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Short MA minus long MA swings tenfold between calm and chaos. Divide by ATR times the square root of the bar-distance between the two averages' centers, lag the long one, compress the tails, and the trend gap finally holds still.
Model price as a local sine and its velocity and acceleration are exact derivatives, with no differencing lag. Acceleration flags a swing running out of gas early, but rides on a fragile sine fit.
The raw stochastic lurches every time an old high or low drops out of its window, noise from a denominator that keeps redefining itself. Smooth it once or twice, center at 50, and the same trick fixes the StochRSI.
Model price as one local sine and solve its frequency from four bars: difference out the offset, ratio out the amplitude, read cos(omega). Low lag, high variance, and it lies when no cycle exists.
A two-bar RSI screams on every shock. Regress it on a slow 20-bar RSI, inverse-logistic its U-shaped distribution first, and trade the residual: how far price strayed from where the trend says it belongs.
A wavelet is a band-pass whose width scales, so one Mexican Hat (the 2nd derivative of a Gaussian) scans every cycle at once. Compact support makes it usable; the resolution tradeoff makes it honest.
Estimate slope from a 4-point cubic with skipped taps and the modified EMA turns faster and lags less. But a derivative is no low-pass: the output is not smooth, and that is the real trade.
The zero-lag EMA borrows the Kalman predict step: smooth Price + kappa*velocity, so the slope cancels lag in trends. The same guess overshoots reversals, so it is no turn detector.
A fixed EMA smooths the same in calm and chaos. The adaptive EMA ties alpha to cycle speed: track tight when clean, smooth hard when noisy, at most one bar lag, if you can estimate omega.
The sinc is the exact brick-wall low-pass: full pass below the cutoff, nothing above. But it runs infinitely long and needs future bars, so you build only a truncated, windowed version that rings.
The Butterworth is the maximally flat low-pass: slow cycles pass with no ripple, each pole cuts noise harder. The bill is lag, about one EMA's worth per pole, so two poles is the sweet spot.
Write any linear indicator as a transfer function H(z), a ratio of two short polynomials, and its full behavior reads out: gain at every cycle, lag in bars, and the poles that make it ring.