Estimating the order of sinusoidal models using the adaptively penalized likelihood approach: Large sample consistency properties☆
Introduction
Estimation of one or more integer-valued parameters that specify a signal model is an important problem for the parametric methods of signal processing. Once the integer-valued parameters are estimated, the vector of real-valued parameters characterizing the signal and noise can be estimated. Some examples of such problems are: estimation of the order of an ARMA model, estimation of the number of components of linearly combined superimposed sinusoids in additive noise, and detection of the number of signals in sensor array processing. The most widely used approach for order estimation in the problems referred to above is based on information theoretic criteria, the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) being the most prominent ones. For a review of the various order estimation methods, see [1], [2], [3], [4], [5], [6], [7], [8], [10], [11], [12], [13], [14], [15], [16] and the references cited therein.
In this paper, we consider the problem of estimating the number of components of the following nonlinear signal model in (1) is assumed to consist of m sinusoidal components and be given by is the vector of unknown signal parameters. is a zero mean, white Gaussian process with finite variance σ2. Thus we have the signal modelω k in (2), (3) is the frequency associated with the kth sinusoidal component and αk's and βk's are the unknown amplitudes of the kth sinusoid. Given a sample of size n, the problem is to estimate the integer-valued parameter m, the signal parameters in θ and the noise variance.
Recently, the paper [2] introduced a new approach for order estimation based on penalizing adaptively the likelihood, referred to as the PAL rule. In this paper, we formulate the PAL rule for estimating the number of components of the superimposed sinusoidal model in (3) and study the theoretical asymptotic statistical properties of the PAL rule. We prove the consistency of the PAL rule estimator in large sample scenario.
The rest of the paper is organized as follows. In Section 2, we spell out the PAL rule for estimating the order of the considered sinusoidal model. In Section 3, we establish the strong consistency of the PAL criterion. Finally, simulation studies are presented in Section 4.
Section snippets
The PAL rule for the sinusoidal model
Stoica and Babu [2] proposed the PAL rule in a general model order estimation framework. We adopt the approach of [2] and formulate the PAL rule for estimating the number of components of the superimposed sinusoidal model in (3). For the model (3), defineLet m0 be the true number of sinusoids in the observed signal. Given a sample of size n, , the problem is to estimate m0.
We make the following assumptions: Assumption A1 For
Consistency of the PAL rule
In this section we prove the main result of the paper: we show that the PAL rule based estimator of the order of the model in (3) is consistent.
The model (3), with m superimposed components, is given byDefinewhere and denotes the transpose of a vector or matrix. Also, define the matrix aswhere , denotes the real part and the imaginary part. Using the above
Numerical examples
In this section, we present some numerical simulations to ascertain the small sample performance of the PAL method and compare it with that of three competing order estimation methods.
We consider the following four simulation models:
Conclusion
In this paper, we use the PAL based order estimation method for estimating the number of components of a superimposed nonlinear sinusoidal model and prove that the estimator of the model order using the PAL rule is consistent. Simulation examples are presented to illustrate the performance of the PAL method for small sample sizes and to compare it with three standard methods, namely, the usual AIC, the usual BIC and the asymptotic MAP rule. It may be noted that, the PAL rule can also be
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2020, Digital Signal Processing: A Review JournalCitation Excerpt :In practice, however, this is often not the case. Model order estimation (MOE) in the presence of noise is an important and challenging issue [13–18]. It has been investigated thoroughly for 1-D signals and generally considered separately from the frequency estimation problem [15–17,19].
Estimating the order of multiple sinusoids model using exponentially embedded family rule: Large sample consistency
2018, Signal ProcessingCitation Excerpt :More recently, Stoica and Babu [4] introduced a novel method based on Penalizing Adaptively the Likelihood (PAL) for model order estimation. Surana et al. [5] studied the large sample asymptotic properties of the PAL method for the problem of order estimation in multiple sinusoidal model and established the consistency of the PAL based estimator for model order estimation. The purpose of this paper is mainly to study the theoretical large sample statistical properties of the EEF rule based estimator of model order for superimposed sinusoidal signal model.
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This work was supported in part by the Swedish Research Council (VR) and the European Research Council (ERC).