A genetic algorithms based technique for computing the nonlinear least squares estimates of the parameters of sum of exponentials model

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Abstract

Estimation of the parameters of a nonlinear sum of exponentials model is an important and well studied problem in time series analysis. The sum of exponentials model finds application in modeling various physical phenomena in a wide variety of real life applications. The problem of finding the nonlinear least squares estimates in well known to be numerically difficult. In this paper, we propose an elitist generational genetic algorithm based iterative procedure for computing the nonlinear least squares estimates. Simulation studies and real life data fitting examples indicate satisfactory performance of the proposed technique.

Highlights

► We consider least squares estimation of the parameters of sum of exponential model. ► A genetic algorithm based method for finding the nonlinear least squares estimates is proposed. ► The proposed method uses an elitist generational genetic algorithm. ► Simulation studies and real life data analysis show the usefulness and efficiency of the proposed technique.

Introduction

We consider the problem of estimation of parameters of the following nonlinear sum of exponentials real compartment modely(t)=k=1Mαkexp(βkt)+ε(t),t=t1,,tn.{αk}k=1M and {βk}k=1M are unknown deterministic parameters of the model. M, the number of components in the sum of exponential model is assumed to be known. The additive noise {ε(t)} is assumed to be a sequence of wide sense stationary random variables represented asε(t)=i=1Pρiε(t-i)+δ(t).

The sequence of random variables {δ(t)} is assumed to be a sequence of independent and identically distributed (i.i.d.) random variable with mean 0 and variance σ2<. (ρ1,,ρP) in (2) are such that the sequence {ε(t)} is covariance stationary. We note that ρi = 0, for all i, corresponds to the i.i.d. noise case. Given a sample of size n, the problem is to estimate the unknown parameters {αk}k=1M and {βk}k=1M.

Estimation of the parameters of a nonlinear compartment model (1) is an important and well studied problem in time series analysis. Compartment models in various forms are used for modeling various physical systems. Applications of such models can be found, among others, in the fields of pharmacokinetics (Bates & Watts, 1988), radiotherapy (Holmström, Ahnesjö, & Petersson, 2001), gene expressions (Giurcaneanu, Tabus, & Astola, 2005) and bio-medical statistics (Lindsey, 2001). See also Anderson, 1983, Seber and Wild, 1989, Gallant, 1987, Bates and Watts, 1988 for discussion on and applications of sum of exponentials compartment models.

Estimation of the parameters of sum of exponential models is an important problem. The first reference regarding parameter estimation of this model goes back to the pioneering work of Prony (1795). Nonlinear least squares estimates (NLSE) of the parameters of the model (1) can be obtained asαˆβˆ=argminα,βt=1ny(t)-k=1MαKeβkt2.The problem of finding the least squares estimates is well known to be numerically difficult (Wilson, 1983, Osborne and Smyth, 1995, Kundu and Mitra, 1998a). Several algorithms have been proposed in literature for obtaining the least squares estimates efficiently. Prominent among these are the methods of Barham & Drane, 1972, Golub & Pereyra, 1973, Osborne, 1975, Osborne, 1976, Osborne and Smyth, 1986, Osborne and Smyth, 1995, Kahn et al., 1992 and Kundu and Mitra (1998b).

In recent years, genetic algorithms (GA) have extensively been used to solve problems in varied types of application areas. See for example Aci et al., 2010, Delavar et al., 2010, Mendi et al., 2010, Mitra and Kundu, 2010, Kwong et al., 2009, Tang et al., 2005, Shin & Lee, 2002, Wong et al., 2008. In this paper, we propose a genetic algorithm based procedure for obtaining the nonlinear least squares estimates of the parameters of the sum of exponentials model. The proposed algorithm frames the problem of finding the NLSE of model (1) in a GA framework and uses an elitism based generational genetic algorithm for computing the NLSE. The proposed procedure is applied for estimation of parameters of different classes of sum of exponential models and also for real life data analysis. Stability and performance analysis studies indicate satisfactory behavior of the proposed algorithm. The rest of the paper is organized as follows. In Section 2, we present the proposed GA based NLSE estimation algorithm. Section 3 reports the results of the simulation studies and in Section 4 we present the real life data analysis. Conclusions are given in Section 5.

Section snippets

Generational genetic algorithm based NLSE

In this section, we present the proposed algorithm for estimation of NLSE. The proposed algorithm used an elitism based generational genetic algorithm for obtaining the NLSE of the parameters of model (1). We first note that for the nonlinear model (1), the set of parameters {αk}k=1M are conditionally linear, i.e. given {βk}k=1M, the model is linear and the associated conditionally linear parameters can be estimated using linear regression technique.

The residual sum of squares for model (1) is

Simulation studies

In this section, we present the simulation studies performed to ascertain the performance and stability of the proposed algorithm. In the simulation studies, we consider decay and growth model forms of the sum of exponential model and apply the proposed procedure for finding the NLSE of the model parameters. We also study various aspects of stability of the proposed estimation procedure.

Real life data analysis

In this section, we consider the problem of fitting sum of exponentials model to two standard real life datasets using the proposed technique. The first real life dataset relates to the data on metabolism of sulfisoxazole and was obtained by Kaplan, Weinfeld, Abruzzo, & Lewis (1972). In this biological experiment, sulfisoxazole was administered to a subject intravenously, blood samples were taken at specific times (non-equidistant), and the concentration of sulfisoxazole in the plasma in

Conclusions

In this paper, we propose a generational genetic algorithm based iterative algorithm for computing the nonlinear least squares estimates of sum of exponentials models. The proposed procedure uses an elitism based generational genetic algorithm for estimation of the nonlinear parameters of the model. Extensive simulation studies and the real life data analysis show the usefulness and efficient performance of the proposed procedure. The proposed procedure has a number of advantages over the

Acknowledgement

The work of the second author is supported by Department of Science & Technology, Government of India, grant No. SR/S4/MS:374/06.

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