AkCalculators

Regression and Least Squares — Understanding Fit, Error and Interpretation

Regression analysis estimates relationships between variables. The simplest and most common is linear regression, which finds the best-fit straight line y = a + bx that minimizes the sum of squared residuals between observed values y_i and predicted values ŷ_i. Polynomial regression generalizes this idea to estimate a polynomial relationship. Least squares is robust, intuitive and computationally efficient, forming the core of many statistical and machine-learning pipelines.

Least squares in brief

Given n observations (x_i, y_i), the least-squares estimate chooses coefficients that minimize S = Σ (y_i − ŷ_i)². For linear regression (degree 1), closed-form solutions exist using simple summations: b = Cov(x,y)/Var(x) and a = ȳ − b x̄. For polynomial regression we solve a normal equation (XᵀX)β = Xᵀy for coefficient vector β where X is the Vandermonde/design matrix.

Key statistics

• Slope / coefficients: parameters that define the fitted relationship.
• Residuals: e_i = y_i − ŷ_i, indicating errors of the fit per point.
• Sum of squared residuals (SSR): Σ e_i² — objective minimized by least squares.
• Correlation coefficient (r): measures linear association between x and y (−1 to 1). For linear regression, b = r (s_y/s_x).
• Coefficient of determination (R²): proportion of y-variance explained by the model: R² = 1 − SSR / SST where SST = Σ (y_i − ȳ)².

Computational details

Polynomial regression solves normal equations which can be ill-conditioned for high degrees or poorly scaled x. This tool supports up to degree 4 and up to 50 points; for larger or noisy datasets consider numerical libraries with regularization (ridge, orthogonal polynomials) or piecewise models (splines).

Interpreting results

High R² indicates the model explains much of the variance, but watch for overfitting with high-degree polynomials. Residual plots help detect non-random patterns: residuals should be roughly randomly scattered — patterns suggest model misspecification.

Examples

Example 1 (linear): Points (1,2),(2,3),(3,5) produce slope b ≈ 1.5 and intercept a ≈ 0.1 — predicted line approximates observed trend. R² tells how tight the points lie around the line.

Example 2 (polynomial): Quadratic fit to ballistic motion data yields coefficients that relate to initial position, velocity and acceleration.

Best practices

1. Plot residuals to check fit quality.
2. Center and scale x when fitting higher-degree polynomials to reduce numerical issues.
3. Prefer lower-degree models unless there is strong theoretical justification.
4. For predictive tasks validate models on held-out data.

This calculator gives you the core tools to run exploratory regression quickly. Use the plot, residuals and summary statistics together to judge model appropriateness. For production analyses, complement this with cross-validation and diagnostic checks.