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*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