Basic Modeling Cycle
The Basic Modeling Cycle is a structured approach used to create mathematical models that simulate real-world scenarios. This process begins with the identification of a specific problem or scenario, necessitating a clear definition of the involved variables. Once the variables are established, mathematical representations of their relationships are constructed, which can take various forms such as algebraic equations or graphical data. The cycle then involves analyzing these relationships through mathematical operations to derive insights or predictions.
A crucial aspect of the modeling cycle is the comparison of the model's outcomes with actual real-world data to assess its accuracy. This step allows for adjustments to be made if the model does not align with observed results. The modeling process is not only prevalent in fields like economics and the hard sciences but also serves as an educational tool, helping individuals understand the applicability of mathematical concepts in everyday life. Furthermore, the creation of models often involves a blend of expertise and creative thinking, with standard practices varying across different disciplines. Overall, the Basic Modeling Cycle highlights the importance of systematic reasoning in both academic and practical contexts.
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Basic Modeling Cycle
Mathematical modeling is a form of simulating hypothetical or real-world scenarios. These scenarios may be as simple as the upkeep costs for a family adopting a new puppy or as complex as the effects of gasoline price decreases on the green jobs market. Because budgeting and saving for specific purposes (college funds, vacation funds) are predicated on models of spending, economic examples are the ones most familiar from everyday life, but mathematical modeling is fundamental to the practice of the hard sciences and featured prominently across the social sciences. Weather forecasting, for example, is based on mathematical models of the behavior of atmospheric phenomena, and transportation planners must model the expected traffic volume of a proposed highway in order to determine how many lanes will be necessary. Modeling is also used as a teaching tool to help understand new concepts and techniques, answering the perennial student question, "When will I use this in the real world?"
The modeling cycle consists of the steps taken in constructing a mathematical model: identification of the scenario or problem and the variables involved; mathematical representations of the relationships among those variables; analysis of those relationships; and comparison of the results or conclusions produced by the model with real world information to verify its accuracy.
Overview
Modeling always involves choices. Debates among economists and scientists often target the choices made in constructing a model, and even family budgets require making an educated guess about the future cost of college, for instance, or assume that commodity prices will remain stable enough for the new puppy's food costs to remain a constant once he reaches his full size. If the model proves inaccurate in its predictions, these initial choices can be reexamined, though they are not always where the fault lies. Different problems require different degrees of certainty or precision in the results. Designing a model is a process that benefits from both expertise in the appropriate areas and creative thinking.
Certain basic norms and standards have been adopted in various disciplines. A meteorologist seeking to forecast the week's weather for his region, for instance, has standard models he can use, rather than having to approach the problem with a blank slate. From time to time these standard models are challenged, or different schools of thought within a discipline prefer different models. All models follow the same basic modeling cycle.
The problem or scenario being modeled must be clearly defined, by first identifying the variables involved and then creating mathematical representations of the relationships among these variables. These representations may be algebraic, geometric, statistical, or take the form of graphs or tables. For instance, in determining the aforementioned puppy's upkeep costs, there is a straightforward relationship between the puppy's expected growth and his annual food costs.
The next step of the cycle is performing the necessary operations implied by those mathematical representations, such as calculating the puppy's food costs based on an appropriate growth rate. Finally, the model is compared with the real world in order to check its accuracy. In the case of forecast models, their results can be compared over time—the puppy cost model may be fact-checked after a year to see if costs have turned out to accord with the model's predictions—and adjusted if need be.
Bibliography
Banerjee, Sandip. Mathematical Modeling: Models, Analysis, and Applications. London: Chapman, 2014.
Bolouri, Hamid. Computational Modeling of Gene Regulatory Networks. London: Imperial College P, 2008.
DiStefano, Joseph, III. Dynamic Systems Biology Modeling and Simulation. Waltham, MA: Academic P, 2013.
Giordano, Frank R, William P. Fox, and Steven B. Horton. A First Course in Mathematical Modeling. 5th ed. Boston: Cengage, 2013.
Ingalls, Brian P. Mathematical Modeling in Systems Biology: An Introduction. Cambridge, MA: MIT P, 2013.
Meerschaert, Mark. Mathematical Modeling. 4th ed.. Waltham, MA: Academic P, 2013.