(sampler_stochastic_method_maximin_lhs_introduction)=
# Introduction

Considering the definition of a LHS sampling, introduced in [](#sampler_stochastic_method_introduction), 
it is clear that permutating a coordinate of two different points, will create a new sampling. If 
one looks at the x-coordinate (corresponding to a normal distribution) in 
{numref}`sampler_srs_lhs_8_points`, one could put the point in the second equi-probable range, in 
the sixth one, and move the point which was in the sixth equi-probable range into the second one, 
without changing the y-coordinate. The results of this permutation is a new sampling with the 
interesting property of remaining a LHS sampling. A follow-up question can then be: what is the 
difference between these two samplings, and would there be any reason to try many permutations ?

This is a very brief introduction to a dedicated field of research: the optimisation of a {{doe}} 
with respect to the goals of the ongoing analysis. In {{uranie}}, a new kind of LHS sampling has been 
recently introduced, called maximin LHS, whose purpose is to maximise the minimal distance between 
two points. The distance under consideration is the **mindist** criterion: let $D=[\mathbf{x}_1, 
\cdots,\mathbf{x}_N] \subset [0,1]^{d}$ be a {{doe}} with $N$ points. The mindist criterion is 
written as: 

```{math}
:label: eq_mindist

\min_{i,j}{||\mathbf{x}_i-\mathbf{x}_j||_{2}}
```

where $||.||_2$ is the euclidian norm. The designs which maximise the mindist criterion are referred 
to as maximin LHS, but generally speaking, a design with a large value of the mindist criterion is 
referred to as maximin LHS as well. It has been observed that the best designs in terms of maximising 
({eq}`eq_mindist`) can be constructed by minimising its $L^{p}$ regularisation instead. It is written 
as 

```{math}
\phi_p := \Big[ \sum_{i<j} || \mathbf{x}_i-\mathbf{x}_j ||_{2}^{-p} \Big]^{\frac{1}{p}}
```

{numref}`sampler_samplingMaxiMinLHSFromLHSGrid` shows, on the left, an example of a LHS when 
considering a problem with two uniform distributions between 0 and 1 but also, on the right, its 
transformation through the maximin optimisation. The mindist criterion is displayed on top for 
comparison purpose.

{{
    "```{" "figure" "} " + parent_dir +
        "/roottest/build/uranie/doc/sampler/use_cases/python/maxi_min_lhs/samplingMaxiMinLHSFromLHSGrid.png\n"
    ":align: center\n"
    ":name: sampler_samplingMaxiMinLHSFromLHSGrid\n"
    + figure_scale + "\n"
    "\n"
    "Transformation of a classical LHS (left) to its corresponding maximin LHS (right) when 
    considering a problem with two uniform distributions between 0 and 1.\n"
    "```"
}}

From a theoretical perspective, using a maximin LHS to build a Gaussian process (GP) emulator can 
reduce the predictive variance when the distribution of the GP is exactly known. However, it is not 
often the case in real applications where both the variance and the range parameters of the GP are 
actually estimated from a set of learning simulations run over the maximin LHS. Unfortunately, the 
locations of maximin LHS are far from each other, which is not a good feature to estimate these 
parameters with precision. That is why maximin LHS should be used with care. Relevant discussions 
dealing with this issue can be found in {cite}`Pronz12`.