Statistical abstraction for multi-scale spatio-temporal systems

Modelling spatio-temporal systems exhibiting multi-scale behaviour is a powerful tool in many branches of science, yet it still presents significant challenges. Here, we consider a general two-layer (agent-environment) modelling framework, where spatially distributed agents behave according to external inputs and internal computation; this behaviour may include influencing their immediate environment, creating a medium over which agent-agent interaction signals can be transmitted. We propose a novel simulation strategy based on a statistical abstraction of the agent layer, which is typically the most detailed component of the model and can incur significant computational cost in simulation. The abstraction makes use of Gaussian Processes, a powerful class of non-parametric regression techniques from Bayesian Machine Learning, to estimate the agent's behaviour given the environmental input. We show on two biological case studies how this technique can be used to speed up simulations and provide further insights into model behaviour

Statistical abstraction for multi-scale spatio-temporal systems

Spatio-temporal systems exhibiting multi-scale behaviour are common in applications ranging from cyber-physical systems to systems biology, yet they present formidable challenges for computational modelling and analysis. Here we consider a prototypic scenario where spatially distributed agents decide their movement based on external inputs and a fast-equilibrating internal computation. We propose a generally applicable strategy based on statistically abstracting the internal system using Gaussian Processes, a powerful class of non-parametric regression techniques from Bayesian Machine Learning. We show on a running example of bacterial chemotaxis that this approach leads to accurate and much faster simulations in a variety of scenarios.Comment: 14th International Conference on Quantitative Evaluation of SysTems (QEST 2017

Automated Deep Abstractions for Stochastic Chemical Reaction Networks

Predicting stochastic cellular dynamics as emerging from the mechanistic models of molecular interactions is a long-standing challenge in systems biology: low-level chemical reaction network (CRN) models give raise to a highly-dimensional continuous-time Markov chain (CTMC) which is computationally demanding and often prohibitive to analyse in practice. A recently proposed abstraction method uses deep learning to replace this CTMC with a discrete-time continuous-space process, by training a mixture density deep neural network with traces sampled at regular time intervals (which can obtained either by simulating a given CRN or as time-series data from experiment). The major advantage of such abstraction is that it produces a computational model that is dramatically cheaper to execute, while preserving the statistical features of the training data. In general, the abstraction accuracy improves with the amount of training data. However, depending on a CRN, the overall quality of the method -- the efficiency gain and abstraction accuracy -- will also depend on the choice of neural network architecture given by hyper-parameters such as the layer types and connections between them. As a consequence, in practice, the modeller would have to take care of finding the suitable architecture manually, for each given CRN, through a tedious and time-consuming trial-and-error cycle. In this paper, we propose to further automatise deep abstractions for stochastic CRNs, through learning the optimal neural network architecture along with learning the transition kernel of the abstract process. Automated search of the architecture makes the method applicable directly to any given CRN, which is time-saving for deep learning experts and crucial for non-specialists. We implement the method and demonstrate its performance on a number of representative CRNs with multi-modal emergent phenotypes

Generative abstraction of Markov population processes

Markov population models are a widespread formalism used to model the dynamics of complex systems, with applications in systems biology and many other fields. The associated Markov stochastic process in continuous time is often analyzed by simulation, which can be costly for large or stiff systems, particularly when a massive number of simulations has to be performed, e.g. in a multi-scale model. A strategy to reduce computational load is to abstract the population model, replacing it with a simpler stochastic model, faster to simulate. Here we pursue this idea, exploring and comparing state-of-the-art generative models, which are flexible enough to automatically learn distributions over entire trajectories, rather than single simulation steps, from observed realizations of the system. In particular, we compare a Generative Adversarial setting with a Score-based Diffusion approach and show how the latter outperforms the former both in terms of accuracy and stability at the cost of slightly higher simulation times. To improve the accuracy of abstract samples, we develop an active learning framework to enrich our dataset with observations whose expected satisfaction of a temporal requirement differs significantly from the abstract one. We experimentally show how the proposed abstractions are well suited to work on multi-scale and data-driven scenarios, meaning that we can infer a (black-box) dynamical model from a pool of real data.(c) 2023 Elsevier B.V. All rights reserved

Phenomenological modelling: statistical abstraction methods for Markov chains

Continuous-time Markov chains have long served as exemplary low-level models for an array of systems, be they natural processes like chemical reactions and population fluctuations in ecosystems, or artificial processes like server queuing systems or communication networks. Our interest in such systems is often an emergent macro-scale behaviour, or phenomenon, which can be well characterised by the satisfaction of a set of properties. Although theoretically elegant, the fundamental low-level nature of Markov chain models makes macro-scale analysis of the phenomenon of interest difficult. Particularly, it is not easy to determine the driving mechanisms for the emergent phenomenon, or to predict how changes at the Markov chain level will influence the macro-scale behaviour. The difficulties arise primarily from two aspects of such models. Firstly, as the number of components in the modelled system grows, so does the state-space of the Markov chain, often making behaviour characterisation untenable under both simulation-based and analytical methods. Secondly, the behaviour of interest in such systems is usually dependent on the inherent stochasticity of the model, and may not be aligned to the underlying state interpretation. In a model where states represent a low-level, primitive aspect of system components, the phenomenon of interest often varies significantly with respect to this low-level aspect that states represent. This work focuses on providing methodological frameworks that circumvent these issues by developing abstraction strategies, which preserve the phenomena of interest. In the first part of this thesis, we express behavioural characteristics of the system in terms of a temporal logic with Markov chain trajectories as semantic objects. This allows us to group regions of the state-space by how well they satisfy the logical properties that characterise macro-scale behaviour, in order to produce an abstracted Markov chain. States of the abstracted chain correspond to certain satisfaction probabilities of the logical properties, and inferred dynamics match the behaviour of the original chain in terms of the properties. The resulting model has a smaller state-space which is interpretable in terms of an emergent behaviour of the original system, and is therefore valuable to a researcher despite the accuracy sacrifices. Coarsening based on logical properties is particularly useful in multi-scale modelling, where a layer of the model is a (continuous-time) Markov chain. In such models, the layer is relevant to other layers only in terms of its output: some logical property evaluated on the trajectory drawn from the Markov chain. We develop here a framework for constructing a surrogate (discrete-time) Markov chain, with states corresponding to layer output. The expensive simulation of a large Markov chain is therefore replaced by an interpretable abstracted model. We can further use this framework to test whether a posited mechanism could be the driver for a specific macro-scale behaviour exhibited by the model. We use a powerful Bayesian non-parametric regression technique based on Gaussian process theory to produce the necessary elements of the abstractions above. In particular, we observe trajectories of the original system from which we infer the satisfaction of logical properties for varying model parametrisation, and the dynamics for the abstracted system that match the original in behaviour. The final part of the thesis presents a novel continuous-state process approximation to the macro-scale behaviour of discrete-state Markov chains with large state-spaces. The method is based on spectral analysis of the transition matrix of the chain, where we use the popular manifold learning method of diffusion maps to analyse the transition matrix as the operator of a hidden continuous process. An embedding of states in a continuous space is recovered, and the space is endowed with a drift vector field inferred via Gaussian process regression. In this manner, we form an ODE whose solution approximates the evolution of the CTMC mean, mapped onto the continuous space (known as the fluid limit). Our method is general and differs significantly from other continuous approximation methods; the latter rely on the Markov chain having a particular population structure, suggestive of a natural continuous state-space and associated dynamics. Overall, this thesis contributes novel methodologies that emphasize the importance of macro-scale behaviour in modelling complex systems. Part of the work focuses on abstracting large systems into more concise systems that retain behavioural characteristics and are interpretable to the modeller. The final part examines the relationship between continuous and discrete state-spaces and seeks for a transition path between the two which does not rely on exogenous semantics of the system states. Further than the computational and theoretical benefits of these methodologies, they push at the boundaries of various prevalent approaches to stochastic modelling

Replicated computations results (RCR) Report for "statistical abstraction for multi-scale spatio-t Systems"

“Statistical abstraction for multi-scale spatio-temporal systems” proposes a methodology that supports analysis of large-scaled spatio-temporal systems. These are represented via a set of agents whose behaviour depends on a perceived field. The proposed approach is based on a novel simulation strategy based on a statistical abstraction of the agents. The abstraction makes use of Gaussian Processes, a powerful class of non-parametric regression techniques from Bayesian Machine Learning, to estimate the agent’s behaviour given the environmental input. The authors use two biological case studies to show how the proposed technique can be used to speed up simulations and provide further insights into model behaviour. This replicated computation results report focuses on the scripts used in the paper to perform such analysis. The required software was straightforward to install and use. All the experimental results from the paper have been reproduced