Build a lazy version of a Markov chain

Constructs the "lazy" chain associated with a markovchain object: at every step, stay put with probability alpha and otherwise take a step of the original chain.

Usage

lazyChain(object, alpha = 0.5)

# S4 method for class 'markovchain'
lazyChain(object, alpha = 0.5)

Arguments

object: A markovchain object.
alpha: A single number in $[0,1]$, the probability of staying in the current state at each step. The default, 0.5, matches the usual textbook "lazy random walk" construction.

Value

A new markovchain object with transition matrix $L = \alpha I + (1-\alpha) P$, the same states, and the same row/column-stochastic storage convention (byrow) as object.

Details

For a transition matrix $P$ (in whichever storage convention object already uses) and a laziness parameter $\alpha\in[0,1]$, the lazy chain's transition matrix is $$L = \alpha I + (1-\alpha) P.$$

Laziness is a standard device for forcing aperiodicity without changing where the chain can go or its stationary distribution:

$L$ has the *same* stationary distribution as $P$ (if $\pi P=\pi$ then $\pi L = \alpha\pi + (1-\alpha)\pi P = \pi$), and the same communicating classes, since $L_{ij}>0 \iff P_{ij}>0$ for $i\ne j$.
For $0<\alpha<1$, $L$ is aperiodic even if $P$ is periodic, because $L_{ii}=\alpha>0$ for every state $i$ rules out any period greater than $1$. This is why mixingTime, which requires aperiodicity, is often applied to lazyChain(object) rather than to a periodic object directly (see mixingTime's own documentation for why it rejects periodic chains outright rather than lazifying them automatically).
Every non-trivial eigenvalue of $L$ is $\alpha + (1-\alpha)\lambda$ for the corresponding eigenvalue $\lambda$ of $P$: laziness shrinks the whole non-trivial spectrum towards $\alpha$, so slem and impliedTimescales generally get *worse* (mixing gets slower) as alpha increases towards 1.

$\alpha=0$ returns $P$ unchanged; $\alpha=1$ returns the identity matrix (a chain that never moves).

References

Levin, D. A. and Peres, Y. (2017). Markov Chains and Mixing Times, 2nd edition. American Mathematical Society.

Examples

# A 2-cycle is periodic (period 2); its lazy version is aperiodic.
statesNames <- c("a", "b")
cycle2 <- new("markovchain", states = statesNames,
  transitionMatrix = matrix(c(0, 1, 1, 0), byrow = TRUE, nrow = 2,
                            dimnames = list(statesNames, statesNames)))
period(cycle2)
#> [1] 2
lazyCycle2 <- lazyChain(cycle2, alpha = 0.5)
period(lazyCycle2)
#> [1] 1
steadyStates(cycle2)
#>        a   b
#> [1,] 0.5 0.5
steadyStates(lazyCycle2) # unchanged by laziness
#>        a   b
#> [1,] 0.5 0.5