Nonlinear model for ECG R–R interval variation using genetic programming approach

Abstract

This paper proposes a nonlinear system modeling method, which predicts characteristics of the ECG R–R interval variation. For determining model equation, we adopted a genetic programming method in which the chromosome represents the model equation consisting of time-delayed variables, constants, and four arithmetic operators, and determines the fitness function. By genetic programming, sequences of regressive nonlinear equations are produced and evolved until the finding of the optimal model equation, which could simulate the spectral, statistical and nonlinear behavior of the given R–R interval dynamics. Experimental results showed that the evolutionary approach could find the equation which simulates the spectral and chaotic dynamics of the given signal. Therefore, the proposed evolutionary approach is useful for the system identification of the nonlinear biological system.

Introduction

Researchers have often considered that a biological system produces signals, regular and periodic, under the normal state. Recently, some clinical observations have indicated that a biological signal seems to be irregular and nonperiodic for the normal states, but surprisingly becomes periodic and regular for the abnormal state, for instance, EEG of petitmal epilepsy, heart rate [7], [12], [13]. Therefore, the chaotic biological signals can be represented by an equation or a group of equations. In our study, we have taken into consideration a deterministic approach for describing a nonlinear biological system and its intrinsic mechanisms. We had attempted to find a nonlinear equation with some control parameters to describe the chaotic characteristics of a biological system [3].

Traditionally, AR model is the most common method to fit a signal from a linear system. In PAR, a current heart rate HR(k) is defined by a quadratic nonlinear function of sum of former heart rate products

$$\sum _i\sum _j{b}_{ij}x(k-i)x(k-j)$$

plus the AR component. In bilinear, a current heart rate $x(k)$ is defined by a quadratic nonlinear function of sum of former heart rate products

$$\sum _i\sum _j{b}_{ij}x(k-i)e(k-j)$$

plus the ARMA component, where $e(k-j)=x(k-j)-x(k-j)\text{,}{b}_{ij}$ is the coefficients to be found and $e(j)$ is the error that the difference of the measured value of x from that estimated at the j th time delay. PAR requires long computer running time (input signals normally less than 250 points) and require the predetermined structure of the function for fitting [10], [11].

Unlike the previous PAR and bilinear, the study adopted genetic programming (GP) approach that was not based on AR method. GP does not require the predetermined structure of the function for fitting and finds the structure of the function as well as the values of the coefficients which represent best given input signals [8]. Genetic programming is one of the known searching methods, which evolves solutions iteratively until, by natural selection, it finds the best satisfied solution to given criteria. The study accounted for the R–R intervals of ECG from a normal child which were shown to be irregular and nonperiodic. The aim of the study was to determine a nonlinear equation optimally fitted to the characteristics of the R–R intervals using the genetic programming.