Fit lines and polynomials to data points using least squares. Visualize residuals, R-squared, and the normal equation.
Enter paired x and y observations and select a straight-line or polynomial model. The visualizer finds coefficients that minimize the sum of squared vertical residuals, then shows the fitted curve, residuals, R-squared, and the corresponding normal equation. The most common surprise is that increasing the polynomial degree can improve the in-sample R-squared without producing a more useful model. A close fit may reflect overfitting, especially when there are few points.
A concrete input and expected output from the current implementation.
Input
Model: linear 0, 1 1, 3 2, 5
Expected output
Best-fit line: y = 1 + 2x Residuals: [0, 0, 0] R-squared: 1 Normal equation: [[3, 3], [3, 5]] [c0, c1]^T = [9, 13]^T
All three points lie exactly on y = 1 + 2x, so every residual and the residual sum of squares are zero. Since the y-values have nonzero variation, R-squared is 1 - 0/8 = 1.
For a line, each point contributes a row [1, x] to a design matrix X. A polynomial of degree d extends that row through x^d. The coefficient vector c minimizes the sum of (y - fitted y)^2. The normal equation (X^T X)c = X^T y expresses the condition satisfied by those least-squares coefficients.
For each point, the residual is observed y minus the y-value predicted by the fitted line or curve. Squaring and summing these residuals gives SSE. When the observed y-values have nonzero total variation SST, R-squared is 1 - SSE/SST. A value near 1 indicates a close fit to the entered points, not necessarily reliable prediction.
No. Adding polynomial terms cannot worsen the ordinary in-sample least-squares fit when the models are nested, so R-squared usually rises or stays unchanged. The added terms may merely follow noise between the entered points. Inspect the residual pattern and curve shape, and use separate validation data when predictive performance matters.
No for a polynomial written as y = f(x). Its residuals are vertical differences in y, so a vertical relation x = constant cannot be represented by that model. If both coordinates contain important measurement error, or perpendicular distance is the intended error measure, an orthogonal or errors-in-variables regression is a different calculation.
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