Enhanced Power System Security Assessment Through Intelligent Decision Trees - Complex System Modelling and Control Through Intelligent Soft Computations - page 345

Fig. 8 Boundary region in
the univariate operating
parameter distribution f(x)

proportional to |h(x)|f(x), i.e., focusing on the region S of f(x). In general, the
importance sampling density can be expressed as,

g ð x Þ¼ p f 1 ð x Þ *I(x 2 S Þþð 1 p Þ f 2 ð x Þ *I(x 62 S Þ ð 8 Þ

where p controls the biasing satisfying the probability condition p 1, f 1 f2(x) is the

probability density function of the boundary region, and f 2 f2(x) is the probability
distribution function of the region outside boundary. We can adopt a composition
algorithm to generate samples from the distribution g(x) (Devroye 1986 ; Gentle
1998 ). Setting p = 0.75, 75 % of the points can be expected from region S, thereby
performing an upward scaling of the distribution f(x) towards the boundary region.
In the multivariate case, sampling techniques such as copulas or LHS or sequential
conditional marginal sampling (SCMS) (Papaefthymiou and Kurowicka 2009 ;
Hormann et al. 2004 ) is used to generate correlated multivariate random vectors from
non-parametric distributions f 1 f1(x) and f 2 f2(x). The SCMS method is time consuming
and requires a lot of memory usage for storing the entire historical data, while LHS
and copulas are relatively faster and consume less memory since they work only with
non-parametric marginal distributions and correlation data. We use copulas for their
simpler and elegant approach in handling any non-parametric marginal distributions

≤

Fig. 9 Importance sampling: upward scaling of boundary region probability