Series: Springer Texts in Statistics
2005, XXIV, 432 p., Hardcover
ISBN: 0-387-40271-3
About this book
Longitudinal data are ubiquitous across Medicine, Public Health,
Public Policy, Psychology, Political Science, Biology, Sociology
and Education, yet many longitudinal data sets remain improperly
analyzed. This book teaches the art and statistical science of
modern longitudinal data analysis. The author emphasizes
specifying, understanding, and interpreting longitudinal data
models. He inspects the longitudinal data graphically, analyzes
the time trend and covariates, models the covariance matrix, and
then draws conclusions.
Covariance models covered include random effects, autoregressive,
autoregressive moving average, antedependence, factor analytic,
and completely unstructured models among others. Longer
expositions explore: an introduction to and critique of simple
non-longitudinal analyses of longitudinal data, missing data
concepts, diagnostics, and simultaneous modeling of two
longitudinal variables. Applications and issues for random
effects models cover estimation, shrinkage, clustered data,
models for binary and count data and residuals and residual plots.
Shorter sections include a general discussion of how
computational algorithms work, handling transformed data, and
basic design issues.
This book requires a solid regression course as background and is
particularly intended for the final year of a Biostatistics or
Statistics Masters degree curriculum. The mathematical
prerequisite is generally low, mainly assuming familiarity with
regression analysis in matrix form. Doctoral students in
Biostatistics or Statistics, applied researchers and quantitative
doctoral students in disciplines such as Medicine, Public Health,
Public Policy, Psychology, Political Science, Biology, Sociology
and Education will find this book invaluable. The book has many
figures and tables illustrating longitudinal data and numerous
homework problems. The associated web site contains many
longitudinal data sets, examples of computer code, and labs to re-enforce
the material.
Table of contents
Introduction to Longitudinal Data.- Plots.- Simple Analyses.-
Critiques of Simple Analyses.- The Multivariate Normal Linear
Model.- Tools and Concepts.- Specifying Covariates.- Modeling the
Covariance Matrix.- Random Effects Models.- Residuals and Case
Diagnostics.- Discrete Longitudinal Data.- Missing Data.-
Analyzing Two Longitudinal Variables.- Further Reading.
Series: Springer Monographs in Mathematics
2005, Approx. 300 p., Hardcover
ISBN: 3-540-26136-2
About this book
"Theory of Association Schemes" is the first concept-oriented
treatment of the structure theory of association schemes. It
contains several recent results which appear for the first time
in text book form. The generalization of Sylow’s group-theoretical
theorems to scheme theory arises as a consequence of arithmetical
considerations about quotient schemes. The theory of Coxeter
schemes (equivalent to the theory of buildings) emerges naturally
and yields a purely algebraic proof of Tits’ main theorem on
buildings of spherical type. Also a scheme-theoretical
characterization of Glauberman’s Z*-involutions is included.
The text is self-contained and accessible for advanced
undergraduate students.
Table of contents
Basic Facts.- Basic Technics.- Quotient Schemes.- Morphisms.-
Normal Closed Subsets.- Products.- Thin Schemes.- Scheme Algebras.-
Dihedral Closed Subsets.- Constrained Sets of Involutions.- The
Exchange Condition.- Spherical Coxeter Schemes.- Historical Notes.
Series: The IMA Volumes in Mathematics and its Applications,
Vol. 140
2005, Approx. 265 p. 22 illus., Hardcover
ISBN: 0-387-25879-5
About this book
The IMA Summer Program on Probability and Partial Differential
Equations in Modern Applied Mathematics took place July 21-August
1, 2003. The program was devoted to the role of probabilistic
methods in modern applied mathematics from perspectives of both a
tool for analysis and as a tool in modeling. There is a growing
recognition in the applied mathematics research community that
stochastic methods are playing an increasingly prominent role in
the formulation and analysis of diverse problems of contemporary
interest in the sciences and engineering.
A probabilistic representation of solutions to partial
differential equations that arise as deterministic models allows
one to exploit the power of stochastic calculus and probabilistic
limit theory in analysis, as well as offer new perspectives on
the phenomena for modeling purposes. In addition, such approaches
can be effective in sorting out multiple scale structure and in
the development of both non-Monte Carlo as well as Monte Carlo
type numerical methods.
There is also a growing recognition of a role in the inclusion of
stochastic terms in the modeling of complex flows, and the
addition of such terms has led to interesting new mathematical
problems at the interface of probability, dynamical systems,
numerical analysis, and partial differential equations.
This volume consists of original contributions by researchers
with a common interest in the problems, but with diverse
mathematical expertise and perspective. The volume will be useful
to researchers and graduate students who are interested in
probabilistic methods, dynamical systems approaches and numerical
analysis for mathematical modeling in engineering and sciences.
Table of contents
Forward - Preface - Nonnegative Markov chains with appliations -
Phase changes with time and multi-scale homogenizations of a
class of anomalous diffusions - Semi-Markov cascade
representations of local solutions to 3-D incompressible Navier-Stokes
- Amplitude equations for SPDEs: Approximate centre manifolds and
invariant measures - Enstrophy and ergodicity of gravity currents
- Stochastic heat and Burgers equations and their singularities -
A gentle introduction to cluster expansions - Continuity of the
Ito-map for Holder rough paths with applications to the Support
Theorem in Holder norm - Data-driven stochastic processes in
fully developed turbulence - Stochastic flows on the circle -
Path integration: connecting pure jump and Wiener processes -
Random dynamical systems in economics - A geometric cascade for
the spectral approximation of the Navier-Stokes equations -
Inertial manifolds for random differential equations - Existence
and uniqueness of classical, nonnegative, smooth solutions of a
class of semi-linear SPDEs - Nonlinear PDE's driven by Levy
diffusions and related statistical issues - List of workshop
participants
Series: Springer Series in Statistics
2005, XVIII, 646 p. 78 illus., Hardcover
ISBN: 0-387-40264-0
About this book
Hidden Markov models have become a widely used class of
statistical models with applications in diverse areas such as
communications engineering, bioinformatics, finance and many more.
This book is a comprehensive treatment of inference for hidden
Markov models, including both algorithms and statistical theory.
Topics range from filtering and smoothing of the hidden Markov
chain to parameter estimation, Bayesian methods and estimation of
the number of states.
In a unified way the book covers both models with finite state
spaces, which allow for exact algorithms for filtering,
estimation etc. and models with continuous state spaces (also
called state-space models) requiring approximate simulation-based
algorithms that are also described in detail. Simulation in
hidden Markov models is addressed in five different chapters that
cover both Markov chain Monte Carlo and sequential Monte Carlo
approaches. Many examples illustrate the algorithms and theory.
The book also carefully treats Gaussian linear state-space models
and their extensions and it contains a chapter on general Markov
chain theory and probabilistic aspects of hidden Markov models.
This volume will suit anybody with an interest in inference for
stochastic processes, and it will be useful for researchers and
practitioners in areas such as statistics, signal processing,
communications engineering, control theory, econometrics, finance
and more. The algorithmic parts of the book do not require an
advanced mathematical background, while the more theoretical
parts require knowledge of probability theory at the measure-theoretical
level.
Table of contents
Introduction.- Main Definitions and Notations.- Filtering and
Smoothing Recursions.- Advanced Topics in Smoothing.-
Applications of Smoothing.- Monte Carlo Methods.- Sequential
Monte Carlo Methods.- Advanced Topics in Sequential Monte Carlo.-
Analysis of Sequential Monte Carlo Methods.- Maximum Likelihood
Inference.- Part I: Optimization through Exact Smoothing.-
Maximum Likelihood Inference.- Part II: Monte Carlo Optimization.-
Statistical Properties of the Maximum Likelihood Estimator.-
Fully Bayesian Approaches.- Elements of Markov Chain Theory.- An
Information-Theoretic Perspective on Order Estimation.
Series: Universitext
2006, Approx. 370 p., Softcover
ISBN: 3-540-25724-1
About this textbook
The idea of this book is to give an extensive description of the
classical complex analysis, here ''classical'' means roughly that
sheaf theoretical and cohomological methods are omitted.
The first four chapters cover the essential core of complex
analysis presenting their fundamental results. After this
standard material, the authors step forward to elliptic functions
and to elliptic modular functions including a taste of all most
beautiful results af this field. The book is rounded by
applications to analytic number theory including distinguished
pearls of this fascinating subject as for instance the Prime
Number Theorem. Great importance is attached to completeness, all
needed notions are developed, only minimal prerequisites (elementary
facts of calculus and algebra) are required.
More than 400 exercises including hints for solutions and many
figures make this an attractive, indispensable book for students
who would like to have a sound introduction to classical complex
analysis.
Table of contents
Differential Calculus in the Complex Plane C.- Integral Calculus
in the Complex Plane.- Sequences and Series of Analytic
Functions, the Residue Theorem.- Construction of Analytic
Functions.- Elliptic Functions.- Elliptic Modular Forms.-
Analytic Number Theory.- Solutions to the Exercises.- References.-
Index.
Series: Probability and its Applications
2005, Approx. 525 p., Hardcover
ISBN: 0-387-25115-4
About this book
This is the first comprehensive treatment of the three basic
symmetries of probability theory - contractability,
exchangeability, and rotatability - defined as invariance in
distribution under contractions, permutations, and rotations.
Originating with the pioneering work of de Finetti from the 1930's,
the theory has evolved into a unique body of deep, beautiful, and
often surprising results, comprising the basic representations
and invariance properties in one and several dimensions, and
exhibiting some unexpected links between the various symmetries
as well as to many other areas of modern probability. Most
chapters require only some basic, graduate level probability
theory, and should be accessible to any serious researchers and
graduate students in probability and statistics. Parts of the
book may also be of interest to pure and applied mathematicians
in other areas. The exposition is formally self-contained, with
detailed references provided for any deeper facts from real
analysis or probability used in the book.
Olav Kallenberg received his Ph.D. in 1972 from Chalmers
University in Gothenburg, Sweden. After teaching for many years
at Swedish universities, he moved in 1985 to the US, where he is
currently Professor of Mathematics at Auburn University. He is
well known for his previous books Random Measures (4th edition,
1986) and Foundations of Modern Probability (2nd edition, 2002)
and for numerous research papers in all areas of probability. In
1977, he was the second recipient ever of the prestigious Rollo
Davidson Prize from Cambridge University. In 1991?94, he served
as the Editor in Chief of Probability Theory and Related Fields.
Professor Kallenberg is an elected fellow of the Institute of
Mathematical Statistics.
Table of contents
The Basic Symmetries.- Conditioning and Martingales.- Convergence
and Approximation.- Predictable Sampling and Mapping.- Decoupling
Identities.- Homogeneity and Reflections.- Symmetric Arrays.-
Multi-variate Rotations.- Symmetric Measures in the Plane.