Regressão Linear y = ax + b

Simple linear regression — coefficients (a, b)

Ordinary least squares (OLS) fits the line y = a·x + b by making the sum of squared residuals as small as it can be. The slope comes out as a = Σ((xᵢ−x̄)(yᵢ−ȳ)) / Σ(xᵢ−x̄)² and the intercept as b = ȳ − a·x̄. The coefficient of determination R² = SSR/SST tells you how much of the variance in Y the model accounts for. Run X=[1,2,3,4,5], Y=[2.1,3.9,6.2,7.8,10.1] through it and you get a ≈ 2.00, b ≈ 0.04 and R² ≈ 0.999. Every estimate carries a standard error, which is what lets you run t-tests against H₀: coefficient = 0. Legendre published the method in 1805 and Gauss in 1809.

Applications

Forecasting sales, experimental physics (Hooke's law F = k·x, Ohm's law V = R·I), econometrics like the Phillips curve and demand functions, classical machine learning with scikit-learn LinearRegression, calibration curves in pharmacology, and quality control on the engineering side.

FAQ

Gauss-Markov assumptions? Linearity, independent errors, homoscedasticity (constant variance) and normal residuals. Those are what OLS needs to be BLUE, the best linear unbiased estimator.

R² close to 1 is always good? Not necessarily. A high R² can also signal overfitting or a spurious correlation. Look at the residual plots, and when you're comparing models with different numbers of predictors, lean on adjusted R² instead.

What about non-linear data? Transform it first (log, √) or switch to polynomial or non-linear regression. For outliers, robust regression such as Huber or RANSAC handles them well.