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108 lines
5.5 KiB
Markdown
108 lines
5.5 KiB
Markdown
# EEO: Ehlers Elegant Oscillator
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A bounded zero-crossing oscillator that applies the Inverse Fisher Transform to RMS-normalized 2-bar momentum, then smooths the result with a 2-pole Super Smoother filter. Output is approximately bounded to [-1, +1].
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| Property | Value |
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|:-------------- |:--------------------------------------------- |
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| **Category** | Oscillators |
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| **Author** | John F. Ehlers |
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| **Source** | TASC, February 2022 |
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| **Article** | "An Elegant Oscillator: Inverse Fisher Transform Redux" |
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| **Input** | Single series (Close) |
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| **Parameters** | BandEdge (default 20) |
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| **Output** | Bounded ≈ [-1, +1] |
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| **Hot after** | 50 + BandEdge bars |
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## Historical Context
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Ehlers' 2022 "Elegant Oscillator" is a refinement of his earlier DSO (2018). Where DSO applies the Fisher Transform (arctanh) to expand a normalized signal, EEO applies the **Inverse Fisher Transform** (tanh) to compress it. The IFT naturally bounds the output to [-1, +1] without needing the ±0.99 clamping that DSO requires. A Super Smoother post-filter then removes residual noise.
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## Architecture & Physics
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### Stage 1: 2-Bar Momentum (Derivative)
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$$\text{Deriv} = \text{Close} - \text{Close}[2]$$
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This is the same "zeros" whitening used in DSO — it removes DC and Nyquist components, creating a band-limited derivative.
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### Stage 2: RMS Normalization (Fixed 50-Bar Window)
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$$\text{RMS} = \sqrt{\frac{1}{50}\sum_{k=0}^{49}\text{Deriv}[k]^2}$$
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$$\text{NDeriv} = \frac{\text{Deriv}}{\text{RMS}}$$
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The fixed 50-bar window (not parameterized) provides a stable normalization base. The RMS measures the "typical" derivative magnitude, so NDeriv represents "how many standard deviations" the current derivative is from zero.
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### Stage 3: Inverse Fisher Transform (tanh)
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$$\text{IFish} = \tanh(\text{NDeriv}) = \frac{e^{2 \cdot \text{NDeriv}} - 1}{e^{2 \cdot \text{NDeriv}} + 1}$$
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The IFT compresses the normalized derivative into [-1, +1]. Values near ±1 indicate extreme momentum relative to recent history.
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### Stage 4: Super Smoother Filter (2-Pole Butterworth)
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$$a_1 = e^{-1.414\pi / \text{BandEdge}}$$
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$$b_1 = 2 \cdot a_1 \cdot \cos\!\left(\frac{1.414 \cdot 180°}{\text{BandEdge}}\right)$$
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$$c_2 = b_1, \quad c_3 = -a_1^2, \quad c_1 = 1 - c_2 - c_3$$
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$$\text{SS} = \frac{c_1}{2}(\text{IFish} + \text{IFish}[1]) + c_2 \cdot \text{SS}[1] + c_3 \cdot \text{SS}[2]$$
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The Super Smoother removes high-frequency chatter from the IFT output while preserving the phase relationship.
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## Performance Profile
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### Operation Count (Streaming Mode, Scalar)
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| Operation | Count | Notes |
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|:----------------------- |:----- |:------------------------------ |
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| Subtraction (Deriv) | 1 | Close - Close[2] |
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| Multiply (deriv²) | 1 | For RMS buffer |
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| RingBuffer add/remove | 1 | O(1) circular buffer |
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| FMA (running sum) | 1 | SumSq update |
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| Sqrt (RMS) | 1 | √(sum/50) |
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| Division (normalize) | 1 | Deriv / RMS |
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| Exp + tanh (IFT) | 1 | Math.Tanh() |
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| FMA × 2 (SSF) | 2 | 2-pole recursive filter |
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| **Total per bar** | **~9** | Constant O(1) |
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### Batch Mode (SIMD Analysis)
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The algorithm's IIR Super Smoother stage prevents full vectorization. Batch mode processes sequentially but avoids per-bar allocation overhead.
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### Quality Metrics
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| Metric | Score | Notes |
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|:------------------- |:----- |:---------------------------------------- |
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| Lag | Low | SSF has minimal phase distortion |
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| Noise rejection | High | IFT compression + SSF smoothing |
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| Sensitivity | High | 2-bar derivative is very responsive |
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| Bounded output | Yes | ≈ [-1, +1] from tanh |
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| Parameter count | 1 | Only BandEdge |
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## Validation
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EEO is validated through self-consistency tests (streaming ≡ batch ≡ span ≡ eventing) and behavioral tests (constant input → 0, trending → non-zero, symmetry).
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### Behavioral Test Summary
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| Test | Expected Result |
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|:----------------------- |:------------------------- |
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| Constant input | Output → 0 |
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| Strong uptrend | Output > 0 |
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| Strong downtrend | Output < 0 |
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| Ascending vs descending | Opposite signs |
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| NaN/Inf input | Finite output (fallback) |
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| Bar correction (isNew) | State restored correctly |
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## Common Pitfalls
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1. **Fixed RMS window**: The 50-bar window is hardcoded per Ehlers' specification. Do not parameterize it — it provides a stable normalization base independent of BandEdge.
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2. **BandEdge vs Period**: BandEdge is the Super Smoother cutoff, not an RMS lookback. Higher BandEdge = more smoothing but more lag.
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3. **Bounded output**: Unlike DSO (which uses Fisher Transform producing unbounded output), EEO output is bounded to ≈ [-1, +1]. Signal levels of ±0.5 are typical thresholds, not ±2 as with DSO.
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4. **Warmup**: Requires 50 + BandEdge bars. The first 50 bars fill the RMS window; then BandEdge more bars are needed for SSF convergence.
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