Identifying Information
The Identifying Information or Entropy
Mathematical Definition
The identifying information/entropy can be calculated
from Congeneric Intervals Chains
Let \(CIC\) is Congenerics Intervals Chains defined as matrix
\[
CIC =
\begin{pmatrix}
\Delta_{1,1} & \Delta_{1,2} & \cdots & \Delta_{1,l} \\
\Delta_{2,1} & \Delta_{2,2} & \cdots & \Delta_{2,l} \\
\vdots & \vdots & \ddots & \vdots \\
\Delta_{m,1} & \Delta_{m,2} & \cdots & \Delta_{m,l}
\end{pmatrix}
\]
\[H=\frac {1} {n} * \sum_{j=1}^{m}{\Bigg(n_j \times \log_2 \bigg( \sum_{i=1}^{l}{\Bigg\{\begin{array}{l}
\Delta_{i,j}, & \Delta_{i,j} \notin \{-\} \\
0, & \Delta_{i,j} \in \{ - \}
\end{array}}\bigg) - \log_2 n_j^{n_j} \Bigg)}\]
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
\[
CID =
\begin{pmatrix}
cid_{1,1} & cid_{1,2} & \cdots & cid_{1,l} \\
cid_{2,1} & cid_{2,2} & \cdots & cid_{2,l} \\
\vdots & \vdots & \ddots & \vdots \\
cid_{m,1} & cid_{m,2} & \cdots & cid_{m,l}
\end{pmatrix}
\]
\[H=\frac {1} {n} * \sum_{j=1}^{m}{\Bigg(n_j \times \log_2 \sum_{i=1}^{l}{ \frac{i \times cid_{i,j}}{n_j}} \Bigg)}\]
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.
\[n=\sum_{j=1}^{m}{n_j}\]