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 ad...
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| Main Author | |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
Boca Raton, Fla
CRC Press
2023
CRC Press LLC Chapman & Hall |
| Edition | 1 |
| Subjects | |
| Online Access | Get full text |
| ISBN | 1032261811 9781032261812 9781032259482 1032259485 |
| DOI | 10.1201/9781003286974 |
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| Abstract | 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. |
|---|---|
| AbstractList | This book 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. 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. Provides a concise overview of differential equations focused on "modelling first" perspective. Introduces concepts from statistics, data science, and other mathematics courses in an accessible way, encouraging further study in mathematics, statistics, and data science. Includes models from biology, chemistry, physics, economics, and the social sciences. Integrates R and the tidyverse throughout the text without assuming extensive previous subject knowledge of the two. Includes a cohesive R package (demodelr) to supplement the material that is accessible for novice R users. |
| Author | Zobitz, John M. |
| Author_xml | – sequence: 1 givenname: John M. surname: Zobitz fullname: Zobitz, John M. |
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| ContentType | eBook Book |
| Copyright | 2023 John M. Zobitz |
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| DOI | 10.1201/9781003286974 |
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| Keywords | visualization simulation Equilibrium Solutions Unstable Stochastic Differential Equations Markov Chain Monte Carlo Methods Data Science Follow Ensemble Average SDE Jacobian Matrix Phase Plane Likelihood Function Euler's Method tidyverse Solution Curves Differential Equation Dataset Yeast parameterization Data Frame Counterclockwise Scatter Plot computation Solution Trajectories Euler Maruyama Method Spiral Source Wo Nullclines Markov Chain Monte Carlo Parameter Brownian Motion Nonlinear Parameter Estimation |
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| Notes | Includes bibliographical references (p. 351-354) and index |
| OCLC | 1350449041 |
| PQID | EBC7130191 |
| PageCount | 379 |
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| PublicationPlace | Boca Raton, Fla |
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| Publisher | CRC Press CRC Press LLC Chapman & Hall |
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| Snippet | Exploring Modeling with Data and Differential Equations Using R provides a unique introduction to differential equations with applications to the biological... This book provides a unique introduction to differential equations with applications to the biological and other natural sciences. Additionally, model... Provides a concise overview of differential equations focused on "modelling first" perspective. Introduces concepts from statistics, data science, and other... |
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| SubjectTerms | Biological models Biological models -- Data processing Biological models -- Mathematical models Differential equations R (Computer program language) |
| TableOfContents | 24.1. Random walk redux -- 24.2. Simulating Brownian motion -- 24.3. Exercises -- 25. Simulating Stochastic Differential Equations -- 25.1. The stochastic logistic model -- 25.2. The Euler-Maruyama method -- 25.3. Adding stochasticity to parameters -- 25.4. Systems of stochastic differential equations -- 25.5. Concluding thoughts -- 25.6. Exercises -- 26. Statistics of a Stochastic Differential Equation -- 26.1. Expected value of a stochastic process -- 26.2. Birth-death processes -- 26.3. Wrapping up -- 26.4. Exercises -- 27. Solutions to Stochastic Differential Equations -- 27.1. Meet the Fokker-Planck equation -- 27.2. Deterministically the end -- 27.3. Exercises -- References -- Index 8.2. Parameter estimation for global temperature data -- 8.3. Moving beyond linear models for parameter estimation -- 8.4. Parameter estimation with nonlinear models -- 8.5. Towards model-data fusion -- 8.6. Exercises -- 9. Probability and Likelihood Functions -- 9.1. Linear regression on a small dataset -- 9.2. Continuous probability density functions -- 9.3. Connecting probabilities to linear regression -- 9.4. Visualizing likelihood surfaces -- 9.5. Looking back and forward -- 9.6. Exercises -- 10. Cost Functions and Bayes' Rule -- 10.1. Cost functions and model-data residuals -- 10.2. Further extensions to the cost function -- 10.3. Conditional probabilities and Bayes' rule -- 10.4. Bayes' rule in action -- 10.5. Next steps -- 10.6. Exercises -- 11. Sampling Distributions and the Bootstrap Method -- 11.1. Histograms and their visualization -- 11.2. Statistical theory: sampling distributions -- 11.3. Summary and next steps -- 11.4. Exercises -- 12. The Metropolis-Hastings Algorithm -- 12.1. Estimating the growth of a dog -- 12.2. Likelihood ratios for parameter estimation -- 12.3. The Metropolis-Hastings algorithm for parameter estimation -- 12.4. Exercises -- 13. Markov Chain Monte Carlo Parameter Estimation -- 13.1. The recipe for MCMC -- 13.2. MCMC parameter estimation with an empirical model -- 13.3. MCMC parameter estimation with a differential equation model -- 13.4. Timing your code -- 13.5. Further extensions to MCMC -- 13.6. Exercises -- 14. Information Criteria -- 14.1. Model assessment guidelines -- 14.2. Information criteria for assessing competing models -- 14.3. A few cautionary notes -- 14.4. Exercises -- III. Stability Analysis for Differential Equations -- 15. Systems of Linear Differential Equations -- 15.1. Linear systems of differential equations and matrix notation -- 15.2. Equilibrium solutions -- 15.3. The phase plane 15.4. Non-equilibrium solutions and their stability -- 15.5. Exercises -- 16. Systems of Nonlinear Differential Equations -- 16.1. Introducing nonlinear systems of differential equations -- 16.2. Zooming in on the phase plane -- 16.3. Determining equilibrium solutions with nullclines -- 16.4. Stability of an equilibrium solution -- 16.5. Graphing nullclines in a phase plane -- 16.6. Exercises -- 17. Local Linearization and the Jacobian -- 17.1. Competing populations -- 17.2. Tangent plane approximations -- 17.3. The Jacobian matrix -- 17.4. Exercises -- 18. What are Eigenvalues? -- 18.1. Introduction -- 18.2. Straight line solutions -- 18.3. Computing eigenvalues and eigenvectors -- 18.4. What do eigenvalues tell us? -- 18.5. Concluding thoughts -- 18.6. Exercises -- 19. Qualitative Stability Analysis -- 19.1. The characteristic polynomial (again) -- 19.2. Stability with the trace and determinant -- 19.3. A workflow for stability analysis -- 19.4. Stability for higher-order systems of differential equations -- 19.5. Exercises -- 20. Bifurcation -- 20.1. A series of equations -- 20.2. Bifurcations with systems of equations -- 20.3. Functions as equilibrium solutions: limit cycles -- 20.4. Bifurcations as analysis tools -- 20.5. Exercises -- IV. Stochastic Differential Equations -- 21. Stochastic Biological Systems -- 21.1. Introducing stochastic effects -- 21.2. A discrete dynamical system -- 21.3. Environmental stochasticity -- 21.4. Discrete systems of equations -- 21.5. Exercises -- 22. Simulating and Visualizing Randomness -- 22.1. Ensemble averages -- 22.2. Repeated iteration -- 22.3. Exercises -- 23. Random Walks -- 23.1. Random walk on a number line -- 23.2. Iteration and ensemble averages -- 23.3. Random walk mathematics -- 23.4. Continuous random walks and diffusion -- 23.5. Exercises -- 24. Diffusion and Brownian Motion Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Contents -- List of Figures -- Welcome -- I. Models with Differential Equations -- 1. Models of Rates with Data -- 1.1. Rates of change in the world: a model is born -- 1.2. Modeling in context: the spread of a disease -- 1.3. Model solutions -- 1.4. Which model is best? -- 1.5. Start here -- 1.6. Exercises -- 2. Introduction to R -- 2.1. R and RStudio -- 2.2. First steps: getting acquainted with R -- 2.3. Increasing functionality with packages -- 2.4. Working with R: variables, data frames, and datasets -- 2.5. Visualization with R -- 2.6. Defining functions -- 2.7. Concluding thoughts -- 2.8. Exercises -- 3. Modeling with Rates of Change -- 3.1. Competing plant species and equilibrium solutions -- 3.2. The Law of Mass Action -- 3.3. Coupled differential equations: lynx and hares -- 3.4. Functional responses -- 3.5. Exercises -- 4. Euler's Method -- 4.1. The flu and locally linear approximation -- 4.2. A workflow for approximation -- 4.3. Building an iterative method -- 4.4. Euler's method and beyond -- 4.5. Exercises -- 5. Phase Lines and Equilibrium Solutions -- 5.1. Equilibrium solutions -- 5.2. Phase lines for differential equations -- 5.3. A stability test for equilibrium solutions -- 5.4. Exercises -- 6. Coupled Systems of Equations -- 6.1. Flu with quarantine and equilibrium solutions -- 6.2. Nullclines -- 6.3. Phase planes -- 6.4. Generating a phase plane in R -- 6.5. Slope fields -- 6.6. Exercises -- 7. Exact Solutions to Differential Equations -- 7.1. Verify a solution -- 7.2. Separable differential equations -- 7.3. Integrating factors -- 7.4. Applying the verification method to coupled equations -- 7.5. Exercises -- II. Parameterizing Models with Data -- 8. Linear Regression and Curve Fitting -- 8.1. What is parameter estimation? |
| Title | Exploring Modeling with Data and Differential Equations Using R |
| URI | https://www.taylorfrancis.com/books/9781003286974 https://cir.nii.ac.jp/crid/1130015553626874112 https://ebookcentral.proquest.com/lib/[SITE_ID]/detail.action?docID=7130191 https://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9781000776744 |
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