Miha Kralj
90d5638008
- LTMA (Linear Trend Moving Average): Introduces a predictive moving average using dual cascaded EMAs for trend estimation. - MCNMA (McNicholl EMA): Implements a zero-lag TEMA using a cascaded EMA structure for enhanced responsiveness. - NLMA (Non-Lag Moving Average): Utilizes a damped cosine kernel to achieve reduced lag in moving averages. - NMA (Natural Moving Average): Adapts smoothing based on volatility profiles using a square-root kernel. - NYQMA (Nyquist Moving Average): Applies the Nyquist-Shannon theorem to prevent aliasing in cascaded moving averages. - RAIN (Rainbow Moving Average): Combines multiple SMA layers with weighted averages for multi-scale smoothing. - TRAMA (Trend Regularity Adaptive Moving Average): Adapts smoothing based on the frequency of new highs and lows in price data.
4.2 KiB
NMA: Natural Moving Average
"Jim Sloman looked at how volatility distributes across a window and asked: if the most volatile bars are recent, should the filter not respond faster? NMA derives its smoothing constant from the volatility profile itself, weighted by a square-root kernel that emphasizes recent action."
NMA is an adaptive IIR filter whose smoothing ratio is derived from a volatility-weighted square-root kernel analysis of log-price movements over a lookback window. When volatility concentrates in recent bars, the ratio approaches 1.0 (fast tracking). When volatility is spread uniformly, the ratio approaches 1/\sqrt{N} (heavy smoothing). The square-root kernel (\sqrt{i+1} - \sqrt{i}) gives a concave-down weighting that gently emphasizes recency, while the log-price transformation normalizes for price level, making the adaptation scale-invariant.
Historical Context
Jim Sloman introduced the Natural Moving Average in Ocean Theory (pages 63-70), a book that applied chaos and complexity theory metaphors to financial markets. The NMA was designed as a "natural" filter that lets the market's own volatility structure determine the smoothing rate, rather than imposing an arbitrary period.
The core innovation is the square-root differencing kernel \sqrt{i+1} - \sqrt{i} as the weighting function for volatility. This kernel has the property that its cumulative sum \sqrt{N} grows sublinearly, meaning each additional bar in the lookback contributes less weight than the previous one. This creates a "diminishing returns" effect: extending the lookback adds context without drowning out recent information.
The log-price transformation (\ln(\text{price}) \times 1000) serves two purposes: (1) it makes the volatility measure proportional to percentage moves rather than absolute dollar moves, and (2) the scaling factor of 1000 brings typical values into a numerically convenient range for the ratio computation.
NMA belongs to the family of adaptive moving averages alongside KAMA, VIDYA, and ADXVMA, but uses a unique adaptation mechanism based on the spatial distribution of volatility rather than a single efficiency or strength metric.
Architecture & Physics
1. Log-Price Buffer
A circular buffer of size N+1 stores \ln(\text{price}) \times 1000 for each bar, providing the lookback data for volatility computation.
2. Volatility-Weighted Square-Root Ratio
For each bar i in the lookback:
o_i = |\ln_i - \ln_{i+1}|
\text{num} = \sum_{i=0}^{N-1} o_i \cdot \left(\sqrt{i+1} - \sqrt{i}\right)
\text{denom} = \sum_{i=0}^{N-1} o_i
\text{ratio} = \frac{\text{num}}{\text{denom}}
3. Adaptive EMA Step
\text{NMA}_t = \text{NMA}_{t-1} + \text{ratio} \times (x_t - \text{NMA}_{t-1})
Mathematical Foundation
Log-price volatility:
o_i = \left|\ln(x_{t-i}) - \ln(x_{t-i-1})\right| \times 1000
Square-root kernel weights:
\phi_i = \sqrt{i+1} - \sqrt{i} = \frac{1}{\sqrt{i+1} + \sqrt{i}}
Note: \phi_i \approx \frac{1}{2\sqrt{i}} for large i, confirming the 1/\sqrt{i} decay rate.
Adaptive ratio:
r = \frac{\sum_{i=0}^{N-1} o_i \cdot \phi_i}{\sum_{i=0}^{N-1} o_i}
Ratio bounds:
- If all volatility is at
i = 0(most recent):r = \phi_0 = \sqrt{1} - \sqrt{0} = 1 - If volatility is uniform:
r = \frac{\sum \phi_i}{N} = \frac{\sqrt{N}}{N} = \frac{1}{\sqrt{N}} - For
N = 40: uniform ratio\approx 0.158, equivalent to EMA period\approx 11
IIR update:
\text{NMA}_t = \text{NMA}_{t-1} + r_t \cdot (x_t - \text{NMA}_{t-1})
Default parameters: period = 40, minPeriod = 1.
Pseudo-code (streaming):
// Store scaled log-price
lnBuf[head] = log(src) * 1000
// Compute volatility-weighted ratio
num = 0; denom = 0
for i = 0 to bars-1:
oi = |lnBuf[t-i] - lnBuf[t-i-1]|
num += oi * (sqrt(i+1) - sqrt(i))
denom += oi
ratio = denom != 0 ? num/denom : 0
// Adaptive EMA step
result = result + ratio * (src - result)
Resources
- Sloman, J. Ocean Theory. Pages 63-70. (Original NMA description.)
- Kaufman, P.J. (2013). Trading Systems and Methods, 5th ed. Wiley. Chapter 7: Adaptive Moving Averages.
- Chande, T.S. & Kroll, S. (1994). The New Technical Trader. Wiley. (Adaptive filter framework.)