Ehlers Elegant Oscillator (EEO)

// Licensed under the Apache License, Version 2.0
// © mihakralj
//@version=6
indicator("Ehlers Elegant Oscillator (EEO)", "EEO", overlay = false)

//@function Ehlers Elegant Oscillator — an Inverse Fisher Transform (IFT) applied to
//          RMS-normalized 2-bar momentum, smoothed by a Super Smoother filter.
//          Pipeline: Deriv = Close - Close[2] → RMS(50) → NDeriv = Deriv/RMS →
//          IFish = tanh(NDeriv) → SuperSmoother(BandEdge).
//          Output is bounded approximately [-1, +1].
//@param source Series to analyze
//@param bandEdge Super Smoother cutoff period (>= 2)
//@returns EEO oscillator value (bounded ±1)
//@reference Ehlers, J.F. (2022). "An Elegant Oscillator: Inverse Fisher Transform Redux."
//           Technical Analysis of Stocks & Commodities, Feb 2022.
//@optimized O(1) per bar via running sum circular buffer for RMS
eeo(series float source, simple int bandEdge) =>
    if bandEdge < 2
        runtime.error("BandEdge must be at least 2")

    float price = nz(source)

    // --- 2-bar momentum (derivative) ---
    float deriv = price - nz(source[2])

    // --- Rolling RMS of derivative over 50 bars ---
    var array<float> buf = array.new_float(50, 0.0)
    var int head = 0
    var float sum_sq = 0.0

    float deriv_sq = deriv * deriv
    float old_sq = array.get(buf, head)
    array.set(buf, head, deriv_sq)
    sum_sq := sum_sq - old_sq + deriv_sq
    head := (head + 1) % 50

    float rms = math.sqrt(math.max(sum_sq / 50.0, 1e-10))

    // --- Normalize derivative by RMS ---
    float nDeriv = rms > 1e-10 ? deriv / rms : 0.0

    // --- Inverse Fisher Transform (tanh) ---
    float e2n = math.exp(2.0 * nDeriv)
    float iFish = (e2n - 1.0) / (e2n + 1.0)

    // --- Super Smoother coefficients (2-pole Butterworth at BandEdge) ---
    float a1 = math.exp(-1.414 * math.pi / bandEdge)
    float b1 = 2.0 * a1 * math.cos(1.414 * 180.0 / bandEdge)
    float c2 = b1
    float c3 = -(a1 * a1)
    float c1 = 1.0 - c2 - c3

    // --- 2-pole Super Smoother filter ---
    var float ss  = 0.0
    var float ss1 = 0.0
    var float ss2 = 0.0
    float iFish1 = nz(iFish[1])
    ss2 := ss1
    ss1 := ss
    ss := c1 * 0.5 * (iFish + iFish1) + c2 * ss1 + c3 * ss2

    ss

// ── Inputs ──
int   p_bandEdge = input.int(20, "BandEdge", minval = 2)
float p_src      = input.source(close, "Source")

// ── Calculation ──
float out = eeo(p_src, p_bandEdge)

// ── Plot ──
plot(out, "EEO", color.yellow, 2)
hline(0, "Zero", color.gray)
hline(0.5, "+0.5", color.new(color.red, 60))
hline(-0.5, "-0.5", color.new(color.green, 60))