Hazard function Weibull

Figure 9.3 State probabilities and hazard functions with A = 0.5h 1, and u = 1,2, 3 and p = 0.5,1,1.5 for Erlang and Weibull distributions, respectively.

Tbe development thus far has been in terms of a hazard function approach and has been largely mathematical in nature. We now explore possible interpretations of the hazard function and the relationship to Weibull s "crack density." Without loss of generality consider the L-E approach and take as a special case a situation in which g(S) is independent of L, for example, a case of uniform tensile loading. Then the hazard function is... 

As a consequence, since the work of Bufe et al. (1977), a suite of models able to account for earthquake interaction and clustering has been introduced. We have presented those most used in practical applications such as Weibull, gamma, and lognormal and described their main features and differences in terms of their probability density functions and hazard functions. In addition we described a more physical model, that is, the Brownian passage time, which is currently used to perform time-dependent seismic hazard analysis in several areas, tectonic and volcanic areas such as California and Italy. 

One has simply to assume a particular probability distribution for A with the survival function available in a closed form, namely the exponential, Erlang, Rayleigh, and Weibull. Table 9.1 summarizes the probability density functions, survival functions, and hazard rates for the above-mentioned distributions. In these expressions, A is the scale parameter and p and v are shape parameters with k, A, p > 0 and v = 1, 2,.... ... 

Figure 9.3 depicts state probability curves for the Erlang and the Weibull distributions. The hazard rates as functions of time are also illustrated. For v 1 and 1, we obtain the behavior corresponding to an exponential... 

The hazard rate h(t) is another important reliability characteristic which can be used to describe the field failure behaviour when using a Weibull distribution. All three elements of the well known bath-tub-curve can be displayed by this theoretical distribution function. Thus it is possible to describe early failure, random failure and wearout failure. 

We develop our filtering systematic mainly on a Weibull parameter p that explains the hazard rate function s behavior. If p < 1, it indicates a decreasing hazard rate and is usually associated with the early failure region. If p l, it means a constant... 

After determination of the change point, two phase hazard rate function, consisting of Weibull and Exponential distributions, can be constructed directly. In this case, there is a discontinuity problem at the change point overall hazard rate function is not continuous. In order to solve this problem, we... 

Figure 6. Hazard rate function h l (t) of forward analysis with Weibull distribution for Mf = 14 months. Beta 0,715 Eta (Day) 2.44E -t 6.

Earthquake Recurrence, Fig. 2 Probability density functions and hazard rate functions for the Poisson model (panel a), the Weibull model (panel b), the gamma model (panel c), and the lognormal model... 

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