A Clarification of the Cauchy Distribution

Lee, Hwi-Young; Park, Hyoung-Jin; Kim, Hyoung-Moon

doi:10.5351/csam.2014.21.2.183

Cited by 8 publications

(8 citation statements)

References 8 publications

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“…Note that a i,2 /a i,1 obeys the standard Cauchy distribution, i.e., a i,2 /a i,1 ∼ Cauchy(0, 1) [65]. Transformation properties of Cauchy distributions assert that h 1 + ai,2 ai,1 h \1 ∼ Cauchy(h 1 , h \1 ) [66]. Recall that the cdf of a Cauchy distributed random variable w ∼ Cauchy (µ 0 , α) is given by [65] F (w; µ 0 , α) = 1 π arctan w − µ 0 α + 1 2 .…”

Section: Proof Of Lemmamentioning

confidence: 99%

Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow

Wang

Giannakis

Eldar

2018

IEEE Trans. Inform. Theory

310

418

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This paper presents a new algorithm, termed truncated amplitude flow (TAF), to recover an unknown vector x from a system of quadratic equations of the form yi = | ai, x | 2 , where ai's are given random measurement vectors. This problem is known to be NP-hard in general. We prove that as soon as the number of equations is on the order of the number of unknowns, TAF recovers the solution exactly (up to a global unimodular constant) with high probability and complexity growing linearly with both the number of unknowns and the number of equations. Our TAF approach adopts the amplitude-based empirical loss function, and proceeds in two stages. In the first stage, we introduce an orthogonality-promoting initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors x and ai's are real-valued, our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version. Numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations. Furthermore, even under the same initialization, the proposed amplitude-based refinement outperforms existing Wirtinger flow variants, corroborating the superior performance of TAF over state-of-the-art algorithms.

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Section: Proof Of Lemmamentioning

confidence: 99%

Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow

Wang

Giannakis

Eldar

2018

IEEE Trans. Inform. Theory

310

418

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“…If a p-dimensional random vectorX follows a multivariate CD with location vector and scale matrix Σ, that is X∼Cauchy p ( , Σ), then the probability density function (PDF) is presented as [39]: 15) where X, ∈ ℝ p and Σ ∈ ℝ p×p is a positive-definite matrix.…”

Section: Mathematical Properties Of Multivariate Cauchy Distributionmentioning

confidence: 99%

Analytical solution of stochastic real‐time dispatch incorporating wind power uncertainty characterized by Cauchy distribution

Yang

et al. 2021

IET Renewable Power Gen

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Real-time power dispatch can coordinate wind farms, automatic generation control units and non-automatic generation control units. In real-time power dispatch, the probable wind power forecast errors should be appropriately formulated to ensure system security with high probability and minimize operational cost. Previous studies and the authors' onsite tests show that Cauchy distribution effectively fits the "leptokurtic" feature of smalltimescale wind power forecast errors distributions. In this paper, a chance-constrained real-time dispatch model with the wind power forecast errors represented by multivariate Cauchy distribution is proposed. Since the Cauchy distribution is stable and has promising mathematical characteristics, the proposed chance-constrained real-time dispatch model can be analytically transformed to a convex optimization problem considering the dependence among wind farms' outputs. Moreover, the proposed model incorporates an affine control strategy compatible with automatic generation control systems. This strategy makes the chance-constrained real-time dispatch adaptively take into account both the potential power ramping requirement and power variation on transmission lines caused by the generation adjustment to offset the wind power forecast errors in real-time power dispatch stage. Numerical test results show that the proposed method is reliable and effective. Meanwhile it is very efficient and suitable for real-time application.

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“…The Cauchy ZMNL is obtained considering the Cauchy distribution, that, for complex random variables is defined as

f^{C a u} false(z false) = \frac{K}{2 π {((), K^{2} + false| z |^{2})}^{3 false/ 2}} .

…”

Section: Generalized Robust Cafmentioning

confidence: 99%

“…When the input samples follow a Gaussian distribution, the TD samples are also normally distributed. This fact follows from the linearity of the operations implemented in (36). Q k has unit norm and thus Y[k] has the same variance of the original samples.…”

Section: Efficiency Analysismentioning

confidence: 99%

Robust transform domain signal processing for GNSS

Borio

Closas

2019

NAVIGATION

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Robust signal processing techniques are effective mitigation approaches that can enable Global Navigation Satellite System (GNSS) receiver operations even in the presence of strong interference. We extend robust GNSS signal processing to the case where interference admits a sparse representation in a transform domain (TD). More specifically, an orthonormal transformation is used to project the received GNSS samples into an appropriate TD. Robust signal processing strategies are then applied. The robust transform domain (RTD) cross‐ambiguity function (CAF) is obtained by applying a zero‐memory nonlinearity (ZMNL) to the TD samples, which, in turn, are used for the evaluation of the CAF. Different nonlinearities are analyzed and different implementations for the evaluation of the RTD CAF are proposed. The RTD approach is characterized from a theoretical point of view, with the usage of Monte Carlo simulations and through the processing of GNSS signals generated with a hardware simulator.

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A Clarification of the Cauchy Distribution

Cited by 8 publications

References 8 publications

Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow

Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow

Analytical solution of stochastic real‐time dispatch incorporating wind power uncertainty characterized by Cauchy distribution

Robust transform domain signal processing for GNSS

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