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WebED Education, Outreach and TrainingA Short Course: Using Modeling and StatisticsTypes of Models Models can characterized and classified in a number of ways, but a is not really important to the kinds of activities we will do. However, it may help to set out a few broad categories of models to aid in understanding how they work. These categories are not necessarily mutually exclusive since a lot of models involve elements from several types. Physical Models Physical models are scale representations of the same physical entities they represent. They are used primarily in engineering of large-scale projects to examine a limited set of behavioral characteristics in the system. The illustration and site below is a good example of a physical model -- one of a stream and dam used to simulate the rate of outflow from the dam under a variety of circumstances.
Model of river and dam. Click on the picture for more information Mathematical Models Mathematical models use mathematical equations to represent the key relationships among system components. The equations can be derived in a number of ways. Many of them come from extensive scientific studies that have formulated a mathematical relationship and then tested it against real data, just like our "driving to work" example. Some come from laboratory testing of relationships where that is feasible. Sometimes real data are used to derive relationships using statistical techniques to fit a particular relationship to the data and to measure the level of error associated with that representation. Mathematical models of large-scale systems often use a combination of approaches -- inserting tested equations where the relationships are well known and inserting statistical relationships where there is less certainty. Such models can also use probabilistic relationships for events that are random or exhibit some type of variable pattern. For example, models where weather events are critical input analyze the long-term weather records for the area under consideration and calculate the frequency of different weather incidents. In Ohio, for example, there is a high probability of many small rainstorms because they occur frequently. There is a much lower probability of extreme events that produce a large amount of rainfall in a short time -- high winds, high snowfalls, etc. These can be incorporated into mathematical models by applying probability distribution in the model and using a random number generator to choose events based on that distribution. Similar rate based distributions are used in population dynamics models relating to birth and death rates, in the modeling of disease (epidemiology) based on risk factors, for earthquake risks, and for many kinds of accident modeling. This is a population dynamics model of a deer population. It shows the equations for the inflow and outflow of deer related to food supply and deaths. Notice that there is a probability function at the bottom of the screen for deaths from predators where the probability increases as the density of the deer population increases. Simulation Models Simulation models are a special subset of mathematical or physical models that allow the user to ask "what if" questions about the system. Changes are made in the physical conditions or their mathematical representation and the model is run many times to "simulate" the impacts of the changes in the conditions. The model results are then compared to gain insight into the behavior of the system. If we go back to our "trip to work" example, we could imagine a simulation model where we input information on all of the road section links and how long it takes to travel on each link under a variety of circumstances. We could then simulate our trip to work over different routes and different conditions and select the best route to take in each circumstance.
Please contact Al-Azad Iqbal or Steve Gordon for Questions and Comments - Updated 10/2/07 |
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