Regularity

The Regularity

Mathematical Definition

The regularity can be calculated

from Congeneric Intervals Chains

Let \(CIC\) is Congenerics Intervals Chains defined as matrix

\[ CIC = \begin{pmatrix} \Delta_{1,1} & \Delta_{1,2} & \cdots & \Delta_{1,l} \\ \Delta_{2,1} & \Delta_{2,2} & \cdots & \Delta_{2,l} \\ \vdots & \vdots & \ddots & \vdots \\ \Delta_{m,1} & \Delta_{m,2} & \cdots & \Delta_{m,l} \end{pmatrix} \]

\[r= \sqrt[n]{\prod_{j=1}^{m} \frac{ \prod_{j=1}^{l} \Bigg\{\begin{array}{l} \Delta_{i,j}, & \Delta_{i,j} \notin \{-\} \\ 1, & \Delta_{i,j} \in \{ - \} \end{array}} {{\left(\frac{1}{n_j} \times \sum_{i=1}^{l}{\Bigg\{\begin{array}{l} \Delta_{i,j}, & \Delta_{i,j} \notin \{-\} \\ 0, & \Delta_{i,j} \in \{ - \} \end{array}}\right)^{n_j}} } }\]

where \(m\) the power, \(n_j = \sum_{i=1}^{l}{\Bigg\{\begin{array}{l} 1, & \Delta_{i,j} \notin \{-\} \\ 0, & \Delta_{i,j} \in \{ - \} \end{array}}\) is count of non-empty elements in \(j\) congeneric intervals chain, \(\Delta_{i,j}\) the \(i\)-th element of \(j\)-th congeneric intervals chain.

\[n=\sum_{j=1}^{m}{n_j}\]

from Congenerics Intervals Distributions

Let \(CID\) is Congenerics Intervals Distributions defined as matrix

\[ CID = \begin{pmatrix} cid_{1,1} & cid_{1,2} & \cdots & cid_{1,l} \\ cid_{2,1} & cid_{2,2} & \cdots & cid_{2,l} \\ \vdots & \vdots & \ddots & \vdots \\ cid_{m,1} & cid_{m,2} & \cdots & cid_{m,l} \end{pmatrix} \]

\[r= \sqrt[n]{\prod_{j=1}^{m} \frac{ \prod_{j=1}^{l} \Bigg\{\begin{array}{l} i \times cid_{i,j}, & cid_{i,j} \neq 0 \\ 1, & cid_{i,j} = 0 \end{array}} {{\left(\frac{1}{n_j} \times \sum_{i=1}^{l}{(i \times cid_{i,j})}\right)^{n_j}} } }\]

where \(m\) the power, \(n_j = \sum_{i=1}^{l}{cid_{i,j}}\) is count of non-empty elements in \(j\) congeneric intervals distribution, \(cid_{i,j}\) the \(i\)-th element of \(j\)-th congeneric intervals distribution.

\[n=\sum_{j=1}^{m}{n_j}\]