Part of the multi-volume work
Open Dynamical Systems
Language: English
Published/Copyright: 2025
Using the full power of countable alphabet thermodynamic formalism and modern functional analysis, the book presents a systematic treatment of the perturbative approach to open dynamical systems, covering those generated by a large class of naturally occurring holes through which the dynamical medium escapes. It highlights escape rates, conditionally invariant measures, surviving equilibrium states, and Hausdorff dimension of surviving sets.
Breaks new ground and contains original research advances.
Provides a compact yet self-contained account of material needed from functional analysis.
Discusses the powerful approach of singular perturbations.
Time series analysis is one of several branches of statistics whose practical importance has increased with the availability of powerful computational tools. Methodology that was originally developed for specialized applications, for example in finance or geophysics, is now widely available within general statistical packages.
The second edition of Time Series: A Biostatistical Introduction is an introductory account of time series analysis, written from the perspective of applied statisticians whose interests lie primarily in the biomedical and health sciences. This edition has a stronger focus on substantive applications, in which each statistical analysis is directed at a specific research question. Separate chapters cover simple descriptive methods of analysis, including time-plots, smoothing, the correlogram and the periodogram; theory of stationary random processes; discrete-time models for single series; continuous-time models for single series; generalized linear models for time series of counts; models for replicated series; spectral analysis, and bivariate time series.
The book is unique in its focus on biomedical and health science applications, which has been strengthened in this second edition. Nevertheless, the methods described are more widely applicable. It should be useful to teachers and students on masters-level degree courses in statistics, biostatistics and epidemiology, and to biomedical and health scientists with a knowledge of statistical methods at undergraduate level. Throughout, examples based on real datasets show a close interplay between statistical method and substantive science. This book will also describe the implementation of the methods in the R computing environment and provide access to R code and datasets.
An accessible guide to understanding statistics using simulations, with examples from a range of scientific disciplines
Real-world challenges such as small sample sizes, skewed distributions of data, biased sampling designs, and more predictors than data points are pushing the limits of classical statistical analysis. This textbook provides a new tool for the statistical toolkit: data simulations. It shows that using simulation and data-generating models is an excellent way to validate statistical reasoning and to augment study design and statistical analysis with planning and visualization. Although data simulations are not new to professional statisticians, Statistics by Simulation makes the approach accessible to a broader audience, with examples from many fields. It introduces the reasoning behind data simulation and then shows how to apply it in planning experiments or observational studies, developing analytical workflows, deploying model diagnostics, and developing new indices and statistical methods.
? Covers all steps of statistical practice, from planning projects to post-hoc analysis and model checking
? Provides examples from disciplines including sociology, psychology, ecology, economics, physics, and medicine
? Includes R code for all examples, with data and code freely available online
? Offers bullet-point outlines and summaries of each chapter
? Minimizes the use of jargon and requires only basic statistical background and skills
Preface
Acknowledgments
Part I: Propositi: Why and how to simulate
1. General Introduction
1.1 What are simulated data?
1.2 Simulated data are specific
1.3 Yes, scientists really simulate data
1.4 There are many good reasons to simulate data
1.5 Useful background knowledge to use this book most effectively
1.6 Notational conventions
1.7 Structure, organisation, and flow
1.8 Summary
2. The basics of simulating data and the need for computational competence
2.1 A road map for simulation in statistics
2.2 Two simple examples
2.3 More complex examples
2.4 Simulating autocorrelated data
2.5 Simulation versus randomisation techniques
2.6 Summary
Part II: Ante mensuram: Prospective simulations of study designs and their power
3. Think before you act
3.1 The illusion of truth: A case study
3.2 The question comes first
3.3 Setting expectations, defining hypotheses
3.4 Testing hypotheses and assessing their support
3.5 Pre-registration
3.6 Summary
4. Prospective simulation of statistical power
4.1 Simple group comparisons
4.2 How many data points do we need for a simple correlation?
4.3 Is “recruit until significant” problematic?
4.4 How long does a time series have to be?
4.5 Improving estimates: Is the experiment powerful enough?
4.6 Summary
Part III: Post mensuram: Simulations in statistical analysis
5. Assumptions: Is that one important?
5.1 Linear regression requires the data to be normally distributed
5.2 Regression models also assume that errors in predictor variables are negligible or unimportant
5.3 The intended, rather than the realised, manipulation is an admissible predictor variable
5.4 ANOVA requires homoscedasticity
5.5 Multiple testing and the inflation of false positives
5.6 Hyper-distributions in mixed-e?ect models are normal
5.7 Correlations among predictors are the same outside the range of the observed data
5.8 Summary
6. Folklore: Is that rule-of-thumb true or useful?
6.1 Model selection does not always improve interpretation
6.2 Selecting one of two correlated predictors does not mitigate collinearity in regression and machine learning
6.3 It is not OK to categorise continuous predictor variables
6.4 Use Monte Carlo simulation when data are heteroscedastic
6.5 Time series should not be detrended by default
6.6 Machine learning and Big Data do not obviate rules-of-thumb
6.7 Summary
7. Workflows and pipelines can introduce and propagate artefacts
7.1 What can we do about missing data?
7.2 Types of missing data
7.3 Imputation of missing predictors
7.4 Estimating values for censored observations
7.5 Pre-selecting predictors
7.6 Regression on residuals
7.7 Error propagation
7.8 Workflow: Stringing multiple statistical steps into an analytical pipeline
7.9 Summary
Part IV: Post exemplum: Diagnostic simulations
8. Evaluating models: How well do they really fit?
8.1 Learning from the prior
8.2 What does a model tell us, and what does it not tell us?
8.3 Visualising more complex effects: conditional, marginal, and partial plots
8.4 Model diagnostics
8.5 Predicting with confidence is not the same as confidence in prediction
8.6 Iterative learning: New priors from old posteriors
8.7 Outlook
8.8 Summary
9. Post hoc alternatives to retrospective power analysis
9.1 Reprise: Prospective power analysis
9.2 What is retrospective power analysis?
9.3 Post hoc alternatives to retrospective power analysis
9.4 Summary: Most retrospective analyses should be avoided
9.5 Coda: What would a Bayesian do instead?
Part V: In posterum: Simulations for new methods
10. Combining studies: Meta-analysis and federated analysis
10.1 Whence the data?
10.2 From meta-analysis through federated analysis to complete analysis
10.3 Meta-analysis
10.4 Individual participant-level meta-analysis
10.5 One-step federated analysis
10.6 Multi-step federated analysis
10.7 Complete data analysis
10.8 Conclusions and outlook
10.9 Summary
11. Putting it through its paces: Does this new method work?
11.1 Unit testing
11.2 Dimensional analysis
11.3 Comparisons
11.4 Intellectual advancement
11.5 Intuitive understanding
11.6 Model-agnostic number of parameters: Generalised degrees of freedom
11.7 Know your limits
11.8 Summary
12. Outroduction: How far should we push simulations?
12.1 Stochastic weather forecasting
12.2 Infusing fake signals to test the workflow at LIGO
12.3 Virtual LIDAR scanning
12.4 Advanced simulation may be neither possible nor desirable
A: Useful R functions for data simulations
A.1 Drawing random values from a distribution
A.2 Doing things repeatedly: for-loops and replicate
A.3 Shuffling, resampling, and bootstrapping: sample()
A.4 Little helpers
A.5 Dedicated simulation packages
Format: Hardback, 342 pages, height x width: 235x155 mm, XII, 342 p., 1 Hardback
Pub. Date: 03-Apr-2025
ISBN-13: 9783031851056
This book provides a concise and self-contained introduction to the foundations of mathematics. The first part covers the fundamental notions of mathematical logic, including logical axioms, formal proofs and the basics of model theory. Building on this, in the second and third part of the book the authors present detailed proofs of Godels classical completeness and incompleteness theorems. In particular, the book includes a full proof of Godels second incompleteness theorem which states that it is impossible to prove the consistency of arithmetic within its axioms. The final part is dedicated to an introduction into modern axiomatic set theory based on Zermelos axioms, containing also a presentation of Godels constructible universe of sets. A recurring theme in the whole book consists of standard and non-standard models of several theories, such as Peano arithmetic, Presburger arithmetic and the real numbers. In addition, the corrected, revised and extended second edition now provides detailed solutions to all exercises.
The book addresses undergraduate mathematics students and is suitable for a one or two semester introductory course into logic and set theory.
0. A Framework for Metamathematics.- Part I Introduction to First-Order
Logic.- 1 Syntax: The Grammar of Symbols.- 2 The Art of Proof.- 3 Semantics:
Making Sense of the Symbols.- Part II Godels Completeness Theorem.- 4
Maximally Consistent Extensions.- 5 The Completeness Theorem.- 6 Language
Extensions by Definitions.- Part III Godels Incompleteness Theorems.- 7
Countable Models of Peano Arithmetic.- 8 Arithmetic in Peano Arithmetic.- 9
Godelisation of Peano Arithmetic.- 10 The First Incompleteness Theorem.- 11
The Second Incompleteness Theorem.- 12 Completeness of Presburger
Arithmetic.- Part IV The Axiom System ZFC.- 13 The Axioms of Set Theory
(ZFC).- 14 Models of Set Theory.- 15 Models and Ultraproducts.- 16 Models of
Peano Arithmetic.- 17 Models of the Real Numbers.- Tautologies.- Solutions.-
References.- Index.
Format: Hardback, 168 pages, height x width: 235x155 mm, 20 Illustrations, color;
27 Illustrations, black and white; X, 168 p. 47 illus., 20 illus. in color., 1 Hardback
Series: Latin American Mathematics Series
Pub. Date: 20-Jun-2025
ISBN-13: 9783031841279
This book focuses on the computational aspects of graph convexity, with a particular emphasis on path convexity within graphs. It provides a thoughtful introduction to this emerging research field, which originated by adapting concepts from convex geometry to combinatorics and has experienced substantial growth.
The book starts with an introduction of fundamental convexity concepts and then proceeds to discuss convexity parameters. These parameters fall into two categories: one derived from abstract convexity studies and another motivated by computational complexity. Subsequent chapters explore geometric convexity within graphs, examining various graph classes such as interval graphs, proper interval graphs, cographs, chordal graphs, and strongly chordal graphs. The text concludes with a study of the computation of convexity parameters across different convexity types, including practical applications in areas like game theory.
Compact and straightforward, this work serves as an ideal entry point for students and researchers interested in pursuing further research in the field of convexity.
The English translation of this book, originally in Portuguese, was facilitated by artificial intelligence. The content was later revised by the authors for accuracy.
Part I Fundamentals of Convexity in Graphs.- Basic Concepts of
Convexity.- Convexity in Graphs.- Graph Convexity Parameters.- Convex
Geometries in Graphs.- Part II Main Convexities and Applications.- P3 and P3*
Convexities.- Geodesic Convexity.- Other Convexities.- Convexity in Oriental
Graphs.- Applications in Graph Convexities.