Fractal Velocity Accelerator [JOAT]

Fractal Velocity Accelerator [JOAT]

Introduction

The Fractal Velocity Accelerator is an advanced open-source momentum indicator that combines fractal efficiency measurement, adaptive Laguerre filtering, and Gaussian smoothing to create a multi-dimensional momentum oscillator with institutional-grade signal generation. This indicator transforms raw price data into a sophisticated momentum measurement system that reveals not just momentum direction and strength, but also velocity, acceleration, and regime characteristics.

Unlike traditional momentum indicators that simply measure rate of change, this system analyzes the efficiency of price movement through fractal mathematics, applies adaptive lag reduction through Laguerre transforms, and smooths data using 4th-order Gauss filters. The result is a momentum oscillator that responds quickly to genuine momentum shifts while filtering out noise and false signals.

Why This Indicator Exists

This indicator addresses fundamental limitations in traditional momentum analysis by introducing fractal efficiency concepts and adaptive filtering:

• 4th-Order Gauss Filter: Ultra-smooth OHLC data processing that eliminates noise while preserving genuine price movements
  
• Fractal Efficiency Engine: Logarithmic path efficiency measurement that quantifies how directly price moves from point A to point B
  
• Adaptive Laguerre Transform: Dynamic lag reduction that adjusts based on fractal efficiency, responding faster during efficient moves
  
• Percentile-Based Bands: Self-adjusting overbought/oversold zones that adapt to each instrument's unique momentum characteristics
  
• Velocity and Acceleration Tracking: First and second derivative calculations that identify momentum shifts before they're obvious
  
• Momentum Regime Classification: Seven-level regime system from Extreme Bearish to Extreme Bullish with confidence measurements
  
• Divergence Detection: Fractal-based divergence scanner that identifies price-momentum asymmetries
  

Each component provides unique intelligence about momentum dynamics. Gauss filtering ensures clean data, fractal efficiency measures directional clarity, Laguerre adaptation reduces lag, percentile bands provide context, velocity/acceleration track changes, regime classification guides strategy, and divergences reveal hidden shifts.



Core Components Explained

1. 4th-Order Gauss Filter System

The indicator applies a sophisticated Gaussian filter to all OHLC data:

w = (2.0 * math.pi / gaussLength)
beta = (1 - math.cos(w)) / (math.pow(1.414, 2.0 / betaDev) - 1)
alpha = (-beta + math.sqrt(beta * beta + 2 * beta))

Gc := math.pow(alpha, 4) * close + 
      4 * (1.0 - alpha) * nz(Gc[1]) - 
      6 * math.pow(1 - alpha, 2) * nz(Gc[2]) + 
      4 * math.pow(1 - alpha, 3) * nz(Gc[3]) - 
      math.pow(1 - alpha, 4) * nz(Gc[4])

This 4th-order filter provides exceptional smoothing while maintaining responsiveness. The filter uses four previous values with specific weightings that create a bell curve response, eliminating high-frequency noise while preserving genuine price movements.

The beta deviation parameter (default 2.0) controls filter aggressiveness. Higher values create more smoothing but add lag. Lower values maintain responsiveness but allow more noise. The default balances these tradeoffs optimally for most instruments.

2. Fractal Efficiency Calculation

Fractal efficiency measures how efficiently price moves by comparing net displacement to total path length:

sumRange = math.sum((math.max(Gh, nz(Gc[1])) - math.min(Gl, nz(Gc[1]))), fractalLength)
totalRange = ta.highest(Gh, fractalLength) - ta.lowest(Gl, fractalLength)

fractalGamma = if totalRange > 0
    math.log(sumRange / totalRange) / math.log(fractalLength)
else
    0.0

fractalEfficiency = math.max(0, math.min(1, (fractalGamma + 1) / 2))

The calculation uses logarithmic scaling to measure path complexity. When price moves in a straight line (high efficiency), the ratio approaches 1.0. When price moves erratically (low efficiency), the ratio approaches 0.0.

Fractal efficiency is normalized to 0-1 range where:
- 1.0 = Perfect efficiency (straight line movement)
- 0.7-1.0 = High efficiency (strong trending)
- 0.4-0.7 = Moderate efficiency (developing trend)
- 0.0-0.4 = Low efficiency (choppy/ranging)

This measurement is crucial because it determines how aggressively the Laguerre filter adapts.

3. Adaptive Laguerre Transform

The Laguerre filter applies adaptive lag reduction based on fractal efficiency:

gamma = laguerreGamma * (1 - fractalEfficiency) + 0.1 * fractalEfficiency

L0 := (1 - gamma) * Gc + gamma * nz(L0[1])
L1 := -gamma * L0 + nz(L0[1]) + gamma * nz(L1[1])
L2 := -gamma * L1 + nz(L1[1]) + gamma * nz(L2[1])
L3 := -gamma * L2 + nz(L2[1]) + gamma * nz(L3[1])

cu = (L0 > L1 ? L0 - L1 : 0) + (L1 > L2 ? L1 - L2 : 0) + (L2 > L3 ? L2 - L3 : 0)
cd = (L0 < L1 ? L1 - L0 : 0) + (L1 < L2 ? L2 - L1 : 0) + (L2 < L3 ? L3 - L2 : 0)

laguerreRSI = cu + cd != 0 ? 100 * (cu / (cu + cd)) : 50

The Laguerre transform creates four cascading filters (L0-L3) that progressively smooth the data. The gamma parameter controls lag - lower gamma means less lag but more noise, higher gamma means more lag but smoother output.

The adaptive component adjusts gamma based on fractal efficiency:
- High efficiency (trending): Gamma decreases toward 0.1, reducing lag for fast response
- Low efficiency (choppy): Gamma increases toward laguerreGamma setting, adding smoothing to filter noise

The cu (count up) and cd (count down) calculations measure upward vs downward movement across the four Laguerre levels, creating an RSI-like oscillator that's far more responsive than traditional RSI.

4. Fractal Momentum Oscillator

The final momentum value combines Laguerre RSI with fractal efficiency:

rawMomentum = (laguerreRSI - 50) * (1 + fractalEfficiency)
momentumEMA = ta.ema(rawMomentum, 5)
fractalMomentum = math.max(-100, math.min(100, momentumEMA))

This calculation:
1. Centers Laguerre RSI around zero by subtracting 50
2. Amplifies the signal by (1 + fractalEfficiency), giving more weight to efficient moves
3. Smooths with 5-period EMA to reduce jitter
4. Bounds the result to -100 to +100 range

The efficiency amplification is key - during high-efficiency trending moves, momentum readings become more extreme, providing clear signals. During low-efficiency choppy moves, momentum readings stay muted, preventing false signals.

5. Velocity and Acceleration Tracking

The indicator calculates first and second derivatives of momentum:

momentumVelocity = ta.change(fractalMomentum, 1)
momentumAcceleration = ta.change(momentumVelocity, 1)
velocityEMA = ta.ema(momentumVelocity, 3)

Velocity (first derivative) shows the rate of momentum change. Positive velocity means momentum is increasing, negative velocity means momentum is decreasing.

Acceleration (second derivative) shows the rate of velocity change. Positive acceleration means velocity is increasing (momentum gaining speed). Negative acceleration means velocity is decreasing (momentum losing speed).

These metrics provide early warning of momentum shifts:
- Positive momentum + positive velocity + positive acceleration = Strong bullish momentum building
- Positive momentum + positive velocity + negative acceleration = Bullish momentum slowing (potential top)
- Positive momentum + negative velocity = Bullish momentum fading (reversal warning)

6. Momentum Regime Classification

The indicator classifies momentum into seven regimes:

• Extreme Bullish: Momentum > threshold (default 60), very strong upward pressure
  
• Strong Bullish: Momentum 40-60, solid upward pressure
  
• Weak Bullish: Momentum 20-40, mild upward pressure
  
• Neutral: Momentum -20 to +20, balanced conditions
  
• Weak Bearish: Momentum -40 to -20, mild downward pressure
  
• Strong Bearish: Momentum -60 to -40, solid downward pressure
  
• Extreme Bearish: Momentum < -threshold, very strong downward pressure
  

Each regime includes confidence measurement equal to the absolute momentum value. Higher confidence indicates stronger regime conviction.

7. Adaptive Band System

The indicator uses percentile-based bands that adapt to each instrument:

momentumPercentile = ta.percentrank(fractalMomentum, bandLength)
dynamicOB = ta.percentile_linear_interpolation(fractalMomentum, bandLength, obLevel)
dynamicOS = ta.percentile_linear_interpolation(fractalMomentum, bandLength, 100 - obLevel)

These bands automatically adjust to the instrument's typical momentum range. An instrument that frequently reaches ±80 will have wider bands than one that typically stays within ±40. This prevents false overbought/oversold signals on volatile instruments and ensures sensitivity on stable instruments.

8. Fractal Divergence Detection

The indicator detects divergences using fractal pivot analysis:

momentumHigh = ta.pivothigh(fractalMomentum, divLookback, divLookback)
momentumLow = ta.pivotlow(fractalMomentum, divLookback, divLookback)

bullishDiv := lastPrice < prevPrice and lastMomentum > prevMomentum and lastMomentum < 0
bearishDiv := lastPrice > prevPrice and lastMomentum < prevMomentum and lastMomentum > 0

Regular divergences signal potential reversals:
- Bullish: Price makes lower low, momentum makes higher low (selling pressure weakening)
- Bearish: Price makes higher high, momentum makes lower high (buying pressure weakening)

Hidden divergences signal trend continuation:
- Hidden Bullish: Price makes higher low, momentum makes lower low (trend resumption after pullback)
- Hidden Bearish: Price makes lower high, momentum makes higher high (downtrend resumption after bounce)

Visual Elements

• Multi-Layer Momentum Line: Three overlaid plots (white underlay, gradient middle, solid core) creating depth and visibility
  
• Velocity Histogram: Histogram showing momentum velocity scaled 10x for visibility
  
• Adaptive Bands: Dynamic overbought/oversold lines that adjust to instrument characteristics
  
• Zone Fills: Gradient fills between bands and zero line showing bullish/bearish zones
  
• Reference Lines: Horizontal lines at extreme (±60), strong (±40), and weak (±20) levels
  
• Regime Background: Subtle background coloring showing current momentum regime
  
• Divergence Labels: Text labels marking regular and hidden divergences
  
• Reversal Signals: Labels marking extreme momentum reversals
  
• Velocity Signals: Small labels marking velocity acceleration/deceleration
  
• Comprehensive Dashboard: 14-row intelligence panel showing momentum value, regime, velocity, acceleration, efficiency, Laguerre RSI, trend strength, consistency, adaptive bands, and divergence status
  

The dashboard provides complete momentum intelligence with color-coded metrics and status indicators.



Input Parameters

Signal Architecture:

• Extreme Momentum Reversals: Toggle high-confidence exhaustion signals (default enabled)
  
• Fractal Divergence Detection: Toggle price-momentum asymmetry detection (default enabled)
  
• Velocity Acceleration Alerts: Toggle momentum acceleration warnings (default enabled)
  
• Extreme Momentum Threshold: Score required for extreme classification (40-90, default 60)
  

Gauss Filter:

• Gauss Filter Length: Smoothing period (5-100, default 20)
  
• Beta Deviation: Filter aggressiveness (0.5-5.0, default 2.0)
  

Fractal Engine:

• Fractal Efficiency Length: Efficiency calculation period (10-200, default 50)
  

Laguerre Transform:

• Laguerre Gamma: Base lag parameter (0.1-0.99, default 0.7)
  

Adaptive Bands:

• Band Percentile Length: Percentile calculation period (20-500, default 100)
  
• Overbought Level: Upper band percentile (50-95, default 75)
  
• Oversold Level: Lower band percentile (5-50, default 25)
  

Divergence:

• Enable Divergence Scanner: Toggle divergence detection (default enabled)
  
• Divergence Lookback: Pivot detection period (3-20, default 5)
  

Visualization:

• Momentum Intelligence Panel: Toggle dashboard (default enabled)
  
• Momentum Regime Zones: Toggle background coloring (default enabled)
  
• Velocity Histogram: Toggle velocity display (default enabled)
  
• Dashboard Scale: Small/Normal/Large sizing (default Normal)
  

Colors:
All colors fully customizable including bullish momentum (neon cyan), bearish momentum (neon pink), extreme bullish (neon green), extreme bearish (neon red), neutral (gold), and divergence (neon purple).

How to Use This Indicator

Step 1: Assess Momentum Value and Direction
Check dashboard "MOMENTUM" value and direction. Positive values indicate bullish momentum, negative indicate bearish. Values above 60 or below -60 suggest extreme conditions that may precede reversals or strong continuations.

Step 2: Identify Current Regime
Review "REGIME" classification and confidence percentage. Extreme regimes with high confidence (>80%) indicate strong momentum that typically continues. Weak regimes suggest transitional conditions.

Step 3: Monitor Velocity and Acceleration
Check "VELOCITY" and "ACCEL" metrics. Positive velocity with positive acceleration suggests momentum is building. Negative acceleration while momentum is still positive warns of potential momentum exhaustion.

Step 4: Evaluate Fractal Efficiency
Review "EFFICIENCY" percentage. High efficiency (>70%) confirms that momentum is backed by clean, directional price movement. Low efficiency (<40%) suggests choppy conditions where momentum signals may be less reliable.

Step 5: Check Adaptive Bands
Monitor "OB LEVEL" and "OS LEVEL" showing dynamic overbought/oversold thresholds. When momentum exceeds these levels, watch for reversal signals or continuation acceleration.

Step 6: Watch for Divergences
Check "DIVERGENCE" status and look for divergence labels. Regular divergences at extreme momentum levels often precede significant reversals. Hidden divergences in established trends suggest continuation after pullbacks.

Step 7: Identify Extreme Reversals
Watch for "EXTREME REVERSAL" labels when momentum crosses from extreme territory. These high-confidence signals often mark major turning points or trend acceleration phases.

Step 8: Track Velocity Acceleration
Monitor velocity acceleration labels. "VELOCITY ACCEL" signals indicate momentum is gaining speed, often marking optimal entry timing in early trend phases.

Best Practices

• Extreme momentum reversals (>60 or <-60) are most reliable when confirmed by velocity deceleration
  
• High fractal efficiency (>70%) validates momentum signals as backed by clean price action
  
• Divergences at extreme momentum levels offer highest-probability reversal setups
  
• Velocity acceleration signals work best in early trend phases, less reliable in mature trends
  
• Adaptive bands automatically adjust to instrument volatility - respect them as dynamic thresholds
  
• Momentum regime transitions provide clear strategy adjustment points
  
• Combine momentum analysis with price action for optimal entry timing
  
• Laguerre RSI above 70 or below 30 confirms extreme momentum readings
  
• Trend strength above 60 indicates strong momentum persistence
  
• Trend consistency above 70 confirms momentum is directionally stable
  
• Hidden divergences in strong trends (momentum >40 or <-40) suggest continuation opportunities
  
• Neutral regime (-20 to +20) suggests range-bound conditions unsuitable for momentum strategies
  

Indicator Limitations

• Momentum indicators are lagging by nature - they confirm trends rather than predict them
  
• Extreme momentum can persist longer than expected during strong trends
  
• Fractal efficiency requires sufficient price history - may be unreliable on newly listed instruments
  
• Gauss filter adds smoothing which inherently introduces some lag
  
• Adaptive bands require adequate history for percentile calculations
  
• Divergences can persist for extended periods before price responds
  
• The indicator works best on liquid instruments with consistent price action
  
• Very low timeframes may produce excessive noise despite filtering
  
• Velocity and acceleration are sensitive to sudden price spikes
  
• Regime classification is probabilistic, not deterministic
  
• The indicator shows momentum dynamics but cannot predict duration
  

Technical Implementation

Built with Pine Script v6 using:

• 4th-order Gaussian filter with customizable beta deviation
  
• Logarithmic fractal efficiency calculation using path complexity measurement
  
• Adaptive Laguerre transform with four cascading filter levels
  
• Fractal momentum oscillator combining Laguerre RSI with efficiency amplification
  
• First and second derivative calculations for velocity and acceleration
  
• Seven-level momentum regime classification with confidence measurement
  
• Percentile-based adaptive bands using linear interpolation
  
• Fractal pivot-based divergence detection system
  
• Multi-layer gradient visualization with depth effects
  
• Comprehensive dashboard with 14 metrics and color-coded indicators
  
• Alert system for reversals, divergences, and velocity signals
  

The code is fully open-source with extensive comments explaining fractal mathematics and adaptive filtering concepts.

Originality Statement

This indicator is original in its integration of fractal efficiency with adaptive momentum measurement. While individual components exist, this indicator is justified because:

• It combines 4th-order Gauss filtering with fractal efficiency and Laguerre transforms in a unified system
  
• The adaptive Laguerre gamma adjustment based on fractal efficiency is a novel approach to lag reduction
  
• Fractal momentum amplification using efficiency multiplier creates regime-aware momentum measurement
  
• Velocity and acceleration tracking provides multi-dimensional momentum analysis
  
• Seven-level regime classification with confidence measurement guides strategy selection
  
• Percentile-based adaptive bands automatically adjust to each instrument's characteristics
  
• Fractal pivot-based divergence detection identifies asymmetries with statistical precision
  
• The comprehensive dashboard synthesizes 14 distinct metrics into unified momentum intelligence
  
• Multi-layer visualization with gradient effects provides exceptional clarity
  
• Integration of efficiency, velocity, acceleration, and regime creates layered confirmation
  

Each component contributes unique intelligence: Gauss filtering ensures clean data, fractal efficiency measures directional quality, Laguerre adaptation reduces lag, momentum oscillator quantifies strength, velocity tracks changes, acceleration identifies inflections, regime classification guides strategy, bands provide context, and divergences reveal hidden shifts. The indicator's value lies in combining these complementary perspectives into a cohesive, adaptive momentum system.

Disclaimer

This indicator is provided for educational and informational purposes only. It is not financial advice or a recommendation to buy or sell any financial instrument. Trading involves substantial risk of loss and is not suitable for all investors.

Momentum analysis is a tool for understanding price dynamics, not a crystal ball for predicting future movement. Extreme momentum readings do not guarantee reversals. Divergences do not guarantee price response. Past momentum patterns do not guarantee future patterns. Market conditions change, and strategies that worked historically may not work in the future.

The metrics displayed are mathematical calculations based on current market data, not predictions of future price movement. Momentum readings, divergences, and regime classifications do not guarantee profitable trades. Users must conduct their own analysis and risk assessment before making trading decisions.

Always use proper risk management, including stop losses and position sizing appropriate for your account size and risk tolerance. Never risk more than you can afford to lose. Consider consulting with a qualified financial advisor before making investment decisions.

The author is not responsible for any losses incurred from using this indicator. Users assume full responsibility for all trading decisions made using this tool.

-Made with passion by officialjackofalltrades