Ehlers Reverse EMA

// Licensed under the Apache License, Version 2.0
// © mihakralj
// Indicator algorithm (C) 2017 John F. Ehlers
//@version=6
indicator("Ehlers Reverse EMA (REVERSEEMA)", "REVERSEEMA", overlay=false)

//@function Calculates Ehlers Reverse EMA using Z-transform inversion of EMA smoothing
//@param source Series to calculate Reverse EMA from
//@param period Lookback period for the base EMA (>= 1)
//@returns Reverse EMA value with lag removed via 8-stage cascaded inversion
//@optimized Uses warmup-compensated EMA with O(1) cascaded reverse stages per bar
reverseema(series float source, simple int period) =>
    if period <= 0
        runtime.error("Period must be positive")

    float src = na(source) ? 0.0 : source

    // EMA smoothing factor from period: alpha = 2/(period+1)
    float a = 2.0 / (period + 1)
    float cc = 1.0 - a
    float beta = cc

    // Precompute powers of cc for the 8 reverse stages
    // Stage k uses cc^(2^(k-1)): 1, 2, 4, 8, 16, 32, 64, 128
    float cc2   = cc * cc
    float cc4   = cc2 * cc2
    float cc8   = cc4 * cc4
    float cc16  = cc8 * cc8
    float cc32  = cc16 * cc16
    float cc64  = cc32 * cc32
    float cc128 = cc64 * cc64

    // --- Forward EMA with warmup compensator ---
    var bool warmup = true
    var float e = 1.0
    var float ema_raw = 0.0
    var float ema_val = 0.0

    ema_raw := a * (src - ema_raw) + ema_raw
    if warmup
        e *= beta
        float comp = 1.0 / (1.0 - e)
        ema_val := comp * ema_raw
        warmup := e > 1e-10
    else
        ema_val := ema_raw

    // --- 8-stage cascaded reverse EMA ---
    // Each stage: RE_k[n] = cc^(2^(k-1)) * RE_{k-1}[n] + RE_{k-1}[n-1]
    // RE1 uses EMA as input: RE1[n] = cc * EMA[n] + EMA[n-1]
    var float re1 = 0.0
    var float re2 = 0.0
    var float re3 = 0.0
    var float re4 = 0.0
    var float re5 = 0.0
    var float re6 = 0.0
    var float re7 = 0.0
    var float re8 = 0.0

    float prev_ema = nz(ema_val[1])
    float prev_re1 = nz(re1[1])
    float prev_re2 = nz(re2[1])
    float prev_re3 = nz(re3[1])
    float prev_re4 = nz(re4[1])
    float prev_re5 = nz(re5[1])
    float prev_re6 = nz(re6[1])
    float prev_re7 = nz(re7[1])

    re1 := cc    * ema_val + prev_ema
    re2 := cc2   * re1     + prev_re1
    re3 := cc4   * re2     + prev_re2
    re4 := cc8   * re3     + prev_re3
    re5 := cc16  * re4     + prev_re4
    re6 := cc32  * re5     + prev_re5
    re7 := cc64  * re6     + prev_re6
    re8 := cc128 * re7     + prev_re7

    // Signal = EMA - alpha * RE8
    float signal = ema_val - a * re8
    na(source) ? na : signal

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

// Inputs
i_period = input.int(20, "Period", minval=1, tooltip="Lookback period for the base EMA")
i_source = input.source(close, "Source")

// Calculation
reverseema_value = reverseema(i_source, i_period)

// Plot
plot(reverseema_value, "REVERSEEMA", color=color.yellow, linewidth=2)
hline(0, "Zero", color=color.gray)