QuanTAlib/lib/trends_IIR/qema/qema.pine

// Licensed under the Apache License, Version 2.0
// © mihakralj
//@version=6
indicator("QEMA (OptA, progressive α, period-only)", "QEMA OptA", overlay=true)

//@function Calculates Quad EMA with progressive alphas and Option A zero-lag weights
//@param srcIn Series to calculate QEMA from
//@param period Lookback period for alpha calculation (>= 1)
//@returns QEMA value with minimized DC lag
//@optimized Uses 4-stage cascaded EMA with optimal weights, O(1) complexity per bar

// ---------- Inputs ----------
i_period = input.int(15, "Period", minval=1)
i_source = input.source(close, "Source")

// ---------- Helpers ----------
clamp01(x) =>
    math.min(1.0, math.max(x, 1e-12))

lag(alpha) =>
    (1.0 - alpha) / alpha

// ---------- QEMA: progressive alphas + Option A weights ----------
qema_optA(series float srcIn, int period) =>
    // --- dirty-data guard (NA only; Pine doesn't expose isfinite/isinf) ---
    var float lastFinite = na
    float src = srcIn
    if na(src)
        src := nz(lastFinite, src)
    else
        lastFinite := src

    // --- base alpha from period ---
    float a1 = clamp01(2.0 / (period + 1.0))

    // --- progressive alpha ramp (your design) ---
    // r = (1/a1)^(1/4) -> a2=a1^(3/4), a3=a1^(1/2), a4=a1^(1/4)
    float r  = math.pow(1.0 / a1, 0.25)
    float a2 = clamp01(a1 * r)
    float a3 = clamp01(a2 * r)
    float a4 = clamp01(a3 * r)

    float d1 = 1.0 - a1
    float d2 = 1.0 - a2
    float d3 = 1.0 - a3
    float d4 = 1.0 - a4

    // --- unbiased warmup (startup bias compensation) ---
    var float e1 = 1.0
    var float e2 = 1.0
    var float e3 = 1.0
    var float e4 = 1.0
    var bool warmup = true

    // --- EMA accumulators ---
    var float rema1 = 0.0
    var float rema2 = 0.0
    var float rema3 = 0.0
    var float rema4 = 0.0

    float ema1 = na
    float ema2 = na
    float ema3 = na
    float ema4 = na

    // Stage 1
    rema1 := rema1 + a1 * (src - rema1)

    if warmup
        e1 *= d1
        e2 *= d2
        e3 *= d3
        e4 *= d4

        float c1 = 1.0 / (1.0 - e1)
        float c2 = 1.0 / (1.0 - e2)
        float c3 = 1.0 / (1.0 - e3)
        float c4 = 1.0 / (1.0 - e4)

        ema1 := rema1 * c1

        rema2 := rema2 + a2 * (ema1 - rema2)
        ema2  := rema2 * c2

        rema3 := rema3 + a3 * (ema2 - rema3)
        ema3  := rema3 * c3

        rema4 := rema4 + a4 * (ema3 - rema4)
        ema4  := rema4 * c4

        // stage 1 is slowest => if it's unbiased, others are too
        warmup := e1 > 1e-10
    else
        ema1 := rema1
        rema2 := rema2 + a2 * (ema1 - rema2)
        ema2 := rema2
        rema3 := rema3 + a3 * (ema2 - rema3)
        ema3 := rema3
        rema4 := rema4 + a4 * (ema3 - rema4)
        ema4 := rema4

    // --- cumulative lags (mean-lag proxy) ---
    float t1 = lag(a1)
    float t2 = lag(a2)
    float t3 = lag(a3)
    float t4 = lag(a4)

    float L1 = t1
    float L2 = t1 + t2
    float L3 = L2 + t3
    float L4 = L3 + t4

    // --- Option A weights: min-energy with constraints ---
    // Σw = 1
    // Σw*L = δ, with δ=0 here (zero DC lag)
    float delta = 0.0

    float B = L1 + L2 + L3 + L4
    float C = L1*L1 + L2*L2 + L3*L3 + L4*L4
    float D = 4.0*C - B*B

    float w1 = na
    float w2 = na
    float w3 = na
    float w4 = na

    if math.abs(D) < 1e-12
        // degenerate (e.g. a1==1 -> all L=0): safest output is ema1==src
        w1 := 1.0
        w2 := 0.0
        w3 := 0.0
        w4 := 0.0
    else
        float lambda = (C - B*delta) / D
        float mu     = (-B + 4.0*delta) / D
        w1 := lambda + mu*L1
        w2 := lambda + mu*L2
        w3 := lambda + mu*L3
        w4 := lambda + mu*L4

    w1*ema1 + w2*ema2 + w3*ema3 + w4*ema4

// ---------- Main ----------
float q = qema_optA(i_source, i_period)
plot(q, "QEMA OptA", color.new(color.yellow, 0), 2)