HARMONIC.

HARMONIC fits an additive model, y = f₁(x₁) + f₂(x₂) + … + f_p(x_p), using the closed-form Fourier-basis method described in a 1992 paper on Fourier smoothers and additive models (OpenAlex ID W2024658605). Each predictor is rescaled to the interval zero to one, then represented as a linear term plus a fixed set of cosine waves. Fitting the weights on this fixed basis is a single linear solve, not an iterative search.

How smoothness is chosen

Two settings govern how flexible a curve is allowed to be: K, the number of cosine terms, and λ, a ridge penalty on the higher-frequency terms. HARMONIC does not ask the user to tune either. It fits a fixed grid of candidates (K in four, eight, twelve, and twenty-five log-spaced values of λ), scores each by generalised cross-validation, and keeps the candidate with the lowest score. This is a grid search over closed-form fits, not an iterative optimiser.

Two honest limitations

Boundary bias. A cosine basis has zero slope at both ends of the rescaled interval by construction. Where a true effect has a non-zero slope at the edge of the observed data, the fitted curve can flatten out artificially in the outermost values of a predictor. Read the extreme low and high end of each curve with that in mind.

Weak components. When a predictor's true effect is small relative to noise, its fitted curve carries a wide ±2 standard-error band. A flat curve with a wide band is not evidence of a flat effect; it is evidence that this dataset does not pin the effect down. The band is the honest answer, not the curve alone.

Measured against the incumbents

Smoothing n = 100,000 points in the browser, measured 2026-07-13 (protocol and raw numbers in the repository under bench/):

Read fairly in both directions: once Pyodide has paid its 36.8 MB and 58-second entry price, compiled LOWESS wins the warm fit (0.76 s to 1.41 s, with hand-fixed smoothing where HARMONIC selects its own). HARMONIC's case is the other four rows.