A joint model-based design of experiments approach for the identification of Gaussian Process models in geological exploration

When searching for potential mining sites, accurately modelling mineral concentrations or rock qualities in the subsurface is a crucial task. However, drilling in these locations is an expensive process, so reliable interpolation and efficient sampling techniques are required (Rossi & Deutsch, 2014). Gaussian Processes (GPs), also known as Kriging models, were first developed in the mining industry in the 1950s and continue to be widely used in resource modelling (Sahimi, 2011). As the true nature of the subsurface is unknown, assumptions must be made about the kernel function, which describes correlation structures between probable distributions of spatial phenomena, and its parameters must be estimated. This is typically accomplished through expert judgement and exploratory data analysis of preliminary samples. Model predictions are updated iteratively as more drilling data becomes available, with a focus on balancing expected exploitation (high grade intercepts) and exploration (minimising the Kriging variance) (Jafrasteh & Suarez, 2020). However, problems can arise if the chosen kernel is incorrect or if high uncertainty affects parameters. This poster showcases a joint model-based design approach (Galvanin et al., 2016) aiming to optimise three objectives: 1) reducing parametric uncertainty; 2) increasing the exploration of the design space to avoid local optima; 3) maximising the distinguishability of candidate model predictions to identify the most suitable kernel function with the minimum number of samples. Two different kernels in an Ordinary Kriging GP were used as candidate models and in-silico data was generated using one kernel. Starting from some initial samples, the optimal design strategy iteratively determined sampling locations to maximise the distinguishability between model predictions with a constraint ensuring that each iteration reduces prediction variance. The correct model could be distinguished and the data approximated well with a limited number of drilling experiments while satisfactorily estimating kernel parameters

Safe model-based design of experiments using Gaussian processes

The construction of kinetic models has become an indispensable step in developing and scale-up of processes in the industry. Model-based design of experiments (MBDoE) has been widely used to improve parameter precision in nonlinear dynamic systems. Such a framework needs to account for both parametric and structural uncertainty, as the physical or safety constraints imposed on the system may well turn out to be violated, leading to unsafe experimental conditions when an optimally designed experiment is performed. In this work, Gaussian processes are utilized in a two-fold manner: 1) to quantify the uncertainty realization of the physical system and calculate the plant-model mismatch, 2) to compute the optimal experimental design while accounting for the parametric uncertainty. TheOur proposed method, Gaussian process-based MBDoE (GP-MBDoE), guarantees the probabilistic satisfaction of the constraints in the context of the model-based design of experiments. GP-MBDoE is assisted with the use of adaptive trust regions to facilitate a satisfactory local approximation. The proposed method can allow the design of optimal experiments starting from limited preliminary knowledge of the parameter set, leading to a safe exploration of the parameter space. This method’s performance is demonstrated through illustrative case studies regarding the parameter identification of kinetic models in flow reactors

Spatio‐temporal analysis of land use/land cover change dynamics in Paraguai/Jauquara Basin, Brazil

Data was collected from freely available images composites from the catalogs of the United States Geological Survey.Although global climate change is receiving considerable attention, the loss of biodiversity worldwide continues. In this study, dynamics of land use/land cover (LULC) change in the Paraguai/Jauquara Basin, Mato Grosso, Brazil, were investigated. Two analyses were performed using R software. The first was a comparative study of LULC among the LULC classes at the polygon scale, and the second was a spatio-temporal analysis of moving polygons restricted to the agricultural regions in terms of topology, size, distance, and direction of change. The data consisted of Landsat images captured in 1993, 1997, 2001, 2005, 2009, 2013, and 2016 and processed using ArcGIS software. The proposed analytical approach handled complex data structures and allowed for a deeper understanding of LULC change over time. The results showed that there was a statistically significant change from regions of natural vegetation to pastures, agricultural regions, and land for other uses, accompanied by a significant trend of expansion of agricultural regions, appearing to stabilize from 2005. Furthermore, different patterns of LULC change were found according to soil type and elevation. In particular, the purple latosol soil type presented the highest expansion indexes since 2001, and the elevated agricultural areas have been expanding and/or stabilizing since 1997.This work is part of the results of the research projects PTDC/MAT-STA/28243/2017 funded by the FCT (Fundação para a Ciência e Tecnologia) and Analise temporal do uso da terra para definição de cenários de mudança da paisagem natural por intervenções de natureza humana no Pantanal de Caceres/MT funded by Fundação de Amparo a Pesquisa do Estado de Mato Grosso-FAPEMAT. The first author also acknowledges Foundation FCT (Fundação para a Ciência e Tecnologia) for funding this research through Individual Scholarship Ph.D. PD/BD/150535/2019

Mechanistic modelling of separating dispersions in pipes using model-based design of experiments techniques

This work presents a parametric study on a mechanistic model for separating liquid–liquid dispersions in pipes. The model considers drop-settling, drop-interface coalescence and drop-drop coalescence, predicting the evolution of four characteristic layers during separation. Parameter estimation, parametric sensitivity analysis (PSA), and model-based design of experiments (MBDoE) techniques are employed to acquire precise parameter estimates and propose optimal experimental conditions, thereby enhancing the accuracy of existing models. Experimental data from literature using oil-in-water dispersions are used for parameter estimation. PSA reveals regions of high sensitivity of the model outputs to uncertain parameters, which are corresponding to favourable sampling locations. Manipulating the mixture velocity, the dispersed phase fraction, and the layer heights at the inlet influences these sensitive regions. Clustered measurements around highly sensitive regions in the pipe enhance the information content they provide. MBDoE demonstrates that either of the A-, D-, or E-optimal experimental design criteria improves the expected parameter precision

Model-based design of experiments in the presence of structural model uncertainty: an extended information matrix approach

The identification of a parametric model, once a suitable model structure is proposed, requires the estimation of its non-measurable parameters. Model-based design of experiment (MBDoE) methods have been proposed in the literature for maximising the collection of information whenever there is a limited amount of resources available for conducting the experiments. Conventional MBDoE methods do not take into account the structural uncertainty on the model equations and this may lead to a substantial miscalculation of the information in the experimental design stage. In this work, an extended formulation of the Fisher information matrix is proposed as a metric of information accounting for model misspecification. The properties of the extended Fisher information matrix are presented and discussed with the support of two simulated case studies

An evolutionary approach to kinetic modelling inspired by Lamarckian inheritance

The mechanistic description of kinetic phenomena requires the construction of systems of differential and algebraic equations where a high number of parameters and state variables may be involved. The complexity associated with kinetic phenomena frequently leads to the construction of kinetic models characterised by some degree of approximation. Whenever an approximated model is falsified by observations, its mathematical structure should be evolved embracing the available experimental evidence [1]. Nonetheless, improving a model is a time and resource intensive task that heavily relies on the presence of experienced researchers. An evolutionary approach to kinetic modelling is proposed in this work which is inspired by the theory of evolution proposed by Jean-Baptiste Lamarck, namely the theory of Lamarckian inheritance [2]. The approach is illustrated qualitatively in the sketch in Figure 1. Lamarck states that the evolution of living beings is directly driven by their interaction with the environment. The use/disuse of an organ determines the evolution of that organ towards higher/lower complexity. The long neck of the giraffe is frequently reported as an example to explain Lamarck’s theory. Primitive giraffes with a short neck would strive to reach the highest leaves. This behaviour, driven by necessities of adaptation, would lead to an elongation of the neck during the giraffe’s lifetime and the characteristic of the long neck is inherited by the offspring. Lamarck’s theory is now widely dismissed. Nonetheless, the field of epigenetics stemmed directly from Lamarck’s philosophy and aims at explaining the complex mechanisms behind the hereditability of environment-driven phenotype changes [3]. In this work, the principles of Lamarckian inheritance are translated into a framework for kinetic model building. In the proposed framework, the evolution of model structures is data-driven. When the model is over-fitting, model parameters that are irrelevant for representing the data are removed from the model structure. When the model is under-fitting, relevant model parameters are evolved into more complex state-dependent expressions. A statistical index, namely a Model Modification Index (MMI), based on the Lagrange multipliers statistic [4], is proposed as a measure of model misspecification to support the evolution of approximated kinetic models towards higher levels of complexity. The use of the MMI is demonstrated in a simulated case study to diagnose misspecification in an approximated kinetic model of baker’s yeast growth [5]. Please click Additional Files below to see the full abstract

A diagnostic procedure for improving the structure of approximated kinetic models

Kinetic models of chemical and biochemical phenomena are frequently built from simplifying assumptions. Whenever a model is falsified by data, its mathematical structure should be modified embracing the available experimental evidence. A framework based on maximum likelihood inference is illustrated in this work for diagnosing model misspecification and improving the structure of approximated models. In the proposed framework, statistical evidence provides a measure to justify a modification of the model structure, namely a reduction of complexity through the removal of irrelevant parameters and/or an increase of complexity through the replacement of relevant parameters with more complex state-dependent expressions. A tailored Lagrange multipliers test is proposed to support the scientist in the improvement of parametric models when an increase in model complexity is required