Extrapolation

Extrapolation is the practice of estimating values outside the range of observed data by applying a model fitted to the available data. It is distinct from interpolation, which predicts within the observed interval.

Common extrapolation approaches include linear extrapolation, which extends the trend using the last observed slope; polynomial or spline extrapolation, which uses fitted curves; and model-based extrapolation that relies on a theoretical or statistical model such as exponential growth, logistic growth, or time-series models (for example ARIMA).

Extrapolation rests on the assumption that the identified pattern continues beyond the data, an assumption that is often uncertain. Errors generally increase with distance from the known data; nonstationarity, regime changes, or incorrect model choice can lead to unreliable estimates.

Applications include forecasting in economics and finance, engineering design, climate and environmental science, and other fields where predicting beyond observed data is needed. Because of the uncertainty, extrapolations are typically presented with uncertainty ranges or scenario analyses rather than precise predictions.

Related concepts include interpolation, predictive modeling, and model validation. Extrapolation emphasizes the limits of knowledge when extending patterns outside the observed range.