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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.
126 lines
4.4 KiB
Markdown
126 lines
4.4 KiB
Markdown
# SSFDSP: Ehlers SSF Detrended Synthetic Price
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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.
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## Historical Context
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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.
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## Architecture & Physics
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### 1. Filter Periods
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From the user-specified dominant cycle period $P$:
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$$P_{fast} = \max(2, \lfloor P / 4 + 0.5 \rfloor)$$
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$$P_{slow} = \max(3, \lfloor P / 2 + 0.5 \rfloor)$$
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### 2. Super-Smoother Coefficients
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For each filter period $p$:
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$$\alpha = \frac{\pi\sqrt{2}}{p}$$
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$$c_2 = 2 e^{-\alpha} \cos(\alpha)$$
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$$c_3 = -e^{-2\alpha}$$
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$$c_1 = 1 - c_2 - c_3$$
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### 3. SSF Recursion
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$$SSF_t = c_1 \cdot \frac{P_t + P_{t-1}}{2} + c_2 \cdot SSF_{t-1} + c_3 \cdot SSF_{t-2}$$
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The 2-bar input averaging provides an additional anti-aliasing stage.
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### 4. SSFDSP Output
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$$SSFDSP_t = SSF_{fast,t} - SSF_{slow,t}$$
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### 5. Complexity
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$O(1)$ 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.
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## Mathematical Foundation
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### Parameters
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| Parameter | Description | Default | Constraint |
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|-----------|-------------|---------|------------|
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| `period` | Expected dominant cycle period | 40 | $\geq 4$ |
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### Super-Smoother Frequency Response
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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.
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### Pseudo-code
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```
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function SSFDSP(source, period):
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pFast ← max(2, round(period / 4))
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pSlow ← max(3, round(period / 2))
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// Fast SSF coefficients
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αf ← √2·π / pFast
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c2f ← 2·exp(-αf)·cos(αf)
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c3f ← -exp(-2·αf)
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c1f ← 1 - c2f - c3f
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// Slow SSF coefficients
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αs ← √2·π / pSlow
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c2s ← 2·exp(-αs)·cos(αs)
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c3s ← -exp(-2·αs)
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c1s ← 1 - c2s - c3s
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ssfFast_1 ← 0; ssfFast_2 ← 0
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ssfSlow_1 ← 0; ssfSlow_2 ← 0
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p_prev ← 0
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for each price in source:
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// Input averaging
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avg ← (price + p_prev) / 2
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// Fast SSF update
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ssfFast ← c1f·avg + c2f·ssfFast_1 + c3f·ssfFast_2
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// Slow SSF update
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ssfSlow ← c1s·avg + c2s·ssfSlow_1 + c3s·ssfSlow_2
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// SSFDSP
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ssfdsp ← ssfFast - ssfSlow
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// Shift state
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ssfFast_2 ← ssfFast_1; ssfFast_1 ← ssfFast
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ssfSlow_2 ← ssfSlow_1; ssfSlow_1 ← ssfSlow
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p_prev ← price
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emit ssfdsp
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```
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### DSP vs SSFDSP
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| Aspect | DSP (EMA-based) | SSFDSP (Super-Smoother) |
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|--------|-----------------|------------------------|
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| Filter type | 1-pole IIR (exponential) | 2-pole Butterworth |
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| Rolloff | $-6$ dB/octave | $-12$ dB/octave |
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| Phase lag at cutoff | Non-zero | Zero |
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| Noise rejection | Moderate | Superior |
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| Turning points | Rounded | Sharper |
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### Output Interpretation
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| Condition | Meaning |
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|-----------|---------|
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| $SSFDSP > 0$ | Bullish cycle phase |
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| $SSFDSP < 0$ | Bearish cycle phase |
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| Zero crossing | Cycle phase transition |
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| Divergence with price | Cycle energy waning; trend exhaustion |
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| Amplitude shrinking | Cycle losing dominance; transition to trend |
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## Resources
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- **Ehlers, J.F.** *Cybernetic Analysis for Stocks and Futures*. Wiley, 2004.
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- **Ehlers, J.F.** *Cycle Analytics for Traders*. Wiley, 2013.
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- **Butterworth, S.** "On the Theory of Filter Amplifiers." *Experimental Wireless*, 7, 1930.
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