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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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# DSP: Ehlers Detrended Synthetic Price
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> "Remove the trend, reveal the cycles."
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The Detrended Synthetic Price (DSP) indicator creates a zero-centered oscillator by subtracting a half-cycle EMA from a quarter-cycle EMA. Developed by John Ehlers, this "synthetic" price highlights underlying cyclical movement, identifying momentum shifts when the faster EMA crosses the slower one.
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DSP creates a zero-centered oscillator by subtracting a half-cycle EMA from a quarter-cycle EMA, isolating the dominant cyclical component of price while cancelling longer-term trends. Developed by John Ehlers, the indicator is grounded in cycle theory rather than arbitrary period selection, making it a principled alternative to MACD for cycle-aware trading. Bias-corrected EMAs ensure accurate amplitude during warmup.
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## Historical Context
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John Ehlers introduced the DSP as part of his research into cycle analytics for traders. While many indicators (like MACD) use arbitrary periods (12/26), DSP is grounded in cycle theory. Ehlers posits that to effectively isolate a cycle, one should filter data based on the dominant cycle period.
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The use of period/4 and period/2 roughly corresponds to extracting the cycle's momentum while cancelling out longer-term trends. This makes DSP particularly effective for cycle-based trading strategies.
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John Ehlers introduced the Detrended Synthetic Price as part of his cycle analytics framework. While MACD uses fixed periods (12/26), DSP calibrates its two EMAs to specific fractions of the dominant cycle period: quarter-cycle for the fast component and half-cycle for the slow. Subtracting aligned filters at these frequencies effectively bandpass-isolates the cycle of interest while suppressing both high-frequency noise and low-frequency trend. The "synthetic" label reflects that the output is a constructed signal that exposes cyclical energy invisible in raw price.
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## Architecture & Physics
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DSP utilizes a dual EMA architecture, calibrated to specific fractions of the cycle period.
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### 1. Component Periods
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$$
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P_{fast} = \max(2, \text{round}(P / 4))
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$$
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From the user-specified dominant cycle period $P$:
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$$
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P_{slow} = \max(3, \text{round}(P / 2))
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$$
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$$P_{fast} = \max(2, \lfloor P / 4 + 0.5 \rfloor)$$
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$$P_{slow} = \max(3, \lfloor P / 2 + 0.5 \rfloor)$$
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### 2. Alpha Coefficients
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$$
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\alpha_{fast} = \frac{2}{P_{fast} + 1}
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$$
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Standard EMA smoothing factors:
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$$
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\alpha_{slow} = \frac{2}{P_{slow} + 1}
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$$
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$$\alpha_{fast} = \frac{2}{P_{fast} + 1}, \qquad \alpha_{slow} = \frac{2}{P_{slow} + 1}$$
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### 3. EMA Updates (with Bias Correction)
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### 3. EMA Updates with Bias Correction
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$$
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EMA_{raw} = \alpha \cdot Price + (1 - \alpha) \cdot EMA_{raw\_prev}
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$$
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Raw EMA recursion:
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$$
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EMA_{corrected} = \frac{EMA_{raw}}{1 - (1-\alpha)^n}
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$$
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$$EMA_{raw,t} = \alpha \cdot P_t + (1 - \alpha) \cdot EMA_{raw,t-1}$$
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### 4. DSP Calculation
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Warmup bias correction (prevents initial distortion):
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$$
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DSP = EMA_{fast} - EMA_{slow}
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$$
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$$EMA_t = \frac{EMA_{raw,t}}{1 - (1 - \alpha)^n}$$
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## Performance Profile
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where $n$ is the number of bars processed.
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### Operation Count (Streaming Mode, per Bar)
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### 4. DSP Output
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| FMA (EMA updates) | 2 | 4 | 8 |
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| MUL (decay factors) | 2 | 3 | 6 |
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| DIV (bias correction) | 2 | 15 | 30 |
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| SUB (DSP = fast - slow) | 1 | 1 | 1 |
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| **Total** | **7** | — | **~45 cycles** |
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$$DSP_t = EMA_{fast,t} - EMA_{slow,t}$$
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### Complexity Analysis
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### 5. Complexity
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- **Streaming:** O(1) per bar—fixed calculation depth
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- **Memory:** O(1)—only EMA state variables
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- **Warmup:** ~2 × slow period for convergence
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$O(1)$ per bar with $O(1)$ memory. Two EMA state variables plus two bias correction accumulators.
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## Validation
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## Mathematical Foundation
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| Library | Status | Notes |
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| :--- | :---: | :--- |
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| TA-Lib | N/A | Not standard |
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| Skender | N/A | Not standard |
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| PineScript | ✅ | Matches Ehlers' reference logic |
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### Parameters
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## Usage & Pitfalls
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| Parameter | Description | Default | Constraint |
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|-----------|-------------|---------|------------|
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| `period` | Dominant cycle period | 40 | $\geq 4$ |
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- **Zero crossing** indicates cycle phase change—above zero is bullish, below zero is bearish
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- **Period should match market cycle**—if market cycle is 20 bars, use period 20 not 40
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- **Not normalized**—amplitude reflects absolute price difference, varies by asset
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- **Whipsaws** occur in ranging markets with cycles shorter than the setting
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- **Divergence** (higher price highs with lower DSP highs) suggests cycle energy loss
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- **Use FusedMultiplyAdd** for optimal precision in EMA recursion
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### Pseudo-code
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## API
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```
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function DSP(source, period):
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pFast ← max(2, round(period / 4))
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pSlow ← max(3, round(period / 2))
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αFast ← 2 / (pFast + 1)
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αSlow ← 2 / (pSlow + 1)
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```mermaid
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classDiagram
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class Dsp {
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+int Period
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+double Value
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+bool IsHot
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+Dsp(int period)
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+Dsp(ITValuePublisher source, int period)
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+TValue Update(TValue input, bool isNew)
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+void Reset()
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}
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emaFastRaw ← 0
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emaSlowRaw ← 0
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decayFast ← 1.0 // (1 - αFast)^n
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decaySlow ← 1.0 // (1 - αSlow)^n
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for each price in source:
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emaFastRaw ← FMA(αFast, price, (1 - αFast) * emaFastRaw)
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emaSlowRaw ← FMA(αSlow, price, (1 - αSlow) * emaSlowRaw)
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decayFast *= (1 - αFast)
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decaySlow *= (1 - αSlow)
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emaFast ← emaFastRaw / (1 - decayFast)
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emaSlow ← emaSlowRaw / (1 - decaySlow)
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dsp ← emaFast - emaSlow
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emit dsp
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```
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### Class: `Dsp`
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### Output Interpretation
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| Parameter | Type | Default | Range | Description |
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| :--- | :--- | :--- | :--- | :--- |
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| `period` | `int` | `40` | `≥4` | Dominant cycle period |
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| Condition | Meaning |
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|-----------|---------|
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| $DSP > 0$ | Fast EMA above slow: bullish cycle phase |
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| $DSP < 0$ | Fast EMA below slow: bearish cycle phase |
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| Zero crossing | Cycle phase transition point |
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| Divergence from price | Cycle energy waning; potential trend exhaustion |
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### Properties
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## Resources
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- `Value` (`double`): The current DSP value (oscillates around 0)
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- `IsHot` (`bool`): Returns `true` when warmup is complete
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### Methods
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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 DSP for a 40-bar cycle
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var dsp = new Dsp(period: 40);
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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 = dsp.Update(new TValue(bar.Date, bar.Close));
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if (dsp.IsHot)
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{
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Console.WriteLine($"{bar.Date}: DSP = {result.Value:F4}");
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// Cycle phase detection
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if (result.Value > 0)
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Console.WriteLine(" → Bullish cycle phase");
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else
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Console.WriteLine(" → Bearish cycle phase");
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}
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}
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// Batch calculation
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var output = Dsp.Calculate(sourceSeries, period: 40);
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```
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- **Ehlers, J.F.** *Cybernetic Analysis for Stocks and Futures*. Wiley, 2004.
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- **Ehlers, J.F.** *Rocket Science for Traders*. Wiley, 2001.
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