Introduction to Agent-Based Modeling (ABM)

Hypotheses/Models

g1 <- lm(y ~ 1, data=cement)
g2 <- lm(y ~ x1 + x2, data=cement)
g3 <- lm(y ~ x1 + x2 + x1*x2, data=cement)
g4 <- lm(y ~ x3 + x4, data=cement)
g5 <- lm(y ~ x3 + x4 + x3*x4, data=cement)

Hypotheses/Models

aictab(list("Constant" = g1, 
            "x1 + x2" = g2, 
            "x1 + x2 + x1*x2" = g3, 
            "x3 + x4" = g4, 
            "x3 + x4 + x3*x4" = g5), 
       second.ord = TRUE)

Hypotheses/Models

Model selection based on AICc:

                K   AICc Delta_AICc AICcWt Cum.Wt     LL
x1 + x2         4  69.31       0.00   0.94   0.94 -28.16
x1 + x2 + x1*x2 5  74.72       5.40   0.06   1.00 -28.07
x3 + x4         4  83.74      14.43   0.00   1.00 -35.37
x3 + x4 + x3*x4 5  88.22      18.90   0.00   1.00 -34.82
Constant        2 111.54      42.22   0.00   1.00 -53.17

Model averaging (Multimodel Inference)

\[ \mathbf{g} = \sum_{i=1}^{R}w_{i}g_{i}. \]

As an example, create an arbitrary input \( x_{1} \), \( x_{2} \), \( x_{3} \), and \( x_{4} \) and look at the predicted values.

X <- data.frame(x1 = 0.14, x2 = 0.40, x3 = 0.52, x4 = 0.05)
modavgPred(list("Constant" = g1,
                "x1 + x2" = g2,
                "x1 + x2 + x1*x2" = g3,
                "x3 + x4" = g4,
                "x3 + x4 + x3*x4" = g5), 
           second.ord = TRUE, 
           newdata = X)

Model-averaged predictions on the response scale
based on entire model set and 95% confidence interval:

  mod.avg.pred uncond.se lower.CL upper.CL
1       53.176     3.242   46.821   59.531

Statistics have multiple modalities or goals. These include:

• estimation of parameters, [Estimation]
• testing of hypotheses, [Hypothesis Testing]
• exploration of covariation, [Multicollinearity]
• prediction into new conditions, [Statistical Modeling, Regression]
• selecting among choices (e.g. models) [Model Selection]

  Brian McGill (aka Dynamic Ecologist)

“All other things being equal, mechanistic models are more powerful since they tell you about the underlying processes driving patterns. They are more likely to work correctly when extrapolating beyond the observed conditions.”
- Bolker (2008) Ecological models and Data in R, p7

How about some definitions?

Mechanistic model: A hypothesized relationship between the variables in the data set where the nature of the relationship is specified in terms of the biological processes that are thought to have given rise to the data. The parameters in the mechanistic model all have biological definitions and so they can be measured independently of the data set referenced above.

Phenomenological/Statistical model: A hypothesized relationship between the variables in the data set, where the relationship seeks only to best describe the data.

“A statistical model foregoes any attempt to explain why the variables interact the way they do, and simply attempts to describe the relationship, with the assumption that the relationship extends past the measured values. Regression models are the standard form of such descriptions…”
- in “Ecological Detective” by Hilborn and Mangel

1. Model formulation: Construct functional relationships that represent biological hypotheses [e.g. linear or nonlinear models]
2. Model fitting: i.e. parameter estimation [e.g. least squares or maximum likelihood estimation]
3. Model diagnostics: Make sure assumptions of model formulations hold [e.g. analyze residuals]
4. Model selection: Choose the best model or set of models [e.g. AIC, forward/backward selection]
5. Model use: Use the model for prediction or inference [e.g. multimodel inference]

“Because the assumptions in the first version of a model are experimental, we have to test whether they are appropriate and useful. For this, we need criteria for whether the model can be considered a good representation of the real system. These criteria are based on patterns or regularities that let us identify and characterize the real system in the first place.”
- Railsback & Grimm

tl;dr ->

• Statistical models are validated by comparison to quantitative information (data).
  
• Mechanistic models are validated by comparison to qualitative information (patterns).

“Because the assumptions in the first version of a model are experimental, we have to test whether they are appropriate and useful. For this, we need criteria for whether the model can be considered a good representation of the real system. These criteria are based on patterns or regularities that let us identify and characterize the real system in the first place.”
- Railsback & Grimm

Process of modeling (POM) is not “linear”

Just like process of science (POS)

“We need to start with a very clear research question…”

“Very often, even our questions are only experimental and later we might need to reformulate the question, perhaps because it turned out to be not clear enough, or too simple, or too complex.”

…for essential processes and structures

“But whatever technique we prefer, this task has to combine existing knowledge and understanding, a 'brainstorming' phase in which we wildly hypothesize, and, most importantly, a simplification phase.”

“The modeling cycle must be started with the most simple model possible, because we want to develop understanding gradually, while iterating through the cycle.”

“…just our preliminary understanding of a system is not sufficient for deciding whether things are more or less important for a model. It is the very purpose of the model to teach us what is important.”

Definition: A model is a concrete or abstract simplification of objects and their relationships or processes in the real world.

In this step, we choose:

• Objects (spatial scale, entities, state variables, parameters)
• Processes (time scale, parameters)

Verbal formulation of the model!!

Mathematical or computational formulation of the model!!

Doing science with models

• Extract information from model through:

  • Simulation
  • Analysis

• Use results to:

  • Explain
  • Predict
  • Use as evidence for hypotheses