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Tsallis Entropy

Source: R/eeg-complexity.R
tsallisEntropy.Rd

Computes the Tsallis entropy of order q of a one-dimensional signal (Tsallis, 1988) – the non-additive one-parameter generalization of the Shannon entropy of the signal's amplitude distribution, complementary to the (additive) renyiEntropy. The signal is discretized into bins equal-width bins to form a probability distribution \(p\), and \(S_q = \frac{1 - \sum_i p_i^q}{q - 1}\) is returned (equivalently \(\sum_i p_i \ln_q(1/p_i)\) with the q-logarithm). \(q \to 1\) recovers the Shannon entropy (in nats); unlike Renyi, Tsallis is non-additive (\(S_q(A,B) = S_q(A) + S_q(B) + (1-q) S_q(A) S_q(B)\) for independent \(A, B\)), so Renyi and Tsallis agree only at \(q = 1\).

Usage

tsallisEntropy(x, q = 2, bins = 16L)

Arguments

x

A numeric vector (the time series).

q

Order of the Tsallis entropy (default 2); q = 1 returns the Shannon entropy.

bins

Number of equal-width bins for the amplitude histogram (default 16).

Value

A single numeric value, the Tsallis entropy.

References

Tsallis, C. (1988). "Possible generalization of Boltzmann-Gibbs statistics." Journal of Statistical Physics, 52(1-2), 479–487. doi:10.1007/BF01016429

See also

renyiEntropy for the additive generalization, svdEntropy, petrosianFD.

Examples

set.seed(1)
tsallisEntropy(rnorm(1000), q = 2)   # non-additive (collision-like)
#> [1] 0.883062
tsallisEntropy(rnorm(1000), q = 1)   # = Shannon entropy
#> [1] 2.300707