MESLAS: Multivariate Excursion Set Learning by Adaptive Sampling

The goal of the MESLAS package is to provide functionalities for simulation and estimation of physical phenomenon using multivariate random fields.

As an example, consider the Temperature and Salinity in the ocean. These two variables can be modelled as a bivariate Gaussian random field (GRF) with some correlation between the outputs. One can then draw realizations of the GRF to get plausible spatial distributions of oceanographic properties (see Figure).

_images/uncond_real_cropped.png

Simulated realization of Temperature and Salinity field drawn from a GRF model.

Having a probabilistic model of the physical phenomenon of interest allows one to extract more information than a purely deterministic model would allow. For example, the question that triggered the development of MESLAS was the indentification of river plumes in coastal regions ([FTE+20]).

Excursion Sets

One of the main functionality of MESLAS is the estimation of excursion sets of GRFs. Given an \(n\)-dimensional multivariate GRF \(\boldsymbol{Z}\), an excursion set is defined as

\[\Gamma = \lbrace x \in X: t_i \leq \boldsymbol{Z}_x^{i} \leq t'_i,~i=1,...,n \rbrace\]

where the thresholds are allowed to be infinite.

In the context of river plume indentification, the river can be modelled as an excursion set where the salinity and temperature are below a certain threshold.

Data Collection and Conditional Distribution

Data is integrated to the GRF model by considering the conditional distribution of the GRF, conditional on the data. Excursion sets may then be estimated by looking at the pointwise excursion probability, i.e. the probability (under the conditional distribution) that a point lies in the excursion set.

_images/excu_proba_cropped.png

Conditional pointwise excursion probability. Data collection locations in green.

Autonomous Adaptive Sampling

The main application for the MESLAS package is adaptive sampling for (multivariate) excursion set recovery. Considering again the temperature/salinity river plume example, one can use an autonomous underwater vehicle (AUV) to collect temperature and salinity data in the coastal region. The data collection process may then be guided using a (multivariate) GRF model.

The above video shows a simulated autonomous data collection process for the river plume example computed using MESLAS. The data collection locations are chosen by myopic optimization of the integrated Bernoulli variance criterion (IBV) available in MESLAS (see [FTE+20] for more details).

The left-hand plot shows the (simulated) ground truth. One is interested in recovering the red regions. The right-hand plot shows the data collection path travelled by the AUV, computed by myopic optimization of the IBV criterion and the posterior excursion probability (the probability for each point to belong to the target excursion set under the posterior law of the GRF model after data collection). The radar shows how informative the potential next sampling locations are, ranked according to IBV.

(For implementation, see Sensor Module.)

Implementation

MESLAS aims at providing GRF functionalities that are easy to use, fully modular and reusable.

For example, the GRF \(\boldsymbol{Z}\) used in Fig.1 with linear trend

\begin{align} \boldsymbol{\mu}(\boldsymbol{x}) &= \boldsymbol{\beta}_i + \boldsymbol{\beta}_1^t \boldsymbol{x} \end{align}

and factor-covariance model

\begin{align} Cov\left[\boldsymbol{Z}_x^i, \boldsymbol{Z}_y^j\right] &= k(x, y) \gamma(i, j)\\ \end{align}

with Martern 3/2 spatial covariance, and uniform mixing covariance defined by

\begin{align} \gamma(i, j) &= \begin{cases} \sigma_l^2,~ i=j\\ γ_0σ_iσ_j,~i≠ j \end{cases} \end{align}

is straightforward to define in MESLAS and to sample from

# Spatial Covariance.
matern_cov = Matern32(lmbda=0.5, sigma=1.0)

# Cross covariance.
cross_cov = UniformMixing(gamma0=0.2, sigmas=[2.25, 2.25])
covariance = FactorCovariance(matern_cov, cross_cov, n_out=n_out)

# Specify mean function, here it is a linear trend that decreases with the
# horizontal coordinate.
beta0s = np.array([7.8, 24.0])
beta1s = np.array([
        [0, -7.0],
        [0, -5.0]])
mean = LinearMean(beta0s, beta1s)

# Create the GRF.
myGRF = GRF(mean, covariance)

# Create an equilateral tringular grid, with 80 nodes per line.
my_grid = TriangularGrid(80)

# Sample all components (isotopic) at all locations of the grid.
ground_truth = myGRF.sample_isotopic(my_grid.points)

More information about the special concepts and notation which are used in MESLAS to handle multivariate random fields may be found at Notations and Conventions for the MESLAS package.

Main Modules:

[FTE+20]

Trygve Olav Fossum, Cédric Travelletti, Jo Eidsvik, David Ginsbourger, and Kanna Rajan. Learning excursion sets of vector-valued gaussian random fields for autonomous ocean sampling. 2020. arXiv:2007.03722.