27 condition monitoring process Jardine et al., 2001; Love and Guo, 1991; Vlok et al.,
2002. A list of applications of PHM in reliability studies can be found in Kumar 1996. The advantage of this method is that it includes both the age and the condition of the
equipment in the calculation of the hazard at time t Jardine et al., 1998. Needless to say, PHM assumes that the covariates will change the hazards that may be true in some
cases, but in reality it may be the state of the system that causes the change in the observed parameters and not vice versa, such as in the case of vibration monitoring.
This would be a significant problem if PHM were used in CBM, where the relationship between the monitored parameters and the underlying state of the system does not
follow the assumptions made in PHM.
2.8.2 Proportional Intensities Model PIM
Another approach used in modelling CBM is the proportional intensities model PIM, in which the proneness of the unit to failure can be characterized by the failure intensity.
A typical example of this type of problem is a repairable system, in which, when the failure occurs, a small part of the system can be repaired without replacing the entire
system. The system may be repaired several times before being finally replaced; therefore a system that is considered repairable cannot be modelled by the conventional
hazard function, as successive failures are not identically distributed and not independent. Instead, it can be modelled using a counting process in which the hazard
function with time-dependent covariates can be replaced with the intensity function, which may be defined in terms of the number of failures,
t N
over time ,
t . In other words, this describes the failure intensity. Wang 2000 defines the intensity function
with a baseline intensity multiplied by a multiplicative factor involving covariates, which can be written as
exp t
y t
n t
n
2-5
where
t n
is the baseline intensity and t
y is a vector of covariates. A non-
homogeneous Poisson process NHPP forms the basis of this model. Common models used to describe the baseline intensity of an NHPP include the power law model,
1
t t
n
and the exponential model,
t
e t
n
. If
,
t t
N is an NHPP
28 having intensity function
t n
, then t
N is a Poisson random variable having mean
t Z
, where t
Z is the cumulative intensity function.
t
du u
n t
Z
2-6
Consider that the system has an increasing intensity function, which tells us that failures will be likely to occur more frequently and at some time t it will become more
economical to replace the system. To find the optimum value of t ,
t
, we minimize the cost model,
1 ]
[ t
Z C
C t
t C
E
f p
2-7
where
p
C
is the cost of replacingpreventive the system.
f
C
is the cost of a failure.
Such applications can be found in Aven 1996, Ascher et al. 1995, Percy et al. 1998 and Watson et al. 2002. PIM is just an extension of PHM Kumar, 1996, which also
assumes that the covariates will influence the intensity, hence it experiences the same shortcomings as those of PHM discussed above.
2.8.3 Markov Models
