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A note on bimodality in the log-likelihood function for penalized spline mixed models

COMPUTATIONAL STATISTICS & DATA ANALYSIS 2009年第4期53卷 920-931页

作者： Welham, S. J. Thompson, R. Rothamsted Res Ctr Math & Computat Biol Biomath & Bioinformat Harpenden AL5 2JQ Herts England Univ London Sch Math Sci London E1 4NS England

For a smoothing spline or general penalized spline model, the smoothing parameter can be estimated using residual maximum likelihood (REML) methods by expressing the spline in the form of a mixed model. The possibility of bimodality in the profile log-likelihood function for the smoothing parameter of these penalized spline mixed models is demonstrated. A canonical transformation into independent observations is used to provide efficient evaluation of the log-likelihood function and gives insight into the incompatibilities between the model and data that cause bimodality. This transformation can also be used to assess the influence of different frequency components in the data on the estimated smoothing parameter. It is demonstrated that, where bimodality occurs in the log-likelihood, Bayesian penalized spline models may show poor mixing in MCMC chains and be sensitive to the choice of prior distributions for variance components. (C) 2008 Elsevier B.V. All rights reserved.

关键词： INFORMATION REGRESSION PARAMETER VARIANCE likelihood functions log-likelihood function variance components splines (mathematics) smoothing Variance

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Optimal distribution-free concentration for the log-likelihood function of Bernoulli variables

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2023年第1期2023卷 81-81页

作者： Ren, Zhonggui Harbin Univ Commerce Coll Fdn Sci Harbin Peoples R China

This paper aims to establish distribution-free concentration inequalities for the log-likelihood function of Bernoulli variables, which means that the tail bounds are independent of the parameters. Moreover, Bernstein's and Bennett's inequalities with optimal constants are obtained. The simulation study shows significant improvements over the previous results.

关键词： Bernstein's inequality Bennett's inequality log-likelihood function Bernoulli distribution

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Non trivial optimal sampling rate for estimating a Lipschitz-continuous function in presence of mean-reverting Ornstein-Uhlenbeck noise

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STATISTICAL PAPERS 2025年第3期66卷 1-18页

作者： Bernardi, Enrico Lanconelli, Alberto Lauria, Christopher S. A. Percin, Berk Tan Univ Bologna Dipartimento Sci Stat Paolo Fortunati Bologna Italy

We consider a mean-reverting Ornstein-Uhlenbeck process that perturbs an unknown Lipschitz-continuous drift and want to estimate the value of that drift at some fixed time horizon. The estimation is based on a finite number of observations obtained by sampling the path of the process at equally spaced instant of times between the time origin and prescribed time horizon. Aim of the study is to find the optimal sample size/rate for achieving the minimum mean square distance between our estimator and the true value of the drift at the time horizon. The estimation procedure we utilize consists of an online time-varying optimization scheme employed through a stochastic gradient ascent algorithm for the log-likelihood of our observations. We discover a trade-off between the correlation of the observations, which increases with the sample size, and the dynamic nature of the unknown drift, which is weakened by increasing the frequency of observation. The mean square error is shown to be non monotonic in the sample size, attaining a global minimum whose precise description depends on the parameters that rule the model. In the static case, i.e. when the unknown drift is constant, our method performs in highly correlated regimes even better than the arithmetic mean of the observations, which represents in such particular instance a natural candidate estimator. A comparison with the global maximum likelihood estimator is also presented.

关键词： Ornstein-Uhlenbeck process Positive correlation log-likelihood function Stochastic gradient ascent

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Efficient Analysis of Overdispersed Data Using an Accurate Computation of the Dirichlet Multinomial Distribution

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2025年第2期47卷 1181-1189页

作者： Baddar, Sherenaz Al-Haj Languasco, Alessandro Migliardi, Mauro Univ Jordan Dept Comp Sci Amman 11942 Jordan Univ Padua Dipartimento Ingn Informaz I-35131 Padua Italy Univ Padua Dipartimento Ingn Informaz I-35131 Padua Italy

Modeling count data using suitable statistical distributions has been instrumental for analyzing the patterns it conveys. However, failing to address critical aspects, like overdispersion, jeopardizes the effectiveness of such an analysis. In this paper, overdispersed count data is modeled using the Dirichlet Multinomial (DM) distribution by maximizing its likelihood using a fixed-point iteration algorithm. This is achieved by estimating the DM distribution parameters while comparing the recent Languasco-Migliardi (LM), and the Yu-Shaw (YS) procedures, which address the well-known computational difficulties of evaluating its log-likelihood. Experiments were conducted using multiple datasets from different domains spanning polls, images, and IoT network traffic. They all showed the superiority of the LM procedure as it succeeded at estimating the DM parameters at the designated level of accuracy in all experiments, while the YS procedure failed to produce sufficiently accurate results (or any results at all) in several experiments. Moreover, the LM procedure achieved a speedup that ranged from 2-fold to 20-fold over YS.

关键词： Accuracy Data models Computational modeling Polynomials Convergence Instruments Vectors Telecommunication traffic Statistical distributions Reliability Pattern analysis overdispersed data Dirichlet multinomial distribution log-likelihood function fixed-point iteration

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EFFECTIVENESS OF THE likelihood function IN logIC-BASED TRACK FORMATION

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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS 1991年第1期27卷 184-187页

作者： BARSHALOM, Y LI, XR Dept. of Electr. & Syst. Eng. Connecticut Univ. Storrs CT USA

A logic-based track formation procedure was presented in [6, 7] which, in addition to the conventional gating logic, utilized the negative log-likelihood function (goodness of fit or sum of residuals) to further reduce the false track acceptance probability. An analytical technique for the evaluation of the effectiveness of this reduction is presented here. The exact solution in closed form and a Gaussian approximation are given and compared. It is shown that the goodness of fit provides an additional reduction of the false track acceptance probability by a factor of 2 to 3.

关键词： likelihood functions ACCEPTANCE log-likelihood function False goodness of fit Gaussian approximation closed forms track formation

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Estimating the Probability Density function of a Nonstationary Non-Gaussian Noise

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 2010年第4期57卷 1429-1435页

作者： Mukherjee, Arpita Sengupta, Aparajita Bengal Engn & Sci Univ Dept Elect Engn Howrah 711103 India

The problem of estimating the probability density function (pdf) of a nonstationary non-Gaussian noise is addressed. The non-Gaussian noise is modeled using Gaussian mixture pdfs, and an algorithm is proposed to estimate the parameters by maximizing the log-likelihood function. Three simulation results illustrate the validity and utility of the proposed algorithm for stationary or nonstationary, Gaussian or non-Gaussian, zeromean or nonzero mean, and unimodal or multimodal distributed noise.

关键词： Gaussian mixture models log-likelihood function non-Gaussian nonstationary

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Semi-supervised learning for k-dependence Bayesian classifiers

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APPLIED INTELLIGENCE 2022年第4期52卷 3604-3622页

作者： Wang, LiMin Zhang, XinHao Li, Kuo Zhang, Shuai Jilin Univ Minist Educ Key Lab Symbol Computat & Knowledge Engn Changchun 130012 Peoples R China JiLin Univ Coll Math Changchun 130012 Peoples R China JiLin Univ Coll Comp Sci & Technol Changchun 130012 Peoples R China JiLin Univ Coll Software Changchun 130012 Peoples R China

Bayesian network classifiers (BNCs) are powerful tools for graphically encoding the dependency relationships among variables in a directed acyclic graph and reasoning under conditions of uncertainty. Ever increasing data quantity makes ever more urgent the need of BNCs that are highly scalable and can perform significantly better in terms of classification. Numerous approaches have been proposed to mine conditional dependencies among attributes implicated in labeled training data under the framework of supervised learning, whereas the specific characteristics of unlabeled testing instances receive less attention. That may lead to overfitting and degradation in classification performance. In this paper, we argue that the knowledge learned from labeled training dataset and that from unlabeled testing instance are complementary in nature. The testing instance is pre-assigned with any possible label to make it complete, then log-likelihood function is introduced and redefined to measure the extents to which the learned BNC fits training or testing data. Heuristic search strategy is applied to learn two kinds of arbitrary k-dependence BNCs (general BNC for modeling training dataset and local BNC for modeling testing instance), which will work as an ensemble to make the final prediction under the framework of semi-supervised learning. The experimental evaluation on 40 publicly available datasets from the UCI machine learning repository reveals that the proposed algorithm achieves competitive classification performance compared with state-of-the-art BNCs and their variants, such as CFWNB, WATAN, FKDB, SKDB and IWAODE.

关键词： Semi-supervised learning Bayesian network classifiers log-likelihood function

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Adaptive bandwidth mode detection algorithm

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IET IMAGE PROCESSING 2011年第8期5卷 645-660页

作者： Dagher, I. Dahdah, K. Univ Balamand Dept Comp Engn El Koura Lebanon

In this study a new algorithm 'adaptive bandwidth mode detection' (ABMD) algorithm has been developed to recover the correct density function without the need to either specify the correct number of Gaussians in the model or the correct bandwidth. The ABMD is employed in modelling visual features in applications such as image segmentation and real-time visual tracking. A simple type of model for these visual features are the Gaussian mixtures, where the number of Gaussian components is variable, thus, making it a flexible method for multimodal representation. This algorithm is used at initialisation for target modelling, where the target update will be done based on the mode propagation with adaptive bandwidth tracker method. It is based on an optimisation technique where a gradient ascent method is used and the optimal solution is selected based on a log-likelihood function. The mode detection ability of ABMD algorithm is compared with both the expectation maximisation and mean-shift algorithms. Furthermore, different video sequences have been employed to show how this approach has the ability to track an object regardless of whether the target model is corrupted with unwanted data at new frames.

关键词： Gaussian component number video sequences Optimisation techniques video signal processing Gaussian mixtures Computer vision and image processing techniques density function recovery Gaussian processes Optical, image and video signal processing expectation maximisation algorithms image segmentation object tracking log-likelihood function multimodal representation image sequences optimisation technique mode propagation visual feature modelling ABMD algorithm optimisation Interpolation and function approximation (numerical analysis) mean-shift algorithms real-time visual tracking Video signal processing adaptive bandwidth mode detection algorithm adaptive bandwidth tracker method target modelling Other topics in statistics gradient methods gradient ascent method expectation-maximisation algorithm

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Performance analysis of a joint estimator for timing, frequency, and phase with continuous-phase modulation

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IET COMMUNICATIONS 2016年第3期10卷 263-271页

作者： Zhai, Wenchao Li, Zan Si, Jiangbo Bai, Jun Xidian Univ State Key Lab Integrated Serv Networks Xian 710071 Shaanxi Peoples R China

Performance analysis is presented for the joint estimation of symbol timing, frequency offset, and phase offset with continuous-phase modulation. In this study, undesirable influences on parameters to be estimated, which arise from the inaccuracy of the already estimated parameters, are computed mathematically. Based on performance analysis, a conditional Cramer-Rao bound (CRB) is put forward, and a hypothesise is made that phase offset variance with frequency error present should match with the conditional CRB for any digital phase modulation. Moreover, the hypothesis is manifested by analysing the log-likelihood function in a more general mathematical manner. To alleviate the influences, a data-aided acquisition and phase-locked loop tracking algorithm is proposed. The results of Monte Carlo simulations are presented, showing good agreement with theoretical analysis. Simulation results also show that the proposed algorithm can improve performance in terms of phase error variance at the expense of a high signal-to-noise ratio threshold.

关键词： continuous phase modulation maximum likelihood estimation frequency estimation timing phase locked loops Monte Carlo methods phase estimation symbol timing frequency offset continuous-phase modulation signals optimum training sequence Cramer-Rao bound CRB closed-form expressions maximum-likelihood estimator phase offset variance frequency error joint estimation algorithms digital phase modulation log-likelihood function data-aided acquisition phase-locked loop tracking algorithm Monte Carlo simulations phase error variance signal-to-noise ratio threshold

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Detection and Prognostics on Low-Dimensional Systems

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IEEE TRANSACTIONS ON SYSTEMS MAN AND CYBERNETICS PART C-APPLICATIONS AND REVIEWS 2009年第1期39卷 44-54页

作者： Srivastava, Ashok N. Das, Santanu NASA Ames Res Ctr Intelligent Syst Div Moffett Field CA 94035 USA NASA Ames Res Ctr Intelligent Data Understanding Grp Moffett Field CA 94035 USA Univ Calif Santa Cruz Univ Affiliated Res Ctr Santa Cruz CA 95064 USA

This paper describes the application or known and novel prognostic algorithms on systems that can be described by low-dimensional, potentially nonlinear dynamics. The methods rely on estimating the conditional probability distribution of the output of the system at a future time given knowledge of the current state of the system. We show how to estimate these conditional probabilities using a variety of techniques, including bragged neural networks and kernel methods such as Gaussian process regression (GPR). The results are compared with standard method such as the nearest neighbor, algorithm. We demonstrate the algorithms on a real-world dataset and a simulated dataset. The real-world dataset consists of the intensity of an NH3 laser. The laser dataset has been shown by other authors to exhibit low-dimensional chaos with sudden drops in intensity. The simulated dataset is generated from the Lorenz attractor and has known statistical characteristics. On these datasets, we show the evolution of the estimated conditional probability distribution, the way it can act as a prognostic signal, and its use as an early warning system. We also review a novel approach to perform GPR with large numbers of data points.

关键词： Anomaly detection Gaussian process regression (GPR) k-nearest neighbor log-likelihood function Lorenz model NH3 laser system prediction prognosis

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