meanRecurrenceTimes {smmR}

Method to get the mean recurrence times \mu

Description

Method to get the mean recurrence times \mu.

Usage

meanRecurrenceTimes(x, klim = 10000)

Arguments

x

An object of S3 class smmfit or smm.

klim

Optional. The time horizon used to approximate the series in the computation of the mean sojourn times vector m (cf. meanSojournTimes function).

Details

Consider a system (or a component) S_{ystem} whose possible states during its evolution in time are E = \{1,\dots,s\}.

We are interested in investigating the mean recurrence times of a discrete-time semi-Markov system S_{ystem}. Consequently, we suppose that the evolution in time of the system is governed by an E-state space semi-Markov chain (Z_k)_{k \in N}. The state of the system is given at each instant k \in N by Z_k: the event \{Z_k = i\}.

Let T = (T_{n})_{n \in N} denote the successive time points when state changes in (Z_{n})_{n \in N} occur and let also J = (J_{n})_{n \in N} denote the successively visited states at these time points.

The mean recurrence of an arbitrary state j \in E is given by:

\mu_{jj} = \frac{\sum_{i \in E} \nu(i) m_{i}}{\nu(j)}

where (\nu(1),\dots,\nu(s)) is the stationary distribution of the embedded Markov chain (J_{n})_{n \in N} and m_{i} is the mean sojourn time in state i \in E (see meanSojournTimes function for the computation).

Value

A vector giving the mean recurrence time (\mu_{i})_{i \in [1,\dots,s]}.