What Does Mathematical Modeling Mean?
When you get information from the real world, it can be useful to use this information to predict what can happen in the future. This is what you do when you use regression. It helps you find a mathematical model that describes what has happened so far, which will hopefully also be an accurate prediction for some time to come.
You can visualize a model as a graph that matches your data points from the real world pretty accurately. Below you can see some points and the graph of a quadratic function. Here, the graph is quite close to the points, and this is a hallmark of a good model. The motivation for creating a model is to be able to make informed predictions about things like the future. “Powerful is s/he who knows what the future brings!”
Theory
Theory
The Range of Validity
The range of validity of the model you are using is the domain of your function. Most mathematical models provide a good approximation at a given interval, while indicating wildly inaccurate values in other places.
It is important to critically evaluate the domain of your model. For example, it makes no sense to consider negative -values when you throw a ball into the air, if you’ve set as the height of the ground.
Theory
Limitations
Since models are simplifications of reality, it’s important to be critical of them, and to not trust them blindly. There could be unknown factors that might affect the result that the model doesn’t take into account. It’s important that you ask questions about how the model has been created and what shortcuts have been taken. That way, you can make an informed decision about whether the model is useful or not.
Example 1
The value of a car is $ today. The car has an annual depreciation of %. That means the value of the car can be described by the model
where is the value of the car, is the number of years, is the initial value, and is the growth factor that causes the value to decrease by % each year, .
Let’s discuss whether the domain of this model only consists of non-negative values of . If the car was new, the range of validity would have been only non-negative values, as it would have been safe to assume that the car did not have a higher value two years ago. But if the car was bought used, you can assume that the annual depreciation of % has been happening for a few years before the car was purchased.
Another factor regarding depreciation for cars is that the value tends to decrease faster the first couple of years after it’s bought new, and then the depreciation slows later. That means this model will probably not be valid throughout the lifetime of the car.
Another thing that could affect the validity of the model is whether the buyer maintains the car properly or not. If they don’t, the depreciation will be higher than anticipated. Most likely, the buyer will not have the car for more than 20 years. In addition, this model goes towards zero, but the owner can always take the car to a junkyard and get some money for it as scrap.
As you can see, there are many factors that affect the model and its domain.
