Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

*Quasi-random number generators* (QRNGs)
produce highly uniform samples of the unit hypercube. QRNGs minimize
the *discrepancy* between the distribution of
generated points and a distribution with equal proportions of points
in each sub-cube of a uniform partition of the hypercube. As a result,
QRNGs systematically fill the “holes” in any initial
segment of the generated quasi-random sequence.

Unlike the pseudorandom sequences described in Common Pseudorandom Number Generation Methods, quasi-random sequences fail many statistical tests for randomness. Approximating true randomness, however, is not their goal. Quasi-random sequences seek to fill space uniformly, and to do so in such a way that initial segments approximate this behavior up to a specified density.

QRNG applications include:

**Quasi-Monte Carlo (QMC) integration.**Monte Carlo techniques are often used to evaluate difficult, multi-dimensional integrals without a closed-form solution. QMC uses quasi-random sequences to improve the convergence properties of these techniques.**Space-filling experimental designs.**In many experimental settings, taking measurements at every factor setting is expensive or infeasible. Quasi-random sequences provide efficient, uniform sampling of the design space.**Global optimization.**Optimization algorithms typically find a local optimum in the neighborhood of an initial value. By using a quasi-random sequence of initial values, searches for global optima uniformly sample the basins of attraction of all local minima.

Imagine a simple 1-D sequence that produces the integers from
1 to 10. This is the basic sequence and the first three points are `[1,2,3]`

:

Now look at how `Scramble`

, `Leap`

, and `Skip`

work together:

`Scramble`

— Scrambling shuffles the points in one of several different ways. In this example, assume a scramble turns the sequence into`1,3,5,7,9,2,4,6,8,10`

. The first three points are now`[1,3,5]`

:`Skip`

— A`Skip`

value specifies the number of initial points to ignore. In this example, set the`Skip`

value to 2. The sequence is now`5,7,9,2,4,6,8,10`

and the first three points are`[5,7,9]`

:`Leap`

— A`Leap`

value specifies the number of points to ignore for each one you take. Continuing the example with the`Skip`

set to 2, if you set the`Leap`

to 1, the sequence uses every other point. In this example, the sequence is now`5,9,4,8`

and the first three points are`[5,9,4]`

:

Statistics and Machine Learning Toolbox™ functions support these quasi-random sequences:

**Halton sequences.**Produced by the`haltonset`

function. These sequences use different prime bases to form successively finer uniform partitions of the unit interval in each dimension.**Sobol sequences.**Produced by the`sobolset`

function. These sequences use a base of 2 to form successively finer uniform partitions of the unit interval, and then reorder the coordinates in each dimension.**Latin hypercube sequences.**Produced by the`lhsdesign`

function. Though not quasi-random in the sense of minimizing discrepancy, these sequences nevertheless produce sparse uniform samples useful in experimental designs.

Quasi-random sequences are functions from the positive integers
to the unit hypercube. To be useful in application, an initial *point set* of
a sequence must be generated. Point sets are matrices of size *n*-by-*d*,
where *n* is the number of points and *d* is
the dimension of the hypercube being sampled. The functions `haltonset`

and `sobolset`

construct
point sets with properties of a specified quasi-random sequence. Initial
segments of the point sets are generated by the `net`

method of the `qrandset`

class
(parent class of the `haltonset`

class
and `sobolset`

class),
but points can be generated and accessed more generally using parenthesis
indexing.

Because of the way in which quasi-random sequences are generated,
they may contain undesirable correlations, especially in their initial
segments, and especially in higher dimensions. To address this issue,
quasi-random point sets often *skip*, *leap* over, or *scramble* values in a sequence.
The `haltonset`

and `sobolset`

functions allow you to specify
both a `Skip`

and a `Leap`

property
of a quasi-random sequence, and the `scramble`

method of the `qrandset`

class
allows you apply a variety of scrambling techniques. Scrambling reduces
correlations while also improving uniformity.

This example shows how to use `haltonset`

to construct a 2-D Halton quasi-random point set.

Create a `haltonset`

object `p`

, that skips the first 1000 values of the sequence and then retains every 101st point.

rng default % For reproducibility p = haltonset(2,'Skip',1e3,'Leap',1e2)

p = Halton point set in 2 dimensions (89180190640991 points) Properties: Skip : 1000 Leap : 100 ScrambleMethod : none

The object `p`

encapsulates properties of the specified quasi-random sequence. The point set is finite, with a length determined by the `Skip`

and `Leap`

properties and by limits on the size of point set indices.

Use `scramble`

to apply reverse-radix scrambling.

```
p = scramble(p,'RR2')
```

p = Halton point set in 2 dimensions (89180190640991 points) Properties: Skip : 1000 Leap : 100 ScrambleMethod : RR2

Use `net`

to generate the first 500 points.

X0 = net(p,500);

This is equivalent to

X0 = p(1:500,:);

Values of the point set `X0`

are not generated and stored in memory until you access `p`

using `net`

or parenthesis indexing.

To appreciate the nature of quasi-random numbers, create a scatter plot of the two dimensions in `X0`

.

scatter(X0(:,1),X0(:,2),5,'r') axis square title('{\bf Quasi-Random Scatter}')

Compare this to a scatter of uniform pseudorandom numbers generated by the `rand`

function.

X = rand(500,2); scatter(X(:,1),X(:,2),5,'b') axis square title('{\bf Uniform Random Scatter}')

The quasi-random scatter appears more uniform, avoiding the clumping in the pseudorandom scatter.

In a statistical sense, quasi-random numbers are too uniform to pass traditional tests of randomness. For example, a Kolmogorov-Smirnov test, performed by `kstest`

, is used to assess whether or not a point set has a uniform random distribution. When performed repeatedly on uniform pseudorandom samples, such as those generated by `rand`

, the test produces a uniform distribution of *p*-values.

nTests = 1e5; sampSize = 50; PVALS = zeros(nTests,1); for test = 1:nTests x = rand(sampSize,1); [h,pval] = kstest(x,[x,x]); PVALS(test) = pval; end histogram(PVALS,100) h = findobj(gca,'Type','patch'); xlabel('{\it p}-values') ylabel('Number of Tests')

The results are quite different when the test is performed repeatedly on uniform quasi-random samples.

p = haltonset(1,'Skip',1e3,'Leap',1e2); p = scramble(p,'RR2'); nTests = 1e5; sampSize = 50; PVALS = zeros(nTests,1); for test = 1:nTests x = p(test:test+(sampSize-1),:); [h,pval] = kstest(x,[x,x]); PVALS(test) = pval; end histogram(PVALS,100) xlabel('{\it p}-values') ylabel('Number of Tests')

Small *p*-values call into question the null hypothesis that the data are uniformly distributed. If the hypothesis is true, about 5% of the *p*-values are expected to fall below 0.05. The results are remarkably consistent in their failure to challenge the hypothesis.

Quasi-random *streams*, produced
by the `qrandstream`

function,
are used to generate sequential quasi-random outputs, rather than
point sets of a specific size. Streams are used like pseudoRNGS, such
as `rand`

, when client applications
require a source of quasi-random numbers of indefinite size that can
be accessed intermittently. Properties of a quasi-random stream, such
as its type (Halton or Sobol), dimension, skip, leap, and scramble,
are set when the stream is constructed.

In implementation, quasi-random streams are essentially very
large quasi-random point sets, though they are accessed differently.
The *state* of
a quasi-random stream is the scalar index of the next point to be
taken from the stream. Use the `qrand`

method of
the `qrandstream`

class
to generate points from the stream, starting from the current state.
Use the `reset`

method
to reset the state to `1`

. Unlike point sets, streams
do not support parenthesis indexing.

This example shows how to generate samples from a quasi-random point set.

Use `haltonset`

to create a quasi-random point set `p`

, then repeatedly increment the index into the point set `test`

to generate different samples.

p = haltonset(1,'Skip',1e3,'Leap',1e2); p = scramble(p,'RR2'); nTests = 1e5; sampSize = 50; PVALS = zeros(nTests,1); for test = 1:nTests x = p(test:test+(sampSize-1),:); [h,pval] = kstest(x,[x,x]); PVALS(test) = pval; end

The same results are obtained by using `qrandstream`

to construct a quasi-random stream `q`

based on the point set `p`

and letting the stream take care of increments to the index.

p = haltonset(1,'Skip',1e3,'Leap',1e2); p = scramble(p,'RR2'); q = qrandstream(p); nTests = 1e5; sampSize = 50; PVALS = zeros(nTests,1); for test = 1:nTests X = qrand(q,sampSize); [h,pval] = kstest(X,[X,X]); PVALS(test) = pval; end

Was this topic helpful?