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.
5.0 KiB
SINE: Ehlers Sine Wave
SINE extracts the dominant cycle from price data using cascaded signal processing: a high-pass filter removes the trend, a Super-Smoother filter removes noise, and a Hilbert Transform FIR decomposes the filtered signal into In-Phase and Quadrature components for power-normalized sine wave output. The result oscillates between -1 and +1, representing the normalized position within the current cycle. Unlike HT_SINE which derives phase from the full TA-Lib Hilbert cascade, this Ehlers implementation uses explicit detrending and bandpass stages for cleaner cycle isolation.
Historical Context
John Ehlers introduced the Sine Wave indicator in Cybernetic Analysis for Stocks and Futures (2004) as a refined approach to cycle extraction. The design philosophy separates three signal processing concerns into distinct filter stages: (1) trend removal via high-pass filtering sets the long-wavelength cutoff, (2) aliasing prevention via Super-Smoother sets the short-wavelength cutoff, and (3) cycle extraction via Hilbert Transform generates the quadrature decomposition. This staged approach produces cleaner output than attempting all three simultaneously (as in the HT_SINE). The Sine Wave output at extremes (\pm 1) indicates the cyclical component is stretched and likely to revert, while zero crossings indicate phase transitions. The indicator is particularly valuable for mean-reversion strategies in ranging markets.
Architecture & Physics
1. High-Pass Filter (Detrending)
A single-pole high-pass filter removes low-frequency trends below the cutoff:
\alpha_{HP} = \frac{1 - \sin(2\pi / P_{HP})}{\cos(2\pi / P_{HP})}
HP_t = \frac{1 + \alpha_{HP}}{2}(P_t - P_{t-1}) + \alpha_{HP} \cdot HP_{t-1}
2. Super-Smoother Filter (Noise Removal)
A 2-pole Butterworth low-pass removes high-frequency noise:
a = e^{-\sqrt{2}\pi / P_{SSF}}
b = 2a \cos(\sqrt{2}\pi / P_{SSF})
c_1 = 1 - b + a^2, \quad c_2 = b, \quad c_3 = -a^2
Filt_t = \frac{c_1}{2}(HP_t + HP_{t-1}) + c_2 \cdot Filt_{t-1} + c_3 \cdot Filt_{t-2}
3. Hilbert Transform FIR
Discrete Hilbert approximation extracts quadrature component:
Q_t = 0.0962 \cdot Filt_{t-3} + 0.5769 \cdot Filt_{t-1} - 0.5769 \cdot Filt_{t-5} - 0.0962 \cdot Filt_{t-7}
I_t = Filt_t
4. Power Normalization
Power_t = I_t^2 + Q_t^2
Sine_t = \frac{I_t}{\sqrt{Power_t}}
When Power \approx 0, output is zero.
5. Complexity
O(1) per bar. Fixed filter stages with ring buffers of 2 (source) + 2 (HP) + 8 (filtered) = 12 elements. Warmup: \max(P_{HP}, P_{SSF}) + 8 bars.
Mathematical Foundation
Parameters
| Parameter | Description | Default | Constraint |
|---|---|---|---|
hpPeriod |
High-pass filter cutoff period | 40 | \geq 1 |
ssfPeriod |
Super-smoother filter period | 10 | \geq 1 |
Tuning Relationship
Typically P_{SSF} \approx P_{HP} / 4 to P_{HP} / 2. The high-pass defines the trend/cycle boundary; the super-smoother defines the noise/cycle boundary. Together they create a bandpass that isolates the frequency range of interest.
Pseudo-code
function SINE(source, hpPeriod, ssfPeriod):
// Precompute HP coefficient
α_hp ← (1 - sin(2π/hpPeriod)) / cos(2π/hpPeriod)
// Precompute SSF coefficients
a ← exp(-√2·π / ssfPeriod)
b ← 2·a·cos(√2·π / ssfPeriod)
c₁ ← (1 - b + a²) / 2
hp_prev ← 0; p_prev ← 0
filt_1 ← 0; filt_2 ← 0
filtBuf ← CircularBuffer(8)
for each price in source:
// High-pass
hp ← 0.5·(1 + α_hp)·(price - p_prev) + α_hp·hp_prev
// Super-smoother
filt ← c₁·(hp + hp_prev) + b·filt_1 - a²·filt_2
// Hilbert FIR quadrature
filtBuf.Add(filt)
Q ← 0.0962·filtBuf[3] + 0.5769·filtBuf[1]
- 0.5769·filtBuf[5] - 0.0962·filtBuf[7]
I ← filt
// Power normalization
power ← I² + Q²
sine ← (power > 0) ? I / √power : 0
// Shift state
hp_prev ← hp; p_prev ← price
filt_2 ← filt_1; filt_1 ← filt
emit sine
SINE vs HT_SINE
| Aspect | SINE | HT_SINE |
|---|---|---|
| Detrending | Explicit high-pass filter | Implicit in Hilbert cascade |
| Noise removal | Explicit Super-Smoother | 4-bar WMA only |
| Period tuning | User-configurable (hpPeriod, ssfPeriod) | Fixed (TA-Lib spec) |
| Output | Single (Sine only) | Dual (Sine + LeadSine) |
| Phase source | I/Q power normalization | DFT phase accumulation |
Output Interpretation
| Condition | Meaning |
|---|---|
Sine \approx +1 |
Cycle peak (potential short / mean-reversion) |
Sine \approx -1 |
Cycle trough (potential long / mean-reversion) |
| Zero crossing up | Bullish phase transition |
| Zero crossing down | Bearish phase transition |
| Erratic output | Strong trend overwhelming cycle extraction |
Resources
- Ehlers, J.F. Cybernetic Analysis for Stocks and Futures. Wiley, 2004.
- Ehlers, J.F. Cycle Analytics for Traders. Wiley, 2013.