meanRecurrenceTimes {smmR} | R Documentation |
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 |
klim |
Optional. The time horizon used to approximate the series in the
computation of the mean sojourn times vector |
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]}
.