Assignments

1. Fit a linear model with optim
   • Functions llike.linearmodel, sumsq.linearmodel
   • agb data: log(agb) as a function of log(dbh)
   • Minimize either sum of squares or likelihood
   • How many parameters?
2. Fit a linear model with variable SD
   • cecrin data: growth (gr12) as a function of diameter (dbh1)
   • Minimize either sum of squares or likelihood
   • How many parameters?
3. Fit a non-linear model to quantitative data
   1. Model types
      • treeheight: ht as a function of dbh (extract one species, eg quaras or tri2tu or pri2co)
      • Use data from a single species to estimate the 3 parameters using maximum likelihood and a Gaussian error.
      • y = H_max (it’s in teaching.functions.r)
        H_max, a, and b are parameters, y is height and x is dbh.
   2. grwfull300: growth as a function of dbh
      • y ∼ x + log(x) (write yourself)
   3. Error functions
      • Try first with Gaussian error
      • For growth rates, then use log-normal or Gamma error (but beware that growth must then be > 0)

4. Fit linear and non-linear models with Metropolis method
   1. Linear
      • Test function linearfit.Bayes in file modelfitBayes.intro.r with any linear model
      • Plot parameter runs
      • Find confidence limits
   2. Adapt linearfit.Bayes to asymptotic (or other model of your preference)
5. Simulation: Create a simulated correlation and test how well lm fits the parameters
   • The basics
     1. Define x from normal distribution
     2. Define slope and intercept parameters
     3. Define error with rnorm and sd
     4. Calculate y
     5. Use lm to estimate slope and intercept
   • More information
     1. Evaluate impact of increasing error
     2. Evaluate impact of error in measuring x
     3. Evaluate impact of highly non-Gaussian x
   • Advanced (extra credit)
     1. Test multiple regression, with x₁ and x₂ predictors
     2. Evaluate impact of correlation between x₁ and x₂

6. Program a Gibbs sampler for tree height model
   • Adapt survivalGibbs.r to linear model
   • Test hierarchical linear model of growth in growthGibbsHier.r
   • Graph all species responses on top of the forest-wide response
   • Histogram of species responses (slope, intercept of linear model) in hierarchical vs. non-hierarchical model
7. Run lmer on growth data
   • Linear model of log(growth) vs. log(dbh) with species a random effect
   • Graph all species responses on top of the forest-wide response
   • Compare results of lmer to Bayesian hierarchical model
8. Run lmer on full growth data
   • Linear model of log(growth) vs. log(dbh)
   • Possible random or fixed effects: species, time period
9. Run lmer on treemass data
   • Linear model of log(agb) vs. log(dbh), or with rainfall, or dry season months
   • Possible random or fixed effects: species, forest type