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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.
114 lines
4.9 KiB
Markdown
114 lines
4.9 KiB
Markdown
# ALLIGATOR: Williams Alligator
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The Williams Alligator is a trend-following system that uses three Smoothed Moving Averages (SMMA/RMA) with different periods and forward display offsets to visualize market phases. The Jaw (13-period, offset 8), Teeth (8-period, offset 5), and Lips (5-period, offset 3) create a layered structure where intertwined lines indicate consolidation ("sleeping") and separated, aligned lines indicate trending conditions ("eating"). The metaphor maps directly to position management: stay out when the alligator sleeps, ride when it eats. Each line uses Wilder's smoothing ($\alpha = 1/N$), which is heavier than standard EMA, providing superior noise rejection at the cost of additional lag.
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## Historical Context
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Bill Williams introduced the Alligator in *Trading Chaos* (1995) as part of his broader chaos theory framework for trading. The metaphor is biological: markets alternate between feeding (trending) and sleeping (ranging) states, and the three moving averages at different timescales reveal which phase is active. The Jaw represents the long-term balance line (the "blue line" on most charting platforms), the Teeth the intermediate balance (red), and the Lips the short-term momentum (green). Williams paired the Alligator with Fractals for entry timing and the Awesome Oscillator for momentum confirmation, creating a complete systematic framework. The forward offsets are display-only transformations — the underlying SMMA calculation uses the current bar's price — but they create visual separation that makes trend direction immediately apparent on charts.
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## Architecture & Physics
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### 1. Three-Line SMMA Structure
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Each line is an independent SMMA (Wilder's RMA) with $\alpha = 1/N$:
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| Line | Period ($N$) | Display Offset | Role |
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|------|-------------|----------------|------|
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| Jaw | 13 | 8 bars forward | Long-term trend (slowest) |
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| Teeth | 8 | 5 bars forward | Intermediate trend |
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| Lips | 5 | 3 bars forward | Short-term momentum (fastest) |
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### 2. SMMA Recursion
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$$\text{SMMA}_t = \frac{1}{N} \cdot P_t + \frac{N-1}{N} \cdot \text{SMMA}_{t-1}$$
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Equivalently using FMA notation:
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$$\text{SMMA}_t = \text{FMA}(\text{SMMA}_{t-1},\; \tfrac{N-1}{N},\; \tfrac{1}{N} \cdot P_t)$$
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### 3. Default Input
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Typical price (HLC/3):
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$$\text{Source} = \frac{H + L + C}{3}$$
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### 4. Forward Offset
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The offsets shift plotted values forward in time for display purposes only. The calculation itself is not shifted — the current SMMA value represents the current bar's computation.
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### 5. Complexity
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- **Time:** $O(1)$ per bar — three parallel SMMA updates
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- **Space:** $O(1)$ — three scalar states (no buffers needed)
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- **Warmup:** 13 bars (Jaw period, the slowest line)
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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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| $N_j$ | jawPeriod | 13 | $N_j \geq 1$ |
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| $O_j$ | jawOffset | 8 | $O_j \geq 0$ |
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| $N_t$ | teethPeriod | 8 | $N_t \geq 1$ |
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| $O_t$ | teethOffset | 5 | $O_t \geq 0$ |
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| $N_l$ | lipsPeriod | 5 | $N_l \geq 1$ |
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| $O_l$ | lipsOffset | 3 | $O_l \geq 0$ |
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### Pseudo-code
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```
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Initialize:
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α_jaw = 1 / jawPeriod
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α_teeth = 1 / teethPeriod
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α_lips = 1 / lipsPeriod
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jaw = teeth = lips = first source value
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e_jaw = e_teeth = e_lips = 1.0 // bias compensation
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On each bar (high, low, close, isNew):
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if !isNew: restore previous state
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source = (high + low + close) / 3.0
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// SMMA updates with bias compensation
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jaw = FMA(jaw, 1 - α_jaw, α_jaw × source)
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e_jaw = e_jaw × (1 - α_jaw)
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jaw_compensated = jaw / (1 - e_jaw)
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teeth = FMA(teeth, 1 - α_teeth, α_teeth × source)
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e_teeth = e_teeth × (1 - α_teeth)
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teeth_compensated = teeth / (1 - e_teeth)
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lips = FMA(lips, 1 - α_lips, α_lips × source)
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e_lips = e_lips × (1 - α_lips)
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lips_compensated = lips / (1 - e_lips)
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output:
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Jaw = jaw_compensated (plot at bar + jawOffset)
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Teeth = teeth_compensated (plot at bar + teethOffset)
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Lips = lips_compensated (plot at bar + lipsOffset)
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```
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### Market Phase Detection
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| Phase | Line Configuration | Action |
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|-------|-------------------|--------|
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| Sleeping | Lines intertwined, crossing | No position; market is consolidating |
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| Awakening | Lines begin separating | Prepare for entry |
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| Eating (bullish) | Lips > Teeth > Jaw, all rising | Long; trend is strong |
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| Eating (bearish) | Lips < Teeth < Jaw, all falling | Short; trend is strong |
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| Sated | Lines converging | Take profits; trend weakening |
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### Output Interpretation
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- **Three values per bar:** Jaw, Teeth, Lips (each a smoothed price level)
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- **Separation width:** Proportional to trend strength
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- **Line ordering:** Determines trend direction
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- **Intertwining:** Signals consolidation — the highest-probability losing zone for trend followers
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## Resources
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- Williams, B. — *Trading Chaos* (John Wiley & Sons, 1995)
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- Williams, B. — *New Trading Dimensions* (John Wiley & Sons, 1998)
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- PineScript reference: `alligator.pine` in indicator directory
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