QuanTAlib/lib/dynamics/adxr/Adxr.md

# ADXR: Average Directional Movement Rating

> *ADXR smooths ADX over time, filtering out momentary strength spikes to reveal the underlying trend conviction.*

| Property         | Value                            |
| ---------------- | -------------------------------- |
| **Category**     | Dynamic                        |
| **Inputs**       | OHLCV bar (TBar)                          |
| **Parameters**   | `period`                      |
| **Outputs**      | Single series (Adxr)                       |
| **Output range** | Varies (see docs)                     |
| **Warmup**       | `adx.WarmupPeriod + period - 1` bars                          |
| **PineScript**   | [adxr.pine](adxr.pine)                       |

- The Average Directional Movement Rating is a smoothed version of ADX that dampens short-term fluctuations in trend strength by averaging the curren...
- **Similar:** [ADX](../adx/Adx.md), [DX](../dx/Dx.md) | **Complementary:** Aroon for trend timing | **Trading note:** Smoothed ADX; slower but fewer false signals. Used in Wilder's Directional Movement System.
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.

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.

## Historical Context

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.

## Architecture & Physics

### 1. ADX Dependency

ADXR is a composite indicator that does not interact with price directly. It instantiates and maintains a full ADX pipeline internally:

$$\text{Price} \rightarrow \text{DM/TR} \rightarrow \text{RMA} \rightarrow \text{DI} \rightarrow \text{DX} \rightarrow \text{ADX} \rightarrow \text{ADXR}$$

### 2. Historical Buffer

A circular buffer of size $N$ stores historical ADX values, providing $O(1)$ access to the value from $N-1$ bars ago.

### 3. Rating Calculation

$$ADXR_t = \frac{ADX_t + ADX_{t-(N-1)}}{2}$$

The $N-1$ lag (rather than $N$) matches TA-Lib's reference implementation exactly.

### 4. Complexity

- **Time:** $O(1)$ per bar — ADX update plus one buffer lookup and one average
- **Space:** $O(N)$ — circular buffer for ADX history
- **Warmup:** $\approx 3N$ bars (ADX convergence + buffer fill)

## Mathematical Foundation

### Parameters

| Symbol | Parameter | Default | Constraint |
|--------|-----------|---------|------------|
| $N$ | period | 14 | $N \geq 2$ |

The period controls both the internal ADX calculation and the historical lookback depth.

### Lag Analysis

| Component | Lag Source |
|-----------|-----------|
| DM → RMA | $\approx N$ bars (Wilder smoothing) |
| DX → ADX | $\approx N$ bars (second RMA) |
| ADX → ADXR | $N-1$ bars (historical average) |
| **Total effective lag** | $\approx 3N - 1$ bars |

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.

### Regime Classification

| ADXR Value | Interpretation |
|------------|----------------|
| < 20 | Sustained range-bound; favor mean-reversion |
| 20–25 | Ambiguous regime; reduce position sizing |
| > 25 | Sustained trending; favor momentum strategies |

## Performance Profile

### Operation Count (Streaming Mode)

ADXR is ADX averaged with its value N bars ago — it wraps ADX with a RingBuffer for the lag.

**Post-warmup steady state (per bar):**

| Operation | Count | Cost (cycles) | Subtotal |
| :--- | :---: | :---: | :---: |
| ADX Update (full pipeline) | 1 | ~79 | 79 |
| RingBuffer write + oldest read | 2 | 1 | 2 |
| ADD + MUL×0.5 (average: (ADX + ADX[N]) / 2) | 2 | 3 | 6 |
| CMP (IsHot guard) | 1 | 1 | 1 |
| **Total** | **6+ADX** | — | **~88 cycles** |

ADXR requires 3N bars of warmup: N for ADX initialization, N for ADX smoothing, N for the lookback buffer. For default $N=14$: ~88 cycles per bar.

### Batch Mode (SIMD Analysis)

| Operation | Vectorizable? | Notes |
| :--- | :---: | :--- |
| ADX calculation | Partial | See ADX analysis — recursive RMA blocks |
| Lag-N average | Yes | VADDPD + multiply by 0.5 once ADX array is known |

The final averaging step is trivially vectorizable once the ADX time series is materialized. The bottleneck remains the ADX RMA recursion.

### Quality Metrics

| Metric | Score | Notes |
| :--- | :---: | :--- |
| **Accuracy** | 9/10 | Exact arithmetic; double-smoothing from underlying ADX |
| **Timeliness** | 3/10 | 3N warmup + half-period average adds significant lag |
| **Smoothness** | 9/10 | Averaging two ADX instances makes it the smoothest directional indicator |
| **Noise Rejection** | 9/10 | Triple smoothing (2× RMA in ADX + final average) is highly noise-resistant |

## Resources

- Wilder, J.W. — *New Concepts in Technical Trading Systems* (Trend Research, 1978)
- PineScript reference: `adxr.pine` in indicator directory