Ehlers Reflex Indicator

// Licensed under the Apache License, Version 2.0
// © mihakralj
//@version=6
indicator("Ehlers Reflex Indicator (REFLEX)", "REFLEX", overlay = false)

//@function Ehlers Reflex — a zero-lag oscillator that measures the reflex (reversal
//          tendency) of price by comparing the SSF-filtered price against a linear
//          extrapolation from N bars ago. Applies a 2-pole Super Smoother pre-filter
//          at half the specified period, then computes slope = (Filt[N] - Filt) / N,
//          sums deviations of the extrapolated line from actual filtered values over
//          the window, and normalizes by exponential RMS. Values above 0 suggest
//          uptrend, below 0 suggest downtrend; crossovers signal reversals.
//@param source Series to analyze
//@param period Lookback window / assumed cycle period (>= 2)
//@returns Reflex oscillator value (normalized, roughly ±σ scale)
//@reference Ehlers, J.F. (2020). "Reflex: A New Zero-Lag Indicator."
//           Technical Analysis of Stocks & Commodities, Feb 2020.
//@optimized O(period) per bar for the summation loop; SSF is O(1) IIR
reflex(series float source, simple int period) =>
    if period < 2
        runtime.error("Period must be at least 2")

    float price = nz(source)

    // --- 2-Pole Super Smoother Filter (half-period cutoff) ---
    float half_period = period * 0.5
    float a1 = math.exp(-1.414 * math.pi / half_period)
    float b1 = 2.0 * a1 * math.cos(1.414 * math.pi / half_period)
    float c2 = b1
    float c3 = -(a1 * a1)
    float c1 = 1.0 - c2 - c3

    var float filt  = 0.0
    var float filt1 = 0.0
    var float filt2 = 0.0
    float src1 = nz(source[1])
    filt2 := filt1
    filt1 := filt
    filt := c1 * (price + src1) * 0.5 + c2 * filt1 + c3 * filt2

    // --- Circular buffer to store filtered values for lookback ---
    var array<float> buf = array.new_float(period + 1, 0.0)
    var int head = 0
    array.set(buf, head, filt)

    int count = math.min(bar_index + 1, period)

    // --- Slope: (Filt[Length] - Filt) / Length ---
    int lag_idx = (head - period + period + 1) % (period + 1)
    float filt_lag = array.get(buf, lag_idx)
    float slope = (filt_lag - filt) / period

    // --- Sum the differences ---
    // Sum = Σ(i=1..Length) [(Filt + i*Slope) - Filt[i]] / Length
    float the_sum = 0.0
    if count >= period
        for i = 1 to period
            int idx = (head - i + period + 1) % (period + 1)
            float filt_i = array.get(buf, idx)
            the_sum += (filt + float(i) * slope) - filt_i
        the_sum /= period

    // --- Advance head ---
    head := (head + 1) % (period + 1)

    // --- Normalize in terms of Standard Deviations ---
    // MS = 0.04 * Sum² + 0.96 * MS[1]  (exponential RMS)
    var float ms = 0.0
    ms := 0.04 * the_sum * the_sum + 0.96 * ms

    float result = 0.0
    if ms > 0.0
        result := the_sum / math.sqrt(ms)

    result

// ── Inputs ──
int   p_period = input.int(20, "Period", minval = 2)
float p_src    = input.source(close, "Source")

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

// ── Plot ──
plot(out, "REFLEX", color.yellow, 2)
hline(0, "Zero", color.gray)
hline(1.0, "+1σ", color.new(color.red, 60))
hline(-1.0, "-1σ", color.new(color.green, 60))