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.
4.4 KiB
SSFDSP: Ehlers SSF Detrended Synthetic Price
SSFDSP isolates the dominant cycle by subtracting a half-cycle Super-Smoother from a quarter-cycle Super-Smoother, producing a zero-centered oscillator with superior noise rejection compared to the EMA-based DSP. The 2-pole Butterworth characteristic of the Super-Smoother filter provides zero phase lag at the cutoff frequency and sharper rolloff than exponential smoothing, making SSFDSP the preferred variant for cycle-aware trading when the approximate dominant period is known.
Historical Context
John Ehlers introduced the concept of Detrended Synthetic Price in Cybernetic Analysis for Stocks and Futures (2004) as a principled method for removing the DC (trend) component while preserving cyclical energy. The original DSP used EMAs, which have a gradual frequency rolloff and non-zero phase lag. The SSF variant substitutes Super-Smoother filters, which are 2-pole Butterworth low-pass designs with matched coefficients that eliminate the Gibbs phenomenon (ringing) common in sharper filters. The result is a cleaner cycle extraction: the SSF's steeper rolloff better separates the quarter-cycle and half-cycle frequency bands, producing tighter zero crossings and more reliable turning point identification than EMA-DSP.
Architecture & Physics
1. Filter Periods
From the user-specified dominant cycle period P:
P_{fast} = \max(2, \lfloor P / 4 + 0.5 \rfloor)
P_{slow} = \max(3, \lfloor P / 2 + 0.5 \rfloor)
2. Super-Smoother Coefficients
For each filter period p:
\alpha = \frac{\pi\sqrt{2}}{p}
c_2 = 2 e^{-\alpha} \cos(\alpha)
c_3 = -e^{-2\alpha}
c_1 = 1 - c_2 - c_3
3. SSF Recursion
SSF_t = c_1 \cdot \frac{P_t + P_{t-1}}{2} + c_2 \cdot SSF_{t-1} + c_3 \cdot SSF_{t-2}
The 2-bar input averaging provides an additional anti-aliasing stage.
4. SSFDSP Output
SSFDSP_t = SSF_{fast,t} - SSF_{slow,t}
5. Complexity
O(1) per bar. Two independent 2-pole IIR filters with O(1) memory. Warmup: approximately 2 \times P_{slow} for convergence. Recursive dependencies prevent SIMD vectorization.
Mathematical Foundation
Parameters
| Parameter | Description | Default | Constraint |
|---|---|---|---|
period |
Expected dominant cycle period | 40 | \geq 4 |
Super-Smoother Frequency Response
The SSF has -3 dB attenuation at the cutoff period, -12 dB/octave rolloff (2-pole), and zero phase lag at the cutoff. This is equivalent to a critically-damped Butterworth filter.
Pseudo-code
function SSFDSP(source, period):
pFast ← max(2, round(period / 4))
pSlow ← max(3, round(period / 2))
// Fast SSF coefficients
αf ← √2·π / pFast
c2f ← 2·exp(-αf)·cos(αf)
c3f ← -exp(-2·αf)
c1f ← 1 - c2f - c3f
// Slow SSF coefficients
αs ← √2·π / pSlow
c2s ← 2·exp(-αs)·cos(αs)
c3s ← -exp(-2·αs)
c1s ← 1 - c2s - c3s
ssfFast_1 ← 0; ssfFast_2 ← 0
ssfSlow_1 ← 0; ssfSlow_2 ← 0
p_prev ← 0
for each price in source:
// Input averaging
avg ← (price + p_prev) / 2
// Fast SSF update
ssfFast ← c1f·avg + c2f·ssfFast_1 + c3f·ssfFast_2
// Slow SSF update
ssfSlow ← c1s·avg + c2s·ssfSlow_1 + c3s·ssfSlow_2
// SSFDSP
ssfdsp ← ssfFast - ssfSlow
// Shift state
ssfFast_2 ← ssfFast_1; ssfFast_1 ← ssfFast
ssfSlow_2 ← ssfSlow_1; ssfSlow_1 ← ssfSlow
p_prev ← price
emit ssfdsp
DSP vs SSFDSP
| Aspect | DSP (EMA-based) | SSFDSP (Super-Smoother) |
|---|---|---|
| Filter type | 1-pole IIR (exponential) | 2-pole Butterworth |
| Rolloff | -6 dB/octave |
-12 dB/octave |
| Phase lag at cutoff | Non-zero | Zero |
| Noise rejection | Moderate | Superior |
| Turning points | Rounded | Sharper |
Output Interpretation
| Condition | Meaning |
|---|---|
SSFDSP > 0 |
Bullish cycle phase |
SSFDSP < 0 |
Bearish cycle phase |
| Zero crossing | Cycle phase transition |
| Divergence with price | Cycle energy waning; trend exhaustion |
| Amplitude shrinking | Cycle losing dominance; transition to trend |
Resources
- Ehlers, J.F. Cybernetic Analysis for Stocks and Futures. Wiley, 2004.
- Ehlers, J.F. Cycle Analytics for Traders. Wiley, 2013.
- Butterworth, S. "On the Theory of Filter Amplifiers." Experimental Wireless, 7, 1930.