Hypothesis assessment with qualitative reasoning: Modelling the Fontestorbes fountain

Highlights

• Evaluating accuracy and consistency of QR models vs data allows to test hypotheses.

• Inconsistencies are better spotted with an incremental model building approach.

• A QR model of the siphon-hypothesis for a real periodic spring tested the method.

• QR model produced the expected non-linear oscillatory behaviour.

• QR model produced true negatives that allow to discard the siphon-hypothesis.

Abstract

This paper demonstrates the utility of the Qualitative Reasoning approach for hypothesis testing in the domain of ecology regarding the functioning of ‘black box’ systems. As a test case, we refer to the study performed by Mangin (1969) with scale models to investigate the hidden mechanism of the Fontestorbes fountain, a spring that exhibits a periodic flow situated in the south of France. In our approach, a Qualitative Reasoning method (and hence a qualitative model) is used to test the ‘siphon-hypothesis’, which traditionally explains the oscillations of the flow rate of a periodic spring by the principle of filling and emptying an underground reservoir through a siphon action. Parts of the simulation results show that the hypothesis is qualitatively accurate; in particular the model produces a cyclic behaviour that matches with the observed one. However, the qualitative model also exhibits a contradictory behaviour (true negative) that challenges the hypothesis consistency. The causal account of this true negative denotes and explains a flaw in the siphon-hypothesis. The paper concludes that, with the Qualitative Reasoning method, models can be constructed for hypothesis testing. Such models should generate the desired behaviour as a first and necessary step to support the viability of the hypothesis. However, the occurrence of unexpected behaviours provides information that challenges the hypothesis, and may lead to having to discard it.

Introduction

Ecological knowledge about a natural phenomenon can be sparse, and the underlying mechanism may be hidden or difficult to investigate. Under such conditions the system manifests itself as a black box for which only the inputs and outputs are known. Often, several hypotheses can be envisaged to explain its behaviour. Hence, there is a need to discriminate between the possible explanations in order to further investigate the ones that are causally sound and have the potential to explain the observed phenomenon. Scientific modelling is a natural means to test in silico whether a mechanism proposed by a hypothesis can effectively exhibit the observed behaviours. Often quantitative approaches, e.g. differential equations, are not appropriate as a first approach for hypothesis-testing in the case of a black box system. The need for a precise parameterisation, for instance of the kinetic properties of the processes or of the system geometry, hampers a straightforward application to simulating black box systems. In such situations, other formalisms can be more appropriate to reflect the causal structure of the physical system.

Conceptual models and diagrammatic representations are traditionally used to formalise a hypothesis prior to experimentation, to building mathematical models, and to making concrete small-scale models (Jørgensen and Bendorrichio, 2001). While effective at clarifying ideas, a limitation of conceptual models is that they are typically static representations that do not support automated derivation of the consequences regarding the mechanisms they represent. A causal reasoning procedure, that infers the consequences of the mechanisms captured in the conceptual model, can support and enhance the assessment of the plausibility of a hypothesis.

Approaches to Qualitative Reasoning (QR) allow for generating simulations that provide such causal accounts. Thus, in the domain of biology, there is a growing interest for the use of qualitative models and reasoning to test the compatibility between a model of the system and experimental data. Typical applications concern metabolic pathways and genetic networks (de Jong et al., 2004, Ironi and Panzeri, 2009, King et al., 2005, Menzies and Compton, 1997, Siegel et al., 2006). Regarding ecology, QR approaches have been applied to modelling and simulating ecological processes (e.g. Guerrin, 1991, Kansou et al., 2013, Salles and Bredeweg, 2006). However, amongst the possible tasks envisaged for a QR model in ecology (Bredeweg et al., 2008, Salles and Bredeweg, 2006), hypothesis-testing has rarely been attempted. The work presented here addresses this issue, and investigates whether QR models are appropriate for hypotheses-testing regarding natural phenomena.

QR (Weld and de Kleer, 1990) is an area of Artificial Intelligence that strives for deriving ‘behaviour from system structure’, and for generating qualitative simulation of the behaviour over time. The firm causal and mathematical foundation of the QR approaches (Travé-Massuyès et al., 2004) guarantees the soundness of the automatic reasoning generating the simulation result. Therefore, QR provides means to make a fair comparison between the model behaviours and the observed behaviours in a qualitative description space. This makes QR approaches valuable for critical assessments of candidate hypotheses aiming at explaining ill-known phenomena.

QR modelling frameworks, such as Garp3 (Bredeweg et al., 2009), also contain modelling features that are important for hypotheses-testing. One of those features is the compositional modelling approach (Falkenhainer and Forbus, 1991), that aims at assembling individual model fragments to compose a larger model that fits a scenario. Each model fragment is activated according to constraints, defined by the modeller, according to the state of the system. Hypotheses-testing can benefit from the modularity conferred by this approach, as a hypothesis often concerns the modification of an existing model, and hence poses the question of which model parts can be reused or need modification.

In this paper the utility of the QR method to investigate the mechanism behind a black box system through hypothesis testing is demonstrated. As a test case, the focus is on the hypothesis regarding the functioning of the Fontestorbes fountain (Ariège, France). The Fontestorbes fountain is actually a periodic spring, also called ebb and flow spring, which exhibits a periodic flow rate during the dry season. Periodic springs appear exclusively in karstified terrains, soluble rocks favour the apparition of fissures and conduits that are necessary conditions for their formation (Bonacci and Bojanić, 1991). The outlet of the fountain is located at the basis of an escarpment. The inner structure of the fountain is not accessible and the mechanism that causes the periodic flow can only be abducted from the observations of the flow rate. During the dry season, the Fontestorbes fountain behaves as an oscillator that transforms a steady input signal (the inflow) into an oscillatory periodic signal (the outflow). The traditional explanation of this phenomenon is based on the principle of an underground reservoir that fills and discharges periodically, due to the action of a siphon. This is referred to as the siphon-hypothesis (Bonacci and Bojanić, 1991, Mangin, 1969). Until Mangin's work with scale models, the siphon-hypothesis was considered the most probable explanation.

This paper reports on using QR to test the siphon-hypothesis. Section 2 presents the Fontestorbes case, while Section 3 introduces the methodology for hypotheses-testing using QR. The next sections present the application to the Fontestorbes fountain. Section 4 discusses the mapping of the data about the behaviour of this system onto the qualitative domain, and defines the expected target behaviours for the qualitative model. Section 5 presents the qualitative model of the hydraulic circuit corresponding to the siphon-hypothesis. Section 6 examines the simulation results of this model with respect to the target defined behaviours. Section 7 reports on the hypothesis-testing through the assessment of the model's accuracy and consistency. Section 8 presents the discussion, and Section 9 the conclusion.