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134 lines
4.9 KiB
Markdown
134 lines
4.9 KiB
Markdown
# DSP: Ehlers Detrended Synthetic Price
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Cycle |
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| **Inputs** | Source (close) |
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| **Parameters** | `period` (default 40) |
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| **Outputs** | Single series (Dsp) |
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| **Output range** | Varies (see docs) |
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| **Warmup** | `slowPeriod * 3` bars |
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### TL;DR
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- DSP creates a zero-centered oscillator by subtracting a half-cycle EMA from a quarter-cycle EMA, isolating the dominant cyclical component of price...
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- Parameterized by `period` (default 40).
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- Output range: Varies (see docs).
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- Requires `slowPeriod * 3` bars of warmup before first valid output (IsHot = true).
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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DSP creates a zero-centered oscillator by subtracting a half-cycle EMA from a quarter-cycle EMA, isolating the dominant cyclical component of price while cancelling longer-term trends. Developed by John Ehlers, the indicator is grounded in cycle theory rather than arbitrary period selection, making it a principled alternative to MACD for cycle-aware trading. Bias-corrected EMAs ensure accurate amplitude during warmup.
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## Historical Context
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John Ehlers introduced the Detrended Synthetic Price as part of his cycle analytics framework. While MACD uses fixed periods (12/26), DSP calibrates its two EMAs to specific fractions of the dominant cycle period: quarter-cycle for the fast component and half-cycle for the slow. Subtracting aligned filters at these frequencies effectively bandpass-isolates the cycle of interest while suppressing both high-frequency noise and low-frequency trend. The "synthetic" label reflects that the output is a constructed signal that exposes cyclical energy invisible in raw price.
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## Architecture & Physics
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### 1. Component 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. Alpha Coefficients
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Standard EMA smoothing factors:
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$$\alpha_{fast} = \frac{2}{P_{fast} + 1}, \qquad \alpha_{slow} = \frac{2}{P_{slow} + 1}$$
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### 3. EMA Updates with Bias Correction
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Raw EMA recursion:
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$$EMA_{raw,t} = \alpha \cdot P_t + (1 - \alpha) \cdot EMA_{raw,t-1}$$
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Warmup bias correction (prevents initial distortion):
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$$EMA_t = \frac{EMA_{raw,t}}{1 - (1 - \alpha)^n}$$
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where $n$ is the number of bars processed.
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### 4. DSP Output
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$$DSP_t = EMA_{fast,t} - EMA_{slow,t}$$
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### 5. Complexity
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$O(1)$ per bar with $O(1)$ memory. Two EMA state variables plus two bias correction accumulators.
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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` | Dominant cycle period | 40 | $\geq 4$ |
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### Pseudo-code
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```
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function DSP(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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αFast ← 2 / (pFast + 1)
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αSlow ← 2 / (pSlow + 1)
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emaFastRaw ← 0
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emaSlowRaw ← 0
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decayFast ← 1.0 // (1 - αFast)^n
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decaySlow ← 1.0 // (1 - αSlow)^n
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for each price in source:
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emaFastRaw ← FMA(αFast, price, (1 - αFast) * emaFastRaw)
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emaSlowRaw ← FMA(αSlow, price, (1 - αSlow) * emaSlowRaw)
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decayFast *= (1 - αFast)
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decaySlow *= (1 - αSlow)
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emaFast ← emaFastRaw / (1 - decayFast)
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emaSlow ← emaSlowRaw / (1 - decaySlow)
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dsp ← emaFast - emaSlow
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emit dsp
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```
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### Output Interpretation
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| Condition | Meaning |
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|-----------|---------|
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| $DSP > 0$ | Fast EMA above slow: bullish cycle phase |
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| $DSP < 0$ | Fast EMA below slow: bearish cycle phase |
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| Zero crossing | Cycle phase transition point |
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| Divergence from price | Cycle energy waning; potential trend exhaustion |
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## Performance Profile
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### Operation Count (Streaming Mode)
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| ADD/SUB | 3 | 1 | 3 |
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| MUL | 4 | 3 | 12 |
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| FMA | 2 | 4 | 8 |
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| DIV | 2 | 15 | 30 |
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| **Total** | **11** | — | **~53 cycles** |
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O(1) per bar. Two EMA updates (fast + slow) using FMA, plus warmup bias-correction divisions. After warmup completes, the DIV cost drops to zero, reducing steady-state to ~23 cycles.
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 9/10 | Bias-corrected EMAs eliminate warmup distortion |
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| **Timeliness** | 8/10 | Quarter-cycle EMA responds quickly; half-cycle provides reference |
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| **Smoothness** | 8/10 | Dual EMA differencing inherently smooths noise |
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| **Memory** | 10/10 | O(1) state: 6 scalar values in record struct |
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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.** *Rocket Science for Traders*. Wiley, 2001.
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