extrapolations

Extrapolation is a method of estimating values outside the range of observed data by extending a known relationship. It relies on a model fitted to available data—such as a straight line in linear regression, a polynomial curve, or a time-series model—and uses that model to predict beyond the observed interval.

Common approaches include linear extrapolation, which projects the trend from the most recent data; polynomial extrapolation, which extends fitted polynomials; exponential and logistic extrapolation, which assume multiplicative growth or bounded growth; and regression-based forecasting using methods like ARIMA or exponential smoothing. The choice of method depends on the data characteristics and the expected behavior outside the observed range.

Key considerations include the assumption that the underlying pattern continues beyond the observed data. Prediction intervals or confidence intervals accompany extrapolated estimates to express uncertainty. The further the forecast extends from known data, the greater the potential error. Extrapolation is sensitive to outliers, nonstationarity, regime changes, and structural breaks, which can render the model unreliable.

Applications span many fields, including economics for market or demand forecasts, physics and engineering for material responses beyond tested conditions, biology for population projections, and climate science for future scenarios. Extrapolation is often contrasted with interpolation, which estimates values within the known data range and generally carries stronger justification.

In practice, researchers emphasize the model’s assumptions, validate with back-testing where possible, and communicate the uncertainty associated with extrapolated values.