Optimum linear regression in additive Cauchy–Gaussian noise

Chen, Yuan; Kuruoğlu, Erçan E.; So, Hing Cheung

doi:10.1016/j.sigpro.2014.07.028

Cited by 9 publications

(9 citation statements)

References 20 publications

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“…Therefore, to obtain the closed-form PDF expression, we use the approximated PDF of the SαS noise. Because for a SαS 5) until a large number of iterations (vii) Discard some samples before convergence variable, a ¼ 1 corresponds to the Cauchy distribution, and a ¼ 2 is the Gaussian process, its PDF is rewritten as the sum of a Cauchy (a ¼ 1) PDF and a Gaussian (a ¼ 2) PDF [38]:…”

Section: The Pdf Approximation Of Mixture Noisementioning

confidence: 99%

“…where 0 nðaÞ 1 is the mixed coefficient, f 1 ðejgÞ and f 2 ðejkÞ denote the unnormalized Cauchy and Gaussian processes, with the dispersion η and variance λ 2 [38], respectively. For the sake of the analytical form of nðaÞ, f 1 ðejgÞ and f 2 ðejk 2 Þ, previous works [45][46][47] are developed, among which the most accurate expression is…”

Section: The Pdf Approximation Of Mixture Noisementioning

confidence: 99%

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Robust Frequency Estimation Under Additive Symmetric α-Stable Gaussian Mixture Noise

Wang¹,

Tian²,

Men³

et al. 2023

Intelligent Automation &Amp; Soft Computing

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Here the estimating problem of a single sinusoidal signal in the additive symmetric α-stable Gaussian (ASαSG) noise is investigated. The ASαSG noise here is expressed as the additive of a Gaussian noise and a symmetric α-stable distributed variable. As the probability density function (PDF) of the ASαSG is complicated, traditional estimators cannot provide optimum estimates. Based on the Metropolis-Hastings (M-H) sampling scheme, a robust frequency estimator is proposed for ASαSG noise. Moreover, to accelerate the convergence rate of the developed algorithm, a new criterion of reconstructing the proposal covariance is derived, whose main idea is updating the proposal variance using several previous samples drawn in each iteration. The approximation PDF of the ASαSG noise, which is referred to the weighted sum of a Voigt function and a Gaussian PDF, is also employed to reduce the computational complexity. The computer simulations show that the performance of our method is better than the maximum likelihood and the lp-norm estimators.

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Section: The Pdf Approximation Of Mixture Noisementioning

confidence: 99%

Section: The Pdf Approximation Of Mixture Noisementioning

confidence: 99%

Robust Frequency Estimation Under Additive Symmetric α-Stable Gaussian Mixture Noise

Wang¹,

Tian²,

Men³

et al. 2023

Intelligent Automation &Amp; Soft Computing

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“…Assume that the priors for all unknown parameters θ and the observations y are statistically independent. With use of ( 15)-( 18) and ( 19)-( 20) as well as Bayes' theorem [25], the posterior expression of all unknown parameters θ can be…”

Section: Posterior Of Unknown Parametersmentioning

confidence: 99%

“…According to [20], the PDF of the mixture can be expressed as Voigt profile. Due to the involved form of the Voigt profile, classical frequency estimators [21][22][23][24], such as the maximum likelihood estimator (MLE) and M-estimator [25], has convergence problem and cannot provide the optimal estimation in the case of high noise power. To obtain the estimates accurately, Markov chain Monte Carlo (MCMC) [26][27][28] method is utilized, which samples unknown parameters from a simple proposal distribution instead of from the complicated posterior PDF directly.…”

Section: Introductionmentioning

confidence: 99%

Robust Frequency Estimation Under Additive Mixture Noise

Chen¹,

Tian²,

Zhang³

et al. 2022

Computers, Materials &Amp; Continua

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In many applications such as multiuser radar communications and astrophysical imaging processing, the encountered noise is usually described by the finite sum of α-stable (1 ≤ α<2) variables. In this paper, a new parameter estimator is developed, in the presence of this new heavy-tailed noise. Since the closed-form PDF of the α-stable variable does not exist except α = 1 and α = 2, we take the sum of the Cauchy (α = 1) and Gaussian (α = 2) noise as an example, namely, additive Cauchy-Gaussian (ACG) noise. The probability density function (PDF) of the mixed random variable, can be calculated by the convolution of the Cauchy's PDF and Gaussian's PDF. Because of the complicated integral in the PDF expression of the ACG noise, traditional estimators, e.g., maximum likelihood, are analytically not tractable. To obtain the optimal estimates, a new robust frequency estimator is devised by employing the Metropolis-Hastings (M-H) algorithm. Meanwhile, to guarantee the fast convergence of the M-H chain, a new proposal covariance criterion is also devised, where the batch of previous samples are utilized to iteratively update the proposal covariance in each sampling process. Computer simulations are carried out to indicate the superiority of the developed scheme, when compared with several conventional estimators and the Cramér-Rao lower bound.

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“…To account for the non-stationarity of the noise, a zero mean Gaussian model with unknown varying variance has been considered in [62] and a Cauchy-Gaussian model in [63]. To account for both of these two prior information, we propose to model the noise as a zero mean non-stationary Gaussian with unknown…”

Section: Introductionmentioning

confidence: 99%

Bayesian sparse solutions to linear inverse problems with non-stationary noise with Student-t priors

Mohammad-Djafari

Dumitru

2015

Digital Signal Processing

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Bayesian approach has become a commonly used method for inverse problems arising in signal and image processing. One of the main advantages of the Bayesian approach is the possibility to propose unsupervised methods where the likelihood and prior model parameters can be estimated jointly with the main unknowns. In this paper, we propose to consider linear inverse problems in which the noise may be non-stationary and where we are looking for a sparse solution. To consider both of these requirements, we propose to use Student-t prior model both for the noise of the forward model and the unknown signal or image. The main interest of the Student-t prior model is its Infinite Gaussian Scale Mixture (IGSM) property. Using the resulted hierarchical prior models we obtain a joint posterior probability distribution of the unknowns of interest (input signal or image) and their associated hidden variables. To be able to propose practical methods, we use either a Joint Maximum A Posteriori (JMAP) estimator or an appropriate Variational Bayesian Approximation (VBA) technique to compute the Posterior Mean (PM) values. The proposed method is applied in many inverse problems such as deconvolution, image restoration and computed tomography. In this paper, we show only some results in signal deconvolution and in periodic components determination of some biological signals related to dynamic circadian clock period determination for cancer studies.

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Optimum linear regression in additive Cauchy–Gaussian noise

Cited by 9 publications

References 20 publications

Robust Frequency Estimation Under Additive Symmetric α-Stable Gaussian Mixture Noise

Robust Frequency Estimation Under Additive Symmetric α-Stable Gaussian Mixture Noise

Robust Frequency Estimation Under Additive Mixture Noise

Bayesian sparse solutions to linear inverse problems with non-stationary noise with Student-t priors

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