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.
4.9 KiB
ALLIGATOR: Williams Alligator
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.
Historical Context
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.
Architecture & Physics
1. Three-Line SMMA Structure
Each line is an independent SMMA (Wilder's RMA) with \alpha = 1/N:
| Line | Period (N) |
Display Offset | Role |
|---|---|---|---|
| Jaw | 13 | 8 bars forward | Long-term trend (slowest) |
| Teeth | 8 | 5 bars forward | Intermediate trend |
| Lips | 5 | 3 bars forward | Short-term momentum (fastest) |
2. SMMA Recursion
\text{SMMA}_t = \frac{1}{N} \cdot P_t + \frac{N-1}{N} \cdot \text{SMMA}_{t-1}
Equivalently using FMA notation:
\text{SMMA}_t = \text{FMA}(\text{SMMA}_{t-1},\; \tfrac{N-1}{N},\; \tfrac{1}{N} \cdot P_t)
3. Default Input
Typical price (HLC/3):
\text{Source} = \frac{H + L + C}{3}
4. Forward Offset
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.
5. Complexity
- Time:
O(1)per bar — three parallel SMMA updates - Space:
O(1)— three scalar states (no buffers needed) - Warmup: 13 bars (Jaw period, the slowest line)
Mathematical Foundation
Parameters
| Symbol | Parameter | Default | Constraint |
|---|---|---|---|
N_j |
jawPeriod | 13 | N_j \geq 1 |
O_j |
jawOffset | 8 | O_j \geq 0 |
N_t |
teethPeriod | 8 | N_t \geq 1 |
O_t |
teethOffset | 5 | O_t \geq 0 |
N_l |
lipsPeriod | 5 | N_l \geq 1 |
O_l |
lipsOffset | 3 | O_l \geq 0 |
Pseudo-code
Initialize:
α_jaw = 1 / jawPeriod
α_teeth = 1 / teethPeriod
α_lips = 1 / lipsPeriod
jaw = teeth = lips = first source value
e_jaw = e_teeth = e_lips = 1.0 // bias compensation
On each bar (high, low, close, isNew):
if !isNew: restore previous state
source = (high + low + close) / 3.0
// SMMA updates with bias compensation
jaw = FMA(jaw, 1 - α_jaw, α_jaw × source)
e_jaw = e_jaw × (1 - α_jaw)
jaw_compensated = jaw / (1 - e_jaw)
teeth = FMA(teeth, 1 - α_teeth, α_teeth × source)
e_teeth = e_teeth × (1 - α_teeth)
teeth_compensated = teeth / (1 - e_teeth)
lips = FMA(lips, 1 - α_lips, α_lips × source)
e_lips = e_lips × (1 - α_lips)
lips_compensated = lips / (1 - e_lips)
output:
Jaw = jaw_compensated (plot at bar + jawOffset)
Teeth = teeth_compensated (plot at bar + teethOffset)
Lips = lips_compensated (plot at bar + lipsOffset)
Market Phase Detection
| Phase | Line Configuration | Action |
|---|---|---|
| Sleeping | Lines intertwined, crossing | No position; market is consolidating |
| Awakening | Lines begin separating | Prepare for entry |
| Eating (bullish) | Lips > Teeth > Jaw, all rising | Long; trend is strong |
| Eating (bearish) | Lips < Teeth < Jaw, all falling | Short; trend is strong |
| Sated | Lines converging | Take profits; trend weakening |
Output Interpretation
- Three values per bar: Jaw, Teeth, Lips (each a smoothed price level)
- Separation width: Proportional to trend strength
- Line ordering: Determines trend direction
- Intertwining: Signals consolidation — the highest-probability losing zone for trend followers
Resources
- Williams, B. — Trading Chaos (John Wiley & Sons, 1995)
- Williams, B. — New Trading Dimensions (John Wiley & Sons, 1998)
- PineScript reference:
alligator.pinein indicator directory