Ehlers Phasor Analysis (PHASOR)

# PHASOR: Phasor Analysis (Ehlers)

## 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

## 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.