Quantitative Qualitative Estimation (QQE)

// Licensed under the Apache License, Version 2.0
// © mihakralj
//@version=6
indicator("Quantitative Qualitative Estimation (QQE)", "QQE", overlay=false)

//@function Calculates QQE — smoothed RSI with dynamic volatility bands
//@param source Series to evaluate (typically close)
//@param rsiPeriod RSI lookback period
//@param smoothFactor EMA smoothing factor for RSI (SF)
//@param qqeFactor Multiplier for the ATR-like trailing band (Wilders factor)
//@returns [qqeLine, trailingLevel] QQE smoothed RSI and its trailing signal
//@description QQE applies a multi-stage smoothing pipeline to RSI:
//  Stage 1: Compute RSI using Wilder's RMA (alpha=1/rsiPeriod) with §2 warmup.
//  Stage 2: Smooth RSI with EMA(smoothFactor) → rsiMA. Also §2 warmup.
//  Stage 3: Compute |rsiMA − rsiMA[1]|, smooth with double EMA(2*SF-1)
//           → "dar" (dynamic average range). Both EMAs use §2 warmup.
//  Stage 4: Build trailing level: upper/lower bands at rsiMA ± qqeFactor*dar.
//           Trailing level follows price directionally (like Parabolic SAR logic):
//             if rsiMA > prevTrail and rsiMA[1] > prevTrail → trail = max(trail, lower)
//             if rsiMA < prevTrail and rsiMA[1] < prevTrail → trail = min(trail, upper)
//             else → trail flips to upper or lower based on rsiMA side.
//  The QQE line is rsiMA; the trailing level acts as signal for crossover triggers.
//  Crosses of QQE above/below trailing level indicate momentum shifts.
//  Zero-crossing of (QQE - 50) can also be used for trend direction.
qqe(series float source, simple int rsiPeriod, simple int smoothFactor, simple float qqeFactor) =>
    if rsiPeriod <= 0
        runtime.error("RSI period must be greater than 0")
    if smoothFactor <= 0
        runtime.error("Smooth factor must be greater than 0")
    if qqeFactor <= 0.0
        runtime.error("QQE factor must be greater than 0")

    float EPSILON = 1e-10

    // --- Stage 1: Wilder RSI via RMA (alpha = 1/rsiPeriod) with §2 warmup ---
    var float prevSrc   = na
    var float rmaGain   = 0.0
    var float rmaLoss   = 0.0
    var float eRma      = 1.0
    float rmaAlpha      = 1.0 / float(rsiPeriod)
    float rmaBeta       = 1.0 - rmaAlpha

    float chg = not na(prevSrc) ? nz(source) - prevSrc : 0.0
    prevSrc := nz(source)
    float gain = chg > 0 ? chg : 0.0
    float loss = chg < 0 ? -chg : 0.0

    rmaGain := rmaGain * rmaBeta + gain * rmaAlpha
    rmaLoss := rmaLoss * rmaBeta + loss * rmaAlpha
    eRma *= rmaBeta
    float cRma = eRma > EPSILON ? 1.0 / (1.0 - eRma) : 1.0
    float avgGain = rmaGain * cRma
    float avgLoss = rmaLoss * cRma
    float rs = avgLoss == 0.0 ? 100.0 : avgGain / avgLoss
    float rsiVal = 100.0 - 100.0 / (1.0 + rs)

    // --- Stage 2: EMA smooth of RSI (alpha = 2/(SF+1)) with §2 warmup ---
    var float rawRsiMa = 0.0
    var float eRsiMa   = 1.0
    float sfAlpha       = 2.0 / (float(smoothFactor) + 1.0)
    float sfBeta        = 1.0 - sfAlpha

    rawRsiMa := rawRsiMa * sfBeta + rsiVal * sfAlpha
    eRsiMa *= sfBeta
    float cRsiMa = eRsiMa > EPSILON ? 1.0 / (1.0 - eRsiMa) : 1.0
    float rsiMa = rawRsiMa * cRsiMa

    // --- Stage 3: Double EMA of |delta(rsiMA)| → dar ---
    //  Both EMAs use period = 2*SF - 1 → alpha = 2 / (2*SF)
    var float prevRsiMa = na
    float darPeriod     = float(2 * smoothFactor - 1)
    float darAlpha      = 2.0 / (darPeriod + 1.0)
    float darBeta       = 1.0 - darAlpha

    float absDelta = not na(prevRsiMa) ? math.abs(rsiMa - prevRsiMa) : 0.0
    prevRsiMa := rsiMa

    var float rawDar1 = 0.0
    var float eDar1   = 1.0
    rawDar1 := rawDar1 * darBeta + absDelta * darAlpha
    eDar1 *= darBeta
    float cDar1 = eDar1 > EPSILON ? 1.0 / (1.0 - eDar1) : 1.0
    float dar1 = rawDar1 * cDar1

    var float rawDar2 = 0.0
    var float eDar2   = 1.0
    rawDar2 := rawDar2 * darBeta + dar1 * darAlpha
    eDar2 *= darBeta
    float cDar2 = eDar2 > EPSILON ? 1.0 / (1.0 - eDar2) : 1.0
    float dar = rawDar2 * cDar2

    // --- Stage 4: Trailing level (directional flip logic) ---
    float band = qqeFactor * dar
    float upperBand = rsiMa + band
    float lowerBand = rsiMa - band

    var float trail = 0.0
    var float prevRsiMa2 = 50.0

    float newTrail = trail
    if rsiMa > trail and prevRsiMa2 > trail
        newTrail := math.max(trail, lowerBand)
    else if rsiMa < trail and prevRsiMa2 < trail
        newTrail := math.min(trail, upperBand)
    else
        newTrail := rsiMa > trail ? lowerBand : upperBand

    prevRsiMa2 := rsiMa
    trail := newTrail

    [rsiMa, trail]

// ---------- Main loop ----------

// Inputs
i_source       = input.source(close, "Source")
i_rsiPeriod    = input.int(14, "RSI Period", minval=1)
i_smoothFactor = input.int(5, "Smooth Factor", minval=1)
i_qqeFactor    = input.float(4.236, "QQE Factor", minval=0.001, step=0.001)

// Calculation
[qqeLine, trailLine] = qqe(i_source, i_rsiPeriod, i_smoothFactor, i_qqeFactor)

// Plot
plot(qqeLine, "QQE", color=color.yellow, linewidth=2)
plot(trailLine, "Trailing", color=color.aqua, linewidth=1)
hline(50, "Midline", color=color.gray, linestyle=hline.style_dotted)
hline(70, "Overbought", color=color.red, linestyle=hline.style_dashed)
hline(30, "Oversold", color=color.green, linestyle=hline.style_dashed)