Extrapolation predicts future behavior using known behavior, often through mathematical equations. Linear, polynomial, and exponential equations are used depending on the data collected. The quality of extrapolated data depends on the original data collection and chosen method. An example of everyday extrapolation is pedestrians crossing a street.
Extrapolating means using the known behavior of something to predict its future behavior. An observer can extrapolate using a formula, data laid out on a graph or programmed into a computer model. Following the scientific method, extrapolation is a technique that an analyst applies to generalize from various forms of collected data. The type of mathematical extrapolation used will depend on whether the data collected 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 gather information about the speed of a car coming towards them. For example, the eye can capture the expanding appearance of headlights at different points in time, and then the brain extrapolates or projects the vehicle’s motion into the future, judging whether the vehicle will arrive at the pedestrian’s location sooner, or later, is managed to cross the street.
In applied mathematics, it is possible to find a formula that matches any collected data 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 similar well documented behaviors. Constants and powers of generalized equations can be fitted to the data to predict or extrapolate changes in the data outside the collected range. In computer models, where data is known in specific places 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 solid modeling uses finite element methods for interpolation while programs for of fluids use finite volume methods.
Some forms of extrapolation depend on the terms of the mathematical equations used to fit the data: linear, polynomial, and exponential. If two data sets vary at a constant rate with each other, the extrapolation is linear and can be represented by a line of constant slope. An example of polynomial extrapolation is fitting data to conical and more complex shapes containing third-, fourth-, or higher-order equations. The higher the order of the equation, the more wiggles, curves, or waves the data represents. For example, there are as many highs and lows in the data as the order of its best-fit equation.
Exponential extrapolation covers datasets that grow or decay exponentially. Geometric growth or decay is an example of exponential extrapolation. These types of projections can be visualized as population curves showing birth and death rates, i.e. population growth and decline. 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 grandchildren.
The goodness of the extrapolated data depends both on the original data collection method and on the extrapolation method chosen. Data can be as smooth and continuous as the motion of a bicycle rolling downhill. It can also be jerky like a cyclist forcing his bike uphill in fits and starts. To extrapolate successfully, the analyst must recognize the characteristics of the behavior he intends to model.
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