Fit Y = aX³ + bX² + cX + d with R², inflection point, critical points, end behavior analysis, and residual table. Comparison to linear fit.
Cubic regression fits Y = aX³ + bX² + cX + d to your data — a flexible model that captures S-curves, asymmetric peaks, and inflection behavior that quadratic regression cannot. Unlike quadratics (which are symmetric), cubics can model phenomena that rise steeply, flatten, and then rise or fall differently.
Enter data and get four coefficients, R², adjusted R², and complete curve analysis: inflection point (where curvature flips), critical points (local maxima/minima), end behavior, and comparison against linear fit. The residual table with visual bars shows where the model fits well and where it struggles.
The "S-Curve Growth" preset demonstrates sigmoid-like adoption patterns. The "Chemical Rate" preset shows a reaction that accelerates, peaks, and reverses. Each preset illustrates a different cubic behavior pattern. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results.
Cubic regression bridges the gap between simple curves (quadratic) and arbitrary flexibility (high-degree polynomials). It's the simplest model that captures inflection behavior — critical in growth analysis, pharmacokinetics, economics, and engineering.
The inflection point is often the most valuable output: in product adoption it marks when growth shifts from viral to saturating, in chemical reactions it marks maximum rate, in structural engineering it marks the transition from elastic to plastic deformation.
Y = aX³ + bX² + cX + d (least squares via normal equations). Inflection: X = −b/(3a). Critical points: 3aX² + 2bX + c = 0.
Result: Y = −0.0584X³ + 0.2890X² + 4.9113X + 0.9091, R² = 0.9987, Inflection at X ≈ 1.65
The cubic model fits the S-curve growth pattern with R² = 0.999. The inflection point at X ≈ 1.65 marks where growth transitions from accelerating to decelerating.
The least-squares cubic requires solving a 4×4 system of normal equations involving sums up to x⁶. The matrix is a Vandermonde-like structure that can become ill-conditioned when X values span a wide range. For numerical stability, centering X (subtracting the mean) or using orthogonal polynomials is recommended for production systems.
Both cubics and logistic functions model S-shaped data. Cubics are easier to compute (closed-form solution) but extrapolate terribly. Logistic models (Y = L/(1+e^(-k(x-x0)))) have natural asymptotes and are biologically meaningful but require iterative fitting. For interpolation within the data range, cubic works well. For extrapolation and interpretation, prefer logistic.
A common approach: fit polynomials of increasing degree and track adjusted R². When adjusted R² stops improving (or decreases), you've found the right degree. For n data points, degree ≥ n−1 fits perfectly (interpolation) but captures all noise. The goal is capturing signal, not fitting noise — Occam's razor applied to curves.
Use cubic when: (1) residuals from quadratic show a remaining pattern, (2) data has an S-shape or asymmetric peak, (3) domain knowledge suggests an inflection point (growth → saturation), (4) R² improves meaningfully over quadratic. Use this as a practical reminder before finalizing the result.
The inflection point is where curvature changes sign — from "bending up" to "bending down" or vice versa. In growth modeling, it marks the transition from accelerating growth to decelerating growth. In economics, it's where diminishing returns begin to dominate.
Minimum 4 (matching 4 parameters), but this gives zero error degrees of freedom. Practically, 10+ points are needed for reliable estimates. Sparse data near the extremes causes wild extrapolation.
Yes, more so than linear or quadratic. With only a few more points than parameters, the cubic will fit noise. Check adjusted R² (penalizes complexity) and examine whether the improvement over quadratic is meaningful (>2 percentage points).
Cubics diverge to ±∞ as X grows — the X³ term eventually dominates. A model that fits perfectly for X in [0, 10] may predict absurd values at X = 20. Never extrapolate cubics beyond the data range without domain justification.
Compute both. If adjusted R² barely improves (< 1-2 points), keep the quadratic. If residuals from the quadratic show an S-shaped pattern, the cubic is needed. Parsimony: use the simplest model that fits.