Notations and Conventions for the MESLAS package

The MESLAS package (Multi-variate Excursion Set Learning by Adaptive Sampling) is a toolbox for simulation and prediction of mulitivariate Gaussian random fields.

The setup of the package is the following: \(Z\) is a \(p\)-dimensional random field on a \(d\)-dimensional domain \(D\).

Our philosophy is to always specify spatial location and response indices together. That is, one should always specify where and what.

Spatial locations are denoted by \(s\) and response indices by \(\ell\). We will use boldface (or, in the code, alternatively uppercase or plurals) to denote vectors of such objects.

A generalized sampling location is thus entirely defined by specifying two vectors

\begin{align} \boldsymbol{s} &= \left(s_1, ..., s_n\right)\in D^n\\ \boldsymbol{\ell} &= \left(\ell_1, ..., \ell_n\right) \in \lbrace 1, ..., p \rbrace^n \end{align}

We will refer to \(n\) as the dimension of the generalized sampling location and usually just talk of location, using the word spatial location when we want to specifically refer to points in \(D\). Also, we will use boldface \(x\) as a shortcut to refer to the couple \(\left(\boldsymbol{s}, \boldsymbol{\ell}\right)\) of spatial location vector and response index vector. The shortcut notation \(Z_{\boldsymbol{x}}\) thus refers to the vector

\[Z_{\boldsymbol{x}}:=\left(Z_{s_1}^{\ell_1}, ..., Z_{s_n}^{\ell_n}\right) \in \mathbb{R}^n.\]

Generalized Vectors

Say we have a \(p\)-dimensional random field \(Z\), which we observe at \(n\) different spatial locations \(s_1,...,s_n\). Then the most natural way to organize the observation in a single object is to bundle them into a \(n\times p\) dimensional vector

\[\begin{split}\begin{pmatrix} Z^1_{s_1} & \dots & Z^p_{s_1}\\ \vdots & & \vdots\\ Z^1_{s_n} & \dots & Z^p_{s_n} \end{pmatrix}.\end{split}\]

We will refer to such a (generalized) vector as a vector in isotopic form. This form might seem the most natural one, but it is not always the most appropriate. For example, when sampling such a field \(Z\) at the different locations \(s_1,...,s_n\), correlations between all loations and all response indices have to be considered. It is thus easier to organise the responses in a one dimensional vector

\[\begin{pmatrix} Z^1_{s_1}& \dots & Z^p_{s_1} & \dots & Z^1_{s_n} & \dots & Z^p_{s_n} \end{pmatrix}\]

which can then be multiplied by the appropriate covariance matrix. We call this the list form.

One should note the that the way in which the reordering is performed does not matter, as long as it is consistent throughout. We thus won’t ever mention it again.

In the same fashion, say we want to consider the correlations between a \(p\) dimensional field \(Z\) at locations \(s_1,...,s_n\) and a \(q\) dimensional field \(Y\) at locations \(t_1,...,t_m\).

Then, for sampling purposes, the most natural way to arrange the covariance information is in list form

\[\begin{split}\begin{pmatrix} Cov(Z^1_{s_1}, Y^1_{t_1}) & \dots & Cov(Z^1_{s_1}, Y^q_{t_1}) & \dots & Cov(Z^1_{s_1}, Y^1_{t_m}) & \dots & Cov(Z^1_{s_1}, Y^q_{t_m})\\ \vdots & & & & & & \vdots \\ Cov(Z^p_{s_1}, Y^1_{t_1}) & \dots & Cov(Z^p_{s_1}, Y^q_{t_1}) & \dots & Cov(Z^p_{s_1}, Y^1_{t_m}) & \dots & Cov(Z^p_{s_1}, Y^q_{t_m})\\ \vdots & & & & & & \vdots \\ Cov(Z^1_{s_n}, Y^1_{t_1}) & \dots & Cov(Z^1_{s_n}, Y^q_{t_1}) & \dots & Cov(Z^1_{s_n}, Y^1_{t_m}) & \dots & Cov(Z^1_{s_n}, Y^q_{t_m})\\ \vdots & & & & & & \vdots \\ Cov(Z^p_{s_n}, Y^1_{t_1}) & \dots & Cov(Z^p_{s_n}, Y^q_{t_1}) & \dots & Cov(Z^p_{s_n}, Y^1_{t_m}) & \dots & Cov(Z^p_{s_n}, Y^q_{t_m}) \end{pmatrix}.\end{split}\]

But when considering, for example, pointwise excursion probability, it makes sense to have covariance matrices between couple of points. I.e., to organise covariance information into a \(n\times m \times p \times q\) matrix, such that at index \(i,j\) we have a \(p\times q\) matrix of covariances between spatial locations \(s_i\) and \(t_j\).