Linear Modelling

Bayesian Methods for Ecological and Environmental Modelling

Peter Levy
UKCEH Edinburgh

Beyond linear models

Four types of system

1. Stable
2. Periodic
3. Chaotic
4. Complex

Stable systems

• example: linear models, most statistical modelling
• relationship \(y = f(x, \cdots)\) stable in time
• \(x\) and \(y\) are independent

Periodic/feedback systems

• examples: predator-prey populations, thermostat
• interactions and feedbacks
• move in repeating patterns
• domain of dynamic modelling & ODEs
• \(dy/dt = f(x, \cdots)\)
• \(dx/dt = f(y, \cdots)\)

Chaotic systems

• examples: dice roll, turbulence, weather
• sensitivity to initial conditions
• rapidly become unpredictable
• difficult to model directly
  • but we can model statistical properties

Complex systems

• examples: food webs, ecosystems, the economy
• assemblages of interacting systems
• predictability depends on relative stability, feedbacks & chaotic behaviour

Four types of model

1. Empirical: \(y = f(\theta, x)\)
2. Statistical: \(y = f(\theta, x) + \epsilon\)
   • \(\theta\) and \(\epsilon\) inferred from measurements of \(y\) and \(x\)
   • commonly uses linear models and ordinary least squares
3. Machine Learning: \(y = f(\theta, x) + \epsilon\)
   • form of \(f\) and \(\theta\) inferred from measurements of \(y\) and \(x\)

Four types of model

4. Process-based / mechanistic: \(y = f(\theta, x)\)
   • \(f\) is based on causal understanding, often nonlinear
   • \(\theta\) inferred from measurements of \(y\) and \(x\)
   • calibration often informal; \(\epsilon\) not explicit
     

Four types of model

4. Process-based / mechanistic: \(y = f(\theta, x)\)
   • \(f\) is based on causal understanding, often nonlinear
   • \(\theta\) inferred from measurements of \(y\) and \(x\)
   • calibration often informal; \(\epsilon\) not explicit

Bayesian approach applies to all types of systems and models (in principle)

Causal inference

Simpson’s paradox

Review

Linear models

• simplest type of model
• parameters can be estimated from data using MCMC
• very easy with rstanarm
• informative priors can be specified

Many other kinds of systems & models exist

• Bayesian approach applies to all

Tomorrow

Model selection & comparison

• which \(x\) variables to include?
• what form of model \(f\) to use?
• which model is better?

Tomorrow

Hierarchical modelling

• samples not independent but grouped
  • e.g. species, sites, dates, methods, …
• groups can be nested
  • regions | sites | plots
  • family | genus | species

Questions?