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Interpolation estimates values between two data points. Linear interpolation connects two points with a line, while higher-order polynomials can provide more accurate results. Discrete data points are plotted on an XY graph, with x representing an independent variable and y a dependent variable. Linear interpolation assumes a constant rate of change, but exponential growth requires a more accurate method. Polynomial interpolation connects data points with a polynomial function, with higher orders providing more complex curves.
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 do interpolation: a line connecting two points is used to estimate intermediate values. Higher-order polynomials can substitute for linear functions for more accurate, but more complicated results. Interpolation can be contrasted with extrapolation, which is used to estimate values outside a set of points rather than between them.
A discrete set of data points has points with two or more coordinates. In a typical XY scatterplot, the horizontal variable is x and the vertical variable is y. Data points with an x and y coordinate can be plotted on this graph for easy viewing. 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, data can only be collected at discrete points. In the example of monitoring a country’s population, a census can only be taken at certain times. These measurements can be plotted as discrete data points on an XY graph.
If a census is only taken every five years, it’s 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 you need the population one year after the census, you 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 accurate enough.
Sometimes, however, linear interpolation introduces too many errors into its estimates. Population, for example, shows exponential growth under many scenarios. In exponential growth, the rate of growth itself is increasing: higher population leads to more births, which increases the total rate at which population increases. In an XY scatterplot, this type of behavior would show an “up curve” trend. A more accurate interpolation method may be appropriate for this type of study.
Polynomial interpolation involves connecting several data points with a polynomial function. A linear function is actually a simple manifold of a polynomial function, i.e. a polynomial of order one. Polynomials, however, can have orders higher than one: order two is a parabola, order three is a cubic function, and so on. A set of population data points might fit better with a polynomial function than a linear function because the former can curve up and down to match the data.
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