Learning from Incomplete Data

Real-world learning tasks often involve high-dimensional data sets with complex patterns of missing features. In this paper we review the problem of learning from incomplete data from two statistical perspectives---the likelihood-based and the Bayesian. The goal is two-fold: to place current neural network approaches to missing data within a statistical framework, and to describe a set of algorithms, derived from the likelihood-based framework, that handle clustering, classification, and function approximation from incomplete data in a principled and efficient manner. These algorithms are based on mixture modeling and make two distinct appeals to the Expectation-Maximization (EM) principle (Dempster, Laird, and Rubin 1977)---both for the estimation of mixture components and for coping with the missing data.

Macroeconomic announcements, price discovery, and order flow efects in the stock market: evidence from incomplete data and multiple financial markets

This paper investigates heterogeneity in the market assessment of public macro- economic announcements by exploring (jointly) two main mechanisms through which macroeconomic news might enter stock prices: instantaneous fundamental news im- pacts consistent with the asset pricing view of symmetric information, and permanent order ow e¤ects consistent with a microstructure view of asymmetric information related to heterogeneous interpretation of public news. Theoretical motivation and empirical evidence for the operation of both mechanisms are presented. Signi cant in- stantaneous news impacts are detected for news related to real activity (including em- ployment), investment, in ation, and monetary policy; however, signi cant order ow e¤ects are also observed on employment announcement days. A multi-market analysis suggests that these asymmetric information e¤ects come from uncertainty about long term interest rates due to heterogeneous assessments of future Fed responses to em- ployment shocks.

DIMENSION REDUCTION FOR POWER SYSTEM MODELING USING PCA METHODS CONSIDERING INCOMPLETE DATA READINGS

Principal Component Analysis (PCA) is a popular method for dimension reduction that can be used in many fields including data compression, image processing, exploratory data analysis, etc. However, traditional PCA method has several drawbacks, since the traditional PCA method is not efficient for dealing with high dimensional data and cannot be effectively applied to compute accurate enough principal components when handling relatively large portion of missing data. In this report, we propose to use EM-PCA method for dimension reduction of power system measurement with missing data, and provide a comparative study of traditional PCA and EM-PCA methods. Our extensive experimental results show that EM-PCA method is more effective and more accurate for dimension reduction of power system measurement data than traditional PCA method when dealing with large portion of missing data set.

A stochastic approximation algorithm with Markov chain Monte-Carlo method for incomplete data estimation problems

We propose a general procedure for solving incomplete data estimation problems. The procedure can be used to find the maximum likelihood estimate or to solve estimating equations in difficult cases such as estimation with the censored or truncated regression model, the nonlinear structural measurement error model, and the random effects model. The procedure is based on the general principle of stochastic approximation and the Markov chain Monte-Carlo method. Applying the theory on adaptive algorithms, we derive conditions under which the proposed procedure converges. Simulation studies also indicate that the proposed procedure consistently converges to the maximum likelihood estimate for the structural measurement error logistic regression model.

GTM-based data visualisation with incomplete data

We analyse how the Generative Topographic Mapping (GTM) can be modified to cope with missing values in the training data. Our approach is based on an Expectation -Maximisation (EM) method which estimates the parameters of the mixture components and at the same time deals with the missing values. We incorporate this algorithm into a hierarchical GTM. We verify the method on a toy data set (using a single GTM) and a realistic data set (using a hierarchical GTM). The results show our algorithm can help to construct informative visualisation plots, even when some of the training points are corrupted with missing values.

Statistical methods for polyhedral shape classification with incomplete data - application to cryo-electron tomographic images

A certain type of bacterial inclusion, known as a bacterial microcompartment, was recently identified and imaged through cryo-electron tomography. A reconstructed 3D object from single-axis limited angle tilt-series cryo-electron tomography contains missing regions and this problem is known as the missing wedge problem. Due to missing regions on the reconstructed images, analyzing their 3D structures is a challenging problem. The existing methods overcome this problem by aligning and averaging several similar shaped objects. These schemes work well if the objects are symmetric and several objects with almost similar shapes and sizes are available. Since the bacterial inclusions studied here are not symmetric, are deformed, and show a wide range of shapes and sizes, the existing approaches are not appropriate. This research develops new statistical methods for analyzing geometric properties, such as volume, symmetry, aspect ratio, polyhedral structures etc., of these bacterial inclusions in presence of missing data. These methods work with deformed and non-symmetric varied shaped objects and do not necessitate multiple objects for handling the missing wedge problem. The developed methods and contributions include: (a) an improved method for manual image segmentation, (b) a new approach to 'complete' the segmented and reconstructed incomplete 3D images, (c) a polyhedral structural distance model to predict the polyhedral shapes of these microstructures, (d) a new shape descriptor for polyhedral shapes, named as polyhedron profile statistic, and (e) the Bayes classifier, linear discriminant analysis and support vector machine based classifiers for supervised incomplete polyhedral shape classification. Finally, the predicted 3D shapes for these bacterial microstructures belong to the Johnson solids family, and these shapes along with their other geometric properties are important for better understanding of their chemical and biological characteristics.

Inferential Implications of Over-Parametrization: A Case Study in Incomplete Categorical Data

P>In the context of either Bayesian or classical sensitivity analyses of over-parametrized models for incomplete categorical data, it is well known that prior-dependence on posterior inferences of nonidentifiable parameters or that too parsimonious over-parametrized models may lead to erroneous conclusions. Nevertheless, some authors either pay no attention to which parameters are nonidentifiable or do not appropriately account for possible prior-dependence. We review the literature on this topic and consider simple examples to emphasize that in both inferential frameworks, the subjective components can influence results in nontrivial ways, irrespectively of the sample size. Specifically, we show that prior distributions commonly regarded as slightly informative or noninformative may actually be too informative for nonidentifiable parameters, and that the choice of over-parametrized models may drastically impact the results, suggesting that a careful examination of their effects should be considered before drawing conclusions.Resume Que ce soit dans un cadre Bayesien ou classique, il est bien connu que la surparametrisation, dans les modeles pour donnees categorielles incompletes, peut conduire a des conclusions erronees. Cependant, certains auteurs persistent a negliger les problemes lies a la presence de parametres non identifies. Nous passons en revue la litterature dans ce domaine, et considerons quelques exemples surparametres simples dans lesquels les elements subjectifs influencent de facon non negligeable les resultats, independamment de la taille des echantillons. Plus precisement, nous montrons comment des a priori consideres comme peu ou non-informatifs peuvent se reveler extremement informatifs en ce qui concerne les parametres non identifies, et que le recours a des modeles surparametres peut avoir sur les conclusions finales un impact considerable. Ceci suggere un examen tres attentif de l`impact potentiel des a priori.

Fitting a mixture model to three-mode three-way data with missing information

When the data consist of certain attributes measured on the same set of items in different situations, they would be described as a three-mode three-way array. A mixture likelihood approach can be implemented to cluster the items (i.e., one of the modes) on the basis of both of the other modes simultaneously (i.e,, the attributes measured in different situations). In this paper, it is shown that this approach can be extended to handle three-mode three-way arrays where some of the data values are missing at random in the sense of Little and Rubin (1987). The methodology is illustrated by clustering the genotypes in a three-way soybean data set where various attributes were measured on genotypes grown in several environments.

Comparing diagnostic tests with missing data

When missing data occur in studies designed to compare the accuracy of diagnostic tests, a common, though naive, practice is to base the comparison of sensitivity, specificity, as well as of positive and negative predictive values on some subset of the data that fits into methods implemented in standard statistical packages. Such methods are usually valid only under the strong missing completely at random (MCAR) assumption and may generate biased and less precise estimates. We review some models that use the dependence structure of the completely observed cases to incorporate the information of the partially categorized observations into the analysis and show how they may be fitted via a two-stage hybrid process involving maximum likelihood in the first stage and weighted least squares in the second. We indicate how computational subroutines written in R may be used to fit the proposed models and illustrate the different analysis strategies with observational data collected to compare the accuracy of three distinct non-invasive diagnostic methods for endometriosis. The results indicate that even when the MCAR assumption is plausible, the naive partial analyses should be avoided.

Missing data mechanisms and their implications on the analysis of categorical data

We review some issues related to the implications of different missing data mechanisms on statistical inference for contingency tables and consider simulation studies to compare the results obtained under such models to those where the units with missing data are disregarded. We confirm that although, in general, analyses under the correct missing at random and missing completely at random models are more efficient even for small sample sizes, there are exceptions where they may not improve the results obtained by ignoring the partially classified data. We show that under the missing not at random (MNAR) model, estimates on the boundary of the parameter space as well as lack of identifiability of the parameters of saturated models may be associated with undesirable asymptotic properties of maximum likelihood estimators and likelihood ratio tests; even in standard cases the bias of the estimators may be low only for very large samples. We also show that the probability of a boundary solution obtained under the correct MNAR model may be large even for large samples and that, consequently, we may not always conclude that a MNAR model is misspecified because the estimate is on the boundary of the parameter space.

Beyond filtered backprojection: a reconstruction software package for ion beam microtomography data

A new version of the TomoRebuild data reduction software package is presented, for the reconstruction of scanning transmission ion microscopy tomography (STIMT) and particle induced X-ray emission tomography (PIXET) images. First, we present a state of the art of the reconstruction codes available for ion beam microtomography. The algorithm proposed here brings several advantages. It is a portable, multi-platform code, designed in C++ with well-separated classes for easier use and evolution. Data reduction is separated in different steps and the intermediate results may be checked if necessary. Although no additional graphic library or numerical tool is required to run the program as a command line, a user friendly interface was designed in Java, as an ImageJ plugin. All experimental and reconstruction parameters may be entered either through this plugin or directly in text format files. A simple standard format is proposed for the input of experimental data. Optional graphic applications using the ROOT interface may be used separately to display and fit energy spectra. Regarding the reconstruction process, the filtered backprojection (FBP) algorithm, already present in the previous version of the code, was optimized so that it is about 10 times as fast. In addition, Maximum Likelihood Expectation Maximization (MLEM) and its accelerated version Ordered Subsets Expectation Maximization (OSEM) algorithms were implemented. A detailed user guide in English is available. A reconstruction example of experimental data from a biological sample is given. It shows the capability of the code to reduce noise in the sinograms and to deal with incomplete data, which puts a new perspective on tomography using low number of projections or limited angle.

Knowledge discovery using multiple sources of biological data

The primary aim of this dissertation is to develop data mining tools for knowledge discovery in biomedical data when multiple (homogeneous or heterogeneous) sources of data are available. The central hypothesis is that, when information from multiple sources of data are used appropriately and effectively, knowledge discovery can be better achieved than what is possible from only a single source. ^ Recent advances in high-throughput technology have enabled biomedical researchers to generate large volumes of diverse types of data on a genome-wide scale. These data include DNA sequences, gene expression measurements, and much more; they provide the motivation for building analysis tools to elucidate the modular organization of the cell. The challenges include efficiently and accurately extracting information from the multiple data sources; representing the information effectively, developing analytical tools, and interpreting the results in the context of the domain. ^ The first part considers the application of feature-level integration to design classifiers that discriminate between soil types. The machine learning tools, SVM and KNN, were used to successfully distinguish between several soil samples. ^ The second part considers clustering using multiple heterogeneous data sources. The resulting Multi-Source Clustering (MSC) algorithm was shown to have a better performance than clustering methods that use only a single data source or a simple feature-level integration of heterogeneous data sources. ^ The third part proposes a new approach to effectively incorporate incomplete data into clustering analysis. Adapted from K-means algorithm, the Generalized Constrained Clustering (GCC) algorithm makes use of incomplete data in the form of constraints to perform exploratory analysis. Novel approaches for extracting constraints were proposed. For sufficiently large constraint sets, the GCC algorithm outperformed the MSC algorithm. ^ The last part considers the problem of providing a theme-specific environment for mining multi-source biomedical data. The database called PlasmoTFBM, focusing on gene regulation of Plasmodium falciparum, contains diverse information and has a simple interface to allow biologists to explore the data. It provided a framework for comparing different analytical tools for predicting regulatory elements and for designing useful data mining tools. ^ The conclusion is that the experiments reported in this dissertation strongly support the central hypothesis.^

Mitigating inconsistencies by coupling data cleaning, filtering, and contextual data validation in wireless sensor networks

With the advent of peer to peer networks, and more importantly sensor networks, the desire to extract useful information from continuous and unbounded streams of data has become more prominent. For example, in tele-health applications, sensor based data streaming systems are used to continuously and accurately monitor Alzheimer's patients and their surrounding environment. Typically, the requirements of such applications necessitate the cleaning and filtering of continuous, corrupted and incomplete data streams gathered wirelessly in dynamically varying conditions. Yet, existing data stream cleaning and filtering schemes are incapable of capturing the dynamics of the environment while simultaneously suppressing the losses and corruption introduced by uncertain environmental, hardware, and network conditions. Consequently, existing data cleaning and filtering paradigms are being challenged. This dissertation develops novel schemes for cleaning data streams received from a wireless sensor network operating under non-linear and dynamically varying conditions. The study establishes a paradigm for validating spatio-temporal associations among data sources to enhance data cleaning. To simplify the complexity of the validation process, the developed solution maps the requirements of the application on a geometrical space and identifies the potential sensor nodes of interest. Additionally, this dissertation models a wireless sensor network data reduction system by ascertaining that segregating data adaptation and prediction processes will augment the data reduction rates. The schemes presented in this study are evaluated using simulation and information theory concepts. The results demonstrate that dynamic conditions of the environment are better managed when validation is used for data cleaning. They also show that when a fast convergent adaptation process is deployed, data reduction rates are significantly improved. Targeted applications of the developed methodology include machine health monitoring, tele-health, environment and habitat monitoring, intermodal transportation and homeland security.

Mitigating Inconsistencies by Coupling Data Cleaning, Filtering, and Contextual Data Validation in Wireless Sensor Networks

With the advent of peer to peer networks, and more importantly sensor networks, the desire to extract useful information from continuous and unbounded streams of data has become more prominent. For example, in tele-health applications, sensor based data streaming systems are used to continuously and accurately monitor Alzheimer's patients and their surrounding environment. Typically, the requirements of such applications necessitate the cleaning and filtering of continuous, corrupted and incomplete data streams gathered wirelessly in dynamically varying conditions. Yet, existing data stream cleaning and filtering schemes are incapable of capturing the dynamics of the environment while simultaneously suppressing the losses and corruption introduced by uncertain environmental, hardware, and network conditions. Consequently, existing data cleaning and filtering paradigms are being challenged. This dissertation develops novel schemes for cleaning data streams received from a wireless sensor network operating under non-linear and dynamically varying conditions. The study establishes a paradigm for validating spatio-temporal associations among data sources to enhance data cleaning. To simplify the complexity of the validation process, the developed solution maps the requirements of the application on a geometrical space and identifies the potential sensor nodes of interest. Additionally, this dissertation models a wireless sensor network data reduction system by ascertaining that segregating data adaptation and prediction processes will augment the data reduction rates. The schemes presented in this study are evaluated using simulation and information theory concepts. The results demonstrate that dynamic conditions of the environment are better managed when validation is used for data cleaning. They also show that when a fast convergent adaptation process is deployed, data reduction rates are significantly improved. Targeted applications of the developed methodology include machine health monitoring, tele-health, environment and habitat monitoring, intermodal transportation and homeland security.