The concept of modeling is the use of mathematics to predict the behavior of a system. It can be as simple as finding the trend in car prices, or as complex as modeling the climate of the Earth. In all cases, we describe the system by mathematical equations. The equation can be based on underlying physical principles, like gravity or electromagnetism, or they can be empirical models derived by fitting them to a set of observations. For example, we might measure the temperature at noon everyday. If we start in March, we might notice that after a month or so there is a gradual warming. We could fit a straight line to these data, and then we could predict the average temperature in June. This might work quite well, but the model wouldn’t be realistic for long-term prediction, since it would predict the temperatures keep rising. Come October, the temperatures will have stopped rising and begun cooling, and we would find our linear model isn’t adequate. After a year, we’d notice a cyclical pattern, and could then possibly fit a oscillatory function to the year-long temperatures, and come up with a good model.

The reason our initial linear model failed (in our hypothetical story) is because it extrapolated into uncharted territory. This is always a risky proposition. In general, when using empirical models like our temperature model, it’s a bad idea to extrapolate too far beyond the end of the observed data. Other ways that empirical models can break down is by spurious data. For example, suppose the thermometer was bumped or otherwise changed during the measurements, making the last few measurements colder than the actual temperature. This would skew the best-fit line, and our model would under-predict the future temperatures.

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