Problem 44933. Vogel-Dobbener entropy
Vogel-Dobbener entropy is a measure of dispersion for ordinal variables.
Given an ordered list of distinct observations u_(1), ..., u_(k) with observed relative frequencies f_1, ..., f_k, the Vogel-Dobbener entropy is defined as
VD = -(sum_{i=1}^{k-1} (F_i log2(F_i) + (1 - F_i) log2(1 - F_i))
where F_1, ... F_k are the cumulative relative frequences, i.e.
F_i = sum_{j=1}^i f_i
The Vogel-Dobbener entropy of a sample satisfies 0 <= VD <= (k - 1); the normalized Vogel-Dobbener entropy is thus defined as
VD* = VD / (k - 1)
For example, suppose that your sample is [2.7 3.3 2.0 3.3 1.7 3.7]. Then:
- k = 5 (there are five distinct observations);
- u_(1), ..., u_(k) = [1.7 2.0 2.7 3.3 3.7]; (note that 3.3 only appears once);
- f_1, ..., f_k = [1/6 1/6 1/6 2/6 1/6];
- F_1, ..., F_k = [1/6 2/6 3/6 5/6 6/6];
- VD = 3.2183; (approx.)
- VD* = 0.8046. (approx.)
Your task is to write a function that, given a list of observations (unordered and possibly containing duplicates) computes the normalized Vogel-Dobbener entropy VD* of the sample. Round to four decimal digits.
Hint: if all observations in the sample are the same, then k = 1, the sum in the definition of VD is empty, and VD equals zero.
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