Residual

Residual refers to something that remains after a process or calculation. In statistics and data analysis, a residual is the difference between an observed value and the value predicted by a model.

In regression and predictive modeling, residual r_i equals y_i minus ŷ_i. Residuals are used to assess model fit; if the model is correct, their mean should be near zero, and they should show no systematic pattern. Standardized or studentized residuals help identify outliers and influential observations.

In time series forecasting, residuals are the forecast errors—the differences between observed values and model predictions for each time point. The assumption that residuals behave like white noise (approximately uncorrelated with constant variance) is a common diagnostic target, with residual diagnostics (such as autocorrelation plots) used to evaluate model adequacy.

In chemistry and related industries, residual solvent denotes solvent remaining in a substance after processing. Regulatory guidelines, such as ICH Q3C, set acceptable limits to ensure safety and quality. Residual solvents are monitored by analytical methods to verify compliance.

Other uses include residuals in data compression or signal processing, where the residual is the difference between the original data and its prediction; in materials science, residual stress refers to internal stresses present in a material in the absence of external loads. The term remains versatile across disciplines, always signaling something left over or not fully explained by the main model or process.