Generating Quasi-Random Numbers

Quasi-Random Sequences

Quasi-random number generators(QRNGs) produce highly uniform samples of the unit hypercube. QRNGs minimize thediscrepancybetween 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 inCommon 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.

Example: Using Scramble, Leap, and Skip

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 howScramble,Skip, andLeapwork together:

Scramble— Scrambling shuffles the points in one of several different ways. In this example, assume a scramble turns the sequence into1,3,5,7,9,2,4,6,8,10. The first three points are now[1,3,5]:
Skip— ASkipvalue specifies the number of initial points to ignore. In this example, set theSkipvalue to 2. The sequence is now5,7,9,2,4,6,8,10and the first three points are[5,7,9]:
Leap— ALeapvalue specifies the number of points to ignore for each one you take. Continuing the example with theSkipset to 2, if you set theLeapto 1, the sequence uses every other point. In this example, the sequence is now5,9,4,8and the first three points are[5,9,4]:

Quasi-Random Point Sets

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

Halton sequences.Produced by thehaltonsetfunction. These sequences use different prime bases to form successively finer uniform partitions of the unit interval in each dimension.
Sobol sequences.Produced by thesobolsetfunction. 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 thelhsdesignfunction. 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 initialpoint setof a sequence must be generated. Point sets are matrices of sizen-by-d, wherenis the number of points anddis the dimension of the hypercube being sampled. The functionshaltonsetandsobolsetconstruct point sets with properties of a specified quasi-random sequence. Initial segments of the point sets are generated by thenetmethod of thehaltonsetandsobolsetclasses, 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 oftenskip,leapover, orscramblevalues in a sequence. Thehaltonsetandsobolsetfunctions allow you to specify both aSkipand aLeapproperty of a quasi-random sequence, and thescramblemethod of thehaltonsetandsobolsetclasses allows you apply a variety of scrambling techniques. Scrambling reduces correlations while also improving uniformity.

Generate a Quasi-Random Point Set

Open Live Script

This example shows how to usehaltonsetto construct a 2-D Halton quasi-random point set.

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

rngdefault% For reproducibilityp = haltonset(2,'Skip',1e3,'Leap',1e2)

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

The objectpencapsulates properties of the specified quasi-random sequence. The point set is finite, with a length determined by theSkipandLeapproperties and by limits on the size of point set indices.

Usescrambleto apply reverse-radix scrambling.

p = scramble(p,'RR2')

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

Usenetto generate the first 500 points.

X0 = net(p,500);

This is equivalent to

X0 = p(1:500,:);

Values of the point setX0are not generated and stored in memory until you accesspusingnetor parenthesis indexing.

To appreciate the nature of quasi-random numbers, create a scatter plot of the two dimensions inX0.

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

Compare this to a scatter of uniform pseudorandom numbers generated by therandfunction.

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

拟随机散射显得更为均匀,分iding 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 bykstest, 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 byrand, the test produces a uniform distribution ofp-values.

nTests = 1e5; sampSize = 50; PVALS = zeros(nTests,1);fortest = 1:nTests x = rand(sampSize,1); [h,pval] = kstest(x,[x,x]); PVALS(test) = pval;endhistogram(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);fortest = 1:nTests x = p(test:test+(sampSize-1),:); [h,pval] = kstest(x,[x,x]); PVALS(test) = pval;endhistogram(PVALS,100) xlabel('{\it p}-values') ylabel('Number of Tests')

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

Quasi-Random Streams

Quasi-randomstreams, produced by theqrandstreamfunction, are used to generate sequential quasi-random outputs, rather than point sets of a specific size. Streams are used like pseudoRNGS, such asrand, 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. Thestateof a quasi-random stream is the scalar index of the next point to be taken from the stream. Use theqrandmethod of theqrandstreamclass to generate points from the stream, starting from the current state. Use theresetmethod to reset the state to1. Unlike point sets, streams do not support parenthesis indexing.

Generate a Quasi-Random Stream

Open Live Script

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

Usehaltonsetto create a quasi-random point setp, then repeatedly increment the index into the point settestto generate different samples.

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

The same results are obtained by usingqrandstreamto construct a quasi-random streamqbased on the point setpand 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);fortest = 1:nTests X = qrand(q,sampSize); [h,pval] = kstest(X,[X,X]); PVALS(test) = pval;end