meanSojournTimes {smmR}

Mean Sojourn Times Function

Description

The mean sojourn time is the mean time spent in each state.

Usage

meanSojournTimes(x, states = x$states, klim = 10000)

Arguments

x

An object of S3 class smmfit or smm.

states

Vector giving the states for which the mean sojourn time should be computed. states is a subset of E.

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 sojourn 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 sojourn times vector is defined as follows:

m_{i} = E[T_{1} | Z_{0} = j] = \sum_{k \geq 0} (1 - P(T_{n + 1} - T_{n} \leq k | J_{n} = j)) = \sum_{k \geq 0} (1 - H_{j}(k)),\ i \in E

Value

A vector of length \textrm{card}(E) giving the values of the mean sojourn times for each state i \in E.