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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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# ADXR: Average Directional Movement Rating
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> If ADX is the speedometer, ADXR is the cruise control setting. It smooths out the acceleration to tell you if the trend has staying power.
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The Average Directional Movement Rating (ADXR) is a smoothed version of the ADX. It dampens the volatility of the ADX itself, providing a more stable—albeit significantly more lagging—measure of trend strength. It is primarily used to rate the efficacy of trend-following strategies before capital is committed.
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The Average Directional Movement Rating is a smoothed version of ADX that dampens short-term fluctuations in trend strength by averaging the current ADX with a historical ADX value. This creates a doubly-lagged metric that sacrifices all timing utility in exchange for stable regime classification. ADXR answers one question: does the current market environment reward trend-following strategies? If ADXR is high, deploy momentum logic. If low, deploy mean-reversion. It is a strategic filter, not a tactical signal.
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## Historical Context
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J. Welles Wilder Jr. introduced ADXR alongside ADX in *New Concepts in Technical Trading Systems* (1978). His goal was simple: ADX can be erratic. By averaging the current ADX with a past ADX, he created a metric that ignores short-term fluctuations in trend strength.
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It is effectively a "momentum of momentum" indicator, smoothed to the point of geological stability.
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J. Welles Wilder Jr. introduced ADXR alongside ADX in *New Concepts in Technical Trading Systems* (1978). His reasoning was pragmatic: ADX itself can be erratic during transitions between trending and ranging regimes, producing whipsaw readings that confuse systematic allocation. By averaging the current ADX with its value from $N-1$ bars ago, Wilder created a "momentum of momentum" indicator smoothed to geological stability. The ADXR found its architectural niche not as a trading signal but as a capital allocation filter — determining whether a trend-following system should be active at all. Its double lag (ADX already lags price; ADXR lags ADX) makes it useless for entry timing by design.
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## Architecture & Physics
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ADXR is a composite indicator. It does not interact with price directly; it interacts with the output of the ADX.
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### 1. ADX Dependency
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1. **Dependency**: It instantiates and maintains a full `Adx` indicator internally.
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2. **History**: It maintains a circular buffer of historical ADX values.
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3. **Averaging**: It computes the arithmetic mean of the current ADX and the ADX from `Period - 1` bars ago.
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ADXR is a composite indicator that does not interact with price directly. It instantiates and maintains a full ADX pipeline internally:
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### The Lag Trade-off
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$$\text{Price} \rightarrow \text{DM/TR} \rightarrow \text{RMA} \rightarrow \text{DI} \rightarrow \text{DX} \rightarrow \text{ADX} \rightarrow \text{ADXR}$$
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ADXR is intentionally slow.
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### 2. Historical Buffer
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* **ADX** lags price because of its multiple smoothing layers.
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* **ADXR** lags ADX because it averages the current value with a value from the distant past.
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A circular buffer of size $N$ stores historical ADX values, providing $O(1)$ access to the value from $N-1$ bars ago.
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This double lag makes ADXR useless for entry timing. Its only valid architectural purpose is **regime filtering**: determining *if* a trend-following system should be active, not *when* it should trade.
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### 3. Rating Calculation
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$$ADXR_t = \frac{ADX_t + ADX_{t-(N-1)}}{2}$$
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The $N-1$ lag (rather than $N$) matches TA-Lib's reference implementation exactly.
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### 4. Complexity
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- **Time:** $O(1)$ per bar — ADX update plus one buffer lookup and one average
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- **Space:** $O(N)$ — circular buffer for ADX history
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- **Warmup:** $\approx 3N$ bars (ADX convergence + buffer fill)
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## Mathematical Foundation
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The formula is deceptively simple, but relies on the complex ADX calculation underneath.
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### Parameters
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$$ ADXR_t = \frac{ADX_t + ADX_{t-(n-1)}}{2} $$
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| Symbol | Parameter | Default | Constraint |
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|--------|-----------|---------|------------|
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| $N$ | period | 14 | $N \geq 2$ |
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Where:
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The period controls both the internal ADX calculation and the historical lookback depth.
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* $ADX_t$ is the current ADX value.
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* $n$ is the Period (typically 14).
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* $ADX_{t-(n-1)}$ is the ADX value from `n-1` periods ago.
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### Pseudo-code
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*Note: The `n-1` lag is used to match TA-Lib's implementation exactly. Some sources cite `n`, but standard reference implementations use `n-1`.*
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```
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Initialize:
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adx = new Adx(period)
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adxBuffer = RingBuffer(period)
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bar_count = 0
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## Performance Profile
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On each bar (high, low, close, isNew):
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if !isNew: restore previous state
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The performance cost is dominated by the underlying ADX calculation. The ADXR step itself is trivial.
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// Full ADX pipeline
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adxValue = adx.Update(high, low, close, isNew)
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### Zero-Allocation Design
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// Store in history
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adxBuffer.Add(adxValue)
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bar_count++
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The implementation uses a circular buffer (`RingBuffer`) to store historical ADX values, ensuring O(1) access and zero heap allocations during the update cycle.
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// ADXR = average of current and (N-1)-lagged ADX
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if bar_count >= period:
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historicalAdx = adxBuffer[0] // oldest value in buffer
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ADXR = (adxValue + historicalAdx) / 2.0
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else:
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ADXR = adxValue // insufficient history
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| Metric | Score | Notes |
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| :--- | :--- | :--- |
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| **Throughput** | 6ns | 6ns / bar (Apple M1 Max). |
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| **Allocations** | 0 | Hot path is allocation-free. |
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| **Complexity** | O(1) | Ring buffer access is constant time. |
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| **Accuracy** | 10/10 | Matches TA-Lib to 1e-9. |
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| **Timeliness** | 1/10 | Double lag (ADX + History). |
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| **Overshoot** | 10/10 | Extremely stable. |
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| **Smoothness** | 10/10 | Extremely stable trend rating. |
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output = ADXR
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```
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## Validation
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### Lag Analysis
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Validation is performed against industry-standard libraries.
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| Component | Lag Source |
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|-----------|-----------|
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| DM → RMA | $\approx N$ bars (Wilder smoothing) |
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| DX → ADX | $\approx N$ bars (second RMA) |
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| ADX → ADXR | $N-1$ bars (historical average) |
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| **Total effective lag** | $\approx 3N - 1$ bars |
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| Library | Status | Notes |
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| :--- | :--- | :--- |
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| **QuanTAlib** | ✅ | Validated. |
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| **TA-Lib** | ✅ | Matches `TA_ADXR` to 1e-9. |
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| **Skender** | N/A | Not implemented in Skender. |
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| **Tulip** | ✅ | Matches `ti.adxr`. |
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| **Ooples** | N/A | Not implemented. |
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For the default period of 14, ADXR carries roughly 41 bars of effective lag. This is a feature, not a limitation — it ensures that only sustained regime changes register in the output.
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### Common Pitfalls
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### Regime Classification
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* **Using for Entries**: Do not use ADXR crossovers for entries. The signal is too late.
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* **Short Periods**: Using a short period (e.g., 3) defeats the purpose of ADXR. If you want responsiveness, use ADX. ADXR is for stability.
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| ADXR Value | Interpretation |
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|------------|----------------|
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| < 20 | Sustained range-bound; favor mean-reversion |
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| 20–25 | Ambiguous regime; reduce position sizing |
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| > 25 | Sustained trending; favor momentum strategies |
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## Resources
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- Wilder, J.W. — *New Concepts in Technical Trading Systems* (Trend Research, 1978)
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- PineScript reference: `adxr.pine` in indicator directory
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