timeCorrelations: Time correlations and time relaxations of observed sequences

In markovchain: Easy Handling Discrete Time Markov Chains

Time correlations and time relaxations of observed sequences

Description

timeCorrelations computes the time autocorrelation of an observed sequence of states, or the time cross-correlation of two sequences, at stationarity. timeRelaxations computes how the expected value of the observable defined by a sequence evolves from a given initial distribution. They correspond to time_correlations() and time_relaxations() of PyDTMC.

Usage

timeCorrelations(object, sequence1, sequence2 = NULL, timePoints = 1)

## S4 method for signature 'markovchain'
timeCorrelations(object, sequence1, sequence2 = NULL, timePoints = 1)

timeRelaxations(object, sequence, initial = NULL, timePoints = 1)

## S4 method for signature 'markovchain'
timeRelaxations(object, sequence, initial = NULL, timePoints = 1)

Arguments

object	A markovchain object.
sequence1, sequence	A sequence of states of the chain (character vector or factor).
sequence2	An optional second sequence of states. If NULL (the default), sequence1 is used, which gives the autocorrelation.
timePoints	A vector of non-negative whole numbers, the lags at which the quantities are computed.
initial	The initial distribution: NULL (uniform, the default), a single state, or a numeric probability vector, as in redistribute.

Details

A sequence defines an observable f on the states: f_j is the number of times state j occurs in it. With f from sequence1, g from sequence2, transition matrix P and stationary distribution \pi,

\mathrm{timeCorrelations}(t) = \sum_i \pi_i f_i (P^t g)_i = E_\pi[f(X_0) g(X_t)],

and, with initial distribution \mu,

\mathrm{timeRelaxations}(t) = \mu P^t f = E_\mu[f(X_t)].

For an ergodic chain, both converge as t grows, to E_\pi[f] E_\pi[g] and to E_\pi[f] respectively, at a speed governed by the second largest eigenvalue modulus (slem).

The powers of P are applied by repeated multiplication, and by repeated squaring for long lags, never through an eigendecomposition. PyDTMC 9.0.0 switches to an eigendecomposition as soon as a lag exceeds the number of states; since its left and right eigenvectors are not biorthonormal when P has complex eigenvalues, it then returns wrong values at every lag for such chains (the tests of this function include one, whose correct values were checked with numpy).

timeCorrelations needs a unique stationary distribution, i.e. exactly one recurrent class, and stops otherwise (PyDTMC returns None). timeRelaxations is defined for every chain; unlike PyDTMC, it does not require a unique stationary distribution.

Value

A numeric vector with one value per element of timePoints, named after them.

References

Noe, F., Doose, S., Daidone, I., Loellmann, M., Sauer, M., Chodera, J. D. and Smith, J. C. (2011). Dynamical fingerprints for probing individual relaxation processes in biomolecular dynamics with simulations and kinetic experiments. Proceedings of the National Academy of Sciences, 108(12), 4822-4827.

See Also

redistribute, slem, relaxationTime

Examples

statesNames <- c("a", "b", "c")
mc <- new("markovchain", states = statesNames,
  transitionMatrix = matrix(c(0.5, 0.5, 0, 0.2, 0.3, 0.5, 0.1, 0.1, 0.8),
    byrow = TRUE, nrow = 3, dimnames = list(statesNames, statesNames)))
x <- c("a", "b", "c", "c", "c", "a")
timeCorrelations(mc, x, timePoints = 0:5)
timeRelaxations(mc, x, initial = "a", timePoints = c(0, 1, 10, 100))

markovchain documentation built on Oct. 10, 2026, 9:07 a.m.