CONSTRUCTING AN ADAPTIVE DECISION PROBLEM

 

Until this point, we have illustrated ARM with some prepared (“in the can”) examples, in which we have already decided upon certain of the basic parameters (objective, decisions, model structure, model weights). Now we will give you the opportunity to begin with a problem that, while hypothetical, may resemble common problems you face as gamebird managers. We will set up the general problem; it is up to you to establish some specifics and flesh out and solve the problem. We’ll provide an outline of the process and some verbal cues below.

 

Before we start, and for the modeling faint of heart, remember the following important rules:

1. Don’t panic[1].
2. It’s only a model[2]
3. Have fun.[3]

 

Problem statement: The objective is to maximize bird production (surviving chicks per adult) for the least amount of economic input. The idea is to make management decision over several different management areas, monitor the results, and then apply what is learnt to other areas (on which a decision as to what to do has not been made). The management options (each of which can be applied annually) are 1) do nothing, 2) supplemental feeding ($10 per unit of effort), 3) predator control ($50 per unit of effort), 4) both feeding and predator control. The economic benefit of increasing production is $10 per 1% increase in production. There are 4 plausible biological models:

1. Neither feeding nor predator control in increase production
2. Feeding increases production by 2% per unit (CV*=10%)
3. Predator control increased production by 2% per unit (CV=10%)
4. In combination feeding and predator control increase production by 5% (CV=10%)

Initially these models have equal weight. Monitoring is estimates of fall recruitment from harvest surveys.

 

*Coefficient of variation (expresses SD as % of mean).
Let’s break this problem down into steps:

 

 

STEP 1: DEFINE THE OBJECTIVE

 

Convert the above verbal statement of the objective into math.

 

One form this could take is:

 

“Maximize V = $10*Prod_increase-$10*Feeding-$50*Pred_control”

 

So for instance if the decision was to feed at level 2 and control predators at level 3 and the response was a productivity increase of 50% then

 

V= $500 - $20-$150 = $330

 

 

 

STEP 2: OUTLINE THE DECISION ALTERNATIVES

 

List the alternatives and connect them to the objective via a model (next step)

 

In this example, the alternatives could be all combinations of feeding at levels 0,1,2, 3, and 4 and predator control at levels 0,1,2, 3, 4, for 25 combinations in all.

 

 

 

 

STEP 3: CONSTRUCT 2 OR MORE PLAUSIBLE ALTERNATIVE MODELS

 

Given observations of N, what do you predict under each model? E.g., what is the mean prediction +-Normal CI.

 

We might express the relationships as:

 

Model 1: Production = Current_production+ Normal(0,CV=10%)

 

Model 2: Production = Current_production*(1+0.02Feeding)+Normal(0,CV=10%)

 

Model 3: Production = Current_production*[1+0.02*Predator_control]+Normal(0,CV=10%)

 

Model 4: Production = Current_production*[1+0.05*(Feeding+Predator_control)]+Normal(0,CV=10%)

 

In each case, Production is following management, current production is preceding management, and feeding and predator control are in units of effort (0,1,2,3,4). The Normal (0,CV) term adds a zero-mean normal error to the prediction, with SD = CV * production (the prediction under the model).

 

In Netica, these can be entered into an equation box:

 

 

Note the we need a conditional statement that associates each model in the alternative model node (here Alt) to to the proper equation. This requires the conditional (Alt = = Model1) ? statement as shown below.

 

 

The equation then needs to be compiled to create the conditional probabilities.

 

 

 

STEP 3A: DEFINE THE SYSTEM STATES TO BE MONITORED (WILL DEPEND ON YOUR MODEL!)

 

In this example, fall age ratio.

 

 

STEP 3B: ASSIGN PRIOR WEIGHTS TO EACH MODEL

 

Already specified as equal (0.25 each).

 

 

STEP 4: SELECT A DECISION TO SATISFY YOUR OBJECTIVE

 

If you are using Netica, this will happen automatically; otherwise, you will need to average over the distributions (suggested for simplicity as normal mean +-95%CI).

 

 

 

STEP 5: MONITOR AND UPDATE

 

Again, Netica will automatically implement Bayes’ rule, once you observe N, make a decision, then observe the next N.

 

 

GO TO STEP 4 AND REPEAT

 

 

Once you build your model, try it out. See what decision seem to be optimal. Do these depend on system state (current number of birds, production rate, etc.)? How much difference does model uncertainty make? Conjure up some monitoring outcomes. How do these influence the model weights?

 

 

Updating

 

Once we make a decision for a particular area, and then learn from monitoring, we will want to apply the results to help us make a better decision in the next candidate area.

 

 

 

 

 

Do you want to add things? Go ahead.. Is there an environmental or other variable that you think is 1) important in driving this system, 2) potentially influences decision making.