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Miha Kralj
136 lines
5.5 KiB
Markdown
136 lines
5.5 KiB
Markdown
# DECO: Ehlers Decycler Oscillator
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> *Ehlers' decycler oscillator removes the trend and isolates residual oscillation — what remains when the drift is subtracted.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Oscillator |
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| **Inputs** | Source (close) |
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| **Parameters** | `shortPeriod` (default 30), `longPeriod` (default 60) |
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| **Outputs** | Single series (Deco) |
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| **Output range** | $0$ to $1$ |
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| **Warmup** | 1 bar |
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| **PineScript** | [deco.pine](deco.pine) |
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- The Decycler Oscillator (DECO) is a DSP-based oscillator developed by John F.
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- **Similar:** [DPO](../dpo/Dpo.md), [Reflex](../reflex/Reflex.md) | **Complementary:** Cycle analysis | **Trading note:** Ehlers' Decycler Oscillator; removes trend to isolate cycle component for timing.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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## Overview
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The Decycler Oscillator (DECO) is a DSP-based oscillator developed by John F. Ehlers that isolates intermediate-frequency market cycles. It computes the difference between two 2-pole Butterworth high-pass filters with different cutoff periods, revealing the spectral band between the two cutoff frequencies.
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## Origin
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- **Author:** John F. Ehlers
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- **Source:** "Decyclers", Technical Analysis of Stocks & Commodities, September 2015
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- **Category:** Oscillator / Digital Signal Processing
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## Formula
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The DECO uses two 2-pole Butterworth high-pass filters:
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```
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α = (cos(0.707 × 2π/period) + sin(0.707 × 2π/period) - 1) / cos(0.707 × 2π/period)
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HP[n] = (1 - α/2)² × (x[n] - 2×x[n-1] + x[n-2]) + 2×(1-α) × HP[n-1] - (1-α)² × HP[n-2]
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DECO = HP_long - HP_short
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```
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The 0.707 factor (1/√2) places the filter response at the -3 dB Butterworth design point.
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### Transfer Function
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Each HP filter has the z-domain transfer function:
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```
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H(z) = (1-α/2)² × (1 - 2z⁻¹ + z⁻²) / (1 - 2(1-α)z⁻¹ + (1-α)²z⁻²)
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```
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The DECO output is the difference H_long(z) - H_short(z), which forms a bandpass response isolating cycles between the short and long cutoff periods.
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## Parameters
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| Parameter | Type | Default | Range | Description |
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|-----------|------|---------|-------|-------------|
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| shortPeriod | int | 30 | > 0 | Short HP cutoff period (bars) |
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| longPeriod | int | 60 | > shortPeriod | Long HP cutoff period (bars) |
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## Interpretation
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The Decycler Oscillator provides several analytical perspectives:
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- **Zero-Line Crossovers:**
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- Crossing above zero indicates bullish cycle momentum
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- Crossing below zero indicates bearish cycle momentum
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- The zero-crossing timing is relatively lag-free
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- **Band Isolation:**
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- The oscillator extracts only cycles within the frequency band defined by the two cutoff periods
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- Shorter cycles and longer trends are both rejected
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- This makes the oscillator highly selective
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- **Divergence Analysis:**
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- Bullish divergence: price makes lower lows while DECO makes higher lows
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- Bearish divergence: price makes higher highs while DECO makes lower highs
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- Indicates potential trend reversal
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- **Multiple Instance Analysis:**
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- Ehlers recommends using multiple DECO instances with different period pairs
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- Crossovers between instances with different coefficients can identify trend reversals
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## Warmup Period
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The indicator requires `longPeriod` bars before producing reliable output. The first two bars always output zero (insufficient price history for the 2-pole HP filter).
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## Properties
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- **Range:** Unbounded (oscillates around zero)
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- **Complexity:** O(1) per bar (pure IIR filter, no lookback buffer needed)
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- **Memory:** O(1) — only stores filter state variables
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## Related Indicators
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- **Decycler (DECYCLER):** The low-pass complement — removes cycles, keeps trend
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- **SSF-DSP:** Similar concept using Super Smooth Filters instead of HP filters
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- **Roofing Filter:** HP + SSF combination for cycle isolation
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- **BandPass Filter:** Ehlers' direct bandpass approach
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## Performance Profile
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### Operation Count (Streaming Mode)
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DECO (Detrended Correlation Oscillator) subtracts a linear regression from price then computes correlation.
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| Linear regression (O(N) or O(1) with prefix sums) | ~4 | 1 | 4 |
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| SUB (detrend: price − regression) | 1 | 1 | 1 |
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| Correlation pipeline (see CTI) | ~22 | 7 | 156 |
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| **Total** | **~27** | — | **~161 cycles** |
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Dominated by the correlation computation. ~161 cycles per bar.
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### Batch Mode (SIMD Analysis)
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| Operation | Vectorizable? | Notes |
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| :--- | :---: | :--- |
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| Linear regression | Yes | Prefix-sum dot products; VFMADD |
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| Detrend subtraction | Yes | VSUBPD |
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| Correlation | Yes | See CTI batch analysis |
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Fully vectorizable in batch. Regression and correlation both benefit from AVX2 VFMADD chains.
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 9/10 | Exact linear detrend + Pearson r |
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| **Timeliness** | 5/10 | Two N-bar windows compound lag |
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| **Smoothness** | 8/10 | Detrending removes linear drift; correlation bounded |
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| **Noise Rejection** | 8/10 | Linear detrend + correlation is doubly robust to trend contamination |
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## References
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1. Ehlers, J. F. (2015). "Decyclers." *Technical Analysis of Stocks & Commodities*, September 2015.
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2. Ehlers, J. F. (2013). *Cycle Analytics for Traders*. Wiley. Chapter 4. |