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Miha Kralj
153 lines
6.4 KiB
Markdown
153 lines
6.4 KiB
Markdown
# REFLEX: Ehlers Reflex Indicator
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> *John Ehlers measured how much a filtered price deviates from its own linear extrapolation. The result is a zero-lag oscillator that catches reversals before they happen, because the deviation is largest precisely when the trend is bending.*
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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** | `period` |
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| **Outputs** | Single series (Reflex) |
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| **Output range** | Varies (see docs) |
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| **Warmup** | `period` bars |
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| **PineScript** | [reflex.pine](reflex.pine) |
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- REFLEX is a zero-lag oscillator that measures the reversal tendency of price by comparing a Super-Smoother-filtered price against a linear extrapol...
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- **Similar:** [Trendflex](../trendflex/Trendflex.md), [Deco](../deco/Deco.md) | **Complementary:** Cycle indicators | **Trading note:** Ehlers' Reflex indicator; cycle-mode oscillator using Super Smoother. Leading turns at zero crossings.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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REFLEX is a zero-lag oscillator that measures the reversal tendency of price by comparing a Super-Smoother-filtered price against a linear extrapolation from $N$ bars ago. The filter computes the slope of the filtered series over the lookback window, projects a straight line, and sums the deviations of the actual filtered values from this projected line. The sum is normalized by an exponential RMS estimate to produce values in roughly $\pm \sigma$ scale. Values above 0 indicate uptrend, below 0 indicate downtrend; crossovers signal potential reversals.
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## Historical Context
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John F. Ehlers published REFLEX in "Reflex: A New Zero-Lag Indicator" (*Technical Analysis of Stocks & Commodities*, February 2020). Ehlers' motivation was to create a cycle-based oscillator that responds to trend reversals with zero lag, unlike traditional oscillators (RSI, stochastic) that inherently lag price due to their smoothing components.
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The core idea is that linear extrapolation of a smoothed series will overshoot (undershoot) when the trend is decelerating (accelerating). By measuring the sum of these overshoots, REFLEX detects curvature changes — exactly the inflection points where trends reverse. This is mathematically similar to measuring the second derivative (acceleration), but the linear-extrapolation approach is more numerically stable and naturally adapts to the trend's own slope.
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The 2-pole Super Smoother pre-filter (at half the specified period) removes high-frequency noise before the reflex computation, preventing false signals from bar-to-bar price noise. The exponential RMS normalization ensures the output has consistent scale regardless of the instrument's volatility.
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## Architecture & Physics
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### 1. Super Smoother Pre-Filter
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A 2-pole IIR low-pass filter with cutoff at half the specified period:
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$$
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\text{Filt} = c_1 \cdot \frac{x_t + x_{t-1}}{2} + c_2 \cdot \text{Filt}_{t-1} + c_3 \cdot \text{Filt}_{t-2}
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$$
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where $a_1 = e^{-\sqrt{2}\pi / (N/2)}$, $c_2 = 2a_1\cos(\sqrt{2}\pi/(N/2))$, $c_3 = -a_1^2$, $c_1 = 1-c_2-c_3$.
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### 2. Linear Extrapolation Slope
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$$
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\text{slope} = \frac{\text{Filt}_{t-N} - \text{Filt}_t}{N}
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$$
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### 3. Deviation Summation
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$$
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\text{Sum} = \frac{1}{N}\sum_{i=1}^{N}\left[(\text{Filt}_t + i \cdot \text{slope}) - \text{Filt}_{t-i}\right]
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$$
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### 4. Exponential RMS Normalization
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$$
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\text{MS} = 0.04 \cdot \text{Sum}^2 + 0.96 \cdot \text{MS}_{t-1}
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$$
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$$
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\text{REFLEX} = \frac{\text{Sum}}{\sqrt{\text{MS}}}
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$$
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## Mathematical Foundation
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**Super Smoother coefficients (half-period cutoff):**
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$$
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a_1 = e^{-\sqrt{2}\pi / (N/2)}, \quad c_2 = 2a_1\cos\!\left(\frac{\sqrt{2}\pi}{N/2}\right), \quad c_3 = -a_1^2, \quad c_1 = 1-c_2-c_3
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$$
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**Deviation from linear trend:**
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$$
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D_i = (\text{Filt}_t + i \cdot \text{slope}) - \text{Filt}_{t-i}, \quad i = 1, \ldots, N
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$$
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**Mean deviation:**
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$$
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\text{Sum} = \frac{1}{N}\sum_{i=1}^{N} D_i
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$$
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**Interpretation:**
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- $\text{Sum} > 0$: filtered price is above its linear extrapolation (upward curvature, potential uptrend)
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- $\text{Sum} < 0$: filtered price is below its linear extrapolation (downward curvature, potential downtrend)
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- Zero crossings signal inflection points (trend reversals)
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**Default parameters:** `period = 20`, `minPeriod = 2`. Output is an oscillator (not overlay).
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**Pseudo-code (streaming):**
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```
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// Super Smoother (2-pole IIR)
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filt = c1*(price + price[1])/2 + c2*filt[1] + c3*filt[2]
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// Store in circular buffer
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buf[head] = filt
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// Slope from N-bar-ago to current
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slope = (filt_lag_N - filt) / N
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// Sum deviations from linear extrapolation
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sum = 0
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for i = 1 to N:
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sum += (filt + i*slope) - filt[i]
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sum /= N
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// Normalize by exponential RMS
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ms = 0.04 * sum² + 0.96 * ms[1]
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return ms > 0 ? sum / sqrt(ms) : 0
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```
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## Performance Profile
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### Operation Count (Streaming Mode)
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Reflex (Ehlers) uses a Super Smoother and a slope sum to detect cycles.
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| SSF update × 2 (FMA coefficients) | 2 | 4 | 8 |
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| Running slope sum (add new + subtract oldest) | 2 | 1 | 2 |
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| RMS normalization (variance accumulation) | 4 | 3 | 12 |
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| SQRT (RMS divisor) | 1 | 20 | 20 |
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| DIV (normalize) | 1 | 15 | 15 |
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| **Total** | **10** | — | **~57 cycles** |
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SQRT dominates. ~57 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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| SSF IIR passes × 2 | **No** | Recursive 2-pole IIR — sequential |
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| Slope sum | Partial | Prefix-sum assist after SSF computed |
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| RMS computation | Yes | VFMADD for variance; VSQRTPD |
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IIR dependencies block bar-parallel SIMD; RMS computation in batch is vectorizable.
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 9/10 | RMS normalization keeps scale consistent |
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| **Timeliness** | 6/10 | SSF half-period lag + slope window |
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| **Smoothness** | 9/10 | Super Smoother base + normalized output |
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| **Noise Rejection** | 9/10 | SSF rejects frequencies above cutoff; RMS stabilizes amplitude |
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## Resources
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- Ehlers, J.F. (2020). "Reflex: A New Zero-Lag Indicator." *Technical Analysis of Stocks & Commodities*, February 2020.
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- Ehlers, J.F. (2013). *Cycle Analytics for Traders*. Wiley. Chapter 3: Super Smoothers. |