LAB 4–OPTIMIZATION
Goals:
The goal of this lab is to familiarize you with some of the commonly used optimization routines
and some of the problems you are likely to encounter using these methods.
Parameter estimation
Recall that in Day 2 of the course, we used PopTools to estimate height-diameter regression
equations for a few species. We used the model:
Height  1.35  ( Max.Height 1.35 )x( 1  e( 1xBxDBH ) )
where B and MaxHeight are estimated parameters and Height and DBH are the data. There are a
number of reasons why the calculation of these parameters may not be straightforward: tree
height is often difficult to measure, crowns are often irregularly shaped, or the species of interest
may deviate from this relationship. These problems are often more marked in tropical forests
because the canopy is highly layered and dense making it difficult to measure height and creating
irregular crowns, and palms are often common. In this lab, we will use data from Casearia
arborea and the palm Prestoea acuminata from the Luquillo Forest Dynamic plot in Puerto Rico
to explore some of these issues.
Step 1. Setting up the model
Open the worksheet called Lab 4.Optmization.xls. First, you will need to set up the
model using the structure we used in Day 2. Note that you will have to use the name Y for the
dependent variable (Height) and T for the independent variable (DBH). You will also need to
choose adequate upper and lower bounds for parameter estimates. We suggest that you plot the
data to come up with reasonable values for these bounds. Use this plot to pick lower and upper
bounds for parameters. Now you can use the parameter estimation routines in PopTools to come
up with the MLE estimates. Recall that:
(1)
(2)
(3)
(4)
First, you need to specify your model, initial conditions, and error structure in the
spreadsheet (in green).
Next you will need to specify the parameter names, initial point estimates and ranges
(in blue).
Finally, you need to declare both your predictive variable and the observations (in
yellow). Choose 1000 for maximum iterations and select an output range. You will see
that the parameter estimates are identical to those you obtained with Solver. In
addition, Poptools calculates likelihood, AIC and support intervals for the parameters.
Finally, you will need to choose an estimation method. Four algorithms are available:
MARQ - Marquardt’s method
BFGS - Broyden-Fletcher-Goldfarb-Shannon method
SIMPLEX – Nelder-Mead Simplex method
SIMANNEAL – Simulated annealing
Some of these methods are best suited to solve local optimization problems. Derivative-based
methods such as Marquardt’s can be messed up when the goodness-of-fit function has sharp
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changes in it. When one of these methods gets to a flat place in the likelihood surface, it gets
faked into thinking it's found the answer (since the derivative is zero); when there's a vertical
discontinuity in the graph, it gets faked into extrapolating wrong. Simplex is a simple bruteforce option of getting around this problem. Rather than starting with a single point guess at
what the parameters are, this method picks a number of points which form a simplex |the
simplest shape possible in an n-dimensional space. Simplex, however, will not work if there are
multiple local minima. The best way to get around this is to use stochastic optimization
algorithms, where you actually pick random numbers to add some “noise" and avoid local
minima. The classic stochastic optimization algorithm is the Metropolis algorithm (or simulated
Annealing).
Step 2. Optimization problems.
(1) Go to the spreadsheet for Casearia arborea, use Marquardt’s method, to come up with MLE
estimates. What results do you get? Why? Try the second method, BFGS. Does it work? Try
the other two methods. Are the results consistent? How do you determine which of the results is
“best” in this case? Can you get around this problem by changing bounds for the parameters?
(2) Go through the same process with the spreadsheet that contains data for the palm Prestoea
acuminata. Try the four methods. Do the parameter values make biological sense?
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