Interpolation

Interpolation is the process of estimating unknown values within the range of a discrete set of known data points. It seeks a function that agrees with the data at the given locations and can be used to predict values at intermediate points.

Interpolation differs from regression in that the interpolant is required to pass exactly through the known points, whereas regression aims to approximate the underlying relationship without necessarily hitting the data points.

Common interpolation methods include linear interpolation, which connects adjacent points with straight lines; polynomial interpolation, which constructs a single polynomial that passes through all points (for n points, up to degree n−1, provided the abscissas are distinct); and spline interpolation, which uses piecewise polynomials joined in a smooth way (cubic splines are a typical choice). Other approaches include piecewise constant (nearest-neighbor), and more sophisticated methods such as radial basis functions and kriging for spatial data. In multiple dimensions, interpolation can be extended by tensor-product schemes or scattered-data methods.

In practice, high-degree polynomial interpolation can exhibit oscillations near the interval ends (the Runge phenomenon); piecewise methods like splines or local polynomials are often preferred for stability and smoothness.

Applications span numerical analysis, data visualization, digital signal and image processing, computer graphics, geostatistics, and environmental sciences, where one needs values at unobserved points from a set of measurements.