How can we identify a good mathematical model? Over focusing on one aspect of modeling (fitting, inference) may not be the best approach. Here are 6 qualities that defines a good model:
- Reflects data accurately
- Not overly complex
- Has predictive power
- Consistent with general knowledge of the system
- Can be studied mathematically
- Can be simulated
Reflects data accurately
A good model should be able to reasonably capture the dynamics of the system as measured in the available data. Other information may be available to the modeler, so this alone does not make a model good. Using all available information helps prevent ‘over fitting’ data and biased predictions.
Not overly complex
Complexity must be determined according to the amount of data available and the number of unknowns. Other information may be available about the system, but estimation ability breaks down as the size of unknowns increases.
Has predictive power
One of the goals of modeling is to obtain predictions, along with understanding the system. As mentioned above, testing this can be done with cross validation methods.
Consistent with general knowledge of the system
General knowledge includes any non data knowledge. This is fairly common in biology, as we may have relative or qualitative observations about about system that we don’t have the technology to get relevant data on.
Can be studied mathematically
Of course this is obvious. Without an analytical or numerical solution, it’s not really math modeling.
Can be simulated
Simulations require a logical structure and parameter estimates. These should exist if the model can be studied mathematically. The simulations may illustrate the effect of parameter values, or the system dynamics, or predictions.