[1.7]. A j-th second-order smoothed bootstrap sample, ℳ(k,j), is composed of:
[1.34]
where
The IBCa method can be presented in the following steps:
– Step 1. Generate n jackknife samples, estimate kernel CDF-s from the jackknife samples, and evaluate the accelerating constant:
[1.35]
where
– Step 2. Generate B smoothed bootstrap samples of the first order, ℳ(k), k = 1,.., B, and estimate the kernel CDF-s from these samples,
– Step 3. From every first-order bootstrap sample generate BB smoothed bootstrap samples of the second order, ℳ(k,j), k = 1,.., B, j = 1,.., BB, and estimate the kernel CDF-s from these samples,
– Step 4. For every first-order bootstrap sample, ℳ(k), and its offspring second-order samples, ℳ(k,j), j = 1,.., BB, estimate the linked bias-correction parameter:
[1.36]
where Φ−1(•) is the inverse function of the standard normal CDF. Calculate the bias-correction parameter as a mean value of the linked bias-correction parameters:
[1.37]
– Step 5. Calculate the orders of percentiles, α1 and α2:
[1.38]
where Φ(•) is the standard normal CDF, and zα and z1-α are percentiles of the standard normal distribution.
– Step 6. Sort in ascending order, The estimate of the confidence interval of length 1 − 2α for magnitude CDF is:
Orlecka-Sikora (2008) also provided a method to evaluate the optimal number of the first-order bootstrap samples, B. In general, B should be large, amounting tens of thousands for the initial sample of hundreds of elements. The reasonable number of the second-order samples is a few hundred for every first-order bootstrap sample. Hence, we have to generate tens of millions samples to get the confidence intervals [1.39]. Nevertheless, this is not a problem for high-performance computing (HPC).
When we assume that the seismic process is Poissonian and that the exact rate of earthquake occurrence, λ, is known, we readily obtain the confidence intervals for the exceedance probability and the mean return period. For the exceedance probability, R(M,D) (equation [1.17]), it is:
and for the mean return period, T(M) (equation [1.18]), it is:
Figure 1.6, taken from Orlecka-Sikora (2008), shows a practical example of the interval kernel estimation of CDF and the related hazard parameters. The studied seismic events were from Rudna deep copper mine in Poland and were parameterized in terms of seismic energy. Nevertheless, because the magnitude and the logarithm of energy have the same distribution, the specific features of the graphs in Figure 1.6 would remain if magnitude was used.
Orlecka-Sikora and Lasocki (2017) presented a modified version of the interval estimation of R(M,D) (equation [1.40]) and T(M) (equation [1.41]), which also takes into account the uncertainty of earthquake occurrence rate, λ. It turned out that the uncertainty of λ only matters when λD is small, less than 5. For greater λD , the uncertainty of the CDF estimate dominates, and the role of the uncertainty of λ is negligible. In this connection, this modified version should mostly be used in hazard studies of low seismic activity regions.
Figure 1.6. Example of interval kernel estimation of CDF and related hazard parameters. The event size is parametrized by the logarithm of seismic energy. (a) CDF(logE) and its magnified part, (b) exceedance probability of logE=7.0 and (c) mean return period. Solid lines – the point kernel estimates (equations [1.32], [1.17] and [1.18]), dashed lines – the 95% confidence intervals from the Iterated BCa method. Reprinted from Orlecka-Sikora (2008, Figures 11, 14 and 15)
1.6. Transformation to equivalent dimensions
Many studies into the physics of earthquake processes are based on investigations into the similarity of seismic events. The similarity of events means the similarity of their parameters. However, the parameters scale differently, which makes impossible comparisons of events in a multi-parameter space. To overcome this problem, Lasocki (2014) proposed a transformation of any seismic event continuous parameterization to the so-called equivalent dimension (ED). The ED of a parameter X is its CDF, Fx. All parameters transformed to EDs scale in the same way, have the uniform distribution in [0, 1], and every ED-transformed parameter space has the Euclidean metric. A rationale for the transformation to ED can be found in the referenced work.
The probabilistic models for earthquake parameters, Fx, are not known in general, but we can estimate Fx using the kernel estimation method (equations [1.6], [1.7], [1.10],
