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.
4.3 KiB
FCB: Fractal Chaos Bands
Fractal Chaos Bands filter raw price action through Bill Williams' fractal detection logic, tracking the highest confirmed fractal high and lowest confirmed fractal low over a lookback period. Unlike Donchian Channels which use every bar's high and low, FCB uses only structurally significant turning points — bars where the middle element of a 3-bar pattern is a local extremum. The result is a "cleaner" channel that ignores transient spikes and focuses on confirmed support and resistance levels. The bands tend to remain flat during trends and step discretely when new structural pivots form, making them useful for identifying genuine breakouts versus noise.
Historical Context
Bill Williams introduced fractal analysis to financial markets in Trading Chaos (1995) and New Trading Dimensions (1998), drawing inspiration from Benoit Mandelbrot's work on fractal geometry and chaos theory. Williams defined a fractal as a 5-bar pattern (later simplified to 3-bar in many implementations) where the central bar represents a local extremum — a point where supply and demand reached temporary equilibrium.
The Fractal Chaos Bands indicator extends Williams' fractal concept by tracking the monotonic extremes of these structural turning points over a lookback window, rather than raw price extremes. This filtering eliminates noise from transient wicks and gap spikes while preserving meaningful market structure. The 3-bar fractal requires one future bar for confirmation, providing inherent stability at the cost of a 1-bar lag.
Architecture & Physics
1. Fractal Detection (3-Bar Pattern)
Up Fractal (local high at t-1):
\text{UpFractal}_t = (H_{t-1} > H_{t-2}) \;\wedge\; (H_{t-1} > H_t)
Down Fractal (local low at t-1):
\text{DownFractal}_t = (L_{t-1} < L_{t-2}) \;\wedge\; (L_{t-1} < L_t)
2. Fractal Value Tracking
A persistent state variable holds the most recent fractal value:
- On up fractal confirmation:
\text{hiFractal} = H_{t-1} - On down fractal confirmation:
\text{loFractal} = L_{t-1} - Between fractals: values persist (hold last fractal)
3. Band Construction (Sliding Window Max/Min of Fractals)
\text{Upper}_t = \max(\text{hiFractal values over period})
\text{Lower}_t = \min(\text{loFractal values over period})
The monotonic deque operates on fractal values rather than raw prices, using the same O(1) amortized algorithm as Donchian Channels.
4. Structural Filtering Property
FCB bands are always within or equal to Donchian bounds:
\text{FCB}_{\text{Upper}} \leq \text{Donchian}_{\text{Upper}}
\text{FCB}_{\text{Lower}} \geq \text{Donchian}_{\text{Lower}}
This is because fractals are a subset of raw highs/lows. A breakout above FCB upper is more significant than above Donchian upper because it breaks a confirmed structural level.
5. Complexity
Fractal detection is O(1) (3 comparisons). Deque maintenance is O(1) amortized. Total: O(1) amortized per bar.
Mathematical Foundation
Parameters
| Parameter | Description | Default | Constraint |
|---|---|---|---|
period |
Lookback window for highest/lowest fractal values | 20 | > 0 |
Pseudo-code
function FCB(high, low, period):
validate: period > 0
// 3-bar fractal detection (confirmed at current bar)
is_fractal_high = high[1] > high[2] AND high[1] > high[0]
is_fractal_low = low[1] < low[2] AND low[1] < low[0]
// Update persistent fractal values
if is_fractal_high: hi_fractal = high[1]
if is_fractal_low: lo_fractal = low[1]
// Sliding window max/min via monotonic deques
upper = max(hi_fractal over period) // deque-based
lower = min(lo_fractal over period) // deque-based
return [upper, lower]
Output Interpretation
| Output | Description |
|---|---|
upper |
Highest confirmed fractal high over lookback (structural resistance) |
lower |
Lowest confirmed fractal low over lookback (structural support) |
Resources
- Williams, B. Trading Chaos. Wiley, 1995. (Original fractal definition for markets)
- Williams, B. New Trading Dimensions. Wiley, 1998. (Extended fractal analysis)
- Mandelbrot, B. The Fractal Geometry of Nature. W.H. Freeman, 1982. (Mathematical fractal theory)