Incomplete Data Problem Research Articles

Estimating the mixture of a proportional hazards model with three types of failure data

Cox’s Proportional Hazards Model (PHM) has been widely applied in the analysis of lifetime data. It can be characterized in terms of covariates that influence the system lifetime, where the covariates describe the operating environment (e.g., temperature, pressure, humidity). When the covariates are assumed to be random variables, the hazards model becomes the mixed PHM. In this article, a parametric method is proposed to estimate the unknown parameters in the mixed PHM. Three types of data are considered: uncategorized field observations, categorized field ones, and categorized experimental ones. The expectation-maximization algorithm is used to handle the incomplete data problem. Simulation results are presented to illustrate the precision and some properties of the estimation results. Accepted in 2005 for a special issue on Reliability co-edited by Hoang Pham, Rutgers University; Dong Ho Park, Hallym University, Korea; and Richard Cassady, University of Arkansas.

Random covariances and mixed-effects models for imputing multivariate multilevel continuous data

Principled techniques for incomplete-data problems are increasingly part of mainstream statistical practice. Among many proposed techniques so far, inference by multiple imputation (MI) has emerged as one of the most popular. While many strategies leading to inference by MI are available in cross-sectional settings, the same richness does not exist in multilevel applications. The limited methods available for multilevel applications rely on the multivariate adaptations of mixed-effects models. This approach preserves the mean structure across clusters and incorporates distinct variance components into the imputation process. In this paper, I add to these methods by considering a random covariance structure and develop computational algorithms. The attraction of this new imputation modeling strategy is to correctly reflect the mean and variance structure of the joint distribution of the data, and allow the covariances differ across the clusters. Using Markov Chain Monte Carlo techniques, a predictive distribution of missing data given observed data is simulated leading to creation of multiple imputations. To circumvent the large sample size requirement to support independent covariance estimates for the level-1 error term, I consider distributional impositions mimicking random-effects distributions assigned a priori. These techniques are illustrated in an example exploring relationships between victimization and individual and contextual level factors that raise the risk of violent crime.

Active Learning for Solving the Incomplete Data Problem in Facial Age Classification by the Furthest Nearest-Neighbor Criterion

Facial age classification is an approach to classify face images into one of several predefined age groups. One of the difficulties in applying learning techniques to the age classification problem is the large amount of labeled training data required. Acquiring such training data is very costly in terms of age progress, privacy, human time, and effort. Although unlabeled face images can be obtained easily, it would be expensive to manually label them on a large scale and getting the ground truth. The frugal selection of the unlabeled data for labeling to quickly reach high classification performance with minimal labeling efforts is a challenging problem. In this paper, we present an active learning approach based on an online incremental bilateral two-dimension linear discriminant analysis (IB2DLDA) which initially learns from a small pool of labeled data and then iteratively selects the most informative samples from the unlabeled set to increasingly improve the classifier. Specifically, we propose a novel data selection criterion called the furthest nearest-neighbor (FNN) that generalizes the margin-based uncertainty to the multiclass case and which is easy to compute, so that the proposed active learning algorithm can handle a large number of classes and large data sizes efficiently. Empirical experiments on FG-NET and Morph databases together with a large unlabeled data set for age categorization problems show that the proposed approach can achieve results comparable or even outperform a conventionally trained active classifier that requires much more labeling effort. Our IB2DLDA-FNN algorithm can achieve similar results much faster than random selection and with fewer samples for age categorization. It also can achieve comparable results with active SVM but is much faster than active SVM in terms of training because kernel methods are not needed. The results on the face recognition database and palmprint/palm vein database showed that our approach can handle problems with large number of classes. Our contributions in this paper are twofold. First, we proposed the IB2DLDA-FNN, the FNN being our novel idea, as a generic on-line or active learning paradigm. Second, we showed that it can be another viable tool for active learning of facial age range classification.

State of the Multiple Imputation Software.

Owing to its practicality as well as strong inferential properties, multiple imputation has been increasingly popular in the analysis of incomplete data. Methods that are not only computationally elegant but also applicable in wide spectrum of statistical incomplete data problems have also been increasingly implemented in a numerous computing environments. Unfortunately, however, the speed of this development has not been replicated in reaching to "sophisticated" users. While the researchers have been quite successful in developing the underlying software, documentation in a style that would be most reachable to the greater scientific society has been lacking. The main goal of this special volume is to close this gap by articles that illustrate these software developments. Here I provide a brief history of multiple imputation and relevant software and highlight the contents of the contributions. Potential directions for the future of the software development is also provided.

Mice: Multivariate Imputation by Chained Equations inR

The R package <b>mice</b> imputes incomplete multivariate data by chained equations. The software mice 1.0 appeared in the year 2000 as an S-PLUS library, and in 2001 as an R package. mice 1.0 introduced predictor selection, passive imputation and automatic pooling. This article documents mice, which extends the functionality of mice 1.0 in several ways. In <b>mice</b>, the analysis of imputed data is made completely general, whereas the range of models under which pooling works is substantially extended. <b>mice</b> adds new functionality for imputing multilevel data, automatic predictor selection, data handling, post-processing imputed values, specialized pooling routines, model selection tools, and diagnostic graphs. Imputation of categorical data is improved in order to bypass problems caused by perfect prediction. Special attention is paid to transformations, sum scores, indices and interactions using passive imputation, and to the proper setup of the predictor matrix. <b>mice</b> can be downloaded from the Comprehensive R Archive Network. This article provides a hands-on, stepwise approach to solve applied incomplete data problems.

Optimization for limited angle tomography in medical image processing

This paper aims to reduce the problems of incomplete data in computed tomography, which happens frequently in medical image process and analysis, e.g., when the high-density region of objects can only be penetrated by X-rays at a limited angular range. As the projection data are available only in an angular range, the incomplete data problem can be attributed to the limited angle problem, which is an ill-posed inverse problem. Image reconstruction based on total variation (TV) reduces the problem and gives better performance on edge-preserving reconstruction; however, the artificial parameter can only be determined through considerable experimentation. In this paper, an effective TV objective function is proposed to reduce the inverse problem in the limited angle tomography. This novel objective function provides a robust and effective reconstruction without any artificial parameter in the iterative processes, using the TV as a multiplicative constraint. The results demonstrate that this reconstruction strategy outperforms some previous ones.

Scattered data interpolation of Radon data

Linear combinations of translates of a given basis function have long been successfully used to solve scattered data interpolation and approximation problems. We demonstrate how the classical basis function approach can be transferred to the projective space ? d?1. To be precise, we use concepts from harmonic analysis to identify positive definite and strictly positive definite zonal functions on ? d?1. These can then be applied to solve problems arising in tomography since the data given there consists of integrals over lines. Here, enhancing known reconstruction techniques with the use of a scattered data interpolant in the "space of lines", naturally leads to reconstruction algorithms well suited to limited angle and limited range tomography. In the medical setting algorithms for such incomplete data problems are desirable as using them can limit radiation dosage.

Image reconstruction by an alternating minimisation

This paper focuses on the problem of incomplete data in the applications of the circular cone-beam computed tomography. This problem is frequently encountered in medical imaging sciences and some other industrial imaging systems. For example, it is crucial when the high density region of objects can only be penetrated by X-rays in a limited angular range. As the projection data are only available in an angular range, the above mentioned incomplete data problem can be attributed to the limited angle problem, which is an ill-posed inverse problem. This paper reports a modified total variation minimisation method to reduce the data insufficiency in tomographic imaging. This proposed method is robust and efficient in the task of reconstruction by showing the convergence of the alternating minimisation method. The results demonstrate that this new reconstruction method brings reasonable performance.

Scalable tensor factorizations for incomplete data

The problem of incomplete data – i.e., data with missing or unknown values – in multi-way arrays is ubiquitous in biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, communication networks, etc. We consider the problem of how to factorize data sets with missing values with the goal of capturing the underlying latent structure of the data and possibly reconstructing missing values (i.e., tensor completion). We focus on one of the most well-known tensor factorizations that captures multi-linear structure, CANDECOMP/PARAFAC (CP). In the presence of missing data, CP can be formulated as a weighted least squares problem that models only the known entries. We develop an algorithm called CP-WOPT (CP Weighted OPTimization) that uses a first-order optimization approach to solve the weighted least squares problem. Based on extensive numerical experiments, our algorithm is shown to successfully factorize tensors with noise and up to 99% missing data. A unique aspect of our approach is that it scales to sparse large-scale data, e.g., 1000 × 1000 × 1000 with five million known entries (0.5% dense). We further demonstrate the usefulness of CP-WOPT on two real-world applications: a novel EEG (electroencephalogram) application where missing data is frequently encountered due to disconnections of electrodes and the problem of modeling computer network traffic where data may be absent due to the expense of the data collection process.

Estimating unconstrained demand rate functions using customer choice sets

A good demand forecast should be at the heart of every revenue management model. Yet most demand models focus on product demand and do not incorporate customer choice behavior under offered alternatives. We use the ideas of customer choice sets to model the customer's buying behavior. A customer choice set is a set of product classes representing the buying preferences and choice decisions of a certain customer group. In this article we present a demand estimation method for these choice sets. The procedure is based on the maximum likelihood method, and to overcome the problem of incomplete data or information we additionally apply the expectation maximization method. Using this demand information per choice sets, the revenue manager obtains a clear view of the underlying demand. In doing so, the sales consequences from different booking control actions can be compared and the overall revenue maximized.

Analysis of interval‐censored disease progression data via multi‐state models under a nonignorable inspection process

Irreversible multi-state models provide a convenient framework for characterizing disease processes that arise when the states represent the degree of organ or tissue damage incurred by a progressive disease. In many settings, however, individuals are only observed at periodic clinic visits and so the precise times of the transitions are not observed. If the life history and observation processes are not independent, the observation process contains information about the life history process, and more importantly, likelihoods based on the disease process alone are invalid. With interval-censored failure time data, joint models are nonidentifiable and data analysts must rely on sensitivity analyses to assess the effect of the dependent observation times. This paper is concerned, however, with the analysis of data from progressive multi-state disease processes in which individuals are scheduled to be seen at periodic pre-scheduled assessment times. We cast the problem in the framework used for incomplete longitudinal data problems. Maximum likelihood estimation via an EM algorithm is advocated for parameter estimation. Simulation studies demonstrate that the proposed method works well under a variety of situations. Data from a cohort of patients with psoriatic arthritis are analyzed for illustration.

Incomplete data problems in wave equation imaging

We study reflection imaging as an incomplete data problem in frequency domain. It turns out that this amounts to inverting the Fourier transform using only frequencies outside some set. By numerical simulations we show the effect of this incompleteness on concrete reconstruction problems. We try to complete the data by analytic continuation. An explicit formula is obtained by an inversion formula for the exponential Radon transform. We discuss the application to medical ultrasound tomography and to seismic imaging. We describe an alternative method based on the presence of reflectors.

Stochastic Approximation Boosting for Incomplete Data Problems

Boosting is a powerful approach to fitting regression models. This article describes a boosting algorithm for likelihood-based estimation with incomplete data. The algorithm combines boosting with a variant of stochastic approximation that uses Markov chain Monte Carlo to deal with the missing data. Applications to fitting generalized linear and additive models with missing covariates are given. The method is applied to the Pima Indians Diabetes Data where over half of the cases contain missing values.

Adaptive wavelet-Galerkin methods for limited angle tomography

This paper studied incomplete data problems of computed tomography that frequently occur in medical or industrial imaging, for example, when the high-density region of objects can only be penetrated by X-rays at a limited angular range. When projection data are available only in an angular range, the incomplete data problem can be attributed to the limited angle problem, which is a severely ill-posed inverse problem. In this paper, a numerical method for the treatment of inverse problems based on an adaptive wavelet-Galerkin method is introduced and investigated. The paper focuses especially on how to avoid inverse crimes in numerical simulations. The method used here combines numerical simplicity and characteristics of adapting to the unknown smoothness of a reconstructed image, which leads to significant reduction in the computational cost. The reconstruction strategy has a comparable performance with a significant reduction in computational time.

On the instantaneous sentinels in pollution problems of incomplete data

In this work, we discuss how to get instantaneous information (at fixed t = T ∈ [0, +∞[) on pollution terms in distributed systems of incomplete data in some ecology and/or meteorology problems, using the sentinel method of Lions [J.-L. Lions, Sentinelles Pour les Systèmes Distribués à Données Incomplètes, Recherches en mathématiques appliquées RMA No. 21, Masson, Paris, 1992] with control and observation having their supports in two different open sets as by Nakoulima [O. Nakoulima, Contrôlabilité à zéro avec contraintes sur le contrôle, C. R. Acad. Sci. Paris, Ser. I 339 (2004), pp. 405–410]. Here, the ecological system is affected by pollution in the boundary of the domain (e.g. a border of a lake). We verify that if the initial data are completely unknown, the sentinel is null. If there is some information on the initial data, the instantaneous sentinel naturally exists if the control set is bigger than the one where the observation may be defined. Finally, we give the characterization of the instantaneous sentinel with some remarks.

Investors' Risk Attitude and Risky Behavior: A Bayesian Approach with Imperfect Information

In a choice model of risky assets the role of risk aversion is analyzed. The measure of risk preference comes from a direct subjective survey question and it is considered as an imperfect information about the true risk attitude of investors. Misclassification between the true and the observed risk aversion is explicitly taken into account in the empirical model. A Data Augmentation approach, a Bayesian procedure for incomplete-data problems, is applied on data from the 2006 Survey of Household Income and Wealth by the Bank of Italy. Results indicate that when misclassification of investors is taken into account model estimates show the good performance of the subjective question when used as a control in a portfolio choice models. Moreover risk aversion emerges as a strong predictor of the probability to hold risky assets. The analysis also shows that probability of misclassification decreases as latent risk aversion increases, that means that more risk tolerant investors tend to be classified erroneously more often than less risk tolerant investors.

Estimation of Space–Time Branching Process Models in Seismology Using an EM–Type Algorithm

Maximum likelihood estimation of branching point process models via numerical optimization procedures can be unstable and computationally intensive. We explore an alternative estimation method based on the expectation-maximization algorithm. The method involves viewing the estimation of such branching processes as analogous to incomplete data problems. Using an application from seismology, we show how the epidemic-type aftershock sequence (ETAS) model can, in fact, be estimated this way, and we propose a computationally efficient procedure to maximize the expected complete data log-likelihood function. Using a space–time ETAS model, we demonstrate that this method is extremely robust and accurate and use it to estimate declustered background seismicity rates of geologically distinct regions in Southern California. All regions show similar declustered background intensity estimates except for the one covering the southern section of the San Andreas fault system to the east of San Diego in which a substantially higher intensity is observed.

Information loss for 2 × 2 tables with missing cell counts: binomial case

We formulate likelihood-based ecological inference for 2 × 2 tables with missing cell counts as an incomplete data problem and study Fisher information loss by comparing estimation from complete and incomplete data. In so doing, we consider maximum-likelihood (ML) estimators of probabilities governed by two independent binomial distributions and obtain simplified expressions for their covariance. These expressions reflect well the additional uncertainty arising from the unobserved data compared to complete data tables. We also discuss an approximation to the expected conditional variance of the unobserved entries and ML parameter bias correction. An empirical example is used to demonstrate the results.

Chronic Unreduced Anterior Shoulder Dislocation

The uniqueness of anatomic structures and their variations provide the basis for forensic identification. During the identification process of 1991-1995 Croatian war victims, forensic pathologists and forensic anthropologists were faced with the problem of incomplete skeletal remains and insufficient premortem medical data. Here we present a case of anterior shoulder dislocation, which was a key confirmation in the identification of a war victim. In our case, anterior shoulder dislocation was not medically treated for 15 years, and the changes in the glenoid cavity and coracoid process indicated morphologic adaptation of anatomic structures to the new mechanical and physiological demands.

Simple and Globally Convergent Methods for Accelerating the Convergence of Any EM Algorithm

Abstract. The expectation‐maximization (EM) algorithm is a popular approach for obtaining maximum likelihood estimates in incomplete data problems because of its simplicity and stability (e.g. monotonic increase of likelihood). However, in many applications the stability of EM is attained at the expense of slow, linear convergence. We have developed a new class of iterative schemes, called squared iterative methods (SQUAREM), to accelerate EM, without compromising on simplicity and stability. SQUAREM generally achieves superlinear convergence in problems with a large fraction of missing information. Globally convergent schemes are easily obtained by viewing SQUAREM as a continuation of EM. SQUAREM is especially attractive in high‐dimensional problems, and in problems where model‐specific analytic insights are not available. SQUAREM can be readily implemented as an ‘off‐the‐shelf’ accelerator of any EM‐type algorithm, as it only requires the EM parameter updating. We present four examples to demonstrate the effectiveness of SQUAREM. A general‐purpose implementation (written in R) is available.