interpolaatsiooni

Interpolaatsiooni (interpolation) is a method in numerical analysis and statistics for estimating unknown values within the range defined by a set of known data points. It aims to construct a function that agrees with the measured values at their positions and enables predictions at intermediate positions. The technique is used when data are sampled discretely and continuous values are needed between samples.

Common approaches include linear interpolation, which connects neighboring data points with straight segments; polynomial interpolation, which builds a single polynomial that passes through all points; and spline interpolation, which uses piecewise polynomials joined smoothly at the data points, typically cubic splines. Other methods include nearest-neighbor interpolation and more advanced techniques such as barycentric or Newton form Lagrange interpolation. The choice depends on the data’s smoothness, the desired smoothness of the result, and computational considerations.

Interpolation is distinct from extrapolation, which estimates values outside the known range. It can introduce errors if the assumed model does not reflect the true underlying function, and high-degree polynomial interpolation can suffer from Runge’s phenomenon, causing large oscillations. Spline methods often offer a robust balance between fidelity and smoothness. In practice, interpolation may interact with measurement noise, sometimes requiring smoothing or regularization.

Applications span numerical simulation, computer graphics, digital signal processing, weather and geospatial data, and data-driven modeling. A simple example of linear interpolation between (x0, y0) and (x1, y1) yields y = y0 + (x − x0)(y1 − y0)/(x1 − x0).