Math Modeling Concepts (so far!)
- Factors in the construction of models
- All models are wrong; some are useful
- Simplicity versus completeness
- The importance of the linear model
- Constraints: time and resources available to do the modeling
- Useful modeling strategy (Polya's): UPCE
- Understand
- Plan
- Carry Out
- Evaluate
- Recurrence Relations
- Growth Models (first order) and dynamics associated with each
- linear
- affine
- quadratic (logistic)
- general (e.g. can you draw a growth model that has certain
properties, such as a stable fixed point at 1, unstable
at 3)
- Finding fixed points, and testing for stability
- Cobwebbing about fixed points to test for stability
- Compartmental models (flows, stocks)
- Simulation - how to generate populations from discrete models
- Stochastic Models
- Environmental stochasticity
- Demographic stochasticity
- Testing parameters for normality
- Histograms (quick visual)
- QQplots
- Chi-square statistic
- Statistical concepts
- Measures of spread
- Measures of central tendency
- variance and standard deviation
- range
- inter-quartile range
- Probability density functions (pdf)
- Cumulative distribution functions (cdf)
- Aspects of the normal distribution (e.g. pdf; 68, 95, 99%
rule; symmetry; etc.)
- Major aspects of the uniform and binomial distribution
- quantile function (percentiles, quantiles)
- expected value
- "Fuzzy" growth models (i.e., parametrized with parameters that
are stochastic, rather than deterministic).
- State Diagram for Stage/class models
- Matrix operations
- Matrix multiplication
- Identity matrix
- Inverse of a matrix
- Transpose of a matrix
- Determinant of a matrix (and how to compute in 2x2 case)
- Eigenvalues/Eigenvectors of a matrix (and how to find them given
that the eigenvalue is known)
- Markov chain models
- Regular matrices and fixed point solutions ("normalized"
eigenvectors of eigenvalue = 1)
- Absorbing markov chains
- Empirical Models
- Simple Linear Regression
- Fitting a least-squares line
- R2 - variance accounted for
- Correlation/Covariance
- SSRegression/SSResidual/SSTotal
- Regression parameters and standard errors and confidence
intervals
- Regression diagnostics (e.g. T-ratios, F-ratios)
- Degrees of freedom
- Ladder of powers
- Non-linear regression
- Linearizable models
- Exponential models (ln(y) regression)
- Power models (ln(x)/ln(y) regression)
- Curvilinear models
- Fundamentally non-linear models (e.g. Logistic model)
- Detrending data (e.g. using trigonometric functions to pick up
periodic trends)
- Interpolation
- Polynomial interpolation
- Spline interpolation - making smooth, piecewise
interpolants of low-degree polynomials
- Fitting slopes, etc. at boundaries
- Problems with linear and quadratic interpolation
(smoothness issues)
- Cubic interpolation
- Multiple Linear Regression
- Idea of Step-wise regression (adding, removing variables)
- T-ratios
- Differential Equations
- Solving simple D.E.s by separation of variables
- Relationship between difference equations and differential
equations
- Autonomous versus non-autonomous
- Population models (exponential growth)
- Logistic growth model
- Simple example models (e.g. epidemic models)
Website maintained by Andy Long.
Comments appreciated.