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- W1972151365 abstract "For a family Ω of sets in R2 and a finite subset S of R2, let pΩ(S) be the number of distinct sets of the form S∩ω for all ω∈Ω. The maximum pattern complexity pΩ∗(k) is the maximum of pΩ(S) among S with #S=k. The S attaining the maximum is considered as the most effective sampling to distinguish the sets in Ω. We obtain the exact values or at least the order of pΩ∗(k) in k for various classes Ω. We also discuss the dual problem in the case that #Ω=∞, that is, consider the partition of R2 generated by a finite family T⊂Ω. The number of elements in the partition is written as pR2(T) and pR2∗(k) is the maximum of pR2(T) among T with #T=k. Here, pΩ∗(k)=pR2∗(k) does not hold in general. For the general setting that Ω is an infinite subset of AΣ, where A is a finite alphabet, Σ is an arbitrary infinite set, and pΩ∗(k)=max#S=k#Ω∣S, it is known that the entropy h(Ω):=limk→∞logpΩ∗(k)/k exists and takes value in {log1,log2,…,log#A}. In this paper, we prove that the entropy h(Σ) of the dual system coincides with h(Ω)." @default.
- W1972151365 created "2016-06-24" @default.
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- W1972151365 date "2012-10-01" @default.
- W1972151365 modified "2023-10-14" @default.
- W1972151365 title "Maximal pattern complexity, dual system and pattern recognition" @default.
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- W1972151365 doi "https://doi.org/10.1016/j.tcs.2012.06.033" @default.
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