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Interpolation involves discovering a pattern in a set of data points to estimate a value between two points. Linear interpolation is one of the simplest ways to interpolate—a line connecting two points is used to estimate intermediate values. Higher-order polynomials can replace linear functions for more accurate, but more complicated, results. Interpolation can be contrasted with extrapolation, which is used to estimate values outside of a set of points instead of between them.
A discrete set of data points has points with two or more coordinates. In a typical XY scatter plot, the horizontal variable is x and the vertical variable is y. Data points with both an x and y coordinate can be plotted on this graph for easy visualization. In practical applications, both x and y represent finite real-world quantities. X generally represents an independent variable, such as time or space, while y represents a dependent variable, such as population.
Often times, data can only be gathered at discrete points. In the example of monitoring a country’s population, a census can only be taken at certain times. These measurements could be plotted as discrete data points on an XY chart.
If a census is only taken every five years, it is impossible to know the exact population between censuses. In linear interpolation, two data points are connected with a linear function. This means that the dependent variable (population) is assumed to change at a constant rate to reach the next data point. If the population one year after a census is needed, one could linearly interpolate the two data points to estimate an intermediate value based on the connecting line. It is typically obvious that the real variable does not change linearly between data points, but this simplification is often sufficiently accurate.
Sometimes, however, linear interpolation introduces too much error in its estimates. Population, for example, exhibits exponential growth in many scenarios. In exponential growth, the rate of growth itself is increasing—a higher population leads to more births, which increases the total rate by which the population increases. In an XY scatter plot, this kind of behavior would show a trend that “curved upward.” A more accurate method of interpolation may be appropriate for this kind of study.
Polynomial interpolation involves connecting numerous data points with a polynomial function. A linear function is actually a simple variety of a polynomial function—namely, a polynomial of order one. Polynomials, though, can have higher orders than one: order two is a parabola, order three is a cubic function, and so on. A set of population data points might be better interpolated with a polynomial function than a linear function because the former can curve up and down to match the data.