On a Segment Partition for Entropy Estimation
Е. А. Тимофеев · Automatic Control and Computer Sciences · 2021
Let $${{Q}_{n}}$$ be a partition of the interval $$[0,1]$$ constructed by the rule $$\begin{array}{*{20}{c}} {{{Q}_{1}} = \{ 0,{{q}^{2}},q,1\} .} \\ {Q_{{n + 1}}^{'} = q{{Q}_{n}} \cap {{q}^{2}}{{Q}_{n}},Q_{{n + 1}}^{{''}} = {{q}^{2}} + q{{Q}_{n}} \cap q{{Q}_{n}},Q_{{n + 1}}^{{'''}} = {{q}^{2}} + q{{Q}_{n}} \cap q + {{q}^{2}}{{Q}_{n}},} \\ {{{Q}_{{n + 1}}} = Q_{{n + 1}}^{'} \cup Q_{{n + 1}}^{{''}} \cup Q_{{n + 1}}^{{'''}},} \end{array}$$ where $${{q}^{2}} + q = 1$$ . We introduce the sequence of numbers $$d = 1,2,1,0,1,2,1,0,1,0,1,2,1,0,1,2,1, \ldots $$ by setting $$\begin{array}{*{20}{c}} {{{d}_{1}} = 1,{{d}_{2}} = 2,{{d}_{4}} = 0;} \\ {d[2{{F}_{{2n}}} + 1:2{{F}_{{2n + 1}}} + 1] = d[1:2{{F}_{{2n - 1}}} + 1];} \\ {\quad n = 0,1,2, \ldots ;} \\ {d[2{{F}_{{2n + 1}}} + 2:2{{F}_{{2n + 1}}} + 2{{F}_{{2n - 2}}}] = d[2{{F}_{{2n - 1}}} + 2:2{{F}_{{2n}}}];} \\ {d[2{{F}_{{2n + 1}}} + 2{{F}_{{2n - 2}}} + 1:2{{F}_{{2n + 1}}} + 2{{F}_{{2n - 1}}} + 1] = d[1:2{{F}_{{2n - 3}}} + 1];} \\ {d[2{{F}_{{2n + 1}}} + 2{{F}_{{2n - 1}}} + 2:2{{F}_{{2n + 2}}}] = d[2{{F}_{{2n - 1}}} + 2:2{{F}_{{2n}}}];} \\ {\quad n = 1,2,3, \ldots ;} \end{array}$$ where $${{F}_{n}}$$ are the Fibonacci numbers ( $${{F}_{{ - 1}}} = 0,{{F}_{0}} = {{F}_{1}} = 1$$ ). The main result of this paper is the following: $$Q_{n}^{'} = 1 - Q_{n}^{{'''}} = \left\{ {\sum\limits_{i = 1}^k \,{{q}^{{n + {{d}_{i}}}}},k = 0,1, \ldots ,{{m}_{n}}} \right\},$$ $$Q_{n}^{{''}} = 1 - Q_{n}^{{''}} = \left\{ {{{q}^{2}} + \sum\limits_{i = {{m}_{n}}}^k \,{{q}^{{n + {{d}_{i}}}}},k = {{m}_{n}} - 1,{{m}_{n}}, \ldots ,{{m}_{{n + 1}}}} \right\},$$ where $${{m}_{{2n}}} = 2{{F}_{{2n - 2}}},{{m}_{{2n + 1}}} = 2{{F}_{{2n - 1}}} + 1$$ .