 7 3 8\times 10^{ - 3} (\mu /\sigma )^{ 3} + 5. 9 9 7 8\times 10^{ - 5} (\mu /\sigma )^{ 4} \quad(r = 0. 9 9 9 9 6). $$

(20b)

Gamma distribution

The density distribution of a gamma distribution is defined by

$$ f (x )= \left\{ {\begin{array}{ll} {{\frac{{\beta^{ - \alpha } }}{\Upgamma (\alpha )}}x^{\alpha - 1} {\text{e}}^{ - x/\beta } ,\quad{\text{ if }}x \, > 0 ,} \\ {0,\quad{\text{ if }}x \, \le 0.} \\ \end{array} } \right. $$

(21)

Its mean and variance are

$$ \mu = \alpha \beta ,\,\sigma^{2} = \alpha \beta^{2} . $$

(22)

Here, the gamma function Γ(α) in Eq. 21 is expressed as $\Upgamma (\alpha )= \int_{ \, 0}^{ \, \infty } {t^{\alpha - 1} {\text{e}}^{ - t}\,{\text{d}}t} .$ The probability distribution function for the gamma distribution is F(x) = 0 for x ≤ 0 and $F(x)=\gamma ({\frac{x}{\beta }};\alpha )$ for x > 0. Here, γ is the incomplete gamma function, $\gamma (x;\alpha )= {\frac{1}{\Upgamma (\alpha )}}\int_{ \, 0}^{ \, x} {t^{\alpha - 1} {\text{e}}^{ - t}\,{\text{d}}t} ,$ whose value can be found in statistical references or online programs.

From Eq. 22, $\mu /\sigma = \sqrt \alpha ,$ thus $\alpha = (\mu /\sigma )^{2} .$ Hence, the homogeneity

$$ H_{ 0. 1} = F (1.1\mu )- F (0.9\mu )= \gamma (1.1\alpha ;\alpha )- \gamma (0.9\alpha ;\alpha )= \gamma [1.1 (\mu /\sigma )^{2} ; (\mu /\sigma )^{2} ]- \gamma [0.9 (\mu /\sigma )^{2} ; (\mu /\sigma )^{2} ] , $$

(23a)

$$ H_{ 0. 2} = F (1.2\mu )- F (0.8\mu ) = \gamma (1.2\alpha ;\alpha )- \gamma (0.8\alpha ;\alpha )= \, \gamma [1.2 (\mu /\sigma )^{2} ; (\mu /\sigma )^{2} ]- \gamma [0.8 (\mu /\sigma )^{2} ; (\mu /\sigma )^{2} ]. \, $$

(23b)

From these two equations, H _0.1 and H _0.2 are functions of a single variable μ/σ. For convenience, they are regressed as

$$ H_{0. 1} = - 8. 1 3 7 8\times 10^{ - 3} + 8. 2 5 7 6\times 10^{ - 2} (\mu /\sigma ) - 1. 3 3 5\times 10^{ - 4} (\mu /\sigma )^{ 2} - 1. 7 3 8 8\times 10^{ - 4} (\mu /\sigma )^{ 3} + 5. 3 40 8\times 10^{ - 6} (\mu /\sigma )^{ 4} \quad(r = 0. 9 9 9 9 9), $$

(24a)

$$ H_{0. 2} = - 1. 70 3 9\times 10^{ - 2} + 0. 1 6 8 4 4(\mu /\sigma ) - 9. 1 7 1 6\times 10^{ - 4} (\mu /\sigma )^{ 2} - 1. 2 3 4 6\times 10^{ - 3} (\mu /\sigma )^{ 3} + 6. 1 5 7 1\times 10^{ - 5} (\mu /\sigma )^{ 4} \quad(r = 0. 9 9 9 9 7). $$

(24b)

Note that the gamma distribution yields a special case of Erlang distribution when α = k is an integer, or the chi-square distribution when α = ν/2 and β = 2 [24]. As H _0.1 and H _0.2 are only related to the ratio μ/σ but not related to specific α or β, the homogeneity equations, as expressed in Eqs. 23a, 23b and 24a, 24b, remain valid for both Erlang and chi-square distributions.

Weibull distribution

The density distribution of a Weibull distribution is defined by:

$$ f (x )= \left\{ {\begin{array}{ll} {\alpha \beta^{ - \alpha } x^{\alpha - 1} {\text{e}}^{{ - (x/\beta )^{\alpha } }} ,\quad{\text{ if }}x \, > 0 ,} \\ {0,\quad{\text{ if }}x \, \le 0.} \\ \end{array} } \right. $$

(25)

Its mean and variance are

$$ \mu = \beta \, \Upgamma \left({\frac{\alpha + 1}{\alpha }} \right) ,\,\sigma^{2} = \beta^{2} \, \left[ {\Upgamma \left({\frac{\alpha + 2}{\alpha }} \right)- \Upgamma^{2} \left({\frac{\alpha + 1}{\alpha }} \right)} \right]. $$

(26)

Thus, we have

$$ (\mu /\sigma )= {\frac{{\Upgamma \left( {{\frac{\alpha + 1}{\alpha }}} \right)}}{{\sqrt {\Upgamma \left( {{\frac{\alpha + 2}{\alpha }}} \right) - \Upgamma^{2} \left( {{\frac{\alpha + 1}{\alpha }}} \right)} }}},\,{\text{or}}\,{\frac{{1 + (\mu /\sigma )^{2} }}{{ (\mu /\sigma )^{2} }}} = {\frac{{\Upgamma \left( {{\frac{\alpha + 2}{\alpha }}} \right)}}{{\Upgamma^{2} \left( {{\frac{\alpha + 1}{\alpha }}} \right)}}}. $$

(27)

From Eq. 27, the relationship between α and (μ/σ) can be established, and their relationship is expressed as:

$$ \alpha = 8. 8 9 4 7\times 10^{ - 2} + 0. 8 4 3 4(\mu /\sigma ) + 0. 10 30 6(\mu /\sigma )^{ 2} - 1.0 6 8 3\times 10^{ - 2} (\mu /\sigma )^{ 3} + 4.00 4 5\times 10^{ - 4} (\mu /\sigma )^{ 4} \quad(r = 1). $$

(28)

The probability distribution function for the Weibull distribution is F(x) = 0 for x ≤ 0 and $F(x) = 1 - {\text {e}}^{{ - (x/\beta )^{\alpha } }}$ for x > 0. Therefore, the homogeneity degree of the Weibull distribution is:

$$ H_{0. 1} = F (1.1\mu )- F (0.9\mu ) = {\text{e}}^{{ - (0.9\mu /\beta )^{\alpha } }} - {\text{e}}^{{ - (1.1\mu /\beta )^{\alpha } }} = {\text{e}}^{{ - \left[ {0.9{{\Upgamma}}\left( {{\frac{\alpha + 1}{\alpha }}} \right)} \right]^{{_{\alpha } }} }} - {\text{e}}^{{ - \left[ {1.1{{\Upgamma}}\left( {{\frac{\alpha + 1}{\alpha }}} \right)} \right]^{{_{\alpha } }} }} , $$

(29a)

$$ H_{0. 2} = F (1.2\mu )- F (0.8\mu )= {\text{e}}^{{ - (0.8\mu /\beta )^{\alpha } }} - {\text{e}}^{{ - (1.2\mu /\beta )^{\alpha } }} = {\text{e}}^{{ - \left[ {0.8\Upgamma \left( {{\frac{\alpha + 1}{\alpha }}} \right)} \right]^{\alpha } }} - {\text{e}}^{{ - \left[ {1.2\Upgamma \left( {{\frac{\alpha + 1}{\alpha }}} \right)} \right]^{\alpha } }} . $$

(29b)

Here, $\mu /\beta = \, \Upgamma \left( {{\frac{\alpha + 1}{\alpha }}} \right)$ according to Eq. 26. For a given μ/σ, the value α can be evaluated from Eq. 28, and thus H _0.1 and H _0.2 can be calculated. For convenience, these equations are regressed as:

$$ H_{0. 1} = 2.0 5 4 9\times 10^{ - 3} + 6. 9 9 5 4\times 10^{ - 2} (\mu /\sigma ) + 2. 60 8 7\times 10^{ - 3} (\mu /\sigma )^{ 2} - 3. 6 6 3\times 10^{ - 4} (\mu /\sigma )^{ 3} + 1.0 20 6\times 10^{ - 5} (\mu /\sigma )^{ 4} \quad(r = 1), $$

(30a)

$$ H_{0. 2} = 5. 9 7 5\times 10^{ - 3} + 0. 1 3 70 9(\mu /\sigma ) + 7. 1 5 4 2\times 10^{ - 3} (\mu /\sigma )^{ 2} - 1. 8 9 8 2\times 10^{ - 3} (\mu /\sigma )^{ 3} + 7. 6 2 6 8\times 10^{ - 5} (\mu /\sigma )^{ 4} \quad(r = 1). $$

(30b)

Note that the Weibull distribution yields a special case, Rayleigh distribution, by taking α = 2 and $\beta = \sqrt 2 \eta$ [24]. As H _0.1 and H _0.2 are not related to specific α or β values in these H equations, they can also be used for the homogeneity quantification of the Rayleigh distribution.

Beta distribution

The beta distribution is defined as:

$$ f (x )= {\frac{1}{B (\alpha ,\beta )}}x^{\alpha - 1} (1 - x )^{\beta - 1} ,\quad0 < x < 1, $$

(31)

where α and β are two independent fitting parameters, and B(α,β)=Γ(α)Γ(β)/Γ(α+β). Its mean $\mu = \alpha / (\alpha + \beta )$ and variance $\sigma^{2} = \alpha \beta / [ (\alpha + \beta )^{ 2} (\alpha + \beta + 1 ) ] ,$ and thus $(\mu /\sigma )^{2} = \alpha (\alpha + \beta + 1 ) /\beta .$ For a given pair of α and β, particular H values can be computed numerically by integrating $f(x)$ in Eq. 31 over the range of μ ± 0.1μ or μ ± 0.2μ. Figure 8 shows some of the computation results of the beta distribution for 0 < α, β ≤ 10, as plotted as H versus $(\mu /\sigma )= \sqrt {\alpha (\alpha + \beta + 1 ) /\beta } .$ The normal distribution is included for comparison. Different to the previous distributions, the homogeneity parameters H _0.1 and H _0.2 here are functions of the ratio μ/σ as well as α (or β), i.e., for a given μ/σ, H _0.1, and H _0.2 are no longer monotonic but may possess different values depending on α or β. Only when α = β, H _0.1 and H _0.2 are monotonic functions of μ/σ, which as expressed as:

$$ \begin{gathered} H_{0. 1} \, = \, - 0. 1 9 3 9 9+ 0. 2 5 5 7(\mu /\sigma ) - 0.0 7 2 2 2 6(\mu /\sigma )^{ 2} + 1. 5 50 3\times 10^{ - 2} (\mu /\sigma )^{ 3} - 1. 8 4 1 1\times 10^{ - 3} (\mu /\sigma )^{ 4} + 1. 1 1 6 8\times 10^{ - 4} (\mu /\sigma )^{ 5} - \hfill \\ 2. 7 1 9 4\times 10^{ - 6} (\mu /\sigma )^{ 6} \quad(r \, = \, 0. 9 9 9 9 9), \hfill \\ \end{gathered} $$

(32a)

$$ \begin{gathered} H_{0. 2} \, = \, - 0. 3 9 2 3+ 0. 5 1 7 2 2(\mu /\sigma ) - 0. 1 4 4 8 9(\mu /\sigma )^{ 2} + 3.00 5 4\times 10^{ - 2} (\mu /\sigma )^{ 3} - 3. 6 4 2 6\times 10^{ - 3} (\mu /\sigma )^{ 4} + 2. 2 6 4 1\times 10^{ - 4} (\mu /\sigma )^{ 5} - \hfill \\ 5. 60 4\times 10^{ - 6} (\mu /\sigma )^{ 6} \quad(r \, = \, 0. 9 9 9 9 8). \hfill \\ \end{gathered} $$

(32b)

Fig. 8

Homogeneity quantities H _0.1 and H _0.2 of the beta distribution, as compared with the normal distribution. Each dot corresponds to a pair of (α, β) values (not shown), while a special case with α = β is highlighted

As shown in Fig. 8, the H values with α = β are highlighted, which are getting close to the normal distribution at higher ratio μ/σ.

Moreover, the homogeneity H functions of the uniform, exponential, Laplace, Pareto, or any other distributions [24] can be formularized from their specific integration forms, while these models are not commonly used for the size distribution modeling.

Appendix 2: Orientation distribution

Circular data

Wrapped normal distribution

The density distribution of the wrapped normal distribution is described as

$$ f (\theta )= {\frac{1}{2\pi }} + {\frac{1}{\pi }}\sum\limits_{p = 1}^{\infty } {\rho^{{p^{2} }} \cos [p (\theta - \mu ) ]} , \quad 0 \le \, \theta \, < 2\pi , \,\, 0 \le \, \rho \, \le 1. $$

(33)

Its mean is μ, and mean resultant length is ρ. By numerical computation, the relationship of H _R, H _0.1, and H _0.2 with ρ are obtained, as shown in Fig. 9, which are expressed as follows:

$$ H_{\text{R}} = \rho , $$

(34a)

$$ H_{0. 1} \, = \, 0. 100 1 1+ 0.0 4 7 10 9\rho + 4. 4 8 8 6\rho^{ 2} - 4 5. 2 6 7\rho^{ 3} + 2 1 6.0 3\rho^{ 4} - 5 4 9. 1 6\rho^{ 5} + 7 6 6. 2 8\rho^{ 6} - 5 5 3. 3 7\rho^{ 7} + 1 6 1. 9 2\rho^{ 8} \quad \left( {r \, = \, 0. 9 9 9 9 5} \right), $$

(34b)

$$ H_{0. 2} \, = \, 0. 1 9 9 7 9+ 0. 6 2 9 5\rho - 7. 1 1 5 2\rho^{ 2} + 6 7. 7 3 8\rho^{ 3} - 30 5. 8 5\rho^{ 4} + 7 3 8. 8 9\rho^{ 5} - 9 7 7. 1 4\rho^{ 6} + 6 6 7. 1 8\rho^{ 7} - 1 8 3. 5 2\rho^{ 8} \quad \left( {r \, = \, 0. 9 9 9 9 7} \right). $$

(34c)

Fig. 9

Homogeneity H of wrapped normal distribution

Wrapped Cauchy distribution

The density distribution of the wrapped Cauchy distribution, with mean direction μ and mean resultant length ρ, is described as:

$$ f (\theta )= {\frac{1}{2\pi }}{\frac{{1 - \rho^{2} }}{{1 + \rho^{2} - 2\rho \cos (\theta - \mu )}}}, \quad 0 \le \, \theta \, < 2\pi , \,\, 0 \le \, \rho \, \le 1. $$

(35)

The relationships of H _R, H _0.1 and H _0.2 with ρ are shown in Fig. 10, which are expressed as follows:

$$ H_{\text{R}} = \rho , $$

(36a)

$$ H_{0. 1} \, = \, 0. 1+ 0. 2 5 4 4 7\rho - 0. 5 7 7 2 8\rho^{ 2} + 4. 1 1 1 5\rho^{ 3} - 9. 2 6 5 4\rho^{ 4} + 10. 6 1 7\rho^{ 5} - 4. 2 3 9 2\rho^{ 6} \quad(r \, = \, 0. 9 9 9 9 9), $$

(36b)

$$ H_{0. 2} \, = \, 0. 2+ 0. 3 6 2 7 4\rho + 0. 4 5 3 2 9\rho^{ 2} - 0. 5 6 5 5 1\rho^{ 3} + 1. 8 8 3 6\rho^{ 4} -0. 8 6 9 2\rho^{ 5} + 0. 5 3 4 7 9\rho^{ 6}\quad \left( {r \, = 1} \right). $$

(36c)

Fig. 10

Homogeneity H of wrapped Cauchy distribution

von Mises distribution

The widely used von Mises distribution, with mean direction μ, is described by a density function of

$$ f (\theta )= {\frac{1}{{2\pi {\kern 1pt} I_{0} (\kappa )}}}{\text{e}}^{\kappa\cos (\theta - \mu )} , \quad 0 \le \theta < 2\pi , \,\, 0 \le \kappa < \infty , $$

(37)

where I ₀ denotes the modified Bessel function of the first kind, with order 0 (p = 0). In general with the order p, it is defined by

$$ I_{p} (\kappa )= {\frac{1}{2\pi }}\int\limits_{0}^{2\pi } {\cos p\theta \;{\text{e}}^{\kappa \cos \theta } {\text{d}}\theta } ,\;{\text{or}}\;I_{p} (\kappa )= \sum\limits_{r = 0}^{\infty } {{\frac{1}{\Upgamma (p + r + 1 )\Upgamma (r + 1 )}}} \left( {{\frac{\kappa }{2}}} \right)^{2r + p} . $$

(38)

Its mean resultant length is $\rho = I_{1} (\kappa )/I_{0} (\kappa ).$ According to these relationships, H _R, H _0.1, and H _0.2 are numerically computed, as shown in Fig. 11. In the range of κ ≤ 20 that covers most practical values, they are expressed as:

$$ \begin{gathered} H_{\text{R}} = \rho = \, 0. 6 20 2 1\kappa - 0. 1 8 6 40\kappa^ 2 + 3. 1 4 7 7\times 10^{ - 2} \kappa^ 3 - 3. 1 7 7\times 10^{ - 3} \kappa^{ 4} + 1. 9 4 3 3\times 10^{ - 4} \kappa^{ 5} - 6. 9 9 1 4\times 10^{ - 6} \kappa^{ 6} + 1. 3 4 1 7\times 10^{ - 7} \kappa^{ 7} - \hfill \\ 1.0 3 5 5\times 10^{ - 9} \kappa^{ 8} \quad(r \, = \, 0. 9 9 9 5 5), \hfill \\ \end{gathered} $$

(39a)

$$ \begin{gathered} H_{{0. 1 }} = \, 0. 1+ 0. 1 2 2 6 8\kappa - 3. 5 6\times 10^{ - 3} \kappa^{ 2} - 3. 1 7 2 5\times 10^{ - 3} \kappa^{ 3} + 7. 3 7 9 4\times 10^{ - 4} \kappa^{ 4} - 7. 7 4 7 5\times 10^{ - 5} \kappa^{ 5} + 4. 3 6 6 5\times 10^{ - 6} \kappa^{ 6} - \hfill \\ 1. 2 7 8 2\times 10^{ - 7} \kappa^{ 7} + 1. 5 2 7\times 10^{ - 9} \kappa^{ 8} \quad(r \, = \, 0. 9 9 9 9 6), \hfill \\ \end{gathered} $$

(39b)

$$ \begin{gathered} H_{{0. 2 }} = \, 0. 2+ 0. 2 3 8 3 7\kappa - 2. 1 7 4 1\times 10^{ - 2} \kappa^{ 2} - 3. 3 7 6 1\times 10^{ - 3} \kappa^{ 3} + 1. 1 1 3 4\times 10^{ - 3} \kappa^{ 4} - 1. 2 6 8 3\times 10^{ - 4} \kappa^{ 5} + 7. 40 5 3\times 10^{ - 6} \kappa^{ 6} - \hfill \\ 2. 2 10 5\times 10^{ - 7} \kappa^{ 7} + 2. 6 7 4\times 10^{ - 9} \kappa^{ 8} \quad(r \, = \, 0. 9 9 9 7 8). \hfill \\ \end{gathered} $$

(39c)

Fig. 11

Homogeneity H of von Mises distribution

Cardioid distribution

The cardioid distribution, with a mean direction at μ, has the density function as follows:

$$ f (\theta )= {\frac{1}{2\pi }} [1 + 2\rho \cos (\theta - \mu ) ], \quad 0 \le \, \theta \, < 2\pi , \,\, \left| \rho \right| \le {\frac{1}{2}}. $$

(40)

Its mean resultant length is ρ, thus

$$ H_{\text{R}} = \rho . $$

(41a)

According to Eq. 40, the density function at ρ = 0.5, 0, −0.5 are shown in Fig. 12a. When ρ = 0.5, the maximum f is at θ = 0, so μ = 0. Therefore, H _0.1 and H _0.2 include areas on the right side of θ = 0 and left side of θ = 2π, as shown in the shadowed areas in Fig. 12a. While when ρ = −0.5, H _0.1 and H _0.2 are obtained from the areas with μ = π. Along with H _R, the calculated results of H _0.1 and H _0.2 are shown in Fig. 12b, which are computed as:

$$ H_{0. 1} = \, 0. 1 + \, 0. 1 9 6 7 3 \left| \rho \right| \quad(r \, = 1), $$

(41b)

$$ H_{0. 2} = \, 0. 2 + \, 0. 3 7 4 1 9 6\left| \rho \right|\quad(r \, = 1). $$

(41c)

Fig. 12

PDF (a) and homogeneity H (b) of cardioid distribution

Triangle distribution

The triangle distribution, with mean direction at μ, has the density function as follows [33]:

$$ f\left( \theta \right) = {\frac{1}{8\pi }}\left( {4 - \pi^{2} \rho + 2\pi \rho |\pi - \theta } \right),\quad0 \le \, \theta \, < 2\pi ,\,\,0 \le \rho \le {\frac{4}{{\pi^{2} }}}. $$

(42)

From this function, it is computed that

$$ H_{\text{R}} = \, \rho , \, $$

(43a)

$$ H_{0. 1} = \, 0. 1 + \, 0. 2 2 20 9\rho \quad(r \, = 1), $$

(43b)

$$ H_{0. 2} = \, 0. 2 + \, 0. 3 9 5 2 2\rho \quad(r \, = 1). $$

(43c)

For other distribution models, H _R, H _0.1, and H _0.2 then can be computed in similar ways. Even when the ρ of a model is not available, for example, the nonsymmetrical models given by Fu and Lauke [8], the H _0.1 and H _0.2 can be calculated numerically.

Spherical data

Fisher distribution

The density function of Fisher distribution [35], with mean direction along the z axis, is expressed as:

$$ f (\theta ,\varphi )= (\kappa /4\pi { \sinh }\kappa ) {\text{exp(}}\kappa { \cos }\theta ) {\text{sin}}\theta , $$

(44)

where κ is the concentration parameter, which controls the curved shape, as plotted for some values in Fig. 13a. Its

$$ H_{\text{R}} = \rho = { \coth }\kappa - 1/\kappa . $$

(45a)

Accordingly, the H values are calculated for different κ (Fig. 13b), and regressed as:

$$ \begin{gathered} H_{0. 1} \, = 1. 2 9 9 4\times 10^{ - 2} + 4. 4 1 6 3\times 10^{ - 2} \kappa - 5.0 1 8 9\times 10^{ - 4} \kappa^{ 2} - 2. 6 1 7 7\times 10^{ - 5} \kappa^{ 3} + 1. 3 5 1 8\times 10^{ - 6} \kappa^{ 4} - 2. 90 2 8\times 10^{ - 8} \kappa^{ 5} + 3. 3 3 5\times 10^{ - 10} \kappa^{ 6} - \hfill \\ 1. 9 9 1 8\times 10^{ - 1 2} \kappa^{ 7} + 4. 8 5 3 3\times 10^{ - 1 5} \kappa^{ 8} \quad \left( {r \, = \, 0. 9 9 9 9 9} \right), \hfill \\ \end{gathered} $$

(45b)

$$ \begin{gathered} H_{0. 2} \, = \, 0.0 60 2 4 1+ 0. 1 5 8 7 6\kappa - 1. 1 7 7 5\times 10^{ - 2} \kappa^{ 2} + 4. 8 7 5 7\times 10^{ - 4} \kappa^{ 3} - 1. 2 1 6 3\times 10^{ - 5} \kappa^{ 4} + 1. 8 6 3 7\times 10^{ - 7} \kappa^{ 5} - 1. 7 1 3 4\times 10^{ - 9} \kappa^{{ 6 }} + \hfill \\ 8. 6 6 2 1\times 10^{ - 1 2} \kappa^{ 7} - 1. 8 4 9 5\times 10^{ - 1 4} \kappa^{ 8} \quad \left( {r \, = \, 0. 9 9 9 8 7} \right). \hfill \\ \end{gathered} $$

(45c)

Fig. 13

PDF (a) and homogeneity H (b) of Fisher distribution

Watson distribution

The density function of Watson distribution [35], with mean direction along the z-axis, is expressed as

$$ f (\theta ,\varphi )= C_{\text{W}} {\text{exp(}}\kappa { \cos }^{2} \theta ) {\text{sin}}\theta , $$

(46)

where $C_{\text{W}} = 1/ [4\pi \int_{ \, 0}^{ \, 1} {{\text{exp(}}\kappa u^{2} ) {\text{d}}u} ]$. When κ ≥ 0, the distribution f is plotted in Fig. 14a, with H _0.1 and H _0.2 areas indicated. As the another half at the supplementary angle (π − θ) in not included, the maximum H value is 0.5. Its ρ is not available. However, the H quantities are calculated numerically as shown in Fig. 14b for 0 ≤ κ ≤ 40, which are expressed as:

$$ \begin{gathered} H_{0. 1} \, = 2. 4 6 9\times 10^{ - 2} + 1. 2 7 5 4\times 10^{ - 2} \kappa + 7.0 8 6 1\times 10^{ - 3} \kappa^{ 2} - 1. 1 1 1 9\times 10^{ - 3} \kappa^{ 3} + 8. 2 3 9 2\times 10^{ - 5} \kappa^{ 4} - 3. 50 1 3\times 10^{ - 6} \kappa^{ 5} + \hfill \\ 8. 6 5 8 7\times 10^{ - 8} \kappa^{ 6} - 1. 1 5 7 3\times 10^{ - 9} \kappa^{ 7} + 6. 4 60 9\times 10^{ - 1 2} \kappa^{ 8} \, \left( {r \, = 1} \right), \hfill \\ \end{gathered} $$

(47a)

$$ \begin{gathered} H_{0. 2} \, = 9. 5 3 4 5\times 10^{ - 2} + 4. 3 9 6 4\times 10^{ - 2} \kappa + 1. 4 3 1 8\times 10^{ - 2} \kappa^{ 2} - 3. 9 4 3 5\times 10^{ - 3} \kappa^{ 3} + 4. 30 4 7\times 10^{ - 4} \kappa^{ 4} - 2. 6 1 9 3\times 10^{ - 5} \kappa^{ 5} + \hfill \\ 9. 5 6 5 3\times 10^{ - 7} \kappa^{ 6} - 2.0 8 4 4\times 10^{ - 8} \kappa^{ 7} + 2. 500 2\times 10^{ - 10} \kappa^{ 8} - 1. 2 70 9\times 10^{ - 1 2} \kappa^{ 9} \quad \left( {r = \, 0. 9 9 9 9 7} \right). \hfill \\ \end{gathered} $$

(47b)

Fig. 14

PDF (a) and homogeneity H (b) of Watson distribution

However when κ < 0, as plotted in Fig. 14a, the f maximum value is at 0.5π, i.e., at the equator, which is the girdle case. It is more reasonable to define the homogeneity, denoted as H′, around 0.5π, as shown in Fig. 14a. Within −40 ≤ κ < 0, the H′ is expressed as:

$$ H_{0.1}^{'} = \, 0. 1 5 3 9 1- 5. 8 3 3\times 10^{ - 2} \kappa - 3. 2 5 1 7\times 10^{ - 3} \kappa^{ 2} - 1. 3 3 8 4\times 10^{ - 4} \kappa^{ 3} - 3. 4 2 8 6\times 10^{ - 6} \kappa^{ 4} - 4. 7 8 2 7\times 10^{ - 8} \kappa^{ 5} - 2. 7 6 3 4\times 10^{ - 10} \kappa^{{ 6 }} \quad(r \, = 1), $$

(48a)

$$ H_{0.2}^{'} = \, 0. 30 50 1- 0. 10 6 3 9\kappa - 8. 3 4 1 5\times 10^{ - 3} \kappa^{ 2} - 3. 9 3 2 7\times 10^{ - 4} \kappa^{ 3} - 1.0 9 8 3\times 10^{ - 5} \kappa^{ 4} - 1. 6 5 7 2\times 10^{ - 7} \kappa^{ 5} - 1.0 3 5 9\times 10^{ - 9} \kappa^{ 6} \quad(r \, = \, 0. 9 9 9 9 9). $$

(48b)