Just about everyone would like to know what the future holds. Some consult tarot cards, tea leaves, crystal balls, and telephone psychics. Others take a more systematic approach-they examine the recent past to understand processes and determine trends that may give insight into the future. In short, they form ideas about how the world works, and from those ideas generate predictions about what will happen in the future. These ideas constitute an abstraction of the real world and form a "model" of a "system" of interrelated components.
Mathematical modeling is a technique for understanding the dynamics of a system and for predicting future outcomes within the system. From a simplified perspective, any system is composed of two fundamental things:
In another example, consider a household budget. There are elements such as income, expenses, savings, etc. and relationships that allocate certain proportions or fixed amounts of the income to the expenses.
In CRiSP Harvest, the basic elements are the fisheries and the stocks. The relationships include the processes by which fishing reduces the stock, production and growth, etc. The properties of these elements and the relationships between them are controlled by the many parameters in the model such as Harvest Rates and production parameters.
Abstraction allows for the simplification of the system because it is not necessary or even desirable for it to be exact or replicate the exact mechanisms. In CRiSP Harvest, the properties of the fishers and the stocks are explained in simplified mathematical terms so that their essential qualities are characterized in a concrete manner. For example, the fisher is presumed to catch fish at a certain rate and the details of exactly how many are being caught at any given time are unimportant.
In the case of the baseball player A, all we need to know are the odds that the batter will get a hit. Our model is simply his/her average: .323. That is a gross simplification of a huge number of things: A's hand-eye coordination, the types of fields (s)he plays on, A's strength, the pitchers technique, diet, coaching, health, etc. We model A's hitting ability so that we can make some kind of prediction of whether or not A will get a hit the next time at bat.
The balsa-wood plane on the other hand crudely represents a real airplane and may have only a handful of parts, but was designed for function over form.
In the case of the CRiSP Harvest model, the uses and purposes include:
Modeling Concepts and Practice
There are two very important steps in the creation of a model: calibration and validation. They help make the model more usable and believable.
In the case of the household budget, there might be a transportation category where bus fare, gas for the car, parking and automobile maintenance is all consolidated. Each month the household evaluates their expenses related to transportation to see if their budget model is accurate. If it is consistently off the mark and changes to expenses can not be made, then it is time to recalibrate the model.
CRiSP Harvest is recalibrated periodically by fisheries scientists. They use updated catch information, escapement estimates and other data from the field to re-establish parameter values.
A more important type of model validation is the process of determining how well the model represents the real system and, consequently, how useful it is in predicting the future. In the baseball example, we might like to know how well a simple batting average model calibrated at the end of every week predicts the batting average during the coming week. If the batter is very consistent, a simple batting average model probably is valid for predicting future performance. However, if the batter is a streak hitter and goes through cycles of hot and cold hitting, a simple batting average may not be an acceptable model. In this case, a more complicated model may be needed that predicts whether the batter will be in his hot or cold cycle during the coming week.
Fishery models can be validated by comparing future predictions with real outcomes. For example, a model calibrated through 1995 can be used to predict escapements and catches in 1996. Once the 1996 season is over, the predictions can be compared to the real-world outcome to see how well the model performed.
Real world model validation is very difficult given the complexity of the systems involved. If a model can not be validated, sometimes, the individual parts are validated and the whole is deemed acceptable provided that the representation of the mechanisms and processes that hold the parts together is acceptable to the community who are building and/or using the model. This is the case when complete model validation cannot be done for some reason (it may be prohibitively expensive, require too much time, etc.) but the value of a working model is significant.