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Add new moving average implementations: LTMA, MCNMA, NLMA, NMA, NYQMA, RAIN, and TRAMA
- 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.
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Miha Kralj
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# HT_PHASOR: Ehlers Hilbert Transform Phasor Components
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> "Phasors let us measure a cycle's position and strength; trading becomes geometry over time."
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HT_PHASOR decomposes the price signal into two orthogonal components: **InPhase** (I) and **Quadrature** (Q) using the Hilbert Transform. These components form a complex phasor (Z = I + jQ) that describes the instantaneous amplitude and phase of the market cycle.
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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.
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## Historical Context
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John Ehlers introduced the decomposition of market data into phasor components in *Rocket Science for Traders* (2001). This decomposition is fundamental to his entire suite of cycle indicators (SineWave, Homodyne, etc.).
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TA-Lib implements HT_PHASOR to expose these intermediate components directly for advanced analysis. QuanTAlib matches the TA-Lib implementation.
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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.
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## Architecture & Physics
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The calculation pipeline extracts the analytic signal's real and imaginary components.
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### 1. WMA Smoothing
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$$
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SmoothPrice_t = \frac{4P_t + 3P_{t-1} + 2P_{t-2} + P_{t-3}}{10}
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$$
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$$SmoothPrice_t = \frac{4P_t + 3P_{t-1} + 2P_{t-2} + P_{t-3}}{10}$$
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### 2. Hilbert Transform
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### 2. Hilbert Transform FIR
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Applied to smoothed price with adaptive bandwidth to generate fundamental components.
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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 $jQ$.
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### 3. Phasor Components
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$$
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I2_t = I1_t - jQ_t
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$$
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$$I_{2,t} = I_{1,t} - jQ_t, \qquad Q_{2,t} = Q_{1,t} + jI_t$$
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$$
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Q2_t = Q1_t + jI_t
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$$
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Both smoothed with EMA ($\alpha = 0.2$):
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Where:
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$$I_t = 0.2 \cdot I_{2,t} + 0.8 \cdot I_{t-1}$$
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- **InPhase (I)**: Smoothed I2—cycle signal aligned with price
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- **Quadrature (Q)**: Smoothed Q2—rate of change (velocity) of cycle
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*Note: InPhase output is delayed by 3 bars to align with Quadrature's effective lag.*
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$$Q_t = 0.2 \cdot Q_{2,t} + 0.8 \cdot Q_{t-1}$$
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### 4. Phase Relationship
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- Q leads I by 90°
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- When I peaks, Q crosses zero (downward)
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- When I crosses zero (upward), Q peaks
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$Q$ 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 $\phi = \arctan(Q/I)$.
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## Performance Profile
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### 5. Complexity
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### Operation Count (Streaming Mode, per Bar)
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$O(1)$ per bar. Fixed Hilbert cascade with circular buffers. Warmup: 32 bars (TA-Lib lookback).
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| MUL (Hilbert taps) | 28 | 3 | 84 |
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| MUL (phasor calc) | 8 | 3 | 24 |
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| ADD/SUB | 35 | 1 | 35 |
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| EMA smoothing | 4 | 4 | 16 |
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| **Total** | **75** | — | **~159 cycles** |
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## Mathematical Foundation
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### Complexity Analysis
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### Parameters
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- **Streaming:** O(1) per bar—fixed Hilbert cascade
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- **Memory:** ~1.2 KB per instance (circular buffers)
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- **Warmup:** 32 bars (TA-Lib lookback)
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| Parameter | Description | Default | Constraint |
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|-----------|-------------|---------|------------|
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| (none) | No user-configurable parameters | | |
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## Validation
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### Pseudo-code
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| Library | Status | Notes |
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| :--- | :---: | :--- |
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| TA-Lib | ✅ | Matches `TALib.Functions.HtPhasor()` |
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| Skender | N/A | Not implemented |
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| PineScript | ✅ | Matches `phasor.pine` |
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```
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function HT_PHASOR(source):
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A ← 0.0962; B ← 0.5769
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smoothBuf ← CircularBuffer(7)
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detBuf, q1Buf, i1Buf ← CircularBuffers
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## Usage & Pitfalls
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I2 ← 0; Q2 ← 0
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- **Dual output**—InPhase (Value) and Quadrature (property)
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- **32-bar warmup required**—ignore early values
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- **Capture Quadrature immediately after Update()**—property updated on each call
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- **Trending markets** break orthogonality—use HT_TRENDMODE to filter
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- **Phasor crossover**:
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- Buy: Q crosses I from below (anticipates cycle trough)
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- Sell: Q crosses I from above (anticipates cycle peak)
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- **For sine input** sin(ωt): InPhase ≈ sin(ωt), Quadrature ≈ cos(ωt)
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for each price in source:
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// WMA smooth
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smooth ← (4·price + 3·p[1] + 2·p[2] + p[3]) / 10
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smoothBuf.Add(smooth)
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## API
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// Hilbert FIR (adaptive)
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det ← A·smooth[0] + B·smooth[2] - B·smooth[4] - A·smooth[6]
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Q1 ← A·det[0] + B·det[2] - B·det[4] - A·det[6]
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I1 ← det[3]
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```mermaid
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classDiagram
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class HtPhasor {
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+double Value
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+double Quadrature
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+bool IsHot
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+HtPhasor()
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+HtPhasor(ITValuePublisher source)
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+TValue Update(TValue input, bool isNew)
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+void Reset()
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}
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// Hilbert of I1 and Q1
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jI ← A·I1[0] + B·I1[2] - B·I1[4] - A·I1[6]
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jQ ← A·Q1[0] + B·Q1[2] - B·Q1[4] - A·Q1[6]
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// Phasor components (EMA smoothed)
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I2 ← 0.2·(I1 - jQ) + 0.8·I2
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Q2 ← 0.2·(Q1 + jI) + 0.8·Q2
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emit InPhase = I2, Quadrature = Q2
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```
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### Class: `HtPhasor`
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### Phasor Crossover Signals
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| Parameter | Type | Default | Range | Description |
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| :--- | :--- | :--- | :--- | :--- |
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| (none) | — | — | — | No constructor parameters |
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| Condition | Signal |
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|-----------|--------|
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| $Q$ crosses $I$ from below | Bullish (anticipates cycle trough) |
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| $Q$ crosses $I$ from above | Bearish (anticipates cycle peak) |
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| $\sqrt{I^2 + Q^2}$ increasing | Cycle amplitude growing |
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| $\sqrt{I^2 + Q^2}$ decreasing | Cycle amplitude fading (trend or noise) |
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### Properties
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### Output Interpretation
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- `Value` (`double`): InPhase component of phasor
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- `Quadrature` (`double`): Quadrature component (90° shifted)
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- `IsHot` (`bool`): Returns `true` when warmup (32 bars) is complete
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| Output | Range | Meaning |
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|--------|-------|---------|
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| `InPhase` | unbounded | Cycle component aligned with price |
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| `Quadrature` | unbounded | Rate of change (velocity) of cycle |
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### Methods
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## Resources
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- `Update(TValue input, bool isNew)`: Updates the indicator with a new data point
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## C# Example
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```csharp
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using QuanTAlib;
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// Create HT_PHASOR
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var htPhasor = new HtPhasor();
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double prevInPhase = 0, prevQuadrature = 0;
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// Update with streaming data
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foreach (var bar in quotes)
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{
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var result = htPhasor.Update(new TValue(bar.Date, bar.Close));
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double inPhase = result.Value;
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double quadrature = htPhasor.Quadrature; // Capture immediately!
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if (htPhasor.IsHot)
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{
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Console.WriteLine($"{bar.Date}: I = {inPhase:F4}, Q = {quadrature:F4}");
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// Phasor crossover detection
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if (inPhase > quadrature && prevInPhase <= prevQuadrature)
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Console.WriteLine(" → Bullish crossover (anticipate trough)");
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else if (inPhase < quadrature && prevInPhase >= prevQuadrature)
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Console.WriteLine(" → Bearish crossover (anticipate peak)");
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}
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prevInPhase = inPhase;
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prevQuadrature = quadrature;
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}
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// Batch calculation
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var output = HtPhasor.Calculate(sourceSeries);
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```
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- **Ehlers, J.F.** *Rocket Science for Traders*. Wiley, 2001.
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- **TA-Lib** `TA_HT_PHASOR()` reference implementation.
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- **Ehlers, J.F.** *Cybernetic Analysis for Stocks and Futures*. Wiley, 2004.
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