Gaussian Likelihood Research Articles

On strong consistency and asymptotic normality of one-step Gauss-Newton estimators in ARMA time series models

ABSTRACT One-step Gauss-Newton estimators for causal and invertible autoregressive moving-average (ARMA) time series models with an unknown mean are considered. A proof of the strong consistency and asymptotic normality of these estimators is provided. In a simulation study, their empirical properties are illustrated in ARMA models and compared to a other estimators: the estimators based on the innovations algorithm procedure, the estimators based on the spectral methods of [Krampe J, Kreiss J-P, Paparoditis E. Estimated Wold representation and spectral-density-driven bootstrap for time series. J R Stat Soc: Ser B (Statist Method). 2018;80:703–726.], and the one-step Gauss-Newton estimators using these methods as preliminary estimators. In the order-one moving-average model, we included the estimator based on the method of moments. The sample mean is used as a preliminary estimator to estimate the mean. These estimators are compared empirically to the maximum Gaussian likelihood estimator with respect to exact biases and mean squared errors.

How accurately can we measure the baryon acoustic oscillation feature?

ABSTRACT Baryon acoustic oscillations (BAOs) represent one of the cleanest probes of dark energy, allowing for tests of the cosmological model through the measurement of distance and expansion rate from a 3D galaxy distribution. The signal appears at large scales in the correlation function where linear theory applies, allowing for the construction of accurate models. However, due to the lower number of modes available at these scales, sample variance has a significant impact on the signal, and may sharpen or widen the underlying peak. Therefore, equivalent mock realizations of a galaxy survey present different errors in the position of the peak when uncertainties are estimated from the posterior probability distribution corresponding to the individual mocks. Hence, the posterior width, often quoted as the error in BAO survey measurements, is subject to sample noise. A different definition of the error is provided by the asymptotic variance of the maximum likelihood estimator, which involves the average over multiple realizations, and is not subject to sample noise. In this work, we reanalyse the main galaxy survey data available for BAO measurements and quantify the impact of the noise component on the error quoted for BAO measurements. We quantify the difference between three definitions of the error: the confidence region computed from a single posterior, the average of the variances of many realizations, and the Fisher matrix prediction assuming a Gaussian likelihood. We also explore the impact of a ‘detectability prior’ based on the significance of the BAO detection.

First Difference MLE and Dynamic Panel Estimation

First difference maximum likelihood (FDML) seems an attractive estimation methodology in dynamic panel data modeling because differencing eliminates fixed effects and, in the case of a unit root, differencing transforms the data to stationarity, thereby addressing both incidental parameter problems and the possible effects of nonstationarity. This paper draws attention to certain pathologies that arise in the use of FDML that have gone unnoticed in the literature and that affect both finite sample peformance and asymptotics. FDML uses the Gaussian likelihood function for first differenced data and parameter estimation is based on the whole domain over which the log-likelihood is defined. However, extending the domain of the likelihood beyond the stationary region has certain consequences that have a major effect on finite sample and asymptotic performance. First, the extended likelihood is not the true likelihood even in the Gaussian case and it has a finite upper bound of definition. Second, it is often bimodal, and one of its peaks can be so peculiar that numerical maximization of the extended likelihood frequently fails to locate the global maximum. As a result of these pathologies, the FDML estimator is a restricted estimator, numerical implementation is not straightforward and asymptotics are hard to derive in cases where the peculiarity occurs with non-negligible probabilities. We investigate these problems, provide a convenient new expression for the likelihood and a new algorithm to maximize it. The peculiarities in the likelihood are found to be particularly marked in time series with a unit root. In this case, the asymptotic distribution of the FDMLE has bounded support and its density is infinite at the upper bound when the time series sample size T approaching infinity. As the panel width n approaching infinity the pathology is removed and the limit theory is normal. This result applies even for T fixed and we present an expression for the asymptotic distribution which does not depend on the time dimension. When n,T approaching infinity, the FDMLE has smaller asymptotic variance than that of the bias corrected MLE, an outcome that is explained by the restricted nature of the FDMLE.

QML estimation with non-summable weight matrices

This paper revisits the theory of asymptotic behaviour of the well-known Gaussian Quasi-Maximum Likelihood estimator of parameters in mixed regressive, high-order autoregressive spatial models. We generalise the approach previously published in the econometric literature by weakening the assumptions imposed on the spatial weight matrix. This allows consideration of interaction patterns with a potentially larger degree of spatial dependence. Moreover, we broaden the class of admissible distributions of model residuals. As an example application of our new asymptotic analysis we also consider the large sample behaviour of a general group effects design.

The likelihood for LSS: stochasticity of bias coefficients at all orders

In the EFT of biased tracers the noise field ϵg is not exactly uncorrelated with the nonlinear matter field δ. Its correlation with δ is effectively captured by adding stochasticities to each bias coefficient. We show that if these stochastic fields are Gaussian (the impact of their non-Gaussianity being subleading on quasi-linear scales anyway) it is possible to resum exactly their effect on the conditional likelihood P[δg|δ] to observe a galaxy field δg given an underlying δ. This resummation allows to take them into account in EFT-based approaches to Bayesian forward modeling. We stress that the resulting corrections to a purely Gaussian conditional likelihood with white-noise covariance are the most relevant on scales where the EFT is under control: they are more important than any non-Gaussianity of the noise ϵg.

Speaker Identity Recognition by Acoustic and Visual Data Fusion through Personal Privacy for Smart Care and Service Applications

Abstract With rapid developments in techniques related to the internet of things, smart service applications such as voice-command-based speech recognition and smart care applications such as context-aware-based emotion recognition will gain much attention and potentially be a requirement in smart home or office environments. In such intelligence applications, identity recognition of the specific member in indoor spaces will be a crucial issue. In this study, a combined audio-visual identity recognition approach was developed. In this approach, visual information obtained from face detection was incorporated into acoustic Gaussian likelihood calculations for constructing speaker classification trees to significantly enhance the Gaussian mixture model (GMM)-based speaker recognition method. This study considered the privacy of the monitored person and reduced the degree of surveillance. Moreover, the popular Kinect sensor device containing a microphone array was adopted to obtain acoustic voice data from the person. The proposed audio-visual identity recognition approach deploys only two cameras in a specific indoor space for conveniently performing face detection and quickly determining the total number of people in the specific space. Such information pertaining to the number of people in the indoor space obtained using face detection was utilized to effectively regulate the accurate GMM speaker classification tree design. Two face-detection-regulated speaker classification tree schemes are presented for the GMM speaker recognition method in this study—the binary speaker classification tree (GMM-BT) and the non-binary speaker classification tree (GMM-NBT). The proposed GMM-BT and GMM-NBT methods achieve excellent identity recognition rates of 84.28% and 83%, respectively; both values are higher than the rate of the conventional GMM approach (80.5%). Moreover, as the extremely complex calculations of face recognition in general audio-visual speaker recognition tasks are not required, the proposed approach is rapid and efficient with only a slight increment of 0.051 s in the average recognition time.

Lightning Location System Detections as Evidence: A Unique Bayesian Framework

Modern lightning location systems (LLSs) remotely detect propagated electromagnetic fields and geolocate them for a better understanding of the lightning phenomenon. In recent years, they have become powerful tools for assessing situations where lightning may be the cause of damage. However, due to the random errors that occur in the remote sensing of electromagnetic propagation, such systems have median location accuracies between 50 and 100 m. This means that there is always some uncertainty in reported geolocations of lightning flashes. To date, there is no effective or standardized method for quantifying this uncertainty and using LLS reports as evidence. This article presents a unique solution to this problem by developing a Bayesian framework. In the field of forensic investigation and engineering, the Bayesian approaches to reporting evidence are widely accepted in legal forums. This article describes the necessary prior probability and likelihood functions (Gaussian mixture model, bivariate Gaussian, and Students’ t-distributions, respectively), and the framework is assessed by comparison with ground-truth events—photographed lightning events to a known location, the Brixton Tower in Johannesburg, South Africa. The framework has a true positive rate between 97% and 99% and 66% and 100% and a true negative rate between 98% and 99% using the bivariate Gaussian and Students’ t-likelihood functions, respectively. Utilizing the bivariate Students’ t-likelihood function achieves a much lower false-positive rate (0.05% to 0.2%) than the bivariate Gaussian likelihood function (0.7%–2%).

Optimizing Regularized Cholesky Score for Order-Based Learning of Bayesian Networks.

Bayesian networks are a class of popular graphical models that encode causal and conditional independence relations among variables by directed acyclic graphs (DAGs). We propose a novel structure learning method, annealing on regularized Cholesky score (ARCS), to search over topological sorts, or permutations of nodes, for a high-scoring Bayesian network. Our scoring function is derived from regularizing Gaussian DAG likelihood, and its optimization gives an alternative formulation of the sparse Cholesky factorization problem from a statistical viewpoint. We combine simulated annealing over permutation space with a fast proximal gradient algorithm, operating on triangular matrices of edge coefficients, to compute the score of any permutation. Combined, the two approaches allow us to quickly and effectively search over the space of DAGs without the need to verify the acyclicity constraint or to enumerate possible parent sets given a candidate topological sort. The annealing aspect of the optimization is able to consistently improve the accuracy of DAGs learned by greedy and deterministic search algorithms. In addition, we develop several techniques to facilitate the structure learning, including pre-annealing data-driven tuning parameter selection and post-annealing constraint-based structure refinement. Through extensive numerical comparisons, we show that ARCS outperformed existing methods by a substantial margin, demonstrating its great advantage in structure learning of Bayesian networks from both observational and experimental data. We also establish the consistency of our scoring function in estimating topological sorts and DAG structures in the large-sample limit. Source code of ARCS is available at https://github.com/yeqiaoling/arcs_bn.

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Statistical analysis of sparse approximate factor models

We consider the problem of estimating sparse approximate factor models. In a first step, we jointly estimate the factor loading parameters and the error - or idiosyncratic - covariance matrix based on the Gaussian quasi-maximum likelihood method. Conditionally on these first step estimators, using the SCAD, MCP and Lasso regularisers, we obtain a sparse error covariance matrix based on a Gaussian QML and, as an alternative criterion, a least squares loss function. Under suitable regularity conditions, we derive error bounds for the regularised idiosyncratic factor model matrix for both Gaussian QML and least squares losses. Moreover, we establish the support recovery property, including the case when the regulariser is non-convex. These theoretical results are supported by empirical studies.

Analytic Calculation of Covariance between Cosmological Parameters from Correlated Data Sets, with an Application to SPTpol.

Consistency checks of cosmological data sets are an important tool because they may suggest systematic errors or the type of modifications to ΛCDM necessary to resolve current tensions. In this work, we derive an analytic method for calculating the level of correlations between model parameters from two correlated cosmological data sets, which complements more computationally expensive simulations. This method is an extension of the Fisher analysis that assumes a Gaussian likelihood and a known data covariance matrix. We apply this method to the South Pole Telescope Polarimeter (SPTpol) temperature and polarization cosmic microwave background (CMB) spectra (TE and EE). We find weak correlations between ΛCDM parameters with a 9% correlation between the TE-only and EE-only constraints on H 0 and a 25% and 32% correlation for log(A s ) and n s respectively. The TE-EE parameter differences are consistent with zero, with a probability to exceed of 0.53. Using simulations we show that this test is independent of the consistency of the SPTpol TE and EE band powers with the best-fit ΛCDM model spectra. Despite the negative correlations between the TE and EE power spectra, the correlations between TE-only and EE-only ΛCDM parameters are positive. Ignoring correlations in the TT-TE and TE-EE comparisons biases the χ 2 low, artificially making parameters look more consistent. Therefore, we conclude that these correlations need to be accounted for when performing internal consistency checks of the TT versus TE versus EE power spectra for future CMB analyses.

CLASSIFICAÇÃO SUPERVISIONADA POR MÁXIMA VEROSSIMILHANÇA GAUSSIANA COMO FERRAMENTA PARA MONITORAMENTO DE ÁREAS DE VEGETAÇÃO EM REGIÕES METROPOLITANAS

Remote sensing has proven to be a great tool for mapping urban growth in regions with increasing urbanization. The most used methodology is the digital classification of multispectral data, performed supervised or unsupervised. This study aims to highlight the areas of vegetation that suffered reduction due to the implementation of projects, consequently the urban expansion, based on the classification supervised by Maximum Gaussian Likelihood (MAXVER), using digital images from the Landsat 5 satellite, having as its area metropolitan region of Porto Alegre, Rio Grande do Sul. The Landsat 5 orbital images dated in 2005 and 2011 were collected in the image bank of the National Institute for Space Research (INPE) and classified in the MultiSpec software. using the urban classes, vegetation, water, soil and sand. With an overall accuracy of over 90% for all classes, a reduction of 19.8% of the vegetation area and a 16.1% increase in the area of urbanization were observed, showing the impact of urbanization on the reduction of areas. of vegetation.

Outliers and misleading leverage effect in asymmetric GARCH-type models

Abstract This paper illustrates how outliers can affect both the estimation and testing of leverage effect by focusing on the TGARCH model. Three estimation methods are compared through Monte Carlo experiments: Gaussian Quasi-Maximum Likelihood, Quasi-Maximum Likelihood based on the Student-t likelihood and Least Absolute Deviation method. The empirical behavior of the t-ratio and the Likelihood Ratio tests for the significance of the leverage parameter is also analyzed. Our results put forward the unreliability of Gaussian Quasi-Maximum Likelihood methods in the presence of outliers. In particular, we show that one isolated outlier could hide true leverage effect whereas two consecutive outliers bias the estimated leverage coefficient in a direction that crucially depends on the sign of the first outlier and could lead to wrongly reject the null of no leverage effect or to estimate asymmetries of the wrong sign. By contrast, we highlight the good performance of the robust estimators in the presence of one isolated outlier. However, when there are patches of outliers, our findings suggest that the sizes and powers of the tests as well as the estimated parameters based on robust methods may still be distorted in some cases. We illustrate these results with two series of daily returns.

Adaptive inference on pure spatial models

In a general class of semiparametric pure spatial models (having no explanatory variables) allowing nonlinearity in the parameter and the weight matrix, we propose adaptive tests and estimates which are asymptotically efficient in the presence of unknown, nonparametric distributional form. Feasibility of adaptive estimation is verified and its efficiency improvement over Gaussian pseudo maximum likelihood is shown to be either less than, or more than, for models with explanatory variables, depending on properties of the spatial weight matrix. An adaptive Lagrange Multiplier testing procedure for lack of spatial dependence is proposed and this, and our adaptive parameter estimate, are extended to cover regression with spatially correlated errors.

Robust particle filter based on Huber function for underwater terrain‐aided navigation

Terrain-aided navigation (TAN) is a promising technique to determine the location of underwater vehicle by matching terrain measurement against a known map. The particle filter (PF) is a natural choice for TAN because of its ability to handle non-linear, multimodal problems. However, the terrain measurements are vulnerable to outliers, which will cause the PF to degrade or even diverge. Modification of the Gaussian likelihood function by using robust cost functions is a way to reduce the effect of outliers on an estimate. The authors propose to use the Huber function to modify the measurement model used to set importance weights in a PF. They verify their method in simulations of multi-beam sonar in a real underwater digital map. The results demonstrate that the proposed method is more robust to outliers than the standard PF (SPF).

Radio Interferometric Calibration Using a Complex Student’s t-distribution and Wirtinger Derivatives

Abstract Radio interferometric gain calibration can be biased by incomplete sky models and radio frequency interference, resulting in calibration artefacts that can restrict the dynamic range of the resulting images. It has been suggested that calibration algorithms employing heavy-tailed likelihood functions are less susceptible to this due to their robustness against outliers in the data. We present an algorithm based on a Student’s t-distribution which leverages the framework of complex optimisation and Wirtinger calculus for efficient and robust interferometric gain calibration. We integrate this algorithm as an option in the newly released calibration software package, CubiCal. We demonstrate that the algorithm can mitigate some of the biases introduced by incomplete sky models and radio frequency interference by applying it to both simulated and real data. Our results show significant improvements compared to a conventional least-squares solver which assumes a Gaussian likelihood function. Furthermore, we provide some insight into why the algorithm outperforms the conventional solver, and discuss specific scenarios (for both direction-independent and direction-dependent self-calibration) where this is expected to be the case.

Constraining new physics from Higgs measurements with Lilith: update to LHC Run 2 results

Lilith is a public Python library for constraining new physics from Higgs signal strength measurements. We here present version 2.0 of Lilith together with an updated XML database which includes the current ATLAS and CMS Run 2 Higgs results for 36 fb^{-1}−1. Both the code and the database were extended from the ordinary Gaussian approximation employed in Lilith-1.1 to using variable Gaussian and Poisson likelihoods. Moreover, Lilith can now make use of correlation matrices of arbitrary dimension. We provide detailed validations of the implemented experimental results as well as a status of global fits for reduced Higgs couplings, Two-Higgs-doublet models of Type I and Type II, and invisible Higgs decays. Lilith-2.0 is available on GitHub and ready to be used to constrain a wide class of new physics scenarios.

Parameter inference and model comparison using theoretical predictions from noisy simulations

ABSTRACTWhen inferring unknown parameters or comparing different models, data must be compared to underlying theory. Even if a model has no closed-form solution to derive summary statistics, it is often still possible to simulate mock data in order to generate theoretical predictions. For realistic simulations of noisy data, this is identical to drawing realizations of the data from a likelihood distribution. Though the estimated summary statistic from simulated data vectors may be unbiased, the estimator has variance that should be accounted for. We show how to correct the likelihood in the presence of an estimated summary statistic by marginalizing over the true summary statistic in the framework of a Bayesian hierarchical model. For Gaussian likelihoods where the covariance must also be estimated from simulations, we present an alteration to the Sellentin–Heavens corrected likelihood. We show that excluding the proposed correction leads to an incorrect estimate of the Bayesian evidence with Joint Light-Curve Analysis data. The correction is highly relevant for cosmological inference that relies on simulated data for theory (e.g. weak lensing peak statistics and simulated power spectra) and can reduce the number of simulations required.

Tomographic weak lensing bispectrum: a thorough analysis towards the next generation of galaxy surveys

ABSTRACT We address key points for an efficient implementation of likelihood codes for modern weak lensing large-scale structure surveys. Specifically, we focus on the joint weak lensing convergence power spectrum–bispectrum probe and we tackle the numerical challenges required by a realistic analysis. Under the assumption of (multivariate) Gaussian likelihoods, we have developed a high performance code that allows highly parallelized prediction of the binned tomographic observables and of their joint non-Gaussian covariance matrix accounting for terms up to the six-point correlation function and supersample effects. This performance allows us to qualitatively address several interesting scientific questions. We find that the bispectrum provides an improvement in terms of signal-to-noise ratio (S/N) of about 10 per cent on top of the power spectrum, making it a non-negligible source of information for future surveys. Furthermore, we are capable to test the impact of theoretical uncertainties in the halo model used to build our observables; with presently allowed variations we conclude that the impact is negligible on the S/N. Finally, we consider data compression possibilities to optimize future analyses of the weak lensing bispectrum. We find that, ignoring systematics, five equipopulated redshift bins are enough to recover the information content of a Euclid-like survey, with negligible improvement when increasing to 10 bins. We also explore principal component analysis and dependence on the triangle shapes as ways to reduce the numerical complexity of the problem.

Data compression in cosmology: A compressed likelihood for Planck data

We apply the massively optimized parameter estimation and data compression technique (MOPED) to the public Planck 2015 temperature likelihood, reducing the dimensions of the data space to one number per parameter of interest. We present CosMOPED, a lightweight and convenient compressed likelihood code implemented in python. In doing so we show that the $\ensuremath{\ell}<30$ Planck temperature likelihood can be well approximated by two Gaussian-distributed data points, which allows us to replace the map-based low-$\ensuremath{\ell}$ temperature likelihood by a simple Gaussian likelihood. We make available a python implementation of Planck's 2015 Plik_lite temperature likelihood that includes these low-$\ensuremath{\ell}$ binned temperature data (Planck-lite-py). We do not explicitly use the large-scale polarization data in CosMOPED, instead imposing a prior on the optical depth to reionization derived from these data. We show that the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ parameters recovered with CosMOPED are consistent with the uncompressed likelihood to within $0.1\ensuremath{\sigma}$, and test that a 7-parameter extended model performs similarly well.