#### Installation — business terrible - 1 part

September 8th, 2015

1.2 Calibration of the Parameters of RV and GP Models

In order to have a consistent comparison between the RV and GP models, a careful scheme is required for the calibration of the model parameters. In practical situations, typically a sample of lifetime data is available from the past failures records, and the mean ( Цт ) and COV ( Vt ) of the lifetime can be estimated from such a sample.

If we assume that the random lifetime sample is generated by a RV deterioration model, the parameters of random rate can be derived in terms of the mean and COV of the lifetime using Eq.(8) as

2

П — 2 + 1/vT and 8 — —PVt 2 (12)

Цт (1 + Vt)

If however the underlying degradation model is assumed to be a stochastic gamma process, the shape and scale parameters can be obtained by solving the following equations of moments:

jdT = £ P{T > t]dt = £ GA(p; a t, ff)dt

+ vT) = 2£ tP{T > t}dt = 2£ tGA(p; a t, p)dt

The two simultaneous equations must be solved numerically to compute the parameters a and Д

In summary, in the present calibration scheme, both RV and GP models are equivalent in a sense that they have identical mean and variance of the lifetime. The parameters of these two models are given in Tables 1 and 2.

Table 1: Parameters used in the model calibration

The evolution of deterioration is compared in the two equivalent RV and GP models. Figure 1 compares the mean rate of deterioration as a function of COV of the lifetime distribution (vT) by fixing the mean lifetime and failure threshold as pT = 50 and p = 100.

The mean deterioration rates in the RV and GP model are given as pA = цЗ and af, respectively. When vT < 0.6, the mean rates in both models are almost identical. However, in cases of vT > 0.6, the GP deterioration rate accelerates much faster than that in the RV model.

Figure 2 compares the COV of deterioration, VXw , in the two models. It is a time-invariant and nonlinear function (Eq.3 and 9) of vT in the RV model. In contrast, VX w in GP model is a time-dependent parameter (Eq. 10), which is decreasing over time. Nevertheless, VX(t) of GP model is always greater than that of an equivalent RV model.

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Deterioration State X(t)

Figure 3: Probability density functions of deteriorationX(t) for vT = 0.3

The probability density function (PDF) of X(t) is given as ga(x;n, St) and ga(x; a t, в) in the RV and GP models, respectively. The evolution of the PDF with time is displayed in Figure 3 for VT = 0.3 . A key observation is that deterioration in GP model has greater variability than that in an equivalent RV model.