# Scrambled Halton

The Halton sequence, which is one of my favourite algorithms ever, can be used for efficient stratified multi-dimensional sampling. Some references:

It is possible to do stratified sampling of hyper-points in the *s-dimensional* unit hyper-cube by picking one consecutive dimension of the Halton series for each component. A convenient way to do so is to use the first `s`

prime numbers as the basis for each Halton sequence.

It is well-known, however, that while this approach works great for low dimensions, high dimensions often exhibit a great degree of undesired correlation. The following image displays a grid where each cell combines two components of the *32-dimensional* Halton hyper-cube.

*Raw 32-dimensional Halton hyper-cube*

One can easily spot how some pairs exhibit an obvious pattern, while others fill their corresponding 2D area very densely. This happens more aggressively for higher dimensions (*i.e.,* to the right/bottom in the image) and for pairs formed with close components (*i.e.,* near the diagonal in the image). Click to enlarge!

A successful approach to dissolve this problem without losing the good properties of stratification is to do *“random digit scrambling”* (*a.k.a.,* rds). During the construction of a Halton number, digits in the range `[0..base[`

are combined. Given a *Pseudo-Random Permutation* of `length=base`

, all that one must do is use `PRP(digit)`

instead of `digit`

directly. This somewhat shuffles Halton pairs in rows and columns in a strong way so the correlation disappears. However, since the PRP is a bijection, the good properties of stratification are generally preserved.

**[EDIT] While this is true, it has been proven that Owen-Scrambling is a superior method.**

How to build a strong and efficient randomised PRP of an arbitrary length is a very interesting subject which details I won’t get into here. You can build your own carefully combining a number of reversible hashing operations.

Here’s the scrambling strategy in action:

*Scrambled 32-dimensional Halton hyper-cube*

Now all the blocks in the grid look equally dense. Magic!

As long as one picks a good seedable PRP, it is possible to generate any number of different samplings, all with the same good properties.

**[EDIT] These are the guidelines of the QRN system that Arion used for many years.**