Publication: Birleşik bayes model seçimi ve gürültülü sinüzoidallerin kestirimi
Abstract
BİRLEŞİK BAYES MODEL SEÇİMİ VE GÜRÜLTÜLÜ SİNÜZOİDALLERİN KESTİRİMİ Bu tezde gürültülü verilerden sinüzoidal sinyallerin sayısını tespit etme ve parametrelerini kestirme problemi, Bayes mantıksal çıkarımı çerçevesinde ele alındı. Bu amaçla, sinüzoidallerin sayısının ve her bir sinüzoidali karakterize eden parametrelerin (açısal frekans ve genlik) sonsal olasılık yoğunluk fonksiyonunun türetilmesinde Bayes olasılık teorisi kullanıldı. Bu, sinyal parametreleri hakkında mantıksal çıkarımlar yapmada daha sağlam bir matematiksel altyapıyı ve sinyal parametrelerin kestirim değerlerindeki belirsizliklerin hesaplanmasını sağladı. Ne yazık ki Bayes yaklaşımının temel problemi, analitiksel çözümü olmayan kapalı formda çok katlı integrallerin değerlendirilmesini gerektirmesidir. Fakat bu integrallerin analitiksel olarak hesaplanması çok zordur. Bu nedenle, sinüzoidallerin sayısının bilindiği ve bilinmediği iki farklı durumda istatiksel sinyal işleme literatüründe önerilen Bretthorst’ un integral yöntemi, Gibbs örneklemesi, paralel tavlama ve tersinir atlamalı Markov halkalı Monte Carlo yöntemleri araştırılarak algoritmaları geliştirildi. Bu algoritmalar, Mathematica programlama diliyle kodlanarak gürültülü sinüzoidal sinyal modelleri için test edildi. Buna ek olarak, farklı sinyal gürültü oranı ve örneklem sayının değişimi altında tek frekanslı sinüzoitten üretilen gürültülü veriler üzerine belli sayıda simülasyon çalışması yapıldı. Bu simulasyon sonuçları kullanılarak, herhangi bir yansız kestiricinin varyansının alt sınırı olarak bilinen Cramer-Rao sınırına göre yöntemlerin performansları karşılaştırıldı.
JOINT BAYESIAN MODEL SELECTION AND ESTIMATION OF NOISY SINUSOIDALS In this thesis, a detection of number of sinusoids and an estimation of their unknown parameters (angular frequency and amplitude) from noisy data was considered within a Bayesian logical inference framework. For this purpose, Bayesian probability theory was used to derive the posterior probability density function for the number of sinusoids and the parameters that characterize each sinusoid. This provides a rigorous mathematical foundation for making inferences about the signal parameters and a basis for quantifying the uncertainties in the estimated signal parameters. Unfortunately, the main problem of the Bayesian approach is that it typically requires the evaluation of high-dimensional integrals that do not admit any closed form analytical expression. But the evaluation of these integrals is more difficult. Therefore, by investigating some methods, such as Bretthorst’s integral method, Gibbs sampling, parallel tempering and reversible jump Markov Chain Monte Carlo methods which are proposed in the statistical signal processing literature, their algorithms were improved for two cases when the number of sinusoids is known or not. These algorithms were coded in Mathematica programming language and tested for noisy sinusoidal signal models. In addition, a number of simulation studies on synthetic data sets for a single sinusoid were made under the variety of different signal noise ratio and the length of data samples. By using these simulation results, the performances of the methods were compared with respect to the Cramér-Rao bound, which is known as a lower bound on the error variance of any unbiased estimator.
JOINT BAYESIAN MODEL SELECTION AND ESTIMATION OF NOISY SINUSOIDALS In this thesis, a detection of number of sinusoids and an estimation of their unknown parameters (angular frequency and amplitude) from noisy data was considered within a Bayesian logical inference framework. For this purpose, Bayesian probability theory was used to derive the posterior probability density function for the number of sinusoids and the parameters that characterize each sinusoid. This provides a rigorous mathematical foundation for making inferences about the signal parameters and a basis for quantifying the uncertainties in the estimated signal parameters. Unfortunately, the main problem of the Bayesian approach is that it typically requires the evaluation of high-dimensional integrals that do not admit any closed form analytical expression. But the evaluation of these integrals is more difficult. Therefore, by investigating some methods, such as Bretthorst’s integral method, Gibbs sampling, parallel tempering and reversible jump Markov Chain Monte Carlo methods which are proposed in the statistical signal processing literature, their algorithms were improved for two cases when the number of sinusoids is known or not. These algorithms were coded in Mathematica programming language and tested for noisy sinusoidal signal models. In addition, a number of simulation studies on synthetic data sets for a single sinusoid were made under the variety of different signal noise ratio and the length of data samples. By using these simulation results, the performances of the methods were compared with respect to the Cramér-Rao bound, which is known as a lower bound on the error variance of any unbiased estimator.