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| Document Type: | Book |
|---|---|
| All Authors / Contributors: |
Dmitry A Kondrashov |
| ISBN: | 9780226371627 022637162X 9780226371764 022637176X |
| OCLC Number: | 1031875159 |
| Description: | xvi, 417 pages ; 24 cm |
| Contents: | 0.1. What is mathematical modeling? -- 0.2. Purpose of this book -- 0.3. Organization of the book -- 1. Arithmetic and variables: The lifeblood of modeling -- 1.1. Blood circulation and mathematical modeling -- 1.2. Parameters and variables in models -- 1.2.1. Discrete state variables: genetics -- 1.2.2. Continuous state variables: concentration -- 1.3. First steps in R programming -- 1.3.1. Numbers and arithmetic operations -- 1.3.2. Variable assignment -- 1.4. Computational projects -- 2. Functions and their graphs -- 2.1. Dimensions of quantities -- 2.2. Functions and their graphs -- 2.2.1. Linear and exponential functions -- 2.2.2. Rational and logistic functions -- 2.3. Scripts, functions, and plotting in R -- 2.3.1. Writing scripts and calling functions -- 2.3.2. Vector variables -- 2.3.3. Arithmetic with vector variables -- 2.3.4. Plotting graphs -- 2.4. Rates of biochemical reactions -- 2.5. Computational projects -- 3. Describing data sets -- 3.1. Mutations and their rates -- 3.2. Describing data sets -- 3.2.1. Central value of a data set -- 3.2.2. Spread of a data set -- 3.2.3. Graphical representation of data sets -- 3.3. Working with data in R -- 3.4. Computational projects -- 3.4.1. Data description -- 3.4.2. Plotting data and fitting by eye -- 4. Random variables and distributions -- 4.1. Probability distributions -- 4.1.1. Axioms of probability -- 4.1.2. Random variables -- 4.1.3. Expectation (mean) of random variables -- 4.1.4. Variance of random variables -- 4.2. Examples of distributions -- 4.2.1. Uniform distribution -- 4.2.2. Binomial distribution -- 4.3. Testing for mutants -- 4.4. Random numbers and iteration in R -- 4.4.1. Random numbers -- 4.4.2. For loops -- 4.5. Computational projects -- 4.5.1. uniform distribution -- 4.5.2. Binomial distribution -- 5. Estimation from a random sample -- 5.1. Law of Large Numbers -- 5.1.1. Sample mean -- 5.1.2. Sample size and standard error -- 5.2. Central Limit Theorem -- 5.2.1. Normal distribution -- 5.2.2. Confidence intervals -- 5.3. Relative risk -- 5.4. Sampling in R -- 5.4.1. simulated sampling -- 5.4.2. computing confidence intervals -- 5.5. Computational projects -- 6. Independence of random variables -- 6.1. Categorical data sets with two variables -- 6.2. Mathematics of independence -- 6.2.1. Conditional probability and information -- 6.2.2. Independence of events -- 6.2.3. Calculation of expected frequencies -- 6.3. Testing for independence -- 6.3.1. Hypothesis testing -- 6.3.2. Rejecting the null hypothesis -- 6.3.3. The chi-squared statistic -- 6.4. Hypothesis testing in R -- 6.5. Independence in data sets -- 6.5.1. Maternal age and Down syndrome -- 6.5.2. Stop-and-frisk and race -- 6.6. Computational projects -- 6.6.1. Thumb-on-top preference and sex -- 6.6.2. Relationship between species and habitat -- 6.6.3. Independence testing of simulated data -- 7. Bayes' amazing formula -- 7.1. Prior knowledge -- 7.2. Bayes' formula -- 7.2.1. Positive and negative predictive values -- 7.3. Applications of Bayesian thinking -- 7.3.1. When too much testing is bad -- 7.3.2. Reliability of scientific studies -- 7.4. Random simulations -- 7.5. Computational project -- 8. Linear regression and correlation -- 8.1. Linear relationship between two variables -- 8.2. Linear least-squares fitting -- 8.2.1. Sum of squared errors -- 8.2.2. Best-fit slope and intercept -- 8.2.3. Correlation and goodness of fit -- 8.3. Linear regression using R -- 8.4. Regression to the mean -- 8.5. Computational projects -- 8.5.1. parental age and new mutations -- 8.5.2. heart rates on two different days -- 9. Nonlinear data fitting -- 9.1. Nonlinear relationships between variables -- 9.2. Fitting using log transforms -- 9.3. Generalized linear fitting in R -- 9.3.1. Logarithmic transforms -- 9.3.2. Polynomial regression -- 9.4. Allometry and power law scaling -- 9.5. Computational projects -- 10. Markov models with discrete states -- 10.1. Building Markov models -- 10.2. Markov property -- 10.2.1. Transition matrices -- 10.2.2. Probability of a string of states -- 10.3. Simulation of Markov models -- 10.4. Markov models of medical treatment -- 10.5. Computational projects -- 10.5.1. State strings for a two-state model -- 10.5.2. State strings for a three-state model -- 11. Probability distributions of Markov chains -- 11.1. Distributions evolve over time -- 11.1.1. Markov chains -- 11.1.2. matrix multiplication -- 11.1.3. Propagation of probability vectors -- 11.2. Matrix multiplication in R -- 11.3. Mutations in evolution -- 11.4. Computational projects -- 11.4.1. Pobability vectors of a two-state model -- 11.4.2. Probability vectors of a three-state model -- 12. Stationary distributions of Markov chains -- 12.1. The origins of Markov chains: A feud and a poem -- 12.2. Stationary distributions -- 12.2.1. Definition of stationary distribution -- 12.2.2. Condition for unique stationary distribution -- 12.3. Multiple random simulations in R -- 12.4. Bioinformatics and Markov models -- 12.5. Computational projects -- 12.5.1. Multiple two-state model simulations -- 12.5.2. Multiple three-state model simulations -- 13. Dynamics of Markov models -- 13.1. Phylogenetic trees -- 13.2. Eigenvalues of Markov models -- 13.2.1. Basic linear algebra terminology -- 13.2.2. Calculation of eigenvalues on paper -- 13.2.3. Calculation of eigenvectors on paper -- 13.2.4. Rate of convergence -- 13.3. Matrix diagonalization in R -- 13.4. Molecular evolution -- 13.4.1. Jukes-Cantor model -- 13.4.2. Time since divergence -- 13.4.3. Calculation of phylogenetic distance -- 13.4.4. Divergence of human and chimp genomes -- 13.5. Computational projects -- 13.5.1. Eigenvalues of a two-state model -- 13.5.2. Eigenvalues of a three-state model -- 13.5.3. Analysis of the Jukes-Cantor model -- 14. Linear difference equations -- 14.1. Discrete-time population models -- 14.1.1. Static population -- 14.1.2. Exponential population growth -- 14.1.3. Example with birth and death -- 14.1.4. Dimensions of birth and death rates -- 14.2. Solutions of linear difference models -- 14.2.1. Simple linear models -- 14.2.2. Models with a constant term -- 14.3. Population growth and decline -- 14.4. Numerical solutions in R -- 14.4.1. Functions in R -- 14.4.2. Solving difference equations -- 14.5. Computational project -- 15. Linear ordinary differential equations -- 15.1. From discrete time to smooth change -- 15.1.1. Bacteria that divide at arbitrary times -- 15.1.2. Growth proportional to population size -- 15.1.3. Chemical kinetics -- 15.2. Solutions of ordinary differential equations -- 15.2.1. Separate-and-integrate method -- 15.2.2. Solution of inhomogeneous ODES -- 15.2.3. Forward Euler method -- 15.3. Numerical solutions of ODEs -- 15.3.1. Implementation of Forward Euler -- 15.3.2. error in Forward Euler solutions -- 15.4. Applications of linear ODE models -- 15.4.1. model of pharmacokinetics -- 15.4.2. Cole's membrane potential model -- 15.5. Computational projects -- 15.5.1. Error and time step -- 15.5.2. Pharmacokinetics model -- 16. Graphical analysis of ordinary differential equations -- 16.1. ODEs with nonlinear terms -- 16.2. Qualitative analysis of ODEs -- 16.2.1. Graphical analysis of the defining function -- 16.2.2. Fixed points and stability -- 16.3. Modeling infectious disease spread -- 16.4. Computational projects -- 16.4.1. Logistic population growth model -- 16.4.2. SIS epidemic model -- 17. Chaos and bifuractions in difference equations -- 17.1. Logistic model in discrete time -- 17.2. Qualitative analysis of difference -- 17.2.1. Fixed points or equilibria -- 17.2.2. Stability of fixed points -- 17.3. Graphical analysis -- 17.3.1. Graphical analysis using R -- 17.3.2. Cobweb plots -- 17.4. Discrete-time logistic model and chaos -- 17.5. Computational projects -- 17.5.1. Graphical stability analysis -- 17.5.2. Investigation of chaotic dynamics. |
| Responsibility: | Dimitry A. Kondrashov. |
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