Generalized Likelihood Function Research Articles

Effective implementation and improvement of fast labeled multi-Bernoulli filter

Effective implementation of the fast labeled multi-Bernoulli (FLMB) filter is addressed for target tracking with interval measurements. Firstly, a sequential Monte Carlo (SMC) implementation of the FLMB filter, SMC-FLMB filter, is derived based on generalized likelihood function weighting. Then, a box particle (BP) implementation of the FLMB filter, BP-FLMB filter, is developed, with a computational complexity reduction of the SMC-FLMB filter. Finally, an improved version of the BP-FLMB filter, improved BP-FLMB (IBP-FLMB) filter, is proposed, improving its estimation accuracy and real-time performance under the conditions of low detection probability and high clutter. Simulation results show that the BP-FLMB filter has a great improvement of the real-time performance than the SMC-FLMB filter, with similar tracking performance. Compared with the BP-FLMB filter, the IBP-FLMB filter has better estimation performance and real-time performance under the conditions of low detection probability and high clutter.

IMM Robust Cardinality Balance Multi‐Bernoulli Filter for Multiple Maneuvering Target Tracking with Interval Measurement

This paper presents a novel Interacting multi-model (IMM) Robust Cardinality balance multi-target multi-Bernoulli (R-CBMeMBer) filter to solve the maneuvering target tracking problem in the case of interval measurement, unknown detection probability and unknown clutter density. In essence, IMM R-CBMeMBer filter is an extended application of R-CBMeMBer filter. In the IMM R-CBMeMBer filter, the target state is first extended to distinguish clutter from the real target. The detection probability and model probability of the target can be adaptively updated. Then, generalized likelihood function and IMM algorithm are introduced to interactively predict and update the state of the target in the IMM R-CBMeMBer filtering process. In addition, a particle application of the IMM R-CBMeMBer filter is given, and a numerical experiment is designed under nonlinear conditions. Meanwhile, Doppler information of the target is employed to estimate the velocity of each maneuvering target. Numerical experiments also verify that the IMM R-CBMeMBer filter can effectively estimate the target position, target velocity, target detection probability and clutter number in the condition of unknown detection probability, unknown clutter rate and interval measurement.

A distributed averaging-based evidential Expectation-Maximization algorithm for density estimation in unreliable sensor networks

Abstract In this paper, a novel distributed averaging-based evidential Expectation-Maximization algorithm is presented for data clustering and probability density function estimation in unreliable sensor networks in the presence of uncertain observations. All observations of the sensor network are statistically modelled by a Gaussian mixture model. A generalized likelihood function is maximized in a distributed manner for density estimation. For this purpose, the parameters of the mixture model such as weighting coefficients, mean vectors, and covariance matrices are updated by using a distributed averaging method. The proposed algorithm can be applied to any network with a mesh topology. The convergence of algorithm is also investigated and guaranteed by using a mathematical analysis. In the simulation result section, the proposed algorithm has been applied to a simulated uncertain sensor network with a mesh topology. The simulations confirm the favourable performance of the distributed averaging-based evidential EM algorithm.

Bayesian Method for the Generalized Exponential Model Using Fuzzy Data

This paper focuses on Bayesian inference for the parameters of the generalized exponential model under asymmetric and symmetric loss functions when the observations are described in terms of fuzzy numbers. First, a generalized likelihood function based on fuzzy data is derived. Then, considering general entropy, linear exponential and squared error loss functions, the Bayes estimates of the parameters are obtained. Since Bayes estimates could not be expressed in closed forms, Metropolis–Hasting samplers are used to compute the approximate Bayes estimates. For comparison purposes, the maximum likelihood estimates of the parameters are also computed. The proposed inferences are illustrated using three real-world examples. The numerical simulation results demonstrate the superiority of the Bayesian method over the maximum likelihood procedure.

Censoring Outliers in Radar Data: An Approximate ML Approach and its Analysis

This paper deals with the problem of censoring outliers in a class of complex multivariate elliptically contoured distributed radar data, which is a vital issue in radar signal processing applications, such as adaptive radar detection and space-time adaptive processing. The maximum likelihood (ML) estimate of the outlier subset is derived resorting to the generalized likelihood function (GLF) criterion. Since the ML estimate involves the solution of a combinatorial problem, a reduced complexity but approximate ML (AML) procedure is also considered. At the analysis stage, the performance of the AML method is evaluated in the presence of both simulated and real radar data, also in comparison with the conventional generalized inner product (GIP) and the reiterative censored GIP (RCGIP) algorithms. The results highlight that the AML technique achieves a satisfactory performance level and can outperform both GIP and RCGIP in some situations.

Bayesian inference using generalized likelihood for non-traditional residuals

In structural engineering, Bayesian inference has been widely used to quantify uncertainties associated with modeling and measurement. Based on quantified uncertainties, predictions can be made in terms of the probability statement to consider predictive uncertainties for reliable decision-making. Traditionally, residual is assumed to be an independent and identically distributed Gaussian distribution with zero-mean and constant variance. In some situations, the traditional residual assumption cannot be adequate and their predictive capability for quantity of interest may be questionable. Although the residual assumption is important for quantifying the uncertainties in parameter estimation, there is little research for structural engineering to investigate influence of the non-traditional residuals to Bayesian inference. As an illustrative study, this study investigates influence of the non-traditional residuals in a concrete creep prediction. Without an adequate statistical representation of the observed residuals, the predictive performances over extrapolation are unreliable and inconsistent. To address the non-traditional residuals, this study introduces a generalized likelihood function. The generalized likelihood function can accommodate heteroscedasticity, temporal correlation and non-normality in observed residuals. By introducing the generalized likelihood function, the predictive performance over extrapolation can be significantly improved by satisfying with observed residual distribution.

Space-Based Focal Plane Ambiguous Measurement Ballistic Target MeMber Tracking.

Aimed at space-based passive detection and tracking of ballistic targets, a multi-target multi-Bernoulli (MeMber) filtering algorithm based on a focal plane ambiguous measurement model is proposed. The measurement error sources of space-based passive detection are analyzed. It is found that focal plane target tracking is the basis of space target tracking in the framework of the distributed data processing structure, and the main error of focal plane measurement is pixel resolution. Based on the above analysis, the focal plane ambiguous measurement model is established to replace the traditional measurement model and the generalized likelihood function is designed. Finally, the MeMber filter is modified based on ambiguous measurement and generalized likelihood function. The simulation experiment compares the tracking effect of a MeMber filter based on ambiguous measurement and traditional measurement, respectively. The filter based on ambiguous measurement achieves better results. It shows that ambiguous measurement is closer to reality and has more application value.

Extending the Applicability of the Generalized Likelihood Function for Zero‐Inflated Data Series

AbstractProper uncertainty estimation for data series with a high proportion of zero and near zero observations has been a challenge in hydrologic studies. This technical note proposes a modification to the Generalized Likelihood function that accounts for zero inflation of the error distribution (ZI‐GL). We compare the performance of the proposed ZI‐GL with the original Generalized Likelihood function using the entire data series (GL) and by simply suppressing zero observations (GLy>0). These approaches were applied to two interception modeling examples characterized by data series with a significant number of zeros. The ZI‐GL produced better uncertainty ranges than the GL as measured by the precision, reliability and volumetric bias metrics. The comparison between ZI‐GL and GLy>0 highlights the need for further improvement in the treatment of residuals from near zero simulations when a linear heteroscedastic error model is considered. Aside from the interception modeling examples illustrated herein, the proposed ZI‐GL may be useful for other hydrologic studies, such as for the modeling of the runoff generation in hillslopes and ephemeral catchments.

Investigating Prior Parameter Distributions in the Inverse Modelling of Water Distribution Hydraulic Models

Inverse modelling concentrates on estimating water distribution system (WDS) model parameters that are not directly measurable, e.g. pipe roughness coefficients, which can, therefore, only be estimated by indirect approaches, i.e. inverse modelling. Estimation of the parameter and predictive uncertainty of WDS models is an essential part of the inverse modelling process. Recently, Markov Chain Monte Carlo (MCMC) simulations have gained in popularity in uncertainty analyses due to their effective and efficient exploration of posterior parameter probability density functions (pdf). A Bayesian framework is used to infer prior parameter information via a likelihood function to plausible ranges of posterior parameter pdf. Improved parameter and predictive uncertainty are achieved through the incorporation of prior pdf of parameter values and the use of a generalized likelihood function. We used three prior information sampling schemes to infer the pipe roughness coefficients of WDS models. A hypothetical case study and a real-world WDS case study were used to illustrate the strengths and weaknesses of a particular selection of a prior information pdf. The results obtained show that the level of parameter identifiability (i.e. sensitivity) is an important property for prior pdf selection.

Assessment of parametric uncertainty for groundwater reactive transport modeling

The validity of using Gaussian assumptions for model residuals in uncertainty quantification of a groundwater reactive transport model was evaluated in this study. Least squares regression methods explicitly assume Gaussian residuals, and the assumption leads to Gaussian likelihood functions, model parameters, and model predictions. While the Bayesian methods do not explicitly require the Gaussian assumption, Gaussian residuals are widely used. This paper shows that the residuals of the reactive transport model are non-Gaussian, heteroscedastic, and correlated in time; characterizing them requires using a generalized likelihood function such as the formal generalized likelihood function developed by Schoups and Vrugt (2010). For the surface complexation model considered in this study for simulating uranium reactive transport in groundwater, parametric uncertainty is quantified using the least squares regression methods and Bayesian methods with both Gaussian and formal generalized likelihood functions. While the least squares methods and Bayesian methods with Gaussian likelihood function produce similar Gaussian parameter distributions, the parameter distributions of Bayesian uncertainty quantification using the formal generalized likelihood function are non-Gaussian. In addition, predictive performance of formal generalized likelihood function is superior to that of least squares regression and Bayesian methods with Gaussian likelihood function. The Bayesian uncertainty quantification is conducted using the differential evolution adaptive metropolis (DREAM(zs)) algorithm; as a Markov chain Monte Carlo (MCMC) method, it is a robust tool for quantifying uncertainty in groundwater reactive transport models. For the surface complexation model, the regression-based local sensitivity analysis and Morris- and DREAM(ZS)-based global sensitivity analysis yield almost identical ranking of parameter importance. The uncertainty analysis may help select appropriate likelihood functions, improve model calibration, and reduce predictive uncertainty in other groundwater reactive transport and environmental modeling.

Parameter estimation from interval-valued data using the expectation-maximization algorithm

This paper investigates on the problem of parameter estimation in statistical model when observations are intervals assumed to be related to underlying crisp realizations of a random sample. The proposed approach relies on the extension of likelihood function in interval setting. A maximum likelihood estimate of the parameter of interest may then be defined as a crisp value maximizing the generalized likelihood function. Using the expectation-maximization (EM) to solve such maximizing problem therefore derives the so-called interval-valued EM algorithm (IEM), which makes it possible to solve a wide range of statistical problems involving interval-valued data. To show the performance of IEM, the following two classical problems are illustrated: univariate normal mean and variance estimation from interval-valued samples, and multiple linear/nonlinear regression with crisp inputs and interval output.

Bayesian inference of uncertainties in precipitation‐streamflow modeling in a snow affected catchment

Bayesian inference is used to study the effect of precipitation and model structural uncertainty on estimates of model parameters and confidence limits of predictive variables in a conceptual rainfall‐runoff model in the snow‐fed Rudbäck catchment (142 ha) in southern Finland. The IHACRES model is coupled with a simple degree day model to account for snow accumulation and melt. The posterior probability distribution of the model parameters is sampled by using the Differential Evolution Adaptive Metropolis (DREAM(ZS)) algorithm and the generalized likelihood function. Precipitation uncertainty is taken into account by introducing additional latent variables that were used as multipliers for individual storm events. Results suggest that occasional snow water equivalent (SWE) observations together with daily streamflow observations do not contain enough information to simultaneously identify model parameters, precipitation uncertainty and model structural uncertainty in the Rudbäck catchment. The addition of an autoregressive component to account for model structure error and latent variables having uniform priors to account for input uncertainty lead to dubious posterior distributions of model parameters. Thus our hypothesis that informative priors for latent variables could be replaced by additional SWE data could not be confirmed. The model was found to work adequately in 1‐day‐ahead simulation mode, but the results were poor in the simulation batch mode. This was caused by the interaction of parameters that were used to describe different sources of uncertainty. The findings may have lessons for other cases where parameterizations are similarly high in relation to available prior information.

Model validation under epistemic uncertainty

This paper develops a methodology to assess the validity of computational models when some quantities may be affected by epistemic uncertainty. Three types of epistemic uncertainty regarding input random variables – interval data, sparse point data, and probability distributions with parameter uncertainty – are considered. When the model inputs are described using sparse point data and/or interval data, a likelihood-based methodology is used to represent these variables as probability distributions. Two approaches – a parametric approach and a non-parametric approach – are pursued for this purpose. While the parametric approach leads to a family of distributions due to distribution parameter uncertainty, the principles of conditional probability and total probability can be used to integrate the family of distributions into a single distribution. The non-parametric approach directly yields a single probability distribution. The probabilistic model predictions are compared against experimental observations, which may again be point data or interval data. A generalized likelihood function is constructed for Bayesian updating, and the posterior distribution of the model output is estimated. The Bayes factor metric is extended to assess the validity of the model under both aleatory and epistemic uncertainty and to estimate the confidence in the model prediction. The proposed method is illustrated using a numerical example.

A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and non‐Gaussian errors

Estimation of parameter and predictive uncertainty of hydrologic models has traditionally relied on several simplifying assumptions. Residual errors are often assumed to be independent and to be adequately described by a Gaussian probability distribution with a mean of zero and a constant variance. Here we investigate to what extent estimates of parameter and predictive uncertainty are affected when these assumptions are relaxed. A formal generalized likelihood function is presented, which extends the applicability of previously used likelihood functions to situations where residual errors are correlated, heteroscedastic, and non‐Gaussian with varying degrees of kurtosis and skewness. The approach focuses on a correct statistical description of the data and the total model residuals, without separating out various error sources. Application to Bayesian uncertainty analysis of a conceptual rainfall‐runoff model simultaneously identifies the hydrologic model parameters and the appropriate statistical distribution of the residual errors. When applied to daily rainfall‐runoff data from a humid basin we find that (1) residual errors are much better described by a heteroscedastic, first‐order, auto‐correlated error model with a Laplacian distribution function characterized by heavier tails than a Gaussian distribution; and (2) compared to a standard least‐squares approach, proper representation of the statistical distribution of residual errors yields tighter predictive uncertainty bands and different parameter uncertainty estimates that are less sensitive to the particular time period used for inference. Application to daily rainfall‐runoff data from a semiarid basin with more significant residual errors and systematic underprediction of peak flows shows that (1) multiplicative bias factors can be used to compensate for some of the largest errors and (2) a skewed error distribution yields improved estimates of predictive uncertainty in this semiarid basin with near‐zero flows. We conclude that the presented methodology provides improved estimates of parameter and total prediction uncertainty and should be useful for handling complex residual errors in other hydrologic regression models as well.

Improved energy detector for random signals in gaussian noise

New and improved energy detector for random signals in Gaussian noise is proposed by replacing the squaring operation of the signal amplitude in the conventional energy detector with an arbitrary positive power operation. Numerical results show that the best power operation depends on the probability of false alarm, the probability of detection, the average signal-to-noise ratio or the sample size. By choosing the optimum power operation according to different system settings, new energy detectors with better detection performances can be derived. These results give useful guidance on how to improve the performances of current wireless systems using the energy detector. It also confirms that the conventional energy detector based on the generalized likelihood ratio test using the generalized likelihood function is not optimum in terms of the detection performance.

Likelihood function modeling of particle filter in presence of non-stationary non-gaussian measurement noise

A generalized likelihood function model of a sampling importance resampling (SIR) particle filter (PF) has been derived for state estimation of a nonlinear system in the presence of non-stationary, non-Gaussian white measurement noise. The measurement noise is modeled by Gaussian mixture probability density function and the noise parameters are estimated by maximizing the log likelihood function of the noise model. This model is then included in the likelihood function of the SIR particle filter (PF) at each time step for online state estimation of the system. The performance of the proposed algorithm has been evaluated by estimating the states of (i) a non-linear system in the presence of non-stationary Rayleigh distributed noise and (ii) a radar tracking system in the presence of glint noise. The simulation results show that the proposed modified SIR PF offers best performance among the considered algorithms for these examples.

Design and analysis of a knowledge-aided radar detector for doppler processing

In this paper we discuss the combined use of a priori information and adaptive signal processing techniques for the design and the analysis of a knowledge-aided (KA) radar receiver for Doppler processing. To this end, resorting to the generalized likelihood function (GLF) criterion (both one-step and two-step), we design and assess data-adaptive procedures for the selection of training data. Then we introduce a KA radar detector composed of three elements: a geographic-map-based data selector, which exploits some a priori information concerning the topography of the observed scene, a data-adaptive training selector which removes dynamic outliers from the training data, and an adaptive radar detector which performs the final decision about the target presence. The performance of the KA algorithm is analyzed both on simulated as well as on real radar data collected by the McMaster University IPIX radar. The results show that the new KA system achieves a satisfactory performance level and can outperform some previously proposed adaptive detection schemes

Sufficiency, generalized likelihood function and exponential family

The problem of characterizing an exponential family by sufficiency of certain statistics is considered. In distinction to most of the papers on this subject we do not want to characterize an exponential family of order less than or equal to k by the existence of a-dimensional sufficient statis tics in.GepenoenX' oi inc sample size. Since such a characterization is only valid under regularity assumptions, which is shown in the paper, we consider a stronger property of an exponential family which turns out to be a characteristic one. At this the concept of generalized likelihood functions will play an important role.