QuanTAlib/lib/cycles/phasor/phasor.md

PHASOR: Phasor Analysis (Ehlers)

Pine Script Implementation of Phasor

Overview and Purpose

The Phasor Analysis indicator, developed by John Ehlers, represents an advanced cycle analysis tool that identifies the phase of the dominant cycle component in a time series through complex signal processing techniques. This sophisticated indicator uses correlation-based methods to determine the real and imaginary components of the signal, converting them to a continuous phase angle that reveals market cycle progression. Unlike traditional oscillators, the Phasor provides unwrapped phase measurements that accumulate continuously, offering unique insights into market timing and cycle behavior.

Core Concepts

• Complex Signal Analysis — Uses real and imaginary components to determine cycle phase
• Correlation-Based Detection — Employs Ehlers' correlation method for robust phase estimation
• Unwrapped Phase Tracking — Provides continuous phase accumulation without discontinuities
• Anti-Regression Logic — Prevents phase angle from moving backward under specific conditions

Market Applications:

• Cycle Timing — Precise identification of cycle peaks and troughs
• Market Regime Analysis — Distinguishes between trending and cycling market conditions
• Turning Point Detection — Advanced warning system for potential market reversals

Common Settings and Parameters

Parameter	Default	Function	When to Adjust
Period	28	Fixed cycle period for correlation analysis	Match to expected dominant cycle length
Source	Close	Price series for phase calculation	Use typical price or other smoothed series
Show Derived Period	false	Display calculated period from phase rate	Enable for adaptive period analysis
Show Trend State	false	Display trend/cycle state variable	Enable for regime identification

Calculation and Mathematical Foundation

Technical Formula:

Stage 1: Correlation Analysis For period n and source x_t:

Real component correlation with cosine wave:

R = \frac{n \sum x_t \cos\left(\frac{2\pi t}{n}\right) - \sum x_t \sum \cos\left(\frac{2\pi t}{n}\right)}{\sqrt{D_{cos}}}

Imaginary component correlation with negative sine wave:

I = \frac{n \sum x_t \left(-\sin\left(\frac{2\pi t}{n}\right)\right) - \sum x_t \sum \left(-\sin\left(\frac{2\pi t}{n}\right)\right)}{\sqrt{D_{sin}}}

where D_{cos} and D_{sin} are normalization denominators.

Stage 2: Phase Angle Conversion $$\theta_{raw} = \begin{cases} 90° - \arctan\left(\frac{I}{R}\right) \cdot \frac{180°}{\pi} & \text{if } R \neq 0 \ 0° & \text{if } R = 0, I > 0 \ 180° & \text{if } R = 0, I \leq 0 \end{cases}$$

Stage 3: Phase Unwrapping

\theta_{unwrapped}(t) = \theta_{unwrapped}(t-1) + \Delta\theta

where \Delta\theta is the normalized phase difference.

Stage 4: Ehlers' Anti-Regression Condition $$\theta_{final}(t) = \begin{cases} \theta_{final}(t-1) & \text{if regression conditions met} \ \theta_{unwrapped}(t) & \text{otherwise} \end{cases}$$

Derived Calculations:

Derived Period: P_{derived} = \frac{360°}{\Delta\theta_{final}} (clamped to [1, 60])

Trend State: $$S_{trend} = \begin{cases} 1 & \text{if } \Delta\theta \leq 6° \text{ and } |\theta| \geq 90° \ -1 & \text{if } \Delta\theta \leq 6° \text{ and } |\theta| < 90° \ 0 & \text{if } \Delta\theta > 6° \end{cases}$$

🔍 Technical Note: The correlation-based approach provides robust phase estimation even in noisy market conditions, while the unwrapping mechanism ensures continuous phase tracking across cycle boundaries.

Interpretation Details

• Phasor Angle (Primary Output):

  • +90°: Potential cycle peak region
  • 0°: Mid-cycle ascending phase
  • -90°: Potential cycle trough region
  • ±180°: Mid-cycle descending phase

• Phase Progression:

  • Continuous upward movement → Normal cycle progression
  • Phase stalling → Potential cycle extension or trend development
  • Rapid phase changes → Cycle compression or volatility spike

• Derived Period Analysis:

  • Period < 10 → High-frequency cycle dominance
  • Period 15-40 → Typical swing trading cycles
  • Period > 50 → Trending market conditions

• Trend State Variable:

  • +1: Long trend conditions (slow phase change in extreme zones)
  • -1: Short trend or consolidation (slow phase change in neutral zones)
  • 0: Active cycling (normal phase change rate)

Applications

• Cycle-Based Trading:

  • Enter long positions near -90° crossings (cycle troughs)
  • Enter short positions near +90° crossings (cycle peaks)
  • Exit positions during mid-cycle phases (0°, ±180°)

• Market Timing:

  • Use phase acceleration for early trend detection
  • Monitor derived period for cycle length changes
  • Combine with trend state for regime-appropriate strategies

• Risk Management:

  • Adjust position sizes based on cycle clarity (derived period stability)
  • Implement different risk parameters for trending vs. cycling regimes
  • Use phase velocity for stop-loss placement timing

Performance Profile

Operation Count (Streaming Mode, per Bar)

For default period n = 28:

Operation	Count	Cost (cycles)	Subtotal
ADD/SUB	6n + 15 ≈ 183	1	183
MUL	4n + 8 ≈ 120	3	360
DIV	3	15	45
SQRT	2	15	30
SIN	n = 28	40	1,120
COS	n = 28	40	1,120
ATAN	1	80	80
CMP	8	1	8
Total	~390	—	~2,946 cycles

Breakdown:

• Correlation sums (over n bars): O(n) operations
  • sin/cos reference wave generation: n SIN + n COS = ~2,240 cycles
  • Σ(x·cos), Σ(x·sin), Σx, Σcos, Σsin: 4n MUL + 5n ADD = ~364 cycles
• Correlation coefficients (R, I):
  • Numerators: 4 MUL + 4 ADD/SUB = ~16 cycles
  • Denominators (D_cos, D_sin): 6 MUL + 4 SUB + 2 SQRT = ~68 cycles
  • Final division: 2 DIV = ~30 cycles
• Phase angle conversion: 1 DIV + 1 ATAN + quadrant logic = ~98 cycles
• Phase unwrapping: 4 ADD/SUB + comparisons = ~10 cycles
• Anti-regression check: comparisons + conditional = ~5 cycles
• Derived outputs (optional):
  • Derived period: 1 DIV + clamp = ~20 cycles
  • Trend state: comparisons = ~5 cycles

Complexity Analysis

Mode	Complexity	Notes
Streaming	O(n)	Correlation requires full period window scan
Batch	O(m·n)	m bars × n period = quadratic in total work

Memory: ~(8n + 80) bytes ≈ 304 bytes for n=28 (correlation buffers + phase state)

SIMD Analysis

Optimization	Applicable	Notes
AVX2 vectorization	Partial	Correlation sums vectorizable; phase logic scalar
FMA	✅	Correlation products: x[i] * cos[i] + sum
SVML trig	✅	Precompute sin/cos tables for fixed period
Batch parallelism	❌	Phase unwrapping requires sequential processing

Optimization strategies:

1. Precompute trig tables: For fixed period, sin/cos values are constant

   • Reduces 2,240 cycles to ~56 cycles (table lookup)
   • Optimized total: ~762 cycles (3.9× speedup)

2. Running sums: Maintain Σx, Σ(x·cos), Σ(x·sin) incrementally

   • Update: add new term, subtract oldest = O(1) per bar
   • Streaming optimized: ~200 cycles with precomputed trig + running sums

3. SIMD correlation: AVX2 can process 4 doubles simultaneously

   • Correlation sums: ~90 cycles (vs 364 scalar)

Quality Metrics

Metric	Score	Notes
Accuracy	8/10	Robust correlation-based phase estimation
Timeliness	6/10	Inherent lag from period-length lookback
Noise Immunity	8/10	Correlation averaging filters noise well
Adaptability	5/10	Fixed period assumption limits adaptation

Limitations and Considerations

• Parameter Sensitivity:

  • Fixed period assumption may not match actual market cycles
  • Requires cycle period optimization for different markets and timeframes
  • Performance degrades when multiple cycles interfere

• Computational Complexity:

  • Correlation calculations over full period windows
  • Multiple mathematical transformations increase processing requirements
  • Real-time implementation requires efficient algorithms

• Market Conditions:

  • Most effective in markets with clear cyclical behavior
  • May provide false signals during strong trending periods
  • Requires sufficient historical data for correlation analysis

Complementary Indicators:

• MESA Adaptive Moving Average (cycle-based smoothing)
• Dominant Cycle Period indicators
• Detrended Price Oscillator (cycle identification)

References

1. Ehlers, J.F. "Cycle Analytics for Traders." Wiley, 2013.
2. Ehlers, J.F. "Cybernetic Analysis for Stocks and Futures." Wiley, 2004.