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Evolution of a distribution over time

Source: R/dynamics.R
redistribute.Rd

Propagates an initial distribution of the states of a discrete-time Markov chain forward in time and returns the whole trajectory of distributions.

Usage

redistribute(object, steps, initial = NULL, lastOnly = FALSE)

# S4 method for class 'markovchain'
redistribute(object, steps, initial = NULL, lastOnly = FALSE)

Arguments

object

A markovchain object.

steps

A single non-negative whole number: the number of steps to propagate. With steps = 0 only the initial distribution is returned.

initial

The initial distribution. Either NULL (default), for the uniform distribution over the states; a single state name, for a point mass on that state; or a numeric vector of non-negative probabilities summing to one. A named numeric vector is matched to the states by name, an unnamed one by position.

lastOnly

Logical. If TRUE, only the distribution after steps steps is returned. Defaults to FALSE.

Value

If lastOnly = FALSE (default), a numeric matrix with steps + 1 rows and one column per state: row \(t\) (labelled "t", from "0") holds the distribution after \(t\) steps, so the first row is the initial distribution. Otherwise, a named numeric vector with the distribution after steps steps.

Details

If \(\mu_0\) is the initial distribution (a row vector) and \(P\) the row-stochastic transition matrix, the distribution after \(t\) steps is $$\mu_t = \mu_{t-1} P = \mu_0 P^t.$$

The implementation propagates the vector step by step, at a cost of \(O(n^2)\) per step for a dense \(n\)-state chain, instead of forming \(P^t\); this is what makes the whole trajectory available at no extra cost. Both row- and column-stochastic storage are supported. Each distribution is renormalized after every step to prevent round-off from accumulating over long horizons.

This mirrors PyDTMC's redistribute(), with the same defaults (uniform initial distribution, output including the initial one). For chains that converge, the rows approach the stationary distribution (see steadyStates); for periodic chains they do not, which is the expected behaviour and not an error.

See also

steadyStates, mixingTime, autoplot.markovchain

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)))
redistribute(mc, steps = 5, initial = "a")
#>         a       b
#> 0 1.00000 0.00000
#> 1 0.70000 0.30000
#> 2 0.52000 0.48000
#> 3 0.41200 0.58800
#> 4 0.34720 0.65280
#> 5 0.30832 0.69168
redistribute(mc, steps = 50, initial = c(a = 0.2, b = 0.8), lastOnly = TRUE)
#>    a    b 
#> 0.25 0.75