Non-Homogeneous Poisson Process
In the NHPP Power Law Process (PLP), the scale and shape parameters can be estimated by plotting logarithm of cumulative number of failures against logarithm of running time on a simple square graph paper or simply plotting the cumulative number of failures against the running time on a logarithm paper (log-log graph). The slope of the best-fitted line gives the value of the "β" and once "β" is known, the value of "α" can be estimated.
The approach is based on PLP model where failure intensity (t) as given in Equation.
The mean value of the intensity function is
E [ N (t)] = ( t / α) β
Where N (t) is cumulative number of failures at time't' (instantaneous). {Now, generalising by introducing n (t) as cumulative number of failure}, E [n (t)] is estimated by the observed number of failures at time't' and hence, we have
ln [n (t)] = β ln t - β ln α
Above equation is of the form Y = m x + c where Slope = m = "β".
An estimate of "α" can be obtained by using the fact that if't' takes a value such that ln [n (t)] is equal to zero (i.e. ln [N (t)] = 0) equation takes the form
β ln to - β ln α = 0
β ln to = β ln α
and thus to is an estimate of "α".