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.
5.6 KiB
SAM: Smoothed Adaptive Momentum
The Smoothed Adaptive Momentum oscillator measures price momentum over an adaptively determined lookback period equal to the dominant cycle length, then smooths the result with a 2-pole Super Smoother filter. Unlike fixed-period momentum indicators (ROC, TRIX) that use an arbitrary lookback, SAM measures the dominant cycle via Ehlers' Homodyne Discriminator and uses that cycle length as the momentum window, ensuring that the momentum measurement always spans exactly one full cycle. This eliminates the half-cycle phase distortion that plagues fixed-period momentum, producing a zero-lag momentum oscillator that naturally adapts to changing market rhythm.
Historical Context
SAM was developed by John F. Ehlers and presented in Chapter 12 ("Adapting to the Trend") of "Cybernetic Analysis for Stocks and Futures" (2004). It represents the synthesis of two Ehlers innovations: the Homodyne Discriminator for cycle measurement and the Super Smoother for noise reduction.
The key insight is that momentum measured over exactly one dominant cycle period produces a nearly zero-mean oscillator with minimal spectral leakage. If the dominant cycle is 20 bars, then close - close[20] captures one full oscillation, with the difference being zero at the start and end of the cycle (when phase completes 360°). A fixed 14-bar momentum, by contrast, may measure a fractional cycle, producing a biased oscillator with a non-zero mean that varies as the cycle length drifts.
The Homodyne Discriminator (named after the homodyne detection technique from radio engineering) measures the instantaneous frequency by correlating the analytic signal with a one-bar-delayed version of itself. The phase change between bars gives the instantaneous frequency, which is smoothed and clamped to produce a stable period estimate in the 6-50 bar range.
The Super Smoother is Ehlers' preferred final-stage filter: a 2-pole IIR low-pass with better amplitude response than a Butterworth of the same order, specifically designed to minimize lag while suppressing high-frequency noise.
Architecture and Physics
The pipeline has five stages:
Stage 1: 4-bar FIR smoother applies a [1, 2, 2, 1]/6 weighted average to eliminate 2-bar and 3-bar cycle noise. This is a standard Ehlers preprocessing step that removes aliasing artifacts without introducing significant lag.
Stage 2: Hilbert Transform extracts the analytic signal from the smoothed price using Ehlers' modified Hilbert Transform. The detrender and quadrature components I_1 and Q_1 are derived via 7-tap FIR filters with empirically chosen coefficients that approximate the Hilbert Transform over financial cycle frequencies.
Stage 3: Phase advance applies the Hilbert Transform to I_1 and Q_1 themselves, producing JI and JQ (90° phase-advanced versions). The phasor addition I_2 = I_1 - JQ and Q_2 = Q_1 + JI creates the forward-rotated analytic signal needed for homodyne detection.
Stage 4: Homodyne Discriminator correlates the current phasor (I_2, Q_2) with the previous bar's phasor to extract the instantaneous frequency:
\text{Re} = I_2 \cdot I_2[1] + Q_2 \cdot Q_2[1], \quad \text{Im} = I_2 \cdot Q_2[1] - Q_2 \cdot I_2[1]
The period is 2\pi / \arctan(\text{Im}/\text{Re}), clamped to [6, 50] and smoothed via two cascaded EMA stages to produce the dominant cycle period.
Stage 5: Adaptive momentum + Super Smoother computes source - source[dcPeriod] where dcPeriod is the rounded dominant cycle, then applies a 2-pole Super Smoother with the user-specified cutoff period.
Mathematical Foundation
4-bar FIR smoother:
s[n] = \frac{x[n] + 2x[n-1] + 2x[n-2] + x[n-3]}{6}
Ehlers Hilbert Transform (7-tap approximation):
H[n] = 0.0962\,s[n] + 0.5769\,s[n-2] - 0.5769\,s[n-4] - 0.0962\,s[n-6]
scaled by the adaptive gain factor (0.075\,P[n-1] + 0.54).
Homodyne Discriminator:
\text{Re}[n] = I_2[n] \cdot I_2[n-1] + Q_2[n] \cdot Q_2[n-1]
\text{Im}[n] = I_2[n] \cdot Q_2[n-1] - Q_2[n] \cdot I_2[n-1]
P = \frac{2\pi}{\arctan(\text{Im}/\text{Re})}, \quad P \in [6, 50]
Dominant Cycle Period (double-smoothed):
P_{\text{inst}} = 0.33 \cdot P + 0.67 \cdot P_{\text{inst}}[1]
P_{\text{DC}} = 0.15 \cdot P_{\text{inst}} + 0.85 \cdot P_{\text{DC}}[1]
Adaptive momentum: M[n] = x[n] - x[n - \lfloor P_{\text{DC}} \rfloor]
2-pole Super Smoother with cutoff C:
a_1 = e^{-\sqrt{2}\pi/C}, \quad b_1 = 2a_1\cos(\sqrt{2}\pi/C)
\text{filt}[n] = \frac{1 - b_1 + a_1^2}{2}(M[n] + M[n-1]) + b_1\,\text{filt}[n-1] - a_1^2\,\text{filt}[n-2]
Parameter constraints: \alpha \in (0, 1), cutoff \ge 2.
SAM(source, alpha, cutoff):
smooth = (src + 2*src[1] + 2*src[2] + src[3]) / 6
// Hilbert Transform -> I1, Q1
// Phase advance -> JI, JQ
// Phasor addition -> I2, Q2
I2 = I1 - JQ; Q2 = Q1 + JI
I2 = alpha*I2 + (1-alpha)*I2[1] // smooth
Q2 = alpha*Q2 + (1-alpha)*Q2[1]
// Homodyne Discriminator
Re = I2*I2[1] + Q2*Q2[1]
Im = I2*Q2[1] - Q2*I2[1]
period = 2*pi / atan(Im/Re), clamped [6,50]
dcPeriod = double_smooth(period)
// Adaptive momentum + Super Smoother
momentum = source - source[dcPeriod]
return superSmoother(momentum, cutoff)
Resources
- Ehlers, J.F. "Cybernetic Analysis for Stocks and Futures." Wiley, 2004. Chapter 12, p.166.
- Ehlers, J.F. "Rocket Science for Traders." Wiley, 2001. Chapters on Hilbert Transform and cycle measurement.
- Ehlers, J.F. "MESA and Trading Market Cycles." 2nd edition, Wiley, 2002.
- Oppenheim, A.V. & Schafer, R.W. "Discrete-Time Signal Processing." 3rd edition, Pearson, 2010. Chapter on Hilbert Transform.