Lineaariregressiota

Lineaariregressiota is the Finnish term for linear regression, a fundamental statistical technique for modelling the relationship between one or more explanatory variables and a continuous response variable. It rests on the premise that the expected value of the response can be expressed as a linear combination of the predictors plus an error term.

In the simplest case of simple linear regression, the model is y = β0 + β1x + ε, where y denotes the dependent variable, x the independent variable, β0 the intercept, β1 the slope coefficient, and ε the residual error assumed to have mean zero and constant variance. Multiple linear regression generalises this to y = β0 + β1x1 + … + βpxp + ε, allowing for several predictors. Coefficients are typically estimated by the method of least squares, which chooses values that minimise the sum of squared residuals.

The validity of a linear regression analysis depends on several key assumptions. The relationship between predictors and response should be linear, errors should be independently and identically distributed, exhibit constant variance (homoscedasticity), and be normally distributed for inference purposes. Multicollinearity, or high correlation among predictors, can inflate variance estimates and destabilise coefficient estimates.

Model performance is assessed using several metrics. R², the coefficient of determination, quantifies the proportion of variance in the response explained by the predictors. Adjusted R² corrects for the number of predictors, penalising over‑parameterisation. For predictive accuracy, the root mean squared error (RMSE) or mean absolute error (MAE) is commonly reported. Cross‑validation provides a robust estimate of predictive generalisability.

Linear regression finds widespread use in economics, social sciences, biology, engineering, and many other fields where a simple, interpretable relationship is sought. Its ease of implementation, transparent interpretation, and solid theoretical foundation make it one of the most frequently applied techniques in the data‑science toolkit. Potential pitfalls include omitted variable bias, outliers, and violations of assumptions, all of which can be mitigated by diagnostic plots, robust estimation techniques, or model refinement.