// Licensed under the Apache License, Version 2.0
// © mihakralj

//@version=6
indicator("NMA - Natural Moving Average", "NMA", overlay=true)

// ── Functions ──────────────────────────────────────────────────────────
// @function Calculates the Natural Moving Average (Jim Sloman, Ocean Theory pgs 63-70).
//   Adaptive IIR filter where the smoothing ratio derives from volatility-weighted
//   sqrt-kernel analysis of log-price movements over a lookback window.
//   Step 1: ln = log(src) × 1000 (scaled natural log)
//   Step 2: For i=0..period-1, accumulate:
//     oi  = |ln[i] - ln[i+1]|           (bar-to-bar log-price volatility)
//     num += oi × (√(i+1) - √i)         (sqrt-differenced weight emphasizes recent bars)
//     denom += oi                        (total volatility normalization)
//   Step 3: ratio = num / denom          (adaptive smoothing factor ∈ [0,1])
//   Step 4: nma = nma[1] + ratio × (src - nma[1])  (IIR adaptive EMA step)
//   When volatility concentrates in recent bars → ratio ≈ 1 → fast tracking.
//   When volatility is spread uniformly → ratio ≈ 1/√period → heavy smoothing.
// @param source Series to smooth
// @param period Lookback window for volatility analysis (must be > 0)
// @returns NMA value
nma(series float source, simple int period) =>
    float src = nz(source)

    // Step 1: scaled natural log of price
    // Use circular buffer to store log-scaled values for lookback
    var array<float> lnBuf = array.new_float(period + 1, 0.0)
    var int head = 0

    float lnVal = src > 0 ? math.log(src) * 1000.0 : 0.0
    array.set(lnBuf, head, lnVal)

    // Step 2: compute volatility-weighted sqrt ratio over lookback
    float num = 0.0
    float denom = 0.0
    int bars = math.min(bar_index + 1, period)

    for i = 0 to bars - 1
        int idx0 = (head - i + period + 1) % (period + 1)
        int idx1 = (head - i - 1 + period + 1) % (period + 1)
        float oi = math.abs(array.get(lnBuf, idx0) - array.get(lnBuf, idx1))
        num += oi * (math.sqrt(i + 1) - math.sqrt(i))
        denom += oi

    // Advance head for next bar
    head := (head + 1) % (period + 1)

    // Step 3: adaptive ratio
    float ratio = denom != 0.0 ? num / denom : 0.0

    // Step 4: IIR adaptive EMA step
    var float result = 0.0
    if bar_index == 0
        result := src
    else
        result := result + ratio * (src - result)

    result

// ── Inputs ─────────────────────────────────────────────────────────────
int i_period  = input.int(40, "Period", minval=1)
float i_source = input.source(close, "Source")

// ── Calculation ────────────────────────────────────────────────────────
float value = nma(i_source, i_period)

// ── Plot ───────────────────────────────────────────────────────────────
plot(value, "NMA", color.yellow, 2)