## Find a copy in the library

Finding libraries that hold this item...

## Details

Document Type: | Book |
---|---|

All Authors / Contributors: |
J O Ramsay; Giles Hooker |

ISBN: | 9781493971886 1493971883 |

OCLC Number: | 985082294 |

Description: | xvii, 230 pages : illustrations (some color) ; 25 cm. |

Contents: | 1. Introduction to Dynamic Models1.1 Six Examples of Input/Output Dynamics1.1.1 Smallpox in Montreal1.1.2 Spread of Disease Equations1.1.3 Filling a Container 1.1.4 Head Impact and Brain Acceleration1.1.5 Compartment models and pharmacokinetics1.1.6 Chinese handwriting 1.1.7 Where to go for More Dynamical Systems1.2 What This Book Undertakes1.3 Mathematical Requirements1.4 Overview 2 DE notation and types 2.1 Introduction and Chapter Overview 2.2 Notation for Dynamical Systems 2.2.1 Dynamical System Variables 2.2.2 Dynamical System Parameters2.2.3 Dynamical System Data Configurations 2.2.4 Mathematical Background 2.3 The Architecture of Dynamic Systems2.4 Types of Differential Equations2.4.1 Linear Differential Equations2.4.2 Nonlinear Dynamical Systems2.4.3 Partial Differential Equations 2.4.4 Algebraic and Other Equations 2.5 Data Configurations 2.5.1 Initial and Boundary Value Configurations2.5.2 Distributed Data Configurations2.5.3 Unobserved or Lightly Observed Variables 2.5.4 Observational Data and Measurement Models2.6 Differential Equation Transformations 2.7 A Notation Glossary 3 Linear Differential Equations and Systems 3.1 Introduction and Chapter Overview 3.2 The First Order Stationary Linear Buffer 3.3 The Second Order Stationary Linear Equation3.4 The mth Order Stationary Linear Buffer3.5 Systems of Linear Stationary Equations 3.6 A Linear System Example: Feedback Control 3.7 Nonstationary Linear Equations and Systems 3.7.1 The First Order Nonstationary Linear Buffer3.7.2 First Order Nonstationary Linear Systems3.8 Linear Differential Equations Corresponding to Sets of Functions 3.9 Green's Functions for Forcing Function Inputs 4 Nonlinear Differential Equations 4.1 Introduction and Chapter Overview 4.2 The Soft Landing Modification4.3 Existence and Uniqueness Results4.4 Higher Order Equations4.5 Input/Output Systems 4.6 Case Studies 4.6.1 Bounded Variation: The Catalytic Equation4.6.2 Rate Forcing: The SIR Spread of Disease System 4.6.3 From Linear to Nonlinear: The FitzHugh-Nagumo Equations 4.6.4 Nonlinear Mutual Forcing: The Tank Reactor Equations 4.6.5 Modeling Nylon Production 5 Numerical Solutions5.1 Introduction 5.2 Euler Methods5.3 Runge-KuttaMethods5.4 Collocation Methods 5.5 Numerical Problems5.5.1 Stiffness 5.5.2 Discontinuous Inputs5.5.3 Constraints and Transformations< 6 Qualitative Behavior 6.1 Introduction 6.2 Fixed Points 6.2.1 Stability 6.3 Global Analysis and Limit Cycles 6.3.1 Use of Conservation Laws 6.3.2 Bounding Boxes6.4 Bifurcations6.4.1 Transcritical Bifurcations6.4.2 Saddle Node Bifurcations 6.4.3 Pitchfork Bifurcations6.4.4 Hopf Bifurcations 6.5 Some Other Features 6.5.1 Chaos 6.5.2 Fast-Slow Systems 6.6 Non-autonomous Systems6.7 Commentary 7 Trajectory Matching 7.1 Introduction 7.2 Gauss-Newton Minimization7.2.1 Sensitivity Equations 7.2.2 Automatic Differentiation7.3 Inference7.4 Measurements on Multiple Variables 7.4.1 Multivariate Gauss-Newton Method 7.4.2 VariableWeighting using Error Variance7.4.3 Estimating s27.4.4 Example: FitzHugh-NagumoModels 7.4.5 Practical Problems: Local Minima 7.4.6 Initial Parameter Values for the Chemostat Data 7.4.7 Identifiability7.5 Bayesian Methods7.6 Multiple Shooting and Collocation 7.7 Fitting Features7.8 Applications: Head Impacts 8 Gradient Matching 8.1 Introduction 8.2 Smoothing Methods and Basis Expansions 8.3 Fitting the Derivative8.3.1 Optimizing Integrated Squared Error (ISSE)8.3.2 Gradient Matching for the Refinery Data 8.3.3 Gradient Matching and the Chemostat Data8.4 System Mis-specification and Diagnostics8.4.1 Diagnostic Plots 8.5 Conducting Inference 8.5.1 Nonparametric Smoothing Variances8.5.2 Example: Refinery Data 8.6 Related Methods and Extensions 8.6.1 Alternative Smoothing Method8.6.2 Numerical Discretization Methods8.6.3 Unobserved Covariates8.6.4 Nonparametric Models 8.6.5 Sparsity and High Dimensional ODEs8.7 Integral Matching8.8 Applications: Head Impacts 9 Profiling for Linear Systems9.1 Introduction and Chapter Overview9.2 Parameter Cascading9.2.1 Two Classes of Parameters9.2.2 Defining Coefficients as Functions of Parameters 9.2.3 Data/Equation Symmetry9.2.4 Inner Optimization Criterion J9.2.5 The Least Squares Cascade Coefficient Function9.2.6 The Outer Fitting Criterion H9.3 Choosing the Smoothing Parameter r 9.4 Confidence Intervals for Parameters 9.4.1 Simulation Sample Results 9.5 Multi-Variable Systems 9.6 Analysis of the Head Impact Data9.7 A Feedback Model for Driving Speed9.7.1 Two-Variable First Order Cruise Control Model 9.7.2 One-Variable Second Order Cruise Control Model 9.8 The Dynamics of the Canadian Temperature Data9.9 Chinese Handwriting9.10 Complexity Bases 9.11 Software and Computation9.11.1 Rate Function Specifications 9.11.2 Model Term Specifications9.11.3 Memoization 10 Nonlinear Profiling10.1 Introduction and Chapter Overview 10.2 Parameter Cascading for Nonlinear Systems10.2.1 The Setup for Parameter Cascading10.2.2 Parameter Cascading Computations10.2.3 Some Helpful Tips 10.2.4 Nonlinear Systems and Other Fitting Criteria 10.3 Lotka-Volterra 10.4 Head Impact10.5 Compound Model for Blood Ethanol10.6 Catalytic model for growth10.7 Aromate ReactionsReferences Glossary Index |

Series Title: | Springer series in statistics. |

Responsibility: | James Ramsay, Giles Hooker. |

### Abstract:

This text focuses on the use of smoothing methods for developing and estimating differential equations following recent developments in functional data analysis and building on techniques described in Ramsay and Silverman (2005) Functional Data Analysis. The central concept of a dynamical system as a buffer that translates sudden changes in input into smooth controlled output responses has led to applications of previously analyzed data, opening up entirely new opportunities for dynamical systems. The technical level has been kept low so that those with little or no exposure to differential equations as modeling objects can be brought into this data analysis landscape. There are already many texts on the mathematical properties of ordinary differential equations, or dynamic models, and there is a large literature distributed over many fields on models for real world processes consisting of differential equations. However, a researcher interested in fitting such a model to data, or a statistician interested in the properties of differential equations estimated from data will find rather less to work with. This book fills that gap.

## Reviews

*Editorial reviews*

Publisher Synopsis

"This book is intended both for first year graduate students and for researchers in applied mathematics and/or statistics who want to check models with differential equations in data science. These kinds of models have a mechanistic approach, enlarging the classes of models for statisticians, and giving techniques for estimation of parameters, assessing the adequacy of models and planning experiments for applied mathematicians." (Sylvie Viguier-Pla, Mathematical Reviews, August, 2018) Read more...

*User-contributed reviews*

Add a review and share your thoughts with other readers.
Be the first.

Add a review and share your thoughts with other readers.
Be the first.

## Tags

Add tags for "Dynamic data analysis : modeling data with differential equations".
Be the first.