New Computer Experiment Designs with Area-Interaction Point Processes

Abstract

This article presents a novel method for constructing computer experiment designs based on the theory of area-interaction point processes. This method is essential for capturing the interactions between different elements within a modeled system, offering a more flexible and adaptable approach compared with traditional mathematical modeling. Unlike conventional rough models that rely on simplified equations, our method employs the Markov Chain Monte Carlo (MCMC) method and the Metropolis–Hastings algorithm combined with Voronoi tessellations. It uses a new dynamic called homogeneous birth and death dynamics of a set of points to generate the designs. This approach does not require the development of specific mathematical models for each system under study, making it universally applicable while achieving comparable results. Furthermore, we provide an in-depth analysis of the convergence properties of the Markov Chain to ensure the reliability of the generated designs. An expanded literature review situates our work within the context of existing research, highlighting its unique contributions and advancements. A comparison between our approach and other existing computer experiment designs has been performed.

1. Introduction

Recent advancements in modeling, coupled with a significant increase in computer processing power, have enabled the development of simulators of unprecedented complexity. These simulators can accurately and precisely replicate physical phenomena, as highlighted by Tolk (2018) [1]. These simulators, requiring intensive calculations and sophisticated algorithms, as described by Sonnessa (2004) [2], to produce accurate and reliable results, can become challenging to use and manage due to their large size. To address these issues, one can use surrogate functions to replace the simulator. These functions are generally simpler and created using approximation or interpolation methods based on computer experiment designs.

In 2008, Franco [3] first introduced computer experiment designs based on Strauss point processes, which included the idea of pairwise point interaction. In 2020, Elmossaoui et al. [4,5] proposed computer experiment designs based on the use of marked Strauss processes, allowing for the simultaneous achievement of two objectives: one concerning the distribution of points and the other concerning the characterization of the marks associated with these points. Recently, in 2023, Elmossaoui et al. [6] proposed a new method to construct computer experiment designs based on the use of connected component point processes. The simulation of point processes allows for modeling and analyzing the distribution of points in a given space, providing a precise overview of the interactions and patterns present in the data [7,8]. By integrating this technique, we can optimize the experiment design by ensuring comprehensive coverage and representativeness of the experimental space, which improves the reliability and relevance of the obtained results.

In particular, the simulation of area-interaction point processes introduced by Baddeley et al. [9] is a key area in stochastic modeling. It involves generating configurations of points in a given space while respecting specific interaction constraints between the points, such as repulsive or attractive forces [10]. For the simulation of these processes, several software tools are available. In Python, libraries such as scikit-learn [11] and seaborn [12] can be used to create, analyze, and visualize point processes. In C++, libraries like CGAL and Boost.Geometry offer tools for point processes with interactions. In R, the package spatstat [13] is particularly useful for simulating and analyzing spatial interaction point processes, and ppstat provides tools for the statistics of these processes.

Our approach aims to use a method to construct computer experiment designs that effectively cover the parameter space and provide information about the entire experimental domain. To do this, we employ the area-interaction point process to simulate the computer experiment included in the design. This method generates points distributed within the hypercube. The use of this process, which is Markovian in the Ripley–Kelly [14] sense, offers an alternative for modeling patterns with spatial distributions varying from regular to irregular, without imposing constraints on the parameters. To generate these designs, we use Markov chain Monte Carlo (MCMC) simulation techniques and the Metropolis–Hastings algorithm [15] with Voronoi–Dirichlet tessellations [16], which, unlike traditional simulation methods that modify points one by one, allows for the simultaneous modification of multiple points through what is known as “the homogeneous birth and death dynamics of a set of points”.

This document is structured as follows: Section 2 presents the essential definitions and notations. Section 3 outlines the main idea of constructing new computer experiment designs based on the use of area-interaction point processes, using the MCMC method and the Metropolis–Hastings algorithm with Voronoi tessellations. Section 4 details the algorithm for constructing these designs. The convergence of this algorithm is discussed in Section 5, followed by the convergence rate in Section 6. Finally, a comparison is proposed in Section 7.

2. Preliminaries

Let $\left(\Omega ,\mathcal{A},\mathcal{P}\right)$ be a probability space. Consider $\chi$ as a non-empty set equipped with the Euclidean distance d, which makes it a complete separable metric space. If we assume that the model has p continuous factors of interest, where $p\ge 1$, then in most cases, $\chi$ will be equal to ${[0,1]}^p$ (a subset of ${\mathbb{R}}^p$). Let m denote the Lebesgue measure on this space, equipped with its Borel sigma-algebra $\mathcal{A}$.

Definition 1.

A configuration is defined as a countable set, unordered set of points $x=\left(x_1,x_2,\dots ,x_n\right)$, where $x_i\in \left(\chi ,d\right)$, representing the points resulting from a random experiment. A configuration is said to be locally finite if it contains at most a finite number of points within any bounded Borel set A in $\left(\chi ,d\right)$. We denote $N^{lf}$ as the family of locally finite configurations.

Definition 2.

A point process is a mapping X from a probability space (comprising a metric space χ equipped with a sigma-algebra $\mathcal{A}$ and a probability measure $\mathcal{P}$) to the family of locally finite configurations of points in χ. This mapping satisfies the property that, for any Borel set $A\subseteq \chi$, the number of points in A, denoted by $N_X\left(A\right)$, is a finite discrete random variable.

Definition 3.

Let ∼ be a symmetric and reflexive relation on χ. Two points $x_i$ and $x_{i^{\prime }}$ are neighbors if $x_i\sim x_{i^{\prime }}$. The neighborhood of y is given by:

$$\partial x_i=\left\{x_i\in xet{x}_i\notin y:\exists x_{i^{\prime }}\in y\mathrm{such} \mathrm{us}{x}_{i^{\prime }}\sim x_i\right\}$$

Definition 4.

A function $\pi :\Omega \to [0,+\infty [$ is called a Markov function in the sense of Ripley and Kelly, with respect to the relation ∼, if for all $x\in \Omega$:

(a) 

$\pi \left(x\right)>0$ Implies $\pi \left(y\right)>0$ for all $y\subset x$.

(b) 

If $\pi \left(x\right)>0$, then for any $x_i\in \chi$, the ratio ${\displaystyle \frac{\pi \left(x\cup \left\{x_i\right\}\right)}{\pi \left(x\right)}}$ depends only on $x_i$ and $\partial x_i\cap x$.

Definition 5.

Let x be a finite set of n points in ${\left[0,1\right]}^p$, the elements ${\left\{x_i\right\}}_{i\in \left\{1,\dots ,n\right\}}$ of x are called centers, sites, or germs. The Voronoi tessellations (or Voronoi cells or regions) $\vartheta \left(x_i\right)$ associated with an element q from x are the set of points that are closer to q than to any other point in x:

$$\vartheta \left(x_i\right)=\left\{s\in {\left[0,1\right]}^p/\forall q\in x,∥s-x_i∥\le ∥s-q∥\right\}$$

Figure 1 represents an example of Voronoi tessellations for 10 and 20 points on the unit square.

3. Computer Experiments Designs Using Markovian Area-Interaction Point Processes

Each experiment $x_i$ is considered as a point or particle defined on ${[0,1]}^p$, and each configuration x is regarded as a computer experiments design. Hence, the n experiments can be seen as realizations of an area-interaction point process [9]. This interaction corresponds to neighborhood properties, as defined by a Markov field in the sense of Ripley–Kelly [14]. This process is essential for modeling repulsive phenomena. The density of the area-interaction point process is defined by the following:

$$\pi \left(x\right)=\alpha {\beta }^{n\left(x\right)}{\gamma }^{-m\left(U_r\left(x\right)\right)}$$

(1)

where $\alpha >0$ is the normalization constant that makes $\pi$ a density, $n\left(x\right)$ is the number of points in the configuration x, $\beta >0$ is a scale parameter, $\gamma >0$ is a repulsion parameter, $m(·)$ is the Lebesgue measure, $r>0$ is the radius of the ball $B\left(x_i,r\right)$ is defined by $B\left(x_i,r\right)=\left\{a\in {\left[0,1\right]}^p,∥x_i-a∥\le r\right\}$, and $U_r\left(x\right)={\displaystyle \bigcup _{i=1}^n}B\left(x_i,r\right)$ is the union of balls centered at $x_i$ with radius r [9,17,18].

The area of the union of discs may be expressed as the decomposition of the union of grains, $U_r\left(x\right)={\displaystyle \bigcup _{i=1}^n}B\left(x_i,r\right)$, in an inclusion–exclusion style [18,19,20]. This is expressed concisely as follows:

$$m\left(U_r\left(x\right)\right)=\sum _{i=1}^{n\left(x\right)}m\left(B\left(x_i,r\right)\right)-\sum _{i<j}m\left(B\left(x_i,r\right)\cap B\left(x_j,r\right)\right)+\cdots +{\left(-1\right)}^{n\left(x\right)+1}m\left(\bigcap _{i=1}^{n\left(x\right)}B\left(x_i,r\right)\right)$$

Figure 2 illustrates an example of $U_r\left(x\right)$ on square ${[0,1]}^2$. The red points $x_i$ and the blue disks of the center $x_i$ and radius r are shown. The blue shaded area in this figure represents $U_r\left(x\right)$, and $m\left(U_r\left(x\right)\right)$ denotes the area of this region.

Simulation of Point Processes Using the Markov Chain Monte Carlo (MCMC) Method and the Metropolis–Hastings Algorithm

This method involves constructing a chain $\left\{X_0,X_1,\cdots ,X_N\right\}$ that converges to the desired distribution $\pi$. In fact, the Metropolis–Hastings algorithm achieves this construction by using a $\pi$-reversible transition kernel. The algorithm goes through two steps:

• We make a proposal for a state change: from x to y, according to a probability law $Q\left(x,.\right)$ with density $q_m\left(x,y\right)$, called the instrumental density.
• We accept y with probability $a_m\left(x,y\right)$; otherwise, we stay in state x (where $a_m:\Omega \times \Omega \to \left[0,1\right]$).

The transition kernel is given by [21]:

$$P\left(x,B\right)=\sum _m\underset{B}{\int }{a}_m\left(x,y\right)q_m\left(x,y\right)dy+\left\{\sum _m\underset{\Omega }{\int }\left[1-a_m\left(x,y\right)\right]q_m\left(x,y\right)dy-\sum _m{q}_m\left(x,\Omega \right)\right\}{\delta }_{x\in B}$$

In our case, we take $q_m\left(x,y\right)=q\left(x,y\right)$ and $a_m\left(x,y\right)=a\left(x,y\right)$ (i.e., a homogeneous Markov chain). From this, we can deduce the Metropolis–Hastings transition matrix $P_{MH}$, which is given by [22]:

$$P_{MH}\left(x,y\right)=a\left(x,y\right)q\left(x,y\right)+\left[\underset{\Omega }{\int }1-a\left(x,z\right)q\left(x,z\right)dz\right]{\delta }_x\left(y\right)$$

where ${\delta }_x(·)$ represents the mass at point x; for simplification, we use the Dirac measure at x (${\delta }_x\left(y\right)=1$ if $x=y$, and 0 otherwise).

The choice of $(Q,a)$ will ensure the $\pi$-reversibility of $P_{MH}$ if the following equilibrium equation is satisfied:

$$\forall x,y\in \Omega :\pi \left(x\right)\times q\left(x,y\right)\times a\left(x,y\right)=\pi \left(y\right)\times q\left(y,x\right)\times a\left(y,x\right)$$

The choice of the acceptance probability a is more limited and essentially dictated by the objective of simulating (asymptotically) a given probability distribution $\pi$. This is the case for the usual choice, where:

$$a\left(x,y\right)={\displaystyle \frac{\pi \left(y\right)\times q\left(y,x\right)}{\pi \left(x\right)\times q\left(x,y\right)}}$$

It is important to note certain points. Firstly, the calculation of $a(x,y)$ does not require knowing the normalization constant of (1). Secondly, in this work, we consider the case where both configurations, x and y, differ at multiple points, which is called the homogeneous birth and death dynamics of a set of points. As a result, the chosen density q is usually symmetric to make the process computationally manageable: $q\left(x,y\right)=q\left(y,x\right)$. Thus, the acceptance probability is reduced to the following:

$$a\left(x,y\right)={\displaystyle \frac{\pi \left(y\right)}{\pi \left(x\right)}}={\displaystyle \frac{{\beta }^{n\left(y\right)}{\gamma }^{-m\left(U_r\left(y\right)\right)}}{{\beta }^{n\left(x\right)}{\gamma }^{-m\left(U_r\left(x\right)\right)}}}={\displaystyle \frac{{\gamma }^{-m\left(U_r\left(y\right)\right)}}{{\gamma }^{-m\left(U_r\left(x\right)\right)}}}$$

4. Algorithm for Constructing Computer Experiments Designs Using Markovian Area-Interaction Point Processes

The computer experiment designs proposed in this work were generated using Algorithm 1, which is essentially a version of the Metropolis–Hastings algorithm with the use of Voronoi tessellations.

For $N=1000$, Figure 3 and Figure 4 show the convergence towards a configuration that characterizes the realization of an area-interaction point process starting from an initial configuration of 25 points, chosen uniformly in ${\left[0,1\right]}^2$ and ${\left[0,1\right]}^3$, respectively.

Influence of Parameters

Figure 5 shows the impact of the parameter r on the final distribution of points. It is crucial to choose the radius r wisely, as a too small radius results in a distribution without interaction but with numerous gaps. On the other hand, a too large radius leads to a distribution with excessive interaction.

The interaction radius is the most sensitive parameter to adjust, and its value must be carefully selected. For a given criterion, the best solution would likely be to tabulate this value according to the number of points and the dimension of the problem.

It is essential to set the repulsion parameter $\gamma$ to a value strictly greater than 1 to avoid condensation of points in a single location and to generate a distribution that satisfies the criterion of space filling.

5. Convergence of the Proposed Algorithm

For each iteration N of the algorithm described above, the chain of computer experiments designs ${\left(X_N\right)}_{N\ge 0}$ generated is a realization of a Markov chain with the transition kernel:

$$P\left(x,y\right)=P_{MH}\left(x,y\right)$$

The essential question that arises is whether the chain converges to the distribution $\pi \left(x\right)$ defined in (1).

Definition 6.

The chain is convergent to the invariant distribution π if:

$$\underset{n\to \infty }{lim}{P}^n\left(x,A\right)=\pi \left(A\right)$$

where A is a Borel set in $\mathcal{A}$, $P^n\left(X_0,A\right)=P\left(X_n\in A|X_0\right)$ with:

$$P\left(X_n,A\right)=P\left(X_{n+1}\in A|X_0,X_1,\dots ,X_n\right)\mathrm{and}\pi \left(A\right)=\int \pi \left(dx\right)P\left(x,A\right)$$

Proposition 1.

On a finite space, the transition kernel P of the Markov chain ${\left(X_N\right)}_{N\ge 0}$ obtained from the construction algorithm is recurrent positive, π-stationary, aperiodic, and primitive (primitive kernel).

Proof. 

Firstly, we show three important properties for the kernel $P_{MH}$: $\pi$-reversibility, $\pi$-stationarity, and $\pi$-irreducibility.

• $\pi$-reversibility
  The transition $P_{MH}$ is $\pi$-reversible if, for any sets A and B, the probability of transitioning from a state in A to a state in B is equal to the probability of transitioning from a state in B to a state in A ($A,B\subset \Omega$):

  $$\forall x\in A,\forall y\in B:\underset{A}{\int }\underset{B}{\int }\pi \left(x\right)P_{MH}\left(x,y\right)dxdy=\underset{B}{\int }\underset{A}{\int }\pi \left(y\right)P_{MH}\left(y,x\right)dydx$$
  Let $x\in A$ and $y\in B$

  $$P_{MH}\left(x,B\right)=\underset{B}{\int }a\left(x,y\right)q\left(x,y\right)dy+\left[\underset{\Omega }{\int }1-a\left(x,z\right)q\left(x,z\right)dz\right]{\delta }_x\left(y\right)$$

  $$=\underset{B}{\int }a\left(x,y\right)q\left(x,dy\right)+s\left(x\right){\delta }_x\left(y\right)$$
  With $s\left(x\right)={\int }_{\Omega }1-a\left(x,z\right)q\left(x,z\right)dz$
  For any Borel sets A and B in $\Omega$, we have:

  $$\underset{A}{\int }\underset{B}{\int }\pi \left(x\right)P_{MH}\left(x,y\right)dxdy=\underset{A}{\int }\underset{B}{\int }\pi \left(x\right)a(x,y)q(x,y)dydx+\underset{A\bigcap B}{\int }{1}_{y=x}\pi \left(x\right)s\left(x\right)dx$$

  $$=\underset{A}{\int }\underset{B}{\int }\pi \left(x\right)a(x,y)q(x,y)dydx+\underset{A\cap B}{\int }{1}_{y=x}\pi \left(y\right)s\left(y\right)dy$$
  And as:

  $$\pi \left(x\right)a(x,y)q(x,y)=\alpha {\beta }^{n\left(x\right)}{\gamma }^{-m\left(U_r\left(x\right)\right)}min\left(1,{\beta }^{n\left(y\right)-n\left(x_i\right)}{\gamma }^{m\left(U_r\left(x_i\right)\right)-m\left(U_r\left(y\right)\right)}\right)q(x,y)$$

  $$=\alpha min\left({\beta }^{n\left(x\right)}{\gamma }^{-m\left(U_r\left(x\right)\right)},{\beta }^{n\left(y\right)}{\gamma }^{-m\left(U_r\left(y\right)\right)}\right)q(x,y)$$

  $$=\alpha {\beta }^{n\left(y\right)}{\gamma }^{-m\left(U_r\left(y\right)\right)}min\left({\beta }^{n\left(x\right)-n\left(y\right)}{\gamma }^{m\left(U_r\left(y\right)\right)-m\left(U_r\left(x\right)\right)},1\right)q(x,y)$$

  $$=\alpha {\beta }^{n\left(y\right)}{\gamma }^{-m\left(U_r\left(y\right)\right)}min\left(1,{\beta }^{n\left(x\right)-n\left(y\right)}{\gamma }^{m\left(U_r\left(y\right)\right)-m\left(U_r\left(x\right)\right)}\right)q(x,y)$$
  And as $q(x,y)=q(y,x)$ so $\pi \left(y\right)a(y,x)q(y,x)=\pi \left(x\right)a(x,y)q(x,y)$
  Using the Fubini’s theorem, we obtain:

  $$\underset{A}{\int }\underset{B}{\int }\pi \left(x\right)a(x,y)q(x,y)dydx=\underset{B}{\int }\underset{A}{\int }\pi \left(y\right)a(y,x)q(y,x)dxdy$$
  Finally, we have:

  $$\underset{A}{\int }\underset{B}{\int }\pi \left(x\right)P_{MH}\left(x,y\right)dxdy=\underset{B}{\int }\underset{A}{\int }\pi \left(y\right)a(y,x)q(y,x)dxdy+\underset{A\bigcap B}{\int }{1}_{y=x}\pi \left(y\right)s\left(y\right)dy$$

  $$=\underset{B}{\int }\underset{A}{\int }\pi \left(y\right)P_{MH}\left(y,x\right)dydx$$
  Hence, $P_{MH}$ is $\pi$-reversible.
• $\pi$-stationarity
  A distribution $\pi$ is said to be stationary for the transition kernel $P_{MH}$ if:

  $$\pi \left(x\right)=\underset{\Omega }{\int }\pi \left(y\right)P_{MH}\left(y,x\right)dy$$

  $$\underset{\Omega }{\int }\pi \left(y\right)P_{MH}\left(y,x\right)dy=\underset{\Omega }{\int }\pi \left(y\right)\left[a\left(x,y\right)q\left(x,y\right)+\left[\underset{\Omega }{\int }1-a\left(x,z\right)q\left(x,z\right)dz\right]{\delta }_x\left(y\right)\right]dy$$

  $$=\underset{\Omega }{\int }\pi \left(y\right)a\left(x,y\right)q\left(x,y\right)dy+\underset{\Omega }{\int }\underset{\Omega }{\int }{\delta }_x\left(y\right)\pi \left(y\right)-\pi \left(y\right)a\left(x,z\right)q\left(x,z\right){\delta }_x\left(y\right)dzdy$$

  $$=\underset{\Omega }{\int }\pi \left(y\right)a\left(x,y\right)q\left(x,y\right)dy+\underset{\Omega }{\int }\underset{\Omega }{\int }{\delta }_x\left(y\right)\pi \left(y\right)dzdy-\underset{\Omega }{\int }\underset{\Omega }{\int }\pi \left(y\right)a\left(x,z\right)q\left(x,z\right){\delta }_x\left(y\right)dzdy$$

  $$=\underset{\Omega }{\int }\pi \left(y\right)a\left(x,y\right)q\left(x,y\right)dy+\underset{\Omega }{\int }{\delta }_x\left(y\right)\pi \left(y\right)dy-\underset{\Omega }{\int }\underset{\Omega }{\int }{1}_{x=y}\pi \left(y\right)a\left(x,z\right)q\left(x,z\right)dzdy$$

  $$=\underset{\Omega }{\int }\pi \left(y\right)a\left(x,y\right)q\left(x,y\right)dy+\underset{\Omega }{\int }{\delta }_x\left(y\right)\pi \left(y\right)dy-\underset{\Omega }{\int }\pi \left(y\right)a\left(y,z\right)q\left(y,z\right)dz$$

  $$=\underset{\Omega }{\int }{\delta }_x\left(y\right)\pi \left(y\right)dy=\pi \left(x\right)$$
  So the chain admits $\pi$ as a stationary distribution.
• $\pi$-irreducibility
  The transition $P_{MH}$ is $\pi$-irreducible if:

  $$\forall A\in \mathcal{A},\pi \left(A\right)>0\Rightarrow \exists t,P_{MH}^t\left(x,A\right)>0$$
  For any Borel set A in $\mathcal{A}$ and for $t=1$, we obtain:

  $$\underset{\Omega }{\int }{1}_{B\left(x,A\right)}{P}_{MH}\left(x,A\right)dx=\underset{\Omega }{\int }{1}_{B\left(x,A\right)}a\left(x,A\right)q\left(x,A\right)dx$$

  $$+\underset{\Omega }{\int }{1}_{B\left(x,A\right)}\left[1-\underset{\Omega }{\int }a\left(x,z\right)q\left(x,z\right)dz\right]{\delta }_x\left(A\right)dx$$

  $$=\underset{\Omega }{\int }{1}_{B\left(x,A\right)}a\left(x,A\right)q\left(x,A\right)dx+\underset{\Omega }{\int }{1}_{B\left(x,x\right)}\left[1-\underset{\Omega }{\int }a\left(x,z\right)q\left(x,z\right)dz\right]dx$$

  $$=\underset{\Omega }{\int }{1}_{B\left(x,A\right)}a\left(x,A\right)q\left(x,A\right)dx+1-\underset{\Omega }{\int }\underset{\Omega }{\int }a\left(x,z\right)q\left(x,z\right)dzdx$$
  As $a(x,A)=min\left(1,{\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(A\right)\right)}\right)$ and $a(x,z)=min\left(1,{\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(z\right)\right)}\right)$, so there are four possible cases:

  -

  If $a\left(x,A\right)=1$ and $a(x,z)={\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(z\right)\right)}$ so:

  $$\underset{\Omega }{\int }{1}_{B\left(x,A\right)}{P}_{MH}\left(x,A\right)dx=\underset{\Omega }{\int }{1}_{B\left(x,A\right)}q\left(x,A\right)dx+1-\underset{\Omega }{\int }\underset{\Omega }{\int }{\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(z\right)\right)}q\left(x,z\right)dzdx$$

  $$=\underset{\Omega }{\int }{1}_{B\left(x,A\right)}q\left(x,A\right)dx+1-{\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(z\right)\right)}>0$$

  -

  if $a(x,A)={\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(A\right)\right)}$ and $a\left(x,z\right)=1$ so:

  $$\underset{\Omega }{\int }{1}_{B\left(x,A\right)}{P}_{MH}\left(x,A\right)dx=\underset{\Omega }{\int }{1}_{B\left(x,A\right)}{\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(A\right)\right)}q\left(x,A\right)dx+1-\underset{\Omega }{\int }\underset{\Omega }{\int }q\left(x,z\right)dzdx$$

  $$={\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(A\right)\right)}\underset{\Omega }{\int }{1}_{B\left(x,A\right)}q\left(x,A\right)dx>0$$

  -

  If $a(x,A)={\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(A\right)\right)}$ and $a(x,z)={\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(z\right)\right)}$ so:

  $$\underset{\Omega }{\int }{1}_{B\left(x,A\right)}{P}_{MH}\left(x,A\right)dx=\underset{\Omega }{\int }{1}_{B\left(x,A\right)}{\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(A\right)\right)}q\left(x,A\right)dx+1$$

  $$-\underset{\Omega }{\int }\underset{\Omega }{\int }{\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(z\right)\right)}q\left(x,z\right)dzdx$$

  $$={\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(A\right)\right)}\underset{\Omega }{\int }{1}_{B\left(x,A\right)}q\left(x,A\right)dx+1-{\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(z\right)\right)}\underset{\Omega }{\int }\underset{\Omega }{\int }q\left(x,z\right)dzdx$$

  $$={\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(A\right)\right)}\underset{\Omega }{\int }{1}_{B\left(x,A\right)}q\left(x,A\right)dx+1-{\gamma }^{m\left(U_r\left(x\right)\right)-m\left(U_r\left(z\right)\right)}>0$$

  So $\underset{\Omega }{\int }{1}_{B\left(x,A\right)}{P_{MH}}^t\left(x,A\right)dx>0\forall t\ge 0$, hence, $P_{MH}$ is $\pi$-irreducible.

By construction of $P=P_{MH}$, we have $\pi P=\pi$. If P is $\pi$-irreducible and possesses a $\pi$-invariant distribution, then P is positive recurrent and $\pi$ is the unique invariant distribution of P [23] (see Proposition 1).

On the other hand, the chain created by the construction algorithm will also be aperiodic as long as at least one pair of configurations $(x,y)$ exist, such that $a\left(x,y\right)<1$, because then we will have $P\left(x,x\right)>0$. We can quickly see that the chain is aperiodic as the event $X_{\left(N+1\right)}=X_{\left(N\right)}$ is possible practically at any time. Indeed, each state can be visited at two consecutive iterations, so $P^1\left(x,x\right)>0$, making the period equal to 1. □

Theorem 1.

The Markov chain ${\left(X_N\right)}_{N\ge 0}$ obtained from the proposed construction algorithm is geometrically ergodic, and its kernel P realizes the simulation of the area-interaction point processes with density $\pi \left(x\right)=\alpha {\beta }^{n\left(x\right)}{\gamma }^{-m\left(U_r\left(x\right)\right)}$. In other words, $v{P}^n$ converges to π as n approaches infinity, where v is an initial distribution, and we have:

$$\underset{n\to \infty }{lim}{∥v{P}^n-\pi ∥}_{TV}=0$$

Here, ${∥·∥}_{TV}$ denotes the total variation norm [24] defined by:

$${∥\mu ∥}_{TV}=\underset{\left|f\right|<1}{sup}\left|\mu \left(f\right)\right|$$

Proof. 

Let v be an initial distribution. For any integer n and $\forall x\in N^{lf}$, we have:

$${∥v{P}^n-\pi ∥}_{TV}={∥v{P}^n-\pi P^n∥}_{TV}\le ∥v-\pi ∥.\parallel P^n\parallel \le 2c\left(P^n\right)$$

And we know that $c\left(P^n\right)\le {\left[c\left(P\right)\right]}^n$ [25] (see Lemma 4.2.2 page 71), so ${∥v{P}^n-\pi ∥}_{TV}\le 2{\left[c\left(P\right)\right]}^n$. And $c\left(P\right)$ is the Dobrushin contraction coefficient [26] defined by the following:

$$c\left(P\right)={\displaystyle \frac{1}{2}}\underset{x,y}{sup}{∥P\left(x,.\right)-P\left(y,.\right)∥}_{TV}$$

According to Proposition 5.1, the kernel P is primitive, so $0\le c\left(P\right)<1$ [25] (see Lemma 4.2.3 p. 72). Therefore, as n tends to infinity, ${∥v{P}^n-\pi ∥}_{TV}$ tends to zero. Hence, the chain is geometrically ergodic and converges to the distribution $\pi \left(x\right)=\alpha {\beta }^{n\left(x\right)}{\gamma }^{-m\left(U_r\left(x\right)\right)}$. □

6. Convergence Speed of the Proposed Algorithm

The objective of this section is to study an approach to estimate the speed at which the Markov kernel $P={P_M}_H$, which is $\pi$-reversible, converges to the distribution $\pi$.

Definition 7.

A Markov kernel K on Ω is a mapping from Ω to Borel measures on Ω such that:

• For every $x\in \Omega$, $K(x,dy)$ is a probability measure.
• For any $\mathcal{A}$, $K(x,A)=\underset{A}{\int }K\left(x,dy\right)$ is a measurable function.
• For $f\in L^{\infty }\left(\Omega \right)$, the Markov operator $K\left(f\right)$ is defined as follows:

  $$K\left(f\right)\left(x\right)=\underset{\Omega }{\int }f\left(y\right)K\left(x,dy\right)$$

(Here, we denote $L^{\infty }\left(\Omega \right)$ as the space of bounded functions on Ω, equipped with the norm ${∥f∥}_{\infty }={\displaystyle \underset{x\in \Omega }{sup}}\left|f\left(x\right)\right|$).

Definition 8.

Let π be a probability measure and a Markov operator, we say that is π-reversible, if and only if it is self-adjoint on, which means that:

$${〈f|g〉}_{\pi }=\int K\left(f\right)gd\pi =\int fK\left(g\right)d\pi ={〈g|f〉}_{\pi },\forall f,g,\pi \in L^{\infty }\left(\Omega \right)$$

Theorem 2.

The Markov chain ${\left(X_n\right)}_{n\in \mathbb{N}}$ associated with P in the proposed algorithm satisfies the following properties for all n: P is irreducible and admits a reversible measure π. Therefore, the eigenvalues ${\lambda }_1>{\lambda }_2\cdots >{\lambda }_N$ of P satisfy ${\lambda }_1=1$. Additionally, for any initial distribution ${\pi }_0$, we have:

$${∥{\pi }_n-\pi ∥}_{TV}\le C{e}^{-\rho n}\mathit{with}C={\displaystyle \frac{1}{2}}\sqrt{\underset{x\in \Omega }{max}\left({\displaystyle \frac{1}{\pi \left(x\right)}}-1\right)}$$

where $\rho =min\left(1-\left|{\lambda }_2\right|,1-\left|{\lambda }_N\right|\right)$ is the spectral gap of P, and ${\pi }_n={\pi }_0{P}^n$ represents the marginal distribution of ${\left(X_n\right)}_{n\in \mathbb{N}}$, and P is exponentially ergodic and converges exponentially fast towards the target distribution π.

Proof. 

According to Proposition 5.1, P is $\pi$-irreducible, $\pi$-reversible, and primitive. By the Perron–Frobenius theorem [27], P has a strictly positive real eigenvalue ${\lambda }_1$, and ${\lambda }_1>{\lambda }_2\cdots >{\lambda }_N$. We need to show that these eigenvalues are contained in the interval $\left[-1,1\right]$.

Let $\mu$ be an eigenvector of $\lambda$, then we have the following:

$$\left|{〈\mu |\mu P〉}_{{\displaystyle \frac{1}{\pi }}}\right|=\left|\int \int \mu \left(x\right)\mu \left(y\right)P\left(x,y\right){\displaystyle \frac{dxdy}{\pi \left(x\right)}}\right|=\left|\int \int \pi \left(y\right){\displaystyle \frac{\mu \left(x\right)}{\pi \left(x\right)}}{\displaystyle \frac{\mu \left(y\right)}{\pi \left(y\right)}}P\left(x,y\right)dxdy\right|$$

$$\le {\displaystyle \frac{1}{2}}\int \int \pi \left(y\right){\left({\displaystyle \frac{\mu \left(x\right)}{\pi \left(x\right)}}\right)}^{2}P\left(x,y\right)dxdy+{\displaystyle \frac{1}{2}}{\int \int \pi \left(y\right)\left({\displaystyle \frac{\mu \left(y\right)}{\pi \left(y\right)}}\right)}^{2}P\left(x,y\right)dxdy$$

$$\le {\displaystyle \frac{1}{2}}\int \pi \left(x\right){\left({\displaystyle \frac{\mu \left(x\right)}{\pi \left(x\right)}}\right)}^{2}dx+{\displaystyle \frac{1}{2}}\int \pi \left(y\right){\left({\displaystyle \frac{\mu \left(y\right)}{\pi \left(y\right)}}\right)}^{2}dy\le \int {\left(\mu \left(x\right)\right)}^2{\displaystyle \frac{dx}{\pi \left(x\right)}}={〈\mu |\mu 〉}_{{\displaystyle \frac{1}{\pi }}}$$

And as $\mu$ is an eigenvector of $\lambda$, we have ${\left({∥\mu ∥}_{{\displaystyle \frac{1}{\pi }}}\right)}^2\ge \left|\lambda \right|{\left({∥\mu ∥}_{{\displaystyle \frac{1}{\pi }}}\right)}^2$, which implies $1\ge \left|\lambda \right|$. Therefore, we have:

$$1\ge {\lambda }_1>{\lambda }_2\cdots >{\lambda }_N\ge -1$$

If $\lambda =1$, then:

$${\displaystyle \frac{\mu \left(x\right)}{\pi \left(x\right)}}{\displaystyle \frac{\mu \left(y\right)}{\pi \left(y\right)}}={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mu \left(x\right)}{\pi \left(x\right)}}\right)}^2+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mu \left(y\right)}{\pi \left(y\right)}}\right)}^2\Rightarrow {\displaystyle \frac{\mu \left(x\right)}{\pi \left(x\right)}}={\displaystyle \frac{\mu \left(y\right)}{\pi \left(y\right)}}$$

That is, $\mu =c\pi$, with c being a constant.

Reciprocally, from the definition of an invariant measure, $\pi$ indeed satisfies $\pi =\pi P$. Thus, we have established that the eigenvalue 1 is a simple eigenvalue, which we denote as ${\lambda }_1=1$.

Now, let us show that $-1$ is not an eigenvalue of P. We assume by contradiction that there exists an eigenvector f for the eigenvalue $-1$. Let x be a state in the state space $\Omega$, such that $f\left(x\right)>0$ and $f\left(x\right)={\displaystyle \underset{y\in \Omega }{max}}\left|f\left(y\right)\right|$ (we can take $-f$ instead of f if necessary). As P is aperiodic according to Proposition 5.1, for sufficiently large n, $P^n(x,x)>0$ [28] (see Lemma 3.15, page 39). We have:

$$\begin{array}{cc}& -f\left(x\right)=P^n\left(f\right)\left(x\right)=\int P^n\left(x,y\right)f\left(y\right)dy\hfill \\ & >-\int P^n\left(x,y\right)\left|f\left(y\right)\right|dy\hfill \\ & >-\int P^n\left(x,y\right)f\left(x\right)dy=-f\left(x\right)\hfill \end{array}$$

This implies that $-f\left(x\right)>-f\left(x\right)$. It is a contradiction.

Finally, let us show that ${∥{\pi }_n-\pi ∥}_{TV}\le C{e}^{-\rho n}$. Using the properties of the total variation norm [24], we have the following:

$$2{∥{\pi }_n-\pi ∥}_{TV}={∥{\pi }_n-\pi ∥}_1\le {∥{\pi }_n-\pi ∥}_2$$

$$={\left(\int {\left({\displaystyle \frac{{\pi }_n\left(x\right)}{\pi \left(x\right)}}-1\right)}^2\pi \left(x\right)dx\right)}^{{\displaystyle \frac{1}{2}}}\le {\left(\int {\left({\pi }_n\left(x\right)-{\pi }_n\left(x\right)\right)}^2{\displaystyle \frac{dx}{\pi \left(x\right)}}\right)}^{{\displaystyle \frac{1}{2}}}={∥{\pi }_n-\pi ∥}_{{\mathcal{L}}^2\left(\Omega ,{\displaystyle \frac{1}{\pi }}\right)}$$

As P is a symmetric operator for its right action on ${\mathcal{L}}^2\left(\Omega ,{\displaystyle \frac{1}{\pi }}\right)$, there exists an orthonormal basis $\left({\mu }_1,{\mu }_2,\cdots ,{\mu }_N\right)$ of this space corresponding to the eigenvalues $\left(1,{\lambda }_1,\cdots {\lambda }_N\right)$. The orthonormal eigenvector ${\mu }_1$ for the eigenvalue 1 is simply the invariant measure $\pi$. Let us write the decomposition of the initial distribution ${\pi }_0$ in the eigenbasis of P:

$${\pi }_0=c\pi +\sum _{i=2}^N{c}_i{\mu }_i$$

For certain real coefficients c and $c_i$, $i\in \left[\left[2,N\right]\right]$, with $c_i={〈{\pi }_0|{\mu }_i〉}_{{\displaystyle \frac{1}{\pi }}}$, applying $P^n$ to the formula above yields:

$${\pi }_n=c\pi +\sum _{i=2}^N{c}_i{\left({\lambda }_i\right)}^n{\mu }_i$$

As $\left|{\lambda }_i\right|<1$ for $i\in \left[\left[2,N\right]\right]$, then $\underset{n\to \infty }{lim}{\pi }_n=c\pi +\underset{n\to \infty }{lim}{\displaystyle \sum _{i=2}^N}{c}_i{\left({\lambda }_i\right)}^n{\mu }_i=c\pi$, which implies that $c=1$. Therefore:

$${\pi }_n-\pi =\sum _{i=2}^N{c}_i{\left({\lambda }_i\right)}^n{\mu }_i$$

and

$${\left({∥{\pi }_n-\pi ∥}_{{\displaystyle \frac{1}{\pi }}}\right)}^2=\sum _{i=2}^N{c}_i^2{\left({\lambda }_i\right)}^{2n}\le \sum _{i=2}^N{c}_i^2{\left(1-\rho \right)}^{2n}\le {\left({∥{\pi }_0-\pi ∥}_{{\displaystyle \frac{1}{\pi }}}\right)}^2{e}^{-2n\rho }$$

With $\rho =min\left(1-\left|{\lambda }_2\right|,1-\left|{\lambda }_N\right|\right)$ To conclude, we need to bound the value ${\left({∥{\pi }_0-\pi ∥}_{{\displaystyle \frac{1}{\pi }}}\right)}^2$, which we denote as $F\left({\pi }_0\right)$.

$$F\left({\pi }_0\right)={\left({∥{\pi }_0-\pi ∥}_{{\displaystyle \frac{1}{\pi }}}\right)}^2=\int {\left({\pi }_0\left(x\right)-\pi \left(x\right)\right)}^2{\displaystyle \frac{dx}{\pi \left(x\right)}}$$

The function F is quadratic and, therefore, convex over a convex part of ${\mathbb{R}}^N$. Hence, it reaches its maximum at an extreme point of the convex set formed by probability measures. The extreme points of this convex set are precisely the Dirac measures ${\delta }_{x_0}$ (convex combination of Dirac measures). Thus,

$$\underset{{\pi }_0\mathrm{probability}\mathrm{on}\Omega }{max}{\left({∥{\pi }_0-\pi ∥}_{{\displaystyle \frac{1}{\pi }}}\right)}^2=\underset{x_0\in \Omega }{max}{\left({∥{\delta }_{x_0}-\pi ∥}_{{\displaystyle \frac{1}{\pi }}}\right)}^2$$

For fixed $x_0$, we have the following:

$${\left({∥{\delta }_{x_0}-\pi ∥}_{{\displaystyle \frac{1}{\pi }}}\right)}^2={\displaystyle \frac{{\left(1-\pi \left(x_0\right)\right)}^2}{\pi \left(x_0\right)}}+\underset{\Omega }{\int }\pi \left(x\right){\delta }_{x\ne x_0}dx$$

$$={\displaystyle \frac{{\left(1-\pi \left(x_0\right)\right)}^2}{\pi \left(x_0\right)}}+\left(1-\pi \left(x_0\right)\right)={\displaystyle \frac{1+{\left(\pi \left(x_0\right)\right)}^2-2\pi \left(x_0\right)+\pi \left(x_0\right)-{\left(\pi \left(x_0\right)\right)}^2}{\pi \left(x_0\right)}}$$

$$={\displaystyle \frac{1-\pi \left(x_0\right)}{\pi \left(x_0\right)}}={\displaystyle \frac{1}{\pi \left(x_0\right)}}-1$$

We conclude that ${∥{\pi }_n-\pi ∥}_{TV}\le C{e}^{-\rho n}$ with $C={\displaystyle \frac{1}{2}}\sqrt{\underset{x\in \Omega }{max}\left({\displaystyle \frac{1}{\pi \left(x\right)}}-1\right)}$. □

7. Numerical Results and Quality of the Proposed Disigns

In this section, we conduct a comparison of the point distributions in the suggested computer experiment designs by employing established criteria. This evaluation aims to assess the extent to which the experimental space is effectively covered and the level of uniformity in the distribution of points.

• The minimum distance criterion (Mindist) [29] aims to maximize the minimum distance between two points in the design.

  $$Mindist=\underset{i}{min}\underset{j\ne i}{min}d(x_i,x_j)$$

  where $d(x_i,x_j)$ is the Euclidean distance between points $x_i$ and $x_j$. A higher value of Mindist should correspond to a more regular dispersion of the points in the design.
• Discrepancy criterion (Disc) [30]: The discrepancy measures the difference between the empirical distribution function of the points on the design and that of the uniform distribution. Unlike the previous two criteria, the discrepancy is not based on the distance between points. There are different measures of discrepancy. We retain the $L_2$-norm discrepancy.

  $$Disc={\left({\displaystyle \frac{1}{3}}\right)}^p-{\displaystyle \frac{2^{1-p}}{n}}\sum _{i=1}^n\prod _{j=1}^p\left(1-{\left(x_i^j\right)}^2\right)+{\displaystyle \frac{1}{n^2}}\sum _{i=1}^n\sum _{k=1}^n\prod _{j=1}^p\left(1-max\left(x_i^j,x_k^j\right)\right)$$
• Coverage criterion (Cov) [31]: allows us to measure the difference between the points of the design and those of a regular grid. This criterion is zero for a regular grid. The objective is, therefore, to minimize it to approach a regular grid and, thus, ensure space-filling, without reaching it to respect a uniform distribution, particularly in projection onto the factorial axes:

  $$Cov={\displaystyle \frac{1}{\overline{\delta }}}\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left({\delta }_i-\overline{\delta }\right)}^2}$$
  With ${\delta }_i=\underset{i\ne j}{min}\left(d(x_i,x_j)\right)$ and $\overline{\delta }={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\delta }_i$.
• The R criterion is the ratio between the maximum and minimum distance between the points of the experimental design. For a regular grid, $R=1$. Thus, the closer R is to 1, the closer the points are to those of a regular grid.

  $$R={\displaystyle \frac{{\displaystyle \underset{i\in \left\{1,\dots ,n\right\}}{max}}{\delta }_i}{{\displaystyle \underset{i\in \left\{1,\dots ,n\right\}}{min}}{\delta }_i}}$$

  where ${\delta }_i={\displaystyle \underset{i\ne j}{min}}\left(d(x_i,x_j)\right)$.

Table 1. The values of discrepancy for the proposed designs Area Interaction, Halton sequences, Sobol sequences, and faure sequence for tow and three dimensions.

Number of	Number of	AID	Halton	Sobol	Faure
Factor	Points		Sequence	Sequence	Sequence
2	50	0.0005215	0.001076	0.000496	0.001076
3	100	0.0009192	0.000178	0.000112	0.000127

compares the discrepancy criterion between the designs proposed in this work (referred to as AID: area interaction designs) and several well-known low-discrepancy sequences: the Halton sequence [32] (a quasi-random sequence used for numerical integration), the Sobol sequence [33] (a quasi-random sequence recognized for its low discrepancy properties), and the Faure sequence [34] (a sequence designed to achieve uniform coverage of the space). It is noteworthy that the AID designs exhibit a low discrepancy, comparable to that of these established low-discrepancy sequences.

To evaluate the quality of a numerical experiment design, it is crucial to employ standard criteria that ensure effective space-filling and uniform distribution. The objective of this section is to calculate the values of these criteria for the proposed designs and a number of commonly used designs in computer experiments, known for their good quality of uniformity and/or space filling.

Here are some definitions

• Random Designs (RD): Designs where points are generated according to a uniform distribution over the hypercube ${[0,1]}^p$.
• Latin Hypercube Designs (LH): LH designs are experimental design techniques aimed at sampling the parameter space efficiently and uniformly [35].
• Maximin Latin Hypercube Designs (mLHS): Optimal designs based on the Maxmin criterion, which seeks to maximize the minimum distance between points in the design space [36].
• Maximum Entropy Designs (Dmax): Designs constructed to maximize the determinant of a covariance matrix. These designs are often employed in kriging to fit response surfaces, assuming an underlying model [37].
• Strauss Designs (SD): Designs developed from a Strauss process that incorporates repulsion between points to optimize the coverage of the parameter space [3].
• Marked Strauss Designs: Designs generated from a marked Strauss process, integrating point repulsion to optimize the parameter space coverage while associating each point with a specific mark aimed at minimizing the prediction error function [4].
• Connected Component Designs (CCD): Designs developed from a Markov point process with connected components, characterized by spatial distributions that are more or less regular without constraints on the parameters [6].
• Proposed Designs (AID).

The Figure 6 and Figure 7 represent the results of various criteria in the form of box plots for two and three dimensions.

According to the results presented in the box plots above, in two dimensions, the proposed designs show significant improvements in both the ratio and minimum distance criteria. This indicates that the points in the proposed designs are well-separated from each other, which ensures good coverage of the experimental space, although there is not an optimal overlap according to the overlap criterion. When examining the discrepancy criterion, it can be confirmed that the proposed designs generate points that are more uniformly distributed within the unit hypercube compared to designs of the same category, such as Strauss designs, marked Strauss designs, and Connected Component Designs. The proposed designs are also better than conventional Latin Hypercube Designs (LHD) and Maximin Latin Hypercube Designs in terms of the distance criteria. In two dimensions, they seem to be a good compromise between a well-distributed point set in the space and its projection onto the margins.

In three dimensions, the proposed designs and Latin Hypercube Designs are better than other designs in terms of distance criteria. In three dimensions, they visibly achieve a point distribution that is closer to a regular grid, ensuring good coverage of the space. It is surprising that the proposed designs do not perform as well in terms of discrepancy for three dimensions, especially compared with their performance in two dimensions. This can be explained by a poor choice of interaction radii, which is compensated for by the potential power, avoiding close points according to the usual distance. Thus, in three dimensions, the proposed designs generally have the worst discrepancy, but they stand out as the best in terms of overlap criteria.

8. Conclusions

The use of area interaction point processes, the MCMC method, and the Metropolis–Hastings algorithm with Voronoi tessellations enables the construction of new designs for computer experiments that are specific to this process. This approach offers significant flexibility, as it allows for the manipulation of the process’s governing law through its representation to impose properties such as optimal space filling and a uniform distribution of points.

Furthermore, this experimental design method, combined with area interaction point processes and the MCMC approach, presents a promising alternative to the classical statistical approach of working with independent realizations of the same distribution. The designs developed in this study have been compared with commonly used designs in computer experiments, demonstrating very satisfactory results.

However, several important limitations and challenges must be addressed. Firstly, calculating the hyper-volume of the union of fixed-radius hyper-balls is complex and often requires approximate numerical methods, such as Monte Carlo simulations. Additionally, simulating points within a polytope is challenging due to the geometric complexity of the regions defined by the constraints of the polytope generated by Voronoi tessellations. Moreover, modeling infinite regions in these tessellations with finite approximations can introduce errors into experimental results.

Finally, there are several promising avenues for future research. For example, exploring issues related to points within the infinite regions of Voronoi tessellations in a closed hypercube could be valuable [38]. Additionally, finding efficient numerical methods for calculating the hyper-volume of the union of fixed-radius hyper-balls represents a crucial research direction for advancing these techniques.