# Background and discussions

Latin Hypercube sampling

Consider that the performance of an engineering system, Z, is a function of a set of s random variablesX, X = [Xj,■ ■ ■,Xs],

Z = h( X), (1)

where h(*) is a deterministic function. The mathematical expectation of Zk or the k-th moments of Z, E(Zk), is defined as,

E(Zk) = J(h(x))fx (x)dx, (2)

n

where Q represents the domain of X, and fX(x) is the joint probability density function of X. The mean of Z, mZ, equals E(Z), and the variance of Z, a2z, equals E(z2)-mZ. The probability that Z is less than or equal to a given value zp, Pf, can be expressed as,

Pf = fx (x)dx, (3)

g <o

where g < 0 represents the domain of Z-zp < 0.

Integrals in Eqs. (2) and (3) can be estimated using the simple simulation technique. Alternatively, the more efficient LHS technique can be employed (Iman and Conover 1980). According to this technique, for practical applications with independent random variables, the generation of Latin hypercube samples of size n could be carried out as follows. The domain of each random variable is divided into n mutually exclusive and collectively exhaustive intervals, and one value is selected randomly in each interval according to the probability distribution of the random variable. A value is randomly selected from the n values for each of the random variables to form the first Latin hypercube sample. The remaining n-1 values for each of the random variables are used to form the second Latin hypercube sample. That is, a value is randomly selected from the n-1 remaining values for each of the random variables to form the second Latin hypercube sample. This process is repeated until n Latin hypercube samples are obtained.

In particular, if the n mutually exclusive and collectively exhaustive intervals have equal probability, the k-th moments of Z is estimated as the average of the sum of the function (h(x))k evaluated at each of the sample points; and the probability of failure Pf defined in Eq. (3) is approximated by the ratio of the number of sample points where h(x)-zp < 0 to n.