mirror of
https://github.com/mihakralj/QuanTAlib.git
synced 2026-08-25 22:08:05 +00:00
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.
97 lines
4.0 KiB
Markdown
97 lines
4.0 KiB
Markdown
# HT_PHASOR: Ehlers Hilbert Transform Phasor Components
|
|
|
|
HT_PHASOR decomposes the price signal into two orthogonal components, InPhase ($I$) and Quadrature ($Q$), using the Hilbert Transform. Together these form a complex phasor $Z = I + jQ$ that describes the instantaneous amplitude and phase of the dominant market cycle. Compatible with TA-Lib's `HT_PHASOR` function, this dual-output indicator provides the fundamental building blocks for cycle analysis, phasor crossover timing, and instantaneous amplitude measurement.
|
|
|
|
## Historical Context
|
|
|
|
John Ehlers introduced phasor decomposition of market data in *Rocket Science for Traders* (2001). In electrical engineering, a phasor represents a sinusoidal signal as a rotating complex vector, separating the cycle's "position" (InPhase) from its "velocity" (Quadrature). Ehlers recognized that this decomposition is the mathematical foundation for all his cycle indicators: HT_SINE, HT_DCPERIOD, HT_DCPHASE, and HOMOD all derive from these same I/Q components. TA-Lib exposes HT_PHASOR to give advanced users direct access to the analytic signal for custom cycle analysis. The InPhase output is delayed by 3 bars to align with the Quadrature component's effective lag from the Hilbert Transform FIR.
|
|
|
|
## Architecture & Physics
|
|
|
|
### 1. WMA Smoothing
|
|
|
|
$$SmoothPrice_t = \frac{4P_t + 3P_{t-1} + 2P_{t-2} + P_{t-3}}{10}$$
|
|
|
|
### 2. Hilbert Transform FIR
|
|
|
|
Using Ehlers' coefficients ($A = 0.0962$, $B = 0.5769$), the 4-tap discrete Hilbert approximation generates the detrender, and from it the fundamental In-Phase and Quadrature components ($I_1$, $Q_1$). Further Hilbert transforms of these produce $jI$ and $jQ$.
|
|
|
|
### 3. Phasor Components
|
|
|
|
$$I_{2,t} = I_{1,t} - jQ_t, \qquad Q_{2,t} = Q_{1,t} + jI_t$$
|
|
|
|
Both smoothed with EMA ($\alpha = 0.2$):
|
|
|
|
$$I_t = 0.2 \cdot I_{2,t} + 0.8 \cdot I_{t-1}$$
|
|
|
|
$$Q_t = 0.2 \cdot Q_{2,t} + 0.8 \cdot Q_{t-1}$$
|
|
|
|
### 4. Phase Relationship
|
|
|
|
$Q$ leads $I$ by $90°$. When $I$ peaks, $Q$ crosses zero downward. When $I$ crosses zero upward, $Q$ peaks. The instantaneous amplitude is $A = \sqrt{I^2 + Q^2}$ and the instantaneous phase is $\phi = \arctan(Q/I)$.
|
|
|
|
### 5. Complexity
|
|
|
|
$O(1)$ per bar. Fixed Hilbert cascade with circular buffers. Warmup: 32 bars (TA-Lib lookback).
|
|
|
|
## Mathematical Foundation
|
|
|
|
### Parameters
|
|
|
|
| Parameter | Description | Default | Constraint |
|
|
|-----------|-------------|---------|------------|
|
|
| (none) | No user-configurable parameters | | |
|
|
|
|
### Pseudo-code
|
|
|
|
```
|
|
function HT_PHASOR(source):
|
|
A ← 0.0962; B ← 0.5769
|
|
smoothBuf ← CircularBuffer(7)
|
|
detBuf, q1Buf, i1Buf ← CircularBuffers
|
|
|
|
I2 ← 0; Q2 ← 0
|
|
|
|
for each price in source:
|
|
// WMA smooth
|
|
smooth ← (4·price + 3·p[1] + 2·p[2] + p[3]) / 10
|
|
smoothBuf.Add(smooth)
|
|
|
|
// Hilbert FIR (adaptive)
|
|
det ← A·smooth[0] + B·smooth[2] - B·smooth[4] - A·smooth[6]
|
|
Q1 ← A·det[0] + B·det[2] - B·det[4] - A·det[6]
|
|
I1 ← det[3]
|
|
|
|
// Hilbert of I1 and Q1
|
|
jI ← A·I1[0] + B·I1[2] - B·I1[4] - A·I1[6]
|
|
jQ ← A·Q1[0] + B·Q1[2] - B·Q1[4] - A·Q1[6]
|
|
|
|
// Phasor components (EMA smoothed)
|
|
I2 ← 0.2·(I1 - jQ) + 0.8·I2
|
|
Q2 ← 0.2·(Q1 + jI) + 0.8·Q2
|
|
|
|
emit InPhase = I2, Quadrature = Q2
|
|
```
|
|
|
|
### Phasor Crossover Signals
|
|
|
|
| Condition | Signal |
|
|
|-----------|--------|
|
|
| $Q$ crosses $I$ from below | Bullish (anticipates cycle trough) |
|
|
| $Q$ crosses $I$ from above | Bearish (anticipates cycle peak) |
|
|
| $\sqrt{I^2 + Q^2}$ increasing | Cycle amplitude growing |
|
|
| $\sqrt{I^2 + Q^2}$ decreasing | Cycle amplitude fading (trend or noise) |
|
|
|
|
### Output Interpretation
|
|
|
|
| Output | Range | Meaning |
|
|
|--------|-------|---------|
|
|
| `InPhase` | unbounded | Cycle component aligned with price |
|
|
| `Quadrature` | unbounded | Rate of change (velocity) of cycle |
|
|
|
|
## Resources
|
|
|
|
- **Ehlers, J.F.** *Rocket Science for Traders*. Wiley, 2001.
|
|
- **TA-Lib** `TA_HT_PHASOR()` reference implementation.
|
|
- **Ehlers, J.F.** *Cybernetic Analysis for Stocks and Futures*. Wiley, 2004.
|