Generalized Likelihood Ratio Statistic Research Articles

Changes in Climatological Variables at Stations around Lake Erie and Lake Michigan

Climatological variables undergo changes over time, and it is important to understand such dynamic changes at global, regional, and local levels. While global and regional studies are common in the study of climate, such studies at a local level are not as common. The aim of this article is to study temporal changes in precipitation, snowfall, and temperature variables at specific stations located on the rims of Lake Erie and Lake Michigan. The identification of changes is carried out by applying change-point analysis to precipitation, snowfall, and temperature data from Buffalo, Erie, and Cleveland stations located on the rim of Lake Erie and at Chicago, Milwaukee, and Green Bay stations located on the rim of Lake Michigan. We adopt mainly the Bayesian information criterion (BIC) method to identify the number and locations of change points, and then we apply the generalized likelihood ratio statistic to test for the statistical significance of the identified change points. We follow this up by finding 95% confidence intervals for those change points that were found to be statistically significant. The results from the analysis show that there are significant changes in precipitation, snowfall, and temperature variables at all six rim stations. Changes in precipitation show consistently significant increases, whereas there is no similar consistency in snowfall increases. Temperature increases are generally quite sharp, and they occur consistently around 1985. Overall, upon combining the amounts of changes from all six stations, the average amount of change in annual average temperature is found to be 0.96 °C, the average percentage of change in precipitation is 16%, and the average percentage of change in snowfall is 17%. The changing local climatic conditions identified in the study are important for local city planners, as well as residents, so that they can be well prepared for changing climatic scenarios.

Estimation and inference of multi-effect generalized geographically and temporally weighted regression models

Geographically and temporally weighted regression (GTWR) models have been an effective tool for exploring spatiotemporal heterogeneity of regression relationships. However, they cannot effectively model such response variables that follows discrete distributions. In this study, we first extend the distributions of response variables to one-parameter exponential family of distributions and formulate generalized geographically and temporally weighted regression (GGTWR) models with their unilaterally temporally weighted maximum likelihood estimation method. Furthermore, we propose so-called multi-effect GGTWR (MEGGTWR) models in which spatiotemporally varying, constant, temporally varying, and spatially varying coefficients may simultaneously be included to reflect different effects of explanatory variables. A coefficient-average-based estimation method is suggested to calibrate MEGGTWR models and a generalized likelihood ratio statistic based test is formulated to identify the types of coefficients. Simulation studies are then conducted to assess the performance of the proposed estimation and inference methods with the impact of multicollinearity among explanatory variables also examined. The results show that the estimation method for MEGGTWR models can accurately estimate various types of coefficients and the test method is of valid type I error and satisfactory power. Finally, the relationship between childhood hand, foot, and mouth disease cases and climate factors is analyzed by the proposed models with their estimation and inference methods and some interesting spatiotemporal patterns are uncovered.

Spectral CUSUM for Online Network Structure Change Detection

Detecting abrupt changes in the community structure of a network from noisy observations is a fundamental problem in statistics and machine learning. This paper presents an online change detection algorithm called Spectral-CUSUM to detect unknown network structure changes through a generalized likelihood ratio statistic. We characterize the average run length (ARL) and the expected detection delay (EDD) of the Spectral-CUSUM procedure and prove its asymptotic optimality. Finally, we demonstrate the good performance of the Spectral-CUSUM procedure and compare it with several baseline methods using simulations and real data examples on seismic event detection using sensor network data.

A deep mixed-effects modeling approach for real-time monitoring of metal additive manufacturing process

In-situ process monitoring of additive manufacturing (AM) is critically important and has been attracting significant efforts to achieve desirable process reliability and quality consistency. Despite enormous progress in embedding various sensors into the AM system, effectively fusing these sensor data for anomaly detection is very challenging, due to complex relationship among sensing signals, process condition and environment. In this article we propose a deep mixed-effects modeling approach to monitor the melt pool temperature for anomaly detection. It consists of a deep neural network (DNN) capturing the relationship between the temperature and other sensing data, a random-effect term accounting for the random variation of mean temperature, and a residual term modeling potential autocorrelations. To train the model (Phase I), an efficient hard expectation-maximization (EM) algorithm is developed, which iteratively optimizes the DNN parameters and the ones for the random-effect and residual terms. In online monitoring (Phase II), a T 2 -based and a generalized likelihood ratio (GLR) test-based control charts are developed to timely detect the process anomalies. The asymptotic behaviors of both the T 2 and GLR statistics are further established. The effectiveness of the proposed approach is demonstrated through thorough simulation study and real case study of an AM process.

Confidence Intervals for Seroprevalence

This paper concerns the construction of confidence intervals in standard seroprevalence surveys. In particular, we discuss methods for constructing confidence intervals for the proportion of individuals in a population infected with a disease using a sample of antibody test results and measurements of the test’s false positive and false negative rates. We begin by documenting erratic behavior in the coverage probabilities of standard Wald and percentile bootstrap intervals when applied to this problem. We then consider two alternative sets of intervals constructed with test inversion. The first set of intervals are approximate, using either asymptotic or bootstrap approximation to the finite-sample distribution of a chosen test statistic. We consider several choices of test statistic, including maximum likelihood estimators and generalized likelihood ratio statistics. We show with simulation that, at empirically relevant parameter values and sample sizes, the coverage probabilities for these intervals are close to their nominal level and are approximately equi-tailed. The second set of intervals are shown to contain the true parameter value with probability at least equal to the nominal level, but can be conservative in finite samples.

Sequential Change-Point Detection for Mutually Exciting Point Processes

We present a new CUSUM procedure for sequential change-point detection in self- and mutually-exciting point processes (specifically, Hawkes networks) using discrete events data. Hawkes networks have become a popular model in statistics and machine learning, primarily due to their capability in modeling irregularly observed data where the timing between events carries a lot of information. The problem of detecting abrupt changes in Hawkes networks arises from various applications, including neuroengineering, sensor networks, and social network monitoring. Despite this, there has not been an efficient online algorithm for detecting such changes from sequential data. To this end, we propose an online recursive implementation of the CUSUM statistic for Hawkes processes, which is computationally and memory-efficient and can be decentralized for distributed computing. We first prove theoretical properties of this new CUSUM procedure, then show the improved performance of this approach over existing methods, including the Shewhart procedure based on count data, the generalized likelihood ratio statistic, and the standard score statistic. This is demonstrated via simulation studies and an application to population code change-detection in neuroengineering.

Process Monitoring Using Kernel PCA and Kernel Density Estimation-Based SSGLR Method for Nonlinear Fault Detection

Fault monitoring is often employed for the secure functioning of industrial systems. To assess performance and enhance product quality, statistical process control (SPC) charts such as Shewhart, CUSUM, and EWMA statistics have historically been utilized. When implemented to multivariate procedures, unfortunately, such univariate control charts demonstrate low fault sensing ability. Due to some limitations of univariate charts, numerous process monitoring techniques dependent on multivariate statistical approaches such as principal component analysis (PCA) and partial least squares (PLS) have been designed. Yet, in some challenging scenarios in industrial chemical and biological processes with notably nonlinear properties, PCA works poorly, according to its presumption that the dataset generally be linear. However, Kernel Principal Component Analysis (KPCA) is a reliable and precise nonlinear process control methodology, but the interaction mainly through upper control limits (UCLs) dependent on the Gaussian distribution may weaken its output. This article introduces time-varying statistical error tracking through Kernel Principal Component Analysis (KPCA) based on Generalized Likelihood Ratio statistics (GLR) using a sequential sampling scheme named KPCA-SSGLR for nonlinear fault detection. The main issue of employing just T2 and Q statistic in KPCA is that they cannot correctly give practitioners the change point of the system fault, preventing practitioners from diagnosing the issue. Based on this perspective, this study attempts to incorporate KPCA with sequential sampling Generalized Likelihood Ratio (SSGLR) for monitoring the nonlinear fault in multivariate systems. The KPCA is utilized for dimension reduction, while the SSGLR is employed as a tracking statistic. The kernel density estimation (KDE) was employed to approximate UCLs for variational system operation relying on KPCA. The testing efficiency of the corresponding KPCA-KDE-SSGLR technique was then analyzed and competed with KPCA and kernel locality preserving projection (KLPP), the UCLs of which were focused on the Gaussian distribution. The purpose of this analysis is to enhance the development of KPCA-KDE-SSGLR to accomplish future enhancements and to advance the practical use of the established model by implementing the sequential sampling GLR approach. The fault monitoring efficiency is demonstrated through different simulation scenarios, one utilizing synthetic data, the other from the Tennessee Eastman technique, and lastly through a hot strip mill. The findings indicate the applicability of the KPCA-KDE-based SSGLR system over the KLPP and KPCA-KDE methods by its two T2 and Q charts to recognize the faults.

An adaptive CUSUM chart for drift detection

AbstractIn practice, sequential processes often have gradual changes in their process distributions over time. This is related to the drift detection problem in statistical process control. In the literature, there have been some existing discussions on this problem. But, most existing methods are designed based on the assumption that the related drift is linear or have another specific pattern. In reality, however, such specified patterns may not be valid. In this paper, we suggest an adaptive cumulative sum (CUSUM) chart to handle the drift detection problem with a flexible drift pattern. The new method integrates the general framework to construct a CUSUM chart based on the generalized likelihood ratio statistic and estimation of a shift size by the exponentially weighted least square regression procedure. Simulation studies show that the proposed method is effective in various cases considered. The new method is also illustrated using an example about the exchange rates between Indian Rupees and US Dollars.

Two new termination rules for multidimensional computerized classification testing

<p id="p00005">Computerized classification testing (CCT) is a subset of computerized adaptive testing (CAT), and it aims to classify examinees into one of at least two possible categories that denote results such as pass/fail or non-mastery/partial mastery/mastery. Therefore, CCTs focus on increasing the accuracy of classification which is different from CATs designed for precise measurement. The termination rule is one of the key components of CCT. However, as pointed out by Nydick (<xref xml:base="bibr" rid="b17">2013</xref>), most CCTs (i.e., UCCTs) were designed under unidimensional item response theory (IRT), in which the unidimensionality assumption is easily violated in practice. Thus, researchers then began to construct multidimensional CCT termination rules (i.e., MCCT) based on multidimensional IRT. To date, however, these rules still have some deficiencies in terms of classification accuracy or test efficiency. <p id="p00010">Most current studies on termination rules of MCCT are based on termination rules of UCCT. In UCCTs, termination rules require setting a cut point, <inline-formula>${{\theta }_{0}}$</inline-formula>, of the latent trait to calculate the statistics; and when they are extended from UCCT to MCCT, the cut point will become a classification bound curve or even a surface (i.e., <inline-formula>$g(\theta )=0$</inline-formula>). At this time, a question is how to convert the curve or surface into <inline-formula>${{\theta }_{0}}$</inline-formula>. To this end, the projected sequential probability ratio test (P-SPRT), constrained SPRT (C-SPRT; Nydick, <xref xml:base="bibr" rid="b17">2013</xref>), and multidimensional generalized likelihood ratio (M-GLR) were respectively proposed to solve the problem in different ways. Among them, P-SPRT and C-SPRT choose specific points on g(<italic>θ</italic>) as the approximate cut point, <inline-formula>${{\hat{\theta }}_{0}}$</inline-formula>, by projecting into Euclidean space or constraining on g(<italic>θ</italic>) respectively; as for M-GLR, because the generalized likelihood ratio statistic can be calculated without a cut point, it can be directly employed in MCCT. To overcome the limitation that P-SPRT may lead to unstable results at the beginning of the test, this study proposed the Mahalanobis distance-based SPRT (Mahalanobis-SPRT). <p id="p00015">In addition, stochastic curtailment is a technique for shortening the test length by predicting whether the classification of participants will change as the test continues. This article also combined M-GLR with the stochastic curtailment and proposed M-GLR with stochastic curtailment (M-SCGLR). <p id="p00020">A full-scale simulation study was conducted to (1) compare both the Mahalanobis-SPRT and M-SCGLR with the P-SPRT, C-SPRT, M-GLR, and multidimensional stochastically curtailed SPRT (M-SCSPRT) under varying conditions; (2) compare the classification performance of the above six termination rules for participants with specific abilities to explore whether there is a significant difference in the sensitivity of various rules to classify specific participants. To achieve the first research objective, three levels of correlation between dimensions (<italic>ρ</italic>=0, 0.5, and 0.8), two item bank structures (within-item multidimensionality and between-item multidimensionality), and two kinds of classification boundary (compensatory boundary and non-compensatory boundary) were considered; to achieve the second objective, 36 specific ability points <inline-formula>$({{\theta }_{1}},{{\theta }_{2}})$</inline-formula> were generated where <inline-formula>${{\theta }_{1}},{{\theta }_{2}}\in \{-0.5,-0.3,-0.1,0.1,0.3,0.5\}$</inline-formula>. The results showed that: (1) when the compensatory classification function was used, the Mahalanobis-SPRT led to higher classification accuracy and similar test length to the rules without stochastic curtailment; (2) under almost all conditions, the M-SCGLR not only possessed higher precision but also maintained the short test length, compared to M-SCSPRT that also uses stochastic curtailment; (3) the six termination rules showed a consistent change in the sensitivity of the precision and test length to specific participants. <p id="p00025">To sum up, two new MCCT termination rules (Mahalanobis-SPRT and M-SCGLR) are put forward in this article. Although the simulation results are very promising, several research directions merit further investigation, such as the development of MCCT termination rules for more than two categories, and the construction of MCCT termination rules by incorporating process data like the response time.

Nonparametric Iterated-Logarithm Extensions of the Sequential Generalized Likelihood Ratio Test

We develop a nonparametric extension of the sequential generalized likelihood ratio (GLR) test and corresponding time-uniform confidence sequences for the mean of a univariate distribution. By utilizing a geometric interpretation of the GLR statistic, we derive a simple analytic upper bound on the probability that it exceeds any prespecified boundary; these are intractable to approximate via simulations due to infinite horizon of the tests and the composite nonparametric nulls under consideration. Using time-uniform boundary-crossing inequalities, we carry out a unified nonasymptotic analysis of expected sample sizes of one-sided and open-ended tests over nonparametric classes of distributions (including sub-Gaussian, sub-exponential, sub-gamma, and exponential families). Finally, we present a flexible and practical method to construct time-uniform confidence sequences that are easily tunable to be uniformly close to the pointwise Chernoff bound over any target time interval.

New Approaches for Monitoring Image Data.

In this paper, we develop new techniques for monitoring image processes under a fairly general setting with spatially correlated pixels in the image. Monitoring and handling the pixels directly is infeasible due to an extremely high image resolution. To overcome this problem, we suggest control charts that are based on regions of interest. The regions of interest cover the original image which leads to a dimension reduction. Nevertheless, the data are still high-dimensional. We consider residual charts based on the generalized likelihood ratio approach. Existing control statistics typically depend on the inverse of the covariance matrix of the process, involving high computing times and frequently generating instable results in a high-dimensional setting. As a solution of this issue, we suggest two further control charts that can be regarded as modifications of the generalized likelihood ratio statistic. Within an extensive simulation study, we compare the newly proposed control charts using the median run length as a performance criterion.

Stopping rules for multi-category computerized classification testing.

Computerized classification testing (CCT) aims to classify persons into one of two or more possible categories to make decisions such as mastery/non-mastery or meet most/meet all/exceed. A defining feature of CCT is its stopping criterion: the test terminates when there is enough confidence to make a decision. There is abundant research on CCT with a single cut-off, and two common stopping criteria are the sequential probability ratio test (SPRT) statistic and the generalized likelihood ratio statistic (GLR). However, there is a relative scarcity of research extending the SPRT to the multi-hypothesis case for when there is more than one cut-off. In this paper, we propose a new multi-category GLR (mGLR) statistic as well as a stochastically curtailed version of the CCT with three or more categories. A simulation study was conducted to show that the mGLR statistic outperformed the existing stopping rules by generating shorter average test length without sacrificing classification accuracy. Results also revealed that the stochastically curtailed mGLR successfully increased test efficiency in certain testing conditions.

A MIMO Version of the Reed-Yu Detector and Its Connection to the Wilks Lambda and Hotelling $T^2$ Statistics

In this paper we study the problem of detecting a known signal transmitted over a MIMO channel of unknown complex gains and additive noise of unknown covariance. The problem arises in many contexts, including transmit-receiver synchronization. We derive the exact probability distribution for a generalized likelihood ratio (GLR) statistic, and establish the connection between this statistic and the Wilks Lambda and Hotelling $T^2$ statistics. We give alternatives to the GLR statistic, which include the Bartlett-Nanda-Pillai trace, the Lawley-Hotelling trace, and the Roy maximum eigenvalue statistics, each of which is favored under special conditions on the actual MIMO channel. For example, if the channel is an incoherent scattering channel, then the competition for greatest power is among the Bartlett-Nanda-Pillai, Lawley-Hotelling, and GLR statistics. If it is a coherent channel that supports a propagating wave, then Roy's test is more powerful. We discuss the null distribution theory of the GLR at length to show how it may be used to accurately predict false alarm probabilities.

Adaptive structure inferences on partially linear error-in-function models with error-prone covariates

Model structural inference on semiparametric measurement error models have not been well developed in the existing literature, partially due to the difficulties in dealing with unobservable covariates. In this study, a framework for adaptive structure selection is developed in partially linear error-in-function models with error-prone covariates. Firstly, based on the profile-least-square estimators of the current models, we define two test statistics via generalized likelihood ratio (GLR) test method (Fan et al. in Ann Stat 29(1):153–193, 2001). The proposed test statistics are shown to possess the Wilks-type properties, and a class of new Wilks phenomenon is unveiled in the family of semiparametric measurement error models. Then, we demonstrate that the GLR statistics asymptotically follow chi-squared distributions under null hypotheses. Further, we propose efficient algorithms to implement our methodology and assess the finite sample performance by simulated examples. A real example is given to illustrate the performance of the present methodology.

Sketching for sequential change-point detection

We present sequential change-point detection procedures based on linear sketches of high-dimensional signal vectors using generalized likelihood ratio (GLR) statistics. The GLR statistics allow for an unknown post-change mean that represents an anomaly or novelty. We consider both fixed and time-varying projections, derive theoretical approximations to two fundamental performance metrics: the average run length (ARL) and the expected detection delay (EDD); these approximations are shown to be highly accurate by numerical simulations. We further characterize the relative performance measure of the sketching procedure compared to that without sketching and show that there can be little performance loss when the signal strength is sufficiently large, and enough number of sketches are used. Finally, we demonstrate the good performance of sketching procedures using simulation and real-data examples on solar flare detection and failure detection in power networks.

Assessing Reciprocity in Polarimetric SAR Data

This letter studies the conformity with the reciprocity theorem on the measured polarimetric synthetic aperture radar (SAR) data. The problem is formalized via a binary hypothesis test where the reciprocity assumption is tested versus its alternative (absence of reciprocity). The generalized likelihood ratio (GLR) is used as design criterion and the resulting decision rule ensures the constant false alarm rate (CFAR) property. At the analysis stage, the performance of the GLR statistic is analyzed on the simulated data as well as on two different measured data sets (collected by two systems) thus highlighting the effectiveness of the approach.

Determinants of Farmers’ Choice among Alternative Rice Output Markets in Kano State, Nigeria

The aim was to analyze the output market outlets accessible to rice farmers and determinants of farmers’ choice among alternative rice production in Kano State, Nigeria. Primary data were collected from 164 rice farmers with the aid of structured questionnaire. This study was conducted in Kura, Garun Malam and Bunkure Local Government Areas of Kano State during 2015 cropping season. A multistage sampling techniques were used for data collection through the use of structured questionnaire. The ordered probit model was used to estimate the parameters of the determinants of farmers’ choices among alternative rice output market outlets by rice farmers in Kano state. The generalized likelihood ratio statistics was -113.401. This ratio exceeds the critical chi-square values at p&lt;0.01 level of significance. The log likelihood ratio value represents the value that maximizes the joint densities in the estimated model. This shows that at least one of the predictors' regression coefficient is not equal to zero in the model. The Prob &gt; chi2 was (50.03) and statistically significant at p&lt;0.01 level of probability. The probability of obtaining this chi-square statistic shows the effect of the predictor variables on specified alpha level. This implies that at least one of the regression coefficients in the model is not equal to zero. farmers’ choices among alternative rice output market outlets was significantly determined by educational status of the farmers, access to credit, cooperative membership, distance to market, quantity of output produced by the farmers and market price of rice (P&lt;0.10). Based on the findings of this study, it could be concluded that the most commonly used output markets by rice farmers was rural assembler (82.3%). Despite increasingly competitive markets, pricing issues for rice remains a concern for farmers.

Process monitoring using PCA-based GLR methods: A comparative study

Statistical process monitoring is a key requirement for many industrial processes. Many of these processes utilize Principal Component Analysis (PCA) in order to carry out statistical process monitoring due to its computational simplicity. Two fault detection charts that are commonly used with the PCA method are the Hotelling T2 and Q statistics. Although these charts are reasonably able to detect most shifts in the process mean, they may be unable to accurately detect other process faults, such as shifts in the process variance. Hypothesis testing methods such as the Generalized Likelihood Ratio (GLR) chart have been developed in order to detect different types of deviations from normal operating conditions, i.e., process faults. Although, GLR charts have shown superior performance in terms of fault detection compared to other existing techniques, literature has only examined the performance of a PCA-based GLR chart designed to detect shifts in the mean. Through a simulated synthetic example, this work evaluates the performance of PCA-based GLR charts designed to independently detect a shift in the mean, independently detect a shift in the variance, and simultaneously detect shifts in the mean and/or variance. The results show that in order to detect a shift in the mean or a shift in the variance, the GLR charts designed to independently detect either type of fault need to be implemented in parallel as they provide significantly lower missed detection rates, than a GLR chart designed to detect shifts in both the process mean and variance simultaneously. The GLR chart designed to simultaneously detect both a shift in the mean and variance does not perform as well as the other GLR charts, since two parameters (both the mean and variance) need to be estimated for this method while maximizing the GLR statistic, as opposed to just a single parameter for the other GLR charts. The practical applicability of the PCA-based GLR charts is demonstrated through the well-known benchmark Tennessee Eastman process, and through a case where autocorrelation is present in the data. Therefore, in order to detect shifts in the mean and/or variance, we recommend parallel implementation of the GLR charts designed to independently detect shifts in the mean, and independently detect shifts in the variance Parallel implementation of these two GLR charts aids with fault classification as well. This work also discusses the importance of selecting an appropriate window length of previous data to be used when computing the maximum likelihood estimates (MLEs) and maximizing the GLR statistic, keeping all fault detection criteria in mind in order to ensure that the desired fault detection results are obtained.

An Approach to Asymptotic Robustness Analysis of Sequential Tests for Composite Parametric Hypotheses

We consider the problem of sequential statistical testing of composite parametric hypotheses. We construct asymptotic expansions for the conditional probabilities of errors and conditional mathematical expectations of the sample size in the presence of “outliers” in observed values. To obtain these results we propose and use a new approach based on the approximation of the generalized likelihood ratio statistic by special Markov chains.

Statistic inference for a single-index ARCH-M model

ABSTRACTFor a single-index autoregressive conditional heteroscedastic (ARCH-M) model, estimators of the parametric and non parametric components are proposed by the profile likelihood method. The research results had shown that all the estimators have consistency and the parametric estimators have asymptotic normality. We extend this line of research by deriving the asymptotic normality of the non parametric estimator. Based on the asymptotic properties, we propose Wald statistic and generalized likelihood ratio statistic to investigate the testing problems for ARCH effect and goodness of fit, respectively. A simulation study is conducted to evaluate the finite-sample performance of the proposed estimation methodology and testing procedure.