Dirichlet Process Mixtures for Density Estimation in Dynamic Nonlinear Modeling: Application to GPS Positioning in Urban Canyons

Rabaoui, Asma; Viandier, Nicolas; Duflos, Emmanuel; Marais, Juliette; Vanheeghe, Philippe

doi:10.1109/tsp.2011.2180901

Cited by 39 publications

(36 citation statements)

References 44 publications

(78 reference statements)

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“…Finally, it must be emphasized that the spectral index α determines the estimation of parameters from geodetic time series and their uncertainties as well [27,25,11]. A recent study [28] shown that it also affects the estimation of real-time vehicles positions from Global Navigation Satellite System (GNSS) signals [28] (a different application of GNSS as presented here). These examples highlight the importance of estimating the spectral index and its uncertainty.…”

Section: Discussionmentioning

confidence: 93%

“…Though this might not be important for detrending in processes like case III, for example, geodetic time series, the non-Gaussianity behaviour of those parameters might still affect the estimates in other applications. [28] shown that classical localization methods are less efficient under multipath effects, than methods that estimate the error distribution. Histograms for marginalized likelihoods also provide information about crosscorrelation among parameters.…”

Section: Synthetic Datamentioning

confidence: 99%

“…Other problems with the above published methods arise from pre-analysis detrending (if need be) as this removes significant amounts of low-frequency energy from the original signal, thereby yielding underestimated spectral indices [25,26]. Considering the impact the stochastic nature of the noise has on parameter estimation [27,28], we used the MCMC method for LRD processes that, unlike current optimization methods, simultaneously estimates all parameters (including the spectral index) and their uncertainties. Moreover, MCMC can detect non-Gaussianity on parameter distributions and any correlation between them.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Bayesian Monte Carlo Markov Chain Method for Parameter Estimation of Fractional Differenced Gaussian Processes

Olivares

Teferle

2013

IEEE Trans. Signal Process.

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We present a Bayesian Monte Carlo Markov Chain method to simultaneously estimate the spectral index and power amplitude of a fractional differenced Gaussian process at low frequency, in presence of white noise, and a linear trend and periodic signals. This method provides a sample of the likelihood function and thereby, using Monte Carlo integration, all parameters and their uncertainties are estimated simultaneously. We test this method with simulated and real Global Positioning System height time series and propose it as an alternative to optimization methods currently in use. Furthermore, without any mathematical proof, the results from the simulations suggest that this method is unaffected by the stationary regime and hence, can be used to check whether or not a time series is stationary.

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Section: Discussionmentioning

confidence: 93%

Section: Synthetic Datamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Bayesian Monte Carlo Markov Chain Method for Parameter Estimation of Fractional Differenced Gaussian Processes

Olivares

Teferle

2013

IEEE Trans. Signal Process.

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“…A particular problem to be solved in this work is to estimate a probability density using a nonparametric Bayesian method involving an endless mix model as considered in [42] and [56]; but in the context of stochastic differential equations. We propose to use the following hierarchical structure to flexibly approximate the probability density of the solutions of the equation given in (1):…”

Section: Approximations Of the Solutions Of A Stochastic Differentialmentioning

confidence: 99%

Approximations of the solutions of a stochastic differential equation using Dirichlet process mixtures and Gaussian mixtures

Infante

Sánchez

Hernández

2016

Stat., optim. inf. comput.

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Stochastic differential equations arise in a variety of contexts. There are many techniques for approximation of the solutions of these equations that include statistical methods, numerical methods, and other approximations. This article implements a parametric and a nonparametric method to approximate the probability density of the solutions of stochastic differential equation from observations of a discrete dynamic system. To achieve these objectives, mixtures of Dirichlet process and Gaussian mixtures are considered. The methodology uses computational techniques based on Gaussian mixtures filters, nonparametric particle filters and Gaussian particle filters to establish the relationship between the theoretical processes of unobserved and observed states. The approximations obtained by this proposal are attractive because the hierarchical structures used for modeling are flexible, easy to interpret and computationally feasible. The methodology is illustrated by means of two problems: the synthetic Lorenz model and a real model of rainfall. Experiments show that the proposed filters provide satisfactory results, demonstrating that the algorithms work well in the context of processes with nonlinear dynamics which require the joint estimation of states and parameters. The estimated measure of goodness of fit validates the estimation quality of the algorithms. 290APPROXIMATIONS OF THE SOLUTIONS OF A STOCHASTIC DIFFERENTIAL EQUATION nonlinear stochastic differential equations, for example [60] and [5] have used an approach based on a Gaussian process to model the time evolution of the solution of a general stochastic differential equation, the methodology uses data assimilation techniques ([31]). Recently [57] introduced a nonparametric method to estimate the function of drift in a stochastic differential equation where a measure of random probability was considered to model the drift and a Expectation-Maximization (EM) algorithm was developed to try to establish a link between unobserved and observed states. This paper aims to approximate the solutions of a stochastic differential equation by using the Gaussian mixture distribution model, Dirichlet process mixture models, together with a non-parametric density estimation algorithm and three sequential filters. The most representative references about the Gaussian mixture distribution model are [61]; [3]; [68] and [2]. Dirichlet processes were treated in [16], and they have been identified in the literature as a useful tool for estimating densities in the field of non-parametric models. The idea of using Dirichlet process is not new, and has been considered in previous works such as [15]; [46]; [42]; [45]; [70] among others.The rest of article is summarized as follows: Section 2 presents the formulation of the problem; Section 3 contains the description of the filtering algorithms defined to approximate a probability density of the solutions; In Section 4, the results obtained for two different examples are shown, and lastly, Section 5 contains a final discussion an...

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“…However, these methods are computationally intensive, making a real time implementation very complicated. Finally, it is interesting to mention other techniques based on non-Gaussian error terms, such as Markov processes [12] or Dirichlet process mixtures [13].…”

Section: Introductionmentioning

confidence: 99%

Smooth Bias Estimation for Multipath Mitigation Using Sparse Estimation

Lesouple

Barbiero

Faurie

et al. 2018

2018 21st International Conference on Information Fusion (FUSION)

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Multipath remains the main source of error when using global navigation satellite systems (GNSS) in constrained environment, leading to biased measurements and thus to inaccurate estimated positions. This paper formulates the GNSS navigation problem as the resolution of an overdetermined system, which depends nonlinearly on the receiver position and linearly on the clock bias and drift, and possible biases affecting GNSS measurements. The extended Kalman filter is used to linearize the navigation problem whereas sparse estimation is considered to estimate multipath biases. We assume that only a part of the satellites are affected by multipath, i.e., that the unknown bias vector is sparse in the sense that several of its components are equal to zero. The natural way of enforcing sparsity is to introduce an 1 regularization associated with the bias vector. This leads to a least absolute shrinkage and selection operator (LASSO) problem that is solved using a reweighted-1 algorithm. The weighting matrix of this algorithm is designed carefully as functions of the satellite carrier to noise density ratio and the satellite elevations. The smooth variations of multipath biases versus time are enforced using a regularization based on total variation. An experiment conducted on real data allows the performance of the proposed method to be appreciated.

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Dirichlet Process Mixtures for Density Estimation in Dynamic Nonlinear Modeling: Application to GPS Positioning in Urban Canyons

Cited by 39 publications

References 44 publications

A Bayesian Monte Carlo Markov Chain Method for Parameter Estimation of Fractional Differenced Gaussian Processes

A Bayesian Monte Carlo Markov Chain Method for Parameter Estimation of Fractional Differenced Gaussian Processes

Approximations of the solutions of a stochastic differential equation using Dirichlet process mixtures and Gaussian mixtures

Smooth Bias Estimation for Multipath Mitigation Using Sparse Estimation

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