A sequence of pseudo-random numbers is a sequence of numbers generated in some systematic way such that its statistical properties are as close as possible to those of true random numbers: for example, negligible correlation between consecutive numbers. The most common methods are based on the multiplicative congruential algorithm, see Knuth (1981). The basic algorithm is defined as:

(1)

The integers  are then divided by  to give uniformly distributed pseudo-random numbers lying in the interval .

Alternatively there is a variant known as the Wichmann–Hill algorithm, see Maclaren (1989), defined as:

(2)

This generates pseudo-random numbers , uniformly distributed in the interval .

Either of these algorithms can be selected to generate uniformly distributed pseudo-random numbers. If the basic algorithm (1) is selected then the NAG generator uses the values  and  in (1). This generator gives a cycle length (i.e., the number of random numbers before the sequence starts repeating itself) of . A good rule of thumb is never to use more numbers than the square root of the cycle length in any one experiment as the statistical properties are impaired. For closely related reasons, breaking numbers down into their bit patterns and using individual bits may cause trouble.

If the Wichmann–Hill algorithm is selected then one or more of 273 independent generators are available. Each of these is defined by the set of constants  and  for . The constants  are in the range 112 to 127 and the constants  are prime numbers in the range 16718909 to 16776971, which are close to . These constants have been chosen so that they give good results with the spectral test, see Knuth (1981) and Maclaren (1989). The period of each Wichmann–Hill generator would be at least  if it were not for common factors between , ,  and . However, each generator should still have a period of at least . Further discussion of the properties of these generators is given in Maclaren (1989) where it was shown that the generated pseudo-random sequences are essentially independent of one another according to the spectral test.

The sequence given in (1) needs an initial value , known as the seed, while the sequence given in (2) needs four such seeds. The use of the same seed will lead to the same sequence of numbers when these are computed serially. One method of obtaining a seed is to use the real-time clock; this will give a non-repeatable sequence. It is important to note that the statistical properties of the random numbers are only guaranteed within sequences and not between sequences. Repeated initialization will thus render the numbers obtained less rather than more independent. Similarly the statistical properties of the random numbers are not guaranteed between two sequences generated using the two algorithms.

Quasi-random numbers are intended primarily for use in Monte Carlo integration. Like pseudo-random numbers they are uniformly distributed but they are not statistically independent, rather they are designed to give a more even distribution in multidimensional space (uniformity). Therefore, they are often more efficient than pseudo-random numbers in multidimensional Monte Carlo methods. There are several quasi-random generators, three of which are available in this chapter, they are the Sobol, Faure and Neiderreiter generators.

Random numbers from other distributions may be obtained from the uniform random numbers by the use of transformations and rejection techniques, and for discrete distributions, by table based methods.

A copula is a function that links the univariate marginal distributions with their multivariate distribution. Sklar's theorem Sklar (1973) states that if  is an -dimensional distribution function with continuous margins , then  has a unique copula representation, , such that

The copula, , is a multivariate uniform distribution whose dependence structure is defined by the dependence structure of the multivariate distribution , with

where . This relationship can be used to simulate variates from distributions defined by the dependence structure of one distribution and each of the marginal distributions given by another. For additional information see Nelsen (1998) or Boye (Unpublished manuscript) and the references therein.

In addition to random numbers from various distributions, random compound structures can be generated. These include random time series, random matrices and random samples.

The efficiency of a simulation exercise may often be increased by the use of variance reduction methods (see Morgan (1984)). It is also worth considering whether a simulation is the best approach to solving the problem. For example, low-dimensional integrals are usually more efficiently calculated by functions in Chapter d01 rather than by Monte Carlo integration.

The functions allow the selection of either the basic generator or one of the suite of Wichmann–Hill generators. Information on the generator is passed between functions using the parameters  and . Two utility functions are provided to initialise the generator.

These functions should only be used once for each stream of random numbers created. Each stream generated by a different generator will be independent. To generate high-dimensional variables one approach would be to use a different independent generator for each dimension.

The functions available for quasi-random number each provide the user the option of selecting a Solbol, Faure or Neiderreiter sequence. The maxium number of dimensions is 40. If higher dimensions are required these generators can be combined with the pseudo-random generators described above.