// Licensed under the Apache License, Version 2.0
// © mihakralj
//@version=6
indicator("Leader EMA (LEMA)", "LEMA", overlay=true)

//@function Computes Leader EMA — Siligardos' leading exponential moving average.
//          Adds a smoothed error correction to the standard EMA, making it respond
//          faster than EMA while maintaining smoothness.
//          Formula: Leader = EMA(source, N) + EMA(source - EMA(source, N), N)
//          The second term smooths the estimation error and adds it back, creating
//          a "leading" effect that anticipates price movement.
//@param source Series to smooth
//@param period Lookback period (determines alpha = 2/(period+1))
//@returns Leader EMA value from first bar (with warmup compensation)
//@reference Siligardos, G.E. (2008). "Leader of the MACD." Technical Analysis of
//           Stocks & Commodities, 26(7), 30-37.
//@optimized O(1) per bar — two IIR state variables with warmup compensation
lema(series float source, simple int period) =>
    float price = nz(source)
    float alpha = 2.0 / (period + 1)
    float beta = 1.0 - alpha

    // --- EMA1: standard EMA of source ---
    var float ema1 = 0.0
    var float e1 = 1.0
    var bool warmup1 = true

    ema1 := alpha * (price - ema1) + ema1
    float comp_ema1 = ema1
    if warmup1
        e1 *= beta
        comp_ema1 := ema1 / (1.0 - e1)
        warmup1 := e1 > 1e-10

    // --- Error: source - EMA(source) ---
    float error = price - comp_ema1

    // --- EMA2: EMA of the error series ---
    var float ema2 = 0.0
    var float e2 = 1.0
    var bool warmup2 = true

    ema2 := alpha * (error - ema2) + ema2
    float comp_ema2 = ema2
    if warmup2
        e2 *= beta
        comp_ema2 := ema2 / (1.0 - e2)
        warmup2 := e2 > 1e-10

    // --- Leader EMA = EMA(source) + EMA(error) ---
    comp_ema1 + comp_ema2

// ── Inputs ──────────────────────────────────────────────────────────────
src = input.source(close, "Source")
per = input.int(14, "Period", minval=1)

// ── Plot ────────────────────────────────────────────────────────────────
plot(lema(src, per), "LEMA", color.new(color.yellow, 0), 2)