1st Edition
Exploring Modeling with Data and Differential Equations Using R
Exploring Modeling with Data and Differential Equations Using R provides a unique introduction to differential equations with applications to the biological and other natural sciences. Additionally, model parameterization and simulation of stochastic differential equations are explored, providing additional tools for model analysis and evaluation. This unified framework sits "at the intersection" of different mathematical subject areas, data science, statistics, and the natural sciences. The text throughout emphasizes data science workflows using the R statistical software program and the tidyverse constellation of packages. Only knowledge of calculus is needed; the text’s integrated framework is a stepping stone for further advanced study in mathematics or as a comprehensive introduction to modeling for quantitative natural scientists.
The text will introduce you to:
- modeling with systems of differential equations and developing analytical, computational, and visual solution techniques.
- the R programming language, the tidyverse syntax, and developing data science workflows.
- qualitative techniques to analyze a system of differential equations.
- data assimilation techniques (simple linear regression, likelihood or cost functions, and Markov Chain, Monte Carlo Parameter Estimation) to parameterize models from data.
- simulating and evaluating outputs for stochastic differential equation models.
An associated R package provides a framework for computation and visualization of results. It can be found here: https://cran.r-project.org/web/packages/demodelr/index.html.
1. Models of rates with data
2. Introduction to R
3. Modeling With Rates of Change
4. Euler’s Method
5. Phase lines and equilibrium solutions
6. Coupled Systems of Equations
7. Exact Solutions to Differential Equations
8. Linear Regression and Curve Fitting
9. Probability and Likelihood Functions
10. Cost Functions & Bayes’ Rule
11. Sampling Distributions and the Bootstrap Method
12. The Metropolis-Hastings Algorithm
13. Markov Chain Monte Carlo Parameter Estimation
14. Information Criteria
15. Systems of linear differential equations
16. Systems of nonlinear equations
17. Local Linearization and the Jacobian
18. What are eigenvalues?
19. Qualitative Stability Analysis
20. Bifurcation
21. Stochastic Biological Systems
22. Simulating and Visualizing Randomness
23. Random Walks
24. Diffusion and Brownian Motion
25. Simulating Stochastic Differential Equations
26. Statistics of a Stochastic Differential Equation
27. Solutions to Stochastic Differential Equations
Biography
John Zobitz is a Professor of Mathematics and Data Science at Augsburg University in Minneapolis, Minnesota. His scholarship in environmental data science includes ecosystem models parameterized with datasets from environmental observation networks. He is a member of the Mathematical Association of America (MAA) and previous president of the North Central Section of the MAA. He has served on the editorial board of MAA Notes. He was a recipient of the Fulbright-Saastamoinen Foundation Grant in Health and Environmental Sciences at the University of Eastern Finland in Kuopio, Finland. In addition, he is an affiliated member of the Ecological Forecasting Network and regularly taught at Fluxcourse, an annual summer course for measurements and modeling of ecosystem biogeochemical fluxes.
"What quantitative skills does the modern biologist need? Some might say just enough to run prepackaged programs and collaborate with the experts. In other words, enough to get into trouble. The ambitious new book "Exploring Modeling with Data and Differential Equations in R" by John Zobitz shows that another path is possible. Biologists with a background in calculus can see the foundational methods in mathematical modeling, statistics and programming put to work through the carefully chosen examples that build in this well-organized book. The philosophy is well-expressed by the first word of the title "Exploring", and the actual code and the graphs it produces are invitations to explore data with models, and not to hold the "math" at arms length. The student or researcher who works through this friendly guide will have the tools and confidence needed to get into the kind of good trouble that leads to insight and discovery."
-Fred Adler, Professor of Mathematics and Director of the School of Biological Sciences University of Utah“It will be so exciting to teach from this book, with its thorough incorporation of computing and data sets. From the start, the topics interweave standard differential equations concepts (analytical solutions, Euler’s method, qualitative approaches, single equations, systems of equations, stability analysis, and more) with an introduction to the R programming language, probability, and statistics. There is significant time spent on stochastic differential equations and on determining model parameter values based on data. These are timely, as current-day mathematical modelers frequently incorporate stochastic effects into their models, and connecting with real data sets is absolutely crucial. Exploring Modeling with Data and Differential Equations Using R works well as a main text or supplement for a course in Differential Equations or Mathematical Modeling. Alternately, it is a resource for those learning new methods while pursuing independent research projects.”
- Meredith Greer, Professor of Mathematics, Bates College
