Lynne Billard

Exploring the limits of bootstrap

Partial table of contents: GENERAL PRINCIPLES OF THE BOOTSTRAP. On the Bootstrap of M-Estimators and Other Statistical Functionals (M. Arcones & E. Gine). Bootstrapping Markov Chains (K. Athreya & C. Fuh). Six Questions Raised by the Bootstrap (B. Efron). Efficient Bootstrap Simulation (P. Hall). Bootstrapping Signs (R. LePage). Bootstrap Bandwidth Selection (J. Marron). APPLICATIONS OF THE BOOTSTRAP. A Generalized Bootstrap (E. Bedrick & J. Hill). Bootstrapping Admissible Linear Model Selection Procedures (D. Brownstone). A Hazard Process for Survival Analysis (J. Hsieh). A Nonparametric Density Estimation Based Resampling Algorithm (M. Taylor & J. Thompson). Nonparametric Rank Estimation Using Bootstrap Resampling and Canonical Correlation Analysis (X. Tu, et al.). Index.

From the statistics of data to the statistics of knowledge: Symbolic data analysis

Increasingly, datasets are so large they must be summarized in some fashion so that the resulting summary dataset is of a more manageable size, while still retaining as much knowledge inherent to the entire dataset as possible. One consequence of this situation is that the data may no longer be formatted as single values such as is the case for classical data, but rather may be represented by lists, intervals, distributions, and the like. These summarized data are examples of symbolic data. This article looks at the concept of symbolic data in general, and then attempts to review the methods currently available to analyze such data. It quickly becomes clear that the range of methodologies available draws analogies with developments before 1900 that formed a foundation for the inferential statistics of the 1900s, methods largely limited to small (by comparison) datasets and classical data formats. The scarcity of available methodologies for symbolic data also becomes clear and so draws attention to an enormous need for the development of a vast catalog (so to speak) of new symbolic methodologies along with rigorous mathematical and statistical foundational work for these methods.

Regression Analysis for Interval-Valued Data

When observations in large data sets are aggregated into smaller more manageable data sizes, the resulting classifications of observations invariably involve symbolic data. In this paper, covariance and correlation functions are introduced for interval-valued symbolic data. These and their associated terms are then used to fit linear regression models to such data. The methods are illustrated with an example from cardiology.

Case studies in biometry

Partial table of contents: ENVIRONMENTAL HAZARDS Spatial Pattern Analyses to Detect Rare Disease Clusters (L. Waller, et al.) Prediction Models for Personal Ozone Exposure Assessment (D. Wypij & L.-J Liu) FORESTRY, FISHERIES, GENETICS Estimating Pine Seedling Response to Ozone and Acidic Rain (J. Rawlings & S. Spruill) Survival Analysis for Size Regulation of Atlantic Halibut (S. Smith, et al.) HABITAT AND ANIMAL STUDIES Spatial Association Learning in Hummingbirds (J. Graham & A. Petkau) Time-Series Analyses of Beaver Body Temperatures (P. Reynolds) HEALTH CARE AND PUBLIC HEALTH POLICY Analysis of Attitudes Towards Workplace Smoking Restrictions (S. Bull) CLINICAL TRIALS Early Lung Cancer Detection Studies (B. Flehinger & M. Kimmel) Quality Control for Bone Mineral Density Scans (S. Wong & N. Lane) EPIDEMIOLOGY, TOXICOLOGY Patterns of Lung Cancer Risk in Ex-Smokers (B. Gillespie, et al.) Drug Interactions Between Morphine and Marijuana (C. Gennings, et al.) Appendix References Indexes.

Symbolic Regression Analysis

Billard and Diday (2000) developed procedures for fitting a regression equation to symbolic interval-valued data. The present paper compares that approach with several possible alternative models using classical techniques; the symbolic regression approach is preferred. Thence, a regression approach is provided for symbolic histogram-valued data. The results are illustrated with a medical data set.

The distribution of the incubation period for the acquired immunodeficiency syndrome (AIDS)

This paper contains details of methods and full results of estimates of the incubation period of acquired immunodeficiency syndrome (AIDS) that were outlined in a previous paper by Medley et al. (Nature, Lond. 328, 719 (1987)). The original model is modified to assess the influence of age and sex on the incubation period: age, but not sex, is statistically significant. The difficulties associated with interpretation of these data, and the additional information that would be required to resolve these difficulties are discussed.

Principal component analysis for interval-valued observations

One feature of contemporary datasets is that instead of the single point value in the p-dimensional space ℜp seen in classical data, the data may take interval values thus producing hypercubes in ℜp. This paper studies the vertices principal components methodology for interval-valued data; and provides enhancements to allow for so-called ‘trivial’ intervals, and generalized weight functions. It also introduces the concept of vertex contributions to the underlying principal components, a concept not possible for classical data, but one which provides a visualization method that further aids in the interpretation of the methodology. The method is illustrated in a dataset using measurements of facial characteristics obtained from a study of face recognition patterns for surveillance purposes. A comparison with analyses in which classical surrogates replace the intervals, shows how the symbolic analysis gives more informative conclusions. A second example illustrates how the method can be applied even when the number of parameters exceeds the number of observations, as well as how uncertainty data can be accommodated.

Dependencies and Variation Components of Symbolic Interval-Valued Data

In 1987, Diday added a new dimension to data analysis with his fundamental paper introducing the notions of symbolic data and their analyses. He and his colleagues, among others, have developed innumerable techniques to analyse symbolic data; yet even more is waiting to be done. One area that has seen much activity in recent years involves the search for a measure of dependence between two symbolic random variables. This paper presents a covariance function for interval-valued data. It also discusses how the total, between interval, and within interval variations relate; and in particular, this relationship shows that a covariance function based only on interval midpoints does not capture all the variations in the data. While important in its own right, the covariance function plays a central role in many multivariate methods.

Symbolic Covariance Principal Component Analysis and Visualization for Interval-Valued Data

This article proposes a new approach to principal component analysis (PCA) for interval-valued data. Unlike classical observations, which are represented by single points in p-dimensional space ℜp, interval-valued observations are represented by hyper-rectangles in ℜp, and as such, have an internal structure that does not exist in classical observations. As a consequence, statistical methods for classical data must be modified to account for the structure of the hyper-rectangles before they can be applied to interval-valued data. This article extends the classical PCA method to interval-valued data by using the so-called symbolic covariance to determine the principal component (PC) space to reflect the total variation of interval-valued data. The article also provides a new approach to constructing the observations in a PC space for better visualization. This new representation of the observations reflects their true structure in the PC space. Supplementary materials for this article are available online.

Some Inference Results for Causal Autoregressive Processes on a Plane

We investigate the causal autoregressive process on a plane pioneered by Whittle (On stationary processes on the plane. Biometrika 41 (1954), 434–49) and further studied by Besag (Spatial interaction and statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B 36 (1974), 192–236). We develop test statistics to test composite hypotheses about the parameters and to test if the process is separable. Also, when some data points are missing, we develop a computational procedure obtained by combining the EM algorithm and bootstrap procedures to find estimates of the parameters and hence the distribution of these estimates.