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To extrapolate is to use the known behavior of something to predict its future behavior. An observer can extrapolate by using a formula, data arranged on a graph, or programmed into a computer model. Following the scientific method, extrapolation is one technique an analyst applies to generalize from various forms of data collected. The type of mathematical extrapolation used will depend on whether the data gathered is continuous or periodic.
An everyday example of extrapolation is illustrated by how pedestrians safely cross busy streets. When pedestrians cross a street, they unknowingly collect information about the speed of a car coming toward them. For example, the eye may capture the expanding appearance of the headlights at several different points in time, and then the brain extrapolates, or projects the movement of the vehicle into the future, judging whether the vehicle will arrive at the pedestrian’s location before, or after, he or she has been able to cross the street.
In applied mathematics, a formula can be found that matches any data collected about the behavior of the physical universe — an extrapolation called curve fitting. Each curve fit to the data has an equation known to represent other well-documented, similar behaviors. Constants and powers of the generalized equations can be fit to the data to predict, or extrapolate, changes in the data outside the collected range. In computer models, where data is known in specific locations and not in others, a continuous spectrum of predictive data can be generated. When data is generated between known data points, the process is usually referred to as interpolation, but the same methods apply: computational software for modeling solids use finite elements methods to interpolate while programs for modeling fluids use finite volume methods.
Some forms of extrapolation depend on terms of the mathematical equations used to fit the data — linear, polynomial, and exponential. If two sets of data vary at a constant rate with one another, the extrapolation is linear — it can be represented by a line of constant slope. An example of a polynomial extrapolation is data fit to conic and more complex shapes containing third, fourth, or higher order equations. The higher the order of the equation, the more oscillations, curves, or waves the data represent. For example, there are as many maxima and minima in the data as the order of its best-fit equation.
Exponential extrapolation covers data sets that either grow or decay exponentially. Geometric growth or decay is an example of exponential extrapolation. These types of projections can be visualized as population curves that show birth and death rates — growth and decay of the population. For example, two parents have two children, but those two, each have two, so that in three generations, the number of great grandchildren will be two to the third power, or an exponent of three — two multiplied by itself three times — resulting in eight great grand children.
The goodness of extrapolated data depends on both the method of collection of the original data and the extrapolation method chosen. Data can be smooth and continuous like the movement of a bicycle rolling downhill. It can, also, be jerky as a cyclist forcing his or her bicycle uphill in fits and starts. To extrapolate successfully, the analyst must recognize the characteristics of the behavior he or she intends to model.