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Normalized entropy rate of a Markov chain

Source: R/dynamics.R
normalizedEntropyRate.Rd

The entropy rate of a finite irreducible Markov chain divided by the topological entropy of its graph, a measure of how random the chain is relative to the most random chain with the same possible transitions.

Usage

normalizedEntropyRate(object)

# S4 method for class 'markovchain'
normalizedEntropyRate(object)

Arguments

object

A markovchain object representing a finite, irreducible discrete-time Markov chain.

Value

A numeric scalar in \([0,1]\): 0 for deterministic dynamics, 1 when the entropy rate reaches the topological entropy.

Details

The ratio \(H / h_{top}\) does not depend on the logarithm base, so no base argument is needed. By the variational principle (topologicalEntropy) it never exceeds one; tiny excursions due to round-off are clipped to \([0,1]\). When \(h_{top}=0\) the chain has a single possible path, hence entropy rate zero, and the ratio is defined here to be 0, the convention used by PyDTMC's entropy_rate_normalized.

Irreducibility is required, as in entropyRate, which is what makes the entropy rate well defined.

See also

entropyRate, topologicalEntropy

Examples

statesNames <- c("a", "b")
mc <- new("markovchain",
  states = statesNames,
  transitionMatrix = matrix(c(0.7, 0.3, 0.1, 0.9),
    byrow = TRUE, nrow = 2,
    dimnames = list(statesNames, statesNames)))
normalizedEntropyRate(mc)
#> [1] 0.5720694