All models' residuals conform to

standard normal distribution. In the verification sample, the agreement rate of normal values was over 90%; except for females' VC and FEV[sub]6 and males' FEV[sub]1/VC, the agreement rate for these three parameters was all 88%.

The symmetric distributed source signals are generated based on the

standard normal distribution. The M x N mixing matrix is generated randomly, whose entries are subject to the

standard normal distribution.

For a constant [eta] [member of] (0,1), let [z.sub.n] denote the [eta]th upper-quantile of

standard normal distribution. For example, if [eta] = 0.1, then [z.sub.n] = 1.2816, which means that for a random variable with

standard normal distribution, it is greater than 1.2816 with 10% probability.

where [mathematical expression not reproducible] is an expectation operator with respect to [mathematical expression not reproducible] and [PHI](x) is cdf of the

standard normal distribution.

(2) The

standard normal distribution variables Z = ([Z.sub.1], [Z.sub.2], ..., [Z.sub.n]) were correlated, and their correlation coefficient matrices were [p.sub.0].

The improper model error may be generated when a single

standard normal distribution approximation is used, so Gaussian mixture distribution approximation is proposed.

It can be achieved by two steps: the first step is to generate the

standard normal distribution series by exchanging the random variables of sobol sequence which obeys the U (0, 1) distribution; the second step is to exchange the

standard normal distribution into the corresponding normal distribution based on the mean and variance of necessary

standard normal distribution series.

also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [PHI](t) = [[integral].sup.t.sub.-[infinity]](x) dx are pdf and cdf of the

standard normal distribution, correspondingly.

where [phi] is the cumulative

standard normal distribution,

We then can take a critical value from the

standard normal distribution at a 10 % significance level to determine if a particular model performs significantly better in explaining the data than another.

Figure 1 illustrates a

standard normal distribution with a mean of zero and a standard deviation of one.

Since the fluctuation of demands has a stochastic character, the possible results are random variables equivalent to the

standard normal distribution. Figure 4 shows that if the demand is a random variable, the occurrence of extreme demands has a very low probability, thus the stock shortage can be most likely avoided only with a safety stock with an infinite high level.