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Über dieses Buch

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.



Chapter 1. Introduction to Dynamic Models

The concept of a differential equation as a model is introduced here as a buffer that modulates sharp changes in an input signal by spreading out the change on the output side. Six examples are taken up, which will reappear later in the book. Each of these examples involve data spread out over the interval of change that will be used later to estimate parameters defining the differential equation. The introduction of vaccination for smallpox in Montreal in the 1870’s and the subsequent decline in deaths, followed by the re-introduction of the disease in 1885 and the subsequent epidemic open the story. The final example models the production of a complex handwriting of “statistics” in Mandarin, and demonstrates that a simple step function input, when passed through a spring-like buffer, can closely capture the curves, cusps and lifts in the actual script. The chapter closes with an overview, an outline of the mathematical skill level required to read the book, and the goals motivating the rest of the book.

James Ramsay, Giles Hooker

Chapter 2. Differential Equations: Notation and Architecture

We now take a more detailed look at various types of differential equations, after first laying out the elements of notation and terminology that are used in subsequent chapters. Classifications of differential equations include linear versus nonlinear, homogeneous (no external input) and non-homogeneous or forced (with external input), single equations versus systems of equations, and first order equations and higher order equations. The exposition aims to make the rest of the book more readable for a relative beginner in this topic of dynamical systems, but leaves any mathematical material to the next two chapters. This chapter also discusses various types of data configurations that are found in modelling situations, including issues around the density of observations required to estimate certain features of the model.

James Ramsay, Giles Hooker

Chapter 3. Linear Differential Equations and Systems

This and the next chapters survey the mathematical material that would be found in a textbook on differential equations, with special emphasis on what is useful from a modelling perspective. Linear differential equations do a great deal of the heavy lifting in many fields of application, including, for example, chemical engineering, electrical circuit theory, financial analysis, kinesiology and pharmacology. Stationary linear differential equations are an important subclass of linear equations, and are discussed in some detail. Nonlinear equation systems often begin as fairly obvious changes to a linear system, so that a good grasp if this rather humble part of dynamic systems theory continues to be useful in the next chapter on nonlinear systems.

James Ramsay, Giles Hooker

Chapter 4. Nonlinear Differential Equations and Systems

Autonomous nonlinear first order systems or single higher order nonlinear equations are expressible as

James Ramsay, Giles Hooker

Chapter 5. Numerical Solutions

In general, nonlinearNumerical solutions|( ordinary differential equation models do not have solutions that can be written down explicitly. Instead, solutions to these equations must be approximated numerically. This chapter reviews two classes of numerical methods: Euler and Runge–Kutta methods that solve equations forwards in time starting from an initial value, and collocation methods which approximate solutions using a basis expansion. The chapter also discusses when these methods can break down and ways to transform a differential equation to improve their performance.

James Ramsay, Giles Hooker

Chapter 6. Qualitative Behavior

Within the discipline of applied mathematics, a large part of dynamical systems theory is concerned with the description of the qualitative behavior of dynamical systems.

James Ramsay, Giles Hooker

Chapter 7. Nonlinear Least Squares or Trajectory Matching

This chapter examines a direct means of fitting the parameters ordinary differential equation models to data: solve the ODE at each candidate set of parameter values and choose the ones which lead to best agreement with the data. Often, we need to search over both parameter values and initial conditions. This chapter introduces Gauss–Newton methods for minimizing a squared error criterion along with the sensitivity equations need to calculate derivatives with respect to parameters. We introduce the concepts of statistical inference—confidence intervals and hypothesis tests—as they apply to nonlinear regression and discuss complications when multiple quantities are measured. We also present Bayesian methods for parameter estimation and ODE-specific schemes designed to overcome difficult optimization surfaces.

James Ramsay, Giles Hooker

Chapter 8. Two-Stage Least Squares or Gradient Matching

This chapterGradient matching|( presents indirect methods of fitting parameters to ordinary differential equation models. Rather than solving the ODE, we instead obtain non-parametric estimates of the state trajectory and its derivative. This allows the right hand side of the ODE to be fit to the estimated derivatives, which is often numerically easier than the trajectory matching described in Chap. 7. We discuss the ways in which this approach allows us to diagnose model mis-specification, and develop confidence intervals for parameters. We also examine the related approach of fitting the trajectory to the integral of the right hand side function.

James Ramsay, Giles Hooker

Chapter 9. Profiled Estimation for Linear Systems Estimated by Least Squares Fitting

We now describe an approach to parameter estimation and statistical inference called parameter cascading or generalized profiling that combines the virtues of both the nonlinear least squares method or trajectory matching described in Chap. 7 and the gradient matching approach in Chap. 8, but in a way that avoids the disadvantages of both.

James Ramsay, Giles Hooker

Chapter 10. Profiled Estimation for Nonlinear Systems

We have seen in the last chapter that linear differential equations can provide nice answers for a wide variety of problems, in the same way that multiple linear regression is the workhorse of applied statistics.

James Ramsay, Giles Hooker


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