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
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
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.