THE FOLD / CO-OP / THE BROADCAST / THE SAVITZKY-GOLAY
THE SAVITZKY-GOLAY
smooth the noise without blurring the shape
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Savitzky–Golay filter smooths noisy data without flattening its features: instead of averaging (which blurs peaks), it fits a low-degree polynomial to each sliding window by least squares and takes the fitted value at the center. Because a polynomial can bend, it preserves the height and width of peaks that a moving average would crush. Remarkably, the whole operation collapses to a single fixed convolution kernel. It is standard in spectroscopy and sensor processing.
LIT verified live: a Savitzky–Golay filter of order d reproduces any polynomial of degree ≤ d exactly (it does not distort what it should preserve) over 200 random cases (window.__savitzkygolay). FIG no framing; exact to floating precision.
LIT verified live: a Savitzky–Golay filter of order d reproduces any polynomial of degree ≤ d exactly (it does not distort what it should preserve) over 200 random cases (window.__savitzkygolay). FIG no framing; exact to floating precision.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-broadcast — the noisy stream that must be cleaned without smearing its shape. The Savitzky–Golay filter is that shape-preserving smoother. AVAN (AI) built the instrument: the windowed least-squares fit, the derived convolution coefficients, and the polynomial-reproduction check.
Credit as content: Abraham Savitzky & Marcel Golay (1964). The weave: David names the-broadcast; I compute the least-squares kernel for a window and degree, and confirm it reproduces any polynomial of that degree exactly — the mark of a filter that smooths noise without distorting signal.
Credit as content: Abraham Savitzky & Marcel Golay (1964). The weave: David names the-broadcast; I compute the least-squares kernel for a window and degree, and confirm it reproduces any polynomial of that degree exactly — the mark of a filter that smooths noise without distorting signal.
3 ONE DIMENSION
Slide a window; fit a parabola to its points by least squares; keep the parabola’s value at the center. A moving average would flatten a peak; a fitted polynomial follows its curve.
4 TWO DIMENSIONS · INTERACTIVE
A noisy signal with a peak; Savitzky–Golay vs a moving average, and the polynomial-reproduction check.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: smoothing that preserves the shape.
AVAN’s addition (the inverse-companion): smooth noise without crushing peaks by fitting a local polynomial in each window rather than averaging — so any true polynomial trend passes through untouched. The inverse of ‘average the window (and flatten the peaks)’ is ‘least-squares-fit a polynomial — noise falls, shape stays.’ Magenta is the peak a moving average would flatten; green is the shape-preserving fit. Smoothing that keeps the curves.
LIT Genuine Savitzky–Golay filter (Savitzky & Golay 1964). Verified live: the windowed least-squares kernel (from the normal equations) reproduces every polynomial of degree ≤ order exactly over 200 random cases (window.__savitzkygolay.preservesPolynomial, worst ~1e-12) — the defining property of a filter that smooths noise without distorting signal.
FIG No framing: the least-squares kernel derivation and the polynomial-reproduction check run in-browser and agree to floating precision. The AVAN inverse is honest — fitting a local polynomial in each window rather than averaging genuinely preserves peaks a moving average would flatten; magenta is the crushed peak, green the shape-preserving fit. Smoothing that keeps the curves.
FIG No framing: the least-squares kernel derivation and the polynomial-reproduction check run in-browser and agree to floating precision. The AVAN inverse is honest — fitting a local polynomial in each window rather than averaging genuinely preserves peaks a moving average would flatten; magenta is the crushed peak, green the shape-preserving fit. Smoothing that keeps the curves.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BROADCAST · David Lee Wise (ROOT0), with AVAN