Ehlers Recursive Median Filter (RMED)

// This Pine Script™ code is subject to the terms of the Mozilla Public License 2.0
// https://mozilla.org/MPL/2.0/
// © QuanTAlib

//@version=6
indicator("Ehlers Recursive Median Filter (RMED)", "RMED", overlay = true)

//@function Ehlers Recursive Median Filter — a nonlinear IIR filter that applies
//          exponential smoothing to a 5-bar running median. The median rejects
//          impulsive spike noise that linear filters (SMA, EMA) cannot handle,
//          while the EMA provides smooth recursive tracking. Alpha is derived from
//          the Ehlers cycle-period formula: α = (cos(2π/P) + sin(2π/P) - 1) / cos(2π/P).
//          The combination produces a filter that is both spike-resistant and smooth,
//          with less lag than a standard median filter of equivalent smoothness.
//@param source Series to filter
//@param period Cycle period for EMA constant derivation (>= 1)
//@returns Recursive median filtered value
//@reference Ehlers, J.F. (2018). "Recursive Median Filters."
//           Technical Analysis of Stocks & Commodities, Mar 2018.
//@optimized O(1) per bar — 5-element sort network + EMA update
rmed(series float source, simple int period) =>
    if period < 1
        runtime.error("Period must be at least 1")

    float price = nz(source)

    // --- Ehlers EMA constant from cycle period ---
    // alpha = (cos(2π/P) + sin(2π/P) - 1) / cos(2π/P)
    float angle = 360.0 / period
    float cos_a = math.cos(angle * math.pi / 180.0)
    float sin_a = math.sin(angle * math.pi / 180.0)
    float alpha = (cos_a + sin_a - 1.0) / cos_a

    // Clamp alpha to valid range
    alpha := math.max(0.0, math.min(1.0, alpha))

    // --- 5-bar median via circular buffer ---
    var array<float> buf = array.new_float(5, 0.0)
    var int head = 0
    array.set(buf, head, price)
    head := (head + 1) % 5

    // Copy to temp array for median extraction
    var array<float> temp = array.new_float(5, 0.0)
    for i = 0 to 4
        array.set(temp, i, array.get(buf, i))
    array.sort(temp)
    float med5 = array.get(temp, 2)  // middle element of sorted 5

    // --- Recursive filter: EMA of median ---
    // RM = alpha * Median5 + (1 - alpha) * RM[1]
    var float rm = 0.0
    if bar_index == 0
        rm := price
    else
        rm := alpha * med5 + (1.0 - alpha) * rm

    rm

// ── Inputs ──
int   p_period = input.int(12, "Period", minval = 1)
float p_src    = input.source(close, "Source")

// ── Calculation ──
float out = rmed(p_src, p_period)

// ── Plot ──
plot(out, "RMED", color.yellow, 2)