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https://github.com/mihakralj/QuanTAlib.git
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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.
117 lines
4.6 KiB
Markdown
117 lines
4.6 KiB
Markdown
# DMX: Directional Movement Index (Jurik)
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The DMX is Mark Jurik's modernized overhaul of Wilder's Directional Movement system, replacing the sluggish RMA smoothing with the Jurik Moving Average (JMA) to achieve faster trend detection with superior noise rejection. The core directional movement logic (+DM, -DM, True Range) is preserved faithfully from Wilder, but the three parallel smoothing passes use JMA's adaptive bandwidth instead of RMA's fixed $\alpha = 1/N$. The result is a directional indicator that reacts 3-5 bars earlier to trend changes than standard DMI while filtering out more noise during consolidation. Output is the difference between smoothed directional indicators: $DMX = DI^+ - DI^-$, positive for uptrends and negative for downtrends.
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## Historical Context
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Wilder's original ADX/DMI system (1978) is foundational but mathematically primitive — its RMA smoothing introduces substantial lag that delays trend detection. Jurik's contribution was recognizing that the directional movement decomposition itself is sound; only the smoothing pipeline needed upgrading. JMA is an adaptive filter that tracks signal closely during transitions (low lag) and smooths aggressively during stable periods (high noise reduction). This dynamic behavior means DMX signals trend changes significantly earlier than DMI without the whipsaw penalty typically associated with faster indicators. DMX is not available in standard TA libraries (TA-Lib, Skender, Tulip) since JMA is a proprietary algorithm. The QuanTAlib implementation uses its own JMA recreation.
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## Architecture & Physics
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### 1. Directional Movement (Wilder's Original)
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$$\text{UpMove} = H_t - H_{t-1}, \quad \text{DownMove} = L_{t-1} - L_t$$
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$$+DM = \begin{cases} \text{UpMove} & \text{if UpMove} > \text{DownMove and UpMove} > 0 \\ 0 & \text{otherwise} \end{cases}$$
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$$-DM = \begin{cases} \text{DownMove} & \text{if DownMove} > \text{UpMove and DownMove} > 0 \\ 0 & \text{otherwise} \end{cases}$$
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### 2. True Range
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$$TR = \max(H_t - L_t,\; |H_t - C_{t-1}|,\; |L_t - C_{t-1}|)$$
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### 3. JMA Smoothing (Replaces RMA)
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Three parallel JMA filters replace Wilder's three RMA passes:
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$$+DM_{\text{smooth}} = \text{JMA}(+DM, N)$$
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$$-DM_{\text{smooth}} = \text{JMA}(-DM, N)$$
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$$TR_{\text{smooth}} = \text{JMA}(TR, N)$$
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### 4. Directional Indicators
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$$DI^+ = 100 \times \frac{+DM_{\text{smooth}}}{TR_{\text{smooth}}}, \quad DI^- = 100 \times \frac{-DM_{\text{smooth}}}{TR_{\text{smooth}}}$$
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### 5. DMX Output
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$$DMX = DI^+ - DI^-$$
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Positive values indicate bullish directional dominance; negative values indicate bearish.
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### 6. Complexity
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- **Time:** $O(1)$ per bar — three JMA updates (each $O(1)$)
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- **Space:** $O(1)$ — JMA maintains fixed-size internal state
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- **Warmup:** $\approx N$ bars (JMA converges faster than RMA)
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## Mathematical Foundation
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### Parameters
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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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### Pseudo-code
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```
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Initialize:
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jmaPlusDM = new JMA(period)
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jmaMinusDM = new JMA(period)
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jmaTR = new JMA(period)
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prevHigh = prevLow = prevClose = NaN
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On each bar (high, low, close, isNew):
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if !isNew: restore previous state
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// Wilder's directional movement decomposition
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TR = max(high - low, |high - prevClose|, |low - prevClose|)
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upMove = high - prevHigh
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downMove = prevLow - low
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+DM = (upMove > downMove AND upMove > 0) ? upMove : 0
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-DM = (downMove > upMove AND downMove > 0) ? downMove : 0
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// Jurik smoothing (replaces Wilder's RMA)
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smoothPlusDM = jmaPlusDM.Update(+DM)
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smoothMinusDM = jmaMinusDM.Update(-DM)
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smoothTR = jmaTR.Update(TR)
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// Directional indicators
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if smoothTR > 0:
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DI_plus = 100 × smoothPlusDM / smoothTR
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DI_minus = 100 × smoothMinusDM / smoothTR
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else:
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DI_plus = DI_minus = 0
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DMX = DI_plus - DI_minus
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prevHigh = high
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prevLow = low
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prevClose = close
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output = DMX
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```
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### DMX vs DMI Comparison
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| Property | DMI (Wilder) | DMX (Jurik) |
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|----------|-------------|-------------|
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| Smoothing | RMA ($\alpha = 1/N$) | JMA (adaptive) |
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| Lag | $\approx N$ bars | $\approx N/2$ bars |
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| Whipsaw rejection | Moderate | High |
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| Available in TA-Lib | Yes | No |
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| Overshoot | Low | Can overshoot in extreme volatility |
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### Period Selection
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Because JMA is more efficient than RMA, slightly longer periods (e.g., 20 instead of 14) can be used without incurring a lag penalty, producing smoother results while maintaining responsiveness.
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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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- Jurik, M. — JMA adaptive smoothing methodology
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- PineScript reference: `dmx.pine` in indicator directory
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