Dynamic Data Analysis: Modeling Data with Differential Equations

1.1 Six Examples of Input/Output Dynamics

1.1.1 Smallpox in Montreal

1.1.2 Spread of Disease Equations

1.1.3 Filling a Container

1.1.4 Head Impact and Brain Acceleration

1.1.5 Compartment models and pharmacokinetics

1.1.6 Chinese handwriting

1.1.7 Where to go for More Dynamical Systems

1.2 What This Book Undertakes

1.3 Mathematical Requirements

1.4 Overview

2 DE notation and types

2.2 Notation for Dynamical Systems

2.2.1 Dynamical System Variables

2.2.2 Dynamical System Parameters

2.2.3 Dynamical System Data Configurations

2.2.4 Mathematical Background

2.3 The Architecture of Dynamic Systems

2.4 Types of Differential Equations

2.4.1 Linear Differential Equations

2.4.2 Nonlinear Dynamical Systems

2.4.3 Partial Differential Equations

2.4.4 Algebraic and Other Equations

2.5 Data Configurations

2.5.1 Initial and Boundary Value Configurations

2.5.3 Unobserved or Lightly Observed Variables

2.5.4 Observational Data and Measurement Models

2.6 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 Equation

3.4 The mth Order Stationary Linear Buffer

3.5 Systems of Linear Stationary Equations

3.6 A Linear System Example: Feedback Control

3.7.1 The First Order Nonstationary Linear Buffer

3.7.2 First Order Nonstationary Linear Systems

3.8 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 Modification

4.3 Existence and Uniqueness Results

4.4 Higher Order Equations

4.5 Input/Output Systems

4.6 Case Studies

4.6.2 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 Solutions

5.1 Introduction

5.2 Euler Methods

5.3 Runge-KuttaMethods

5.4 Collocation Methods

5.5 Numerical Problems

5.5.1 Stiffness

5.5.2 Discontinuous Inputs

5.5.3 Constraints and Transformations

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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 Boxes

6.4 Bifurcations

6.4.1 Transcritical Bifurcations

6.4.2 Saddle Node Bifurcations

6.4.3 Pitchfork Bifurcations

6.4.4 Hopf Bifurcations

6.5 Some Other Features

6.5.1 Chaos

6.5.2 Fast-Slow Systems

6.6 Non-autonomous Systems

6.7 Commentary

7 Trajectory Matching

7.1 Introduction

7.2 Gauss-Newton Minimization

7.2.1 Sensitivity Equations

7.2.2 Automatic Differentiation

7.3 Inference

7.4 Measurements on Multiple Variables

7.4.1 Multivariate Gauss-Newton Method

7.4.2 VariableWeighting using Error Variance

7.4.3 Estimating s2

7.4.4 Example: FitzHugh-NagumoModels

7.4.5 Practical Problems: Local Minima

7.4.6 Initial Parameter Values for the Chemostat Data

7.4.7 Identifiability

7.5 Bayesian Methods

7.6 Multiple Shooting and Collocation

7.7 Fitting Features

7.8 Applications: Head Impacts

8 Gradient Matching

8.1 Introduction

8.2 Smoothing Methods and Basis Expansions

8.3 Fitting the Derivative

8.3.1 Optimizing Integrated Squared Error (ISSE)

8.3.2 Gradient Matching for the Refinery Data

8.3.3 Gradient Matching and the Chemostat Data

8.4 System Mis-specification and Diagnostics

8.5 Conducting Inference

8.5.1 Nonparametric Smoothing Variances

8.5.2 Example: Refinery Data

8.6 Related Methods and Extensions

8.6.1 Alternative Smoothing Method

8.6.2 Numerical Discretization Methods

8.6.3 Unobserved Covariates

8.6.4 Nonparametric Models

8.6.5 Sparsity and High Dimensional ODEs

8.7 Integral Matching

8.8 Applications: Head Impacts

9 Profiling for Linear Systems

9.1 Introduction and Chapter Overview

9.2.1 Two Classes of Parameters

9.2.2 Defining Coefficients as Functions of Parameters

9.2.3 Data/Equation Symmetry

9.2.4 Inner Optimization Criterion J

9.2.5 The Least Squares Cascade Coefficient Function

9.2.6 The Outer Fitting Criterion H

9.3 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 Data

9.7 A Feedback Model for Driving Speed

9.7.2 One-Variable Second Order Cruise Control Model

9.8 The Dynamics of the Canadian Temperature Data

9.9 Chinese Handwriting

9.10 Complexity Bases

9.11 Software and Computation

9.11.1 Rate Function Specifications

9.11.2 Model Term Specifications

9.11.3 Memoization

10 Nonlinear Profiling

10.1 Introduction and Chapter Overview

10.2 Parameter Cascading for Nonlinear Systems

10.2.1 The Setup for Parameter Cascading

10.2.3 Some Helpful Tips

10.2.4 Nonlinear Systems and Other Fitting Criteria

10.3 Lotka-Volterra

10.4 Head Impact

10.5 Compound Model for Blood Ethanol

10.6 Catalytic model for growth

10.7 Aromate Reactions

References

Glossary

Index