Qualitative models and fuzzy systems: an integrated approach for learning from data

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Abstract

This paper presents a method for the identification of the dynamics of non-linear systems by learning from data. The key idea which underlies our approach consists of the integration of qualitative modeling techniques with fuzzy logic systems. The resulting hybrid method exploits the a priori structural knowledge on the system to initialize a fuzzy inference procedure which determines, from the available experimental data, a functional approximation of the system dynamics that can be used as a reasonable predictor of the patient's future state. The major advantage which results from such an integrated framework lies in a significant improvement of both efficiency and robustness of identification methods based on fuzzy models which learn an input–output relation from data. As a benchmark of our method, we have considered the problem of identifying the response to the insulin therapy from insulin-dependent diabetic patients: the results obtained are presented and discussed in the paper.

Introduction

The prediction of the evolution over time of the patient's state plays a crucial role both in a diagnostic and therapeutic medical context. A traditional way to approach such a problem deals with both the formulation of mathematical models of the dynamics of patho–physiological systems and the simulation of their behaviour [3]. Such models, which are generally described by ordinary differential equations (ODE), are computationally tractable with classical methods which allow us to derive, either analytically or numerically, meaningful predictions of the behaviour of the considered system. However, for the medical domain as for many other physical domains, quantitative model formulation may not be successfully applicable due to the incompleteness of the available knowledge about either the functional relationships between variables or the numerical values of model parameters, which could be non identifiable both for the lack of adequate experimental settings and for the impossibility of measuring in vivo the values of a few variables. Approaches recently proposed to cope with difficulties in model building in the presence of incomplete knowledge are represented by qualitative modeling methods [10]. Such methods allow us to describe the dynamics of a system through qualitative differential equations (QDE), where functional relationships and numeric values are respectively defined in terms of regions of monotonicity and of their ordinal relations with landmark values. Given a set of QDE’s which model a dynamical system and an initial state, i.e. the qualitative magnitude of variables along with their direction of change, qualitative descriptions of the behaviours of the system are derived through qualitative simulation.

Whilst qualitative predictions of a patho–physiological system behaviour may be properly exploited in the testing phase of diagnostic reasoning [6], they are almost always inadequate to be used in a therapy planning context as the effects of different therapies are requested to be deeply investigated at a quantitative level.

As an alternative to conventional mathematical modeling frameworks, the so-called non-parametric approaches, that are able to describe the dynamics of a real system from input–output data, have been proposed. Neural networks, multi-variate splines and fuzzy logic systems are the best known approximation schemes used for learning an input–output relation from data 7, 8, 17. Although these approaches are successfully applied to a variety of domains, they are affected by two main drawbacks that are particularly serious in medicine: (1) the identification result, a non-linear function, does not capture any structural knowledge; and (2) the model identification procedure usually requires a large amount of data and is often extremely inefficient.

This paper deals with a method which integrates both mathematical and non-parametric modeling frameworks with the goal to preserve the advantages which come from the application of both of them and, at the same time, to overcome their respective drawbacks. Therefore, our goal is to define a robust and efficient method for non-linear dynamical system identification, which exploits both the available a priori knowledge, namely the structural and human expert knowledge, and the experimental one. The structural knowledge is properly represented by qualitative models, whereas the human expert knowledge is represented by rules which describe linguistic information. As a qualitative modeling formalism, we have chosen QSIM [10] due to both its expressive power to represent differential equations in case of incomplete knowledge and its reasonable predictive capacity. Among the non-parametric approaches, fuzzy logic systems seem to be the most suitable ones as they are capable of conjugating experimental data analysis and prior knowledge representation through rules, and, moreover, they can be proven to be universal approximators [17], i.e. they can approximate at any degree of accuracy any non-linear function. In general, fuzzy logic methods are based on fuzzy inference procedures which are initialized with a rule-base predefined by the human expert. When such a base is not available or poorly defined, the fuzzy inference may also become extremely inefficient. Our method aim to solve the problem of the construction of a meaningful rule-base: fuzzy rules are automatically generated by encoding the knowledge of the system dynamics captured by its qualitative simulated behaviours. The basic steps in our method may be summarized as follows:

  • formulation and simulation of a qualitative model of the patho–physiological system being studied;

  • automatic generation of the initial set of Fuzzy Rules (FR) from the outcome of the qualitative simulations;

  • generation of the Fuzzy System (FS) which corresponds to the FR;

  • parameter estimation of the FS from a set of experimental data.

We have applied our method to the identification of the dynamics of the blood glucose metabolism in insulin dependent diabetic patients. In particular, we addressed the problem of predicting the blood glucose level (BGL) time course in response to different perturbations, such as meals and conventional insulin therapies. In such a clinical context, the experimental data usually come from the home-monitoring of patients: their low-quality and the high complexity of the system motivate the use of the approach we have implemented. The assessment of a tool capable of deriving reliable and accurate predictions may allow us to achieve the ultimate goal of improving the control of BGL in diabetes therapy.

Section snippets

Background

This section recalls the basic concepts and definitions of both QSIM and fuzzy logic systems which are relevant to a clear and formal description of our method. A more extensive treatment can be found in 10, 17.

Description of the method

The system identification procedure we propose proceeds in three main phases (Fig. 2):

(1) QSIM model formulation and simulation. The prior patho–physiological knowledge of the system being studied must be organized so that its structural and behavioural model can be defined. More precisely, the variables of interest and the network of interactions between them, along with their mathematical descriptions, must be specified: variables are described by their respective quantity spaces, whereas

An application: identification of BGL dynamics in diabetic patients

The identification and prediction of the BGL dynamics in patients suffering from insulin dependent diabetes mellitus (IDDM) is a well-known problem in the bioengineering field. The blood glucose levels (BGL) in IDDM patients are not kept within acceptable values because of the insufficient production of endogenous insulin, the main glucoregulatory hormone. Herein, we will concentrate on the problem of forecasting the patient's response to conventional exogenous insulin therapy, i.e. several

Discussion

The work proposed in this paper describes a methodological framework that may be applied to a number of different applications. We use quite a complex application problem as a test benchmark, for which our proposed methodology could provide valuable contributions.

Clearly, several problems remain which need further work. Some of these issues are described below:

Conclusions

Medical applications are often characterized by problems of a two-fold nature: (1) the physician usually has a deep patho physiological knowledge of the patient's behaviour; and (2) it is not possible to measure all of the quantities necessary to completely assess the patient's state.

As a consequence, although many mathematical models of physiological systems are available, it is usually not possible to identify such systems, and it is hence necessary to resort to black-box strategies, with a

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