Showing papers by "Bernard W. Silverman published in 2002"

Applied Functional Data Analysis: Methods and Case Studies

Abstract: Introduction- Life Course Data in Criminology- The Nondurable Goods Index- Bone Shapes from a Paleopathology Study- Modeling Reaction Time Distributions- Zooming in on Human Growth- Time Warping Handwriting and Weather Records- How do Bone Shapes Indicate Arthritis?- Functional Models for Test Items- Predicting Lip Acceleration from Electromyography- Variable Seasonal Trend in the Goods Index- The Dynamics of Handwriting Printed Characters- A Differential Equation for Juggling

Posterior probability intervals for wavelet thresholding

Abstract: Summary. We use cumulants to derive Bayesian credible intervals for wavelet regression estimates. The first four cumulants of the posterior distribution of the estimates are expressed in terms of the observed data and integer powers of the mother wavelet functions. These powers are closely approximated by linear combinations of wavelet scaling functions at an appropriate finer scale. Hence, a suitable modification of the discrete wavelet transform allows the posterior cumulants to be found efficiently for any given data set. Johnson transformations then yield the credible intervals themselves. Simulations show that these intervals have good coverage rates, even when the underlying function is inhomogeneous, where standard methods fail. In the case where the curve is smooth, the performance of our intervals remains competitive with established nonparametric regression methods.

Functional Models for Test Items

Abstract: After our bank accounts and our taxes, it is hard to imagine data playing a more central role in our lives than the examinations, opinion surveys, attitude questionnaires, and psychological scales administered to ourselves, our children, and our students. These data may not on first impression appear to be functional, but we show that functional data analysis can reveal how both test takers and test items perform in test situations. To provide a concrete frame of reference, we look at the responses of 5000 examinees to 60 items in a test of mathematics achievement developed by the American College Testing Program. We apply functional principal components analysis to explore variation across test items, and we check the fairness of certain items by comparing male and female performance. Finally, we use a functional property of these data to develop a useful new way of describing the performance of individual examinees. Let us assume that each of n items is given to each of N examinees, and that each item is answered either correctly or incorrectly. We record each response with a value of 1 if examinee j answers item i correctly, and 0 otherwise. We want to use these data, crude as they may seem, to provide a reasonable answer to the question, " What is the probability P ij that examinee j gets item i right? " Since we have only a single 0/1 datum to estimate P ij , we obviously need to make some simplifying assumptions. We can take advantage of the fact that exam performances are not really all that unique; given this many

Zooming in on Human Growth

Abstract: The careful documentation of human growth is essential in order to define what we call normal growth, so that we can detect as early as possible when something is going wrong with the growth process. Auxologists, the scientists that specialize in the study of growth, also need high quality data to advance our understanding of how the body regulates its own growth. It may come as a surprise to learn that human growth at the macro level that we see in our children is not that well understood. Growth data are exceedingly expensive to collect since children must be brought into the laboratory at preassigned ages over about a 20-year span. Meeting this observational regime requires great dedication and persistence by the parents, and the dropout rate is understandably high, even taking for granted the long-term commitment of maintaining a growth laboratory. The Fels Institute in Ohio, for example, has been collecting growth data since 1929, and is now measuring the third generation for some of its original cases. The accurate measurement of height is also difficult, and requires considerable training. Height diminishes throughout the day as the spine compresses, but it also depends on other factors. Infants must be measured lying down, and when the transition is made to measuring their standing height, measurements shrink by around one centimeter. The most careful procedures still exhibit standard deviations over repeated measurements of about three millimeters.