A smoothness priors approach to the modeling of time series with trends and seasonalities is shown. An observed time series is decomposed into local polynomial trend, seasonal, globally stationary autoregressive and observation error components. Each component is characterized by an unknown variance-white noise perturbed difference equation constraint. The constraints or Bayesian smoothness priors are expressed in state-space model form. A Kalman predictor yields the likelihood for the unknown variances (hyperparameters) with a computa- tional complexity, O(N). Likelihoods are computed for different constraint order models in different subsets of constraint equation model classes. Akaike's mini- mum AIC procedure is used to select the best model fitted to the data within and between the alternative model classes. Smoothing is achieved by a smoother algorithm. Examples are shown.

Smoothness Priors Analysis of Time Series addresses some of the problems of modeling stationary and nonstationary time series primarily from a Bayesian stochastic regression "smoothness priors" state space point of view. Prior distributions on model coefficients are parametrized by hyperparameters. Maximizing the likelihood of a small number of hyperparameters permits the robust modeling of a time series with relatively complex structure and a very large number of implicitly inferred parameters. The critical statistical ideas in smoothness priors are the likelihood of the Bayesian model and the use of likelihood as a measure of the goodness of fit of the model. The emphasis is on a general state space approach in which the recursive conditional distributions for prediction, filtering, and smoothing are realized using a variety of nonstandard methods including numerical integration, a Gaussian mixture distribution-two filter smoothing formula, and a Monte Carlo "particle-path tracing" method in which the distributions are approximated by many realizations. The methods are applicable for modeling time series with complex structures.

With the support of the Office of Naval Research Program on Statistics and Probability (Dr. Edward J. Wegman, Director), The Department of Statistics at Texas A&M University hosted a Symposium on Time Series Analysis of Irregularly Observed Data during the period February 10-13, 1983. The symposium aimed to provide a review of the state of the art, define outstanding problems for research by theoreticians, transmit to practitioners recently developed algorithms, and stimulate interaction between statisticians and researchers in subject matter fields. Attendance was limited to actively involved researchers. This volume contains refereed versions of the papers presented at the Symposium. We would like to express our appreciation to the many colleagues and staff members whose cheerful help made the Symposium a successful happening which was enjoyed socially and intellectually by all participants. I would like to especially thank Dr. Donald W. Marquardt whose interest led me to undertake to organize this Symposium. This volume is dedicated to the world wide community of researchers who develop and apply methods of statistical analysis of time series. r:;) \J Picture Caption Participants in Symposium on Time Series Analysis of Irregularly Observed Data at Texas A&M University, College Station, Texas, February 10-13, 1983 First Row: Henry L. Gray, D. W. Marquardt, P. M. Robinson, Emanuel Parzen, Julia Abrahams, E. Masry, H. L. Weinert, R. H. Shumway.

Longitudinal Data Analysis, Volume 50 in the Handbook of Statistics series covers how data consists of a series of repeated observations of the same subjects over an extended time frame and is thus useful for measuring change. Such studies and the data arise in a variety of fields, such as health sciences, genomic studies, experimental physics, sociology, sports and student enrollment in universities. For example, in health studies, intra-subject correlation of responses must be accounted for, covariates vary with time, and bias can arise if patients drop out of the study. - Provides the authority and expertise of leading contributors from an international board of authors - Presents the latest release in the Handbook of Statistics series - Updated release includes the latest information on Modeling and Analysis of Longitudinal Data

This IMA Volume in Mathematics and its Applications NEW DIRECTIONS IN TIME SERIES ANALYSIS, PART II is based on the proceedings of the IMA summer program "New Directions in Time Series Analysis. " We are grateful to David Brillinger, Peter Caines, John Geweke, Emanuel Parzen, Murray Rosenblatt, and Murad Taqqu for organizing the program and we hope that the remarkable excitement and enthusiasm of the participants in this interdisciplinary effort are communicated to the reader. A vner Friedman Willard Miller, Jr. PREFACE Time Series Analysis is truly an interdisciplinary field because development of its theory and methods requires interaction between the diverse disciplines in which it is applied. To harness its great potential, strong interaction must be encouraged among the diverse community of statisticians and other scientists whose research involves the analysis of time series data. This was the goal of the IMA Workshop on "New Directions in Time Series Analysis. " The workshop was held July 2-July 27, 1990 and was organized by a committee consisting of Emanuel Parzen (chair), David Brillinger, Murray Rosenblatt, Murad S. Taqqu, John Geweke, and Peter Caines. Constant guidance and encouragement was provided by Avner Friedman, Director of the IMA, and his very helpful and efficient staff. The workshops were organized by weeks. It may be of interest to record the themes that were announced in the IMA newsletter describing the workshop: l.

Available in English for the first time, this classic and influential book by the late Kohei Ohtsu presents real examples of ships in motion under irregular ocean waves, how to understand the characteristics of fluctuations of stochastic phenomena through spectral analysis methods and statistical modeling. It also explains how to realize prediction and optimal control based on time series models. In recent years, the need to improve safety and reduce environmental impact in ship operations has been increasing, and the statistical methods presented in this book will be increasingly needed in the future. In addition, the recent development of innovative AI technology and highspeed communications will make it possible to adapt this method not only to ship monitoring and control, but also to any field that involves irregular fluctuations, and it is expected to contribute to solving issues that have been difficult to solve in the past. Part 1 describes classical spectral method for the analysis of stochastic phenomena. In Part 2, this book explains methods to construct time series models using the information criterion, to capture the characteristics of ship and engine motions using the model, to design a model-based monitoring system that informs navigators operating the ship and managers ashore. Furthermore, it explains statistical control method to design an autopilot system and the governor of a marine engine, while showing actual examples. Part 3 presents the basic knowledge necessary for understanding these topics of the book, namely, the basic theory of ship motion, probability and statistics, Kalman filter and statistical optimal control theory.