Nonparametric log spectrum estimation using disconnected regression splines and genetic algorithms
Introduction
This article studies the problem of nonparametric log spectral density estimation. Various log spectrum estimation procedures that adopt the idea of smoothing the log periodogram have been proposed. These include kernel smoothing (e.g., [6], [14], [17], [20]), smoothing spline methods (e.g., [19], [23]) and wavelet techniques (e.g., [9], [18], [24]). The new procedure that this article proposes uses a different statistical model to model the target log spectrum: the target log spectrum is modelled by a series of disconnected regression splines that partition the domain of the spectrum (briefly, regression splines are a special kind of polynomial functions and a brief introduction is given in Section 2.2). As will be demonstrated below, such a model is extremely suitable for spectra with inhomogeneous structures.
It will be shown below that the problem of estimating a log spectrum using this disconnected regression spline approach can be posed as a statistical model selection problem, in which different candidate models may have different dimensions. In order to tackle this model selection problem, we employ a modified form of the Akaike's information criterion (AIC) [1] to construct an objective function for which the best fitting model for our problem is defined as its optimizer. However, optimizing this objective function would involve solving a hard and large scale optimization problem. In this work we propose using genetic algorithms for solving such problems (e.g., see [5], [7] and references given therein). Simulation results suggest that the use of genetic algorithms is very effective.
The rest of this article is organized as follows. Background material on the properties of log periodograms and regression splines is given in Section 2. In Section 3 we present our new disconnected regression spline model for log spectra. Section 4 describes the above mentioned AIC model selection method while Section 5 proposes a genetic algorithm for solving the related optimization problem. Section 6 reports simulation results. Finally, conclusions are given in Section 7.
Section snippets
Log periodogram
Suppose that {xt} is a real-valued, zero-mean strictly stationary process with unknown spectral density S, and that a finite-sized realization x0,…,x2n−1 of {xt} is observed. The periodogram is defined asTo simplify notation, write ωl=2πl/(2n). Since the spectral density S is symmetric about ω=π, we shall focus our discussion on S(ωl) for l=0,…,n−1.
Let γr=E(xt−rxt), r=0,1,…, be the autocovariance function. If all moments of xt exist, the sum of all |γr
Log spectrum model: disconnected regression splines
This section presents our model for the log spectrum f. One characteristic of our model is that it is capable of handling f with inhomogeneous structures. The idea is to approximate f by a series of disconnected quadratic regression splines. In this way boundary points between any two adjacent quadratic regression splines can serve as locations of sudden changes in f; see Fig. 2 for an illustration. In the sequel we shall call these boundary points break points. Despite that regression splines
Model selection and parameter estimation
If f is modelled by the above disconnected regression splines , , then an estimate of f can be obtained by first estimating and then plugging the resulting estimate into (3) and (4). Hence using the disconnected regression spline approach, our original log spectrum estimation problem can be posed as a model selection problem, with each candidate model specified by a . The goal, then, is to select a “best” . Notice that different
Optimization by genetic algorithms
When the number of data points is large, finding the best estimate defined by the above AIC/BIC criterion would involve solving a hard, large scale minimization problem. Common techniques for dealing with these types of problems include knot addition, knot deletion, knot movement or combinations of them; e.g., [8], [13], [16]. However, these techniques do not provision the inclusion of break points in our model. In this article we suggest using genetic algorithms, which are also known as
Numerical experiments
This section reports results of those numerical experiments that were conducted for evaluating and comparing the practical performance of the proposed log spectrum estimation procedure with some other procedures appearing in the literature.
Conclusion
In this article an automatic log spectrum estimation procedure based on the disconnected regression spline model is proposed. Numerical experiments have been conducted to compare the practical performances of the proposed procedure with some other log spectral density estimation procedures commonly found in the literature. Empirical results from the experiments suggest that the proposed procedure, despite of its high complexity, is a very promising and reliable procedure, especially when the
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