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Miha Kralj
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- 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.
80 lines
3.0 KiB
Markdown
80 lines
3.0 KiB
Markdown
# CG: Ehlers Center of Gravity
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CG identifies potential turning points using the physics concept of weighted center of mass applied to a price window. Developed by John Ehlers, the oscillator measures where the "weight" of prices is concentrated within a lookback period, producing a leading indicator that oscillates around zero with minimal lag compared to traditional moving average crossover systems.
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## Historical Context
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John Ehlers introduced the Center of Gravity oscillator in *Cybernetic Analysis for Stocks and Futures* (2002). Drawing from classical mechanics, the indicator applies the concept that the center of mass of a distribution reveals its balance point. In the price context, the CG identifies where momentum is concentrated within a sliding window. Unlike momentum oscillators that differentiate price (and amplify noise), CG integrates position-weighted price, providing smoother turning point detection. The indicator's leading characteristic arises from the weighting scheme: as new prices shift the balance point, the CG responds before the window's simple average would.
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## Architecture & Physics
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### 1. Weighted Sum (Numerator)
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Position-weighted accumulation over the lookback window:
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$$Num = \sum_{i=1}^{n} i \cdot P_{t-n+i}$$
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where $i$ ranges from 1 (oldest) to $n$ (newest), giving linearly increasing weight to more recent data.
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### 2. Simple Sum (Denominator)
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$$Den = \sum_{i=1}^{n} P_{t-n+i}$$
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### 3. Center of Gravity
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$$CG_t = \frac{Num}{Den} - \frac{n + 1}{2}$$
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The term $\frac{n + 1}{2}$ is the geometric center of the window, centering the output around zero. When recent prices dominate, $CG > 0$ (bullish); when older prices dominate, $CG < 0$ (bearish).
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### 4. Complexity
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Streaming uses running sums for both numerator and denominator: $O(1)$ per bar with $O(n)$ memory for the ring buffer.
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## Mathematical Foundation
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### Parameters
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| Parameter | Description | Default | Constraint |
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|-----------|-------------|---------|------------|
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| `period` | Lookback window length | 10 | $> 0$ |
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### Pseudo-code
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```
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function CG(source, period):
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buffer ← RingBuffer(period)
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runNum ← 0 // weighted sum
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runDen ← 0 // simple sum
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for each price in source:
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buffer.Add(price)
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if buffer.Count < period: continue
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// Compute from buffer (or maintain running sums)
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num = 0
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den = 0
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for i = 0 to period-1:
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w = i + 1
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num += w * buffer[i]
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den += buffer[i]
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cg = (den ≠ 0) ? (num / den) - (period + 1) / 2.0 : 0
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emit cg
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```
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### Output Interpretation
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| Condition | Meaning |
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|-----------|---------|
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| $CG > 0$ | Weight concentrated in recent prices (bullish momentum) |
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| $CG < 0$ | Weight concentrated in older prices (bearish momentum) |
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| Zero crossing up | Momentum shifting bullish |
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| Zero crossing down | Momentum shifting bearish |
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| Hanging at extremes | Strong trend in progress |
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## Resources
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- **Ehlers, J.F.** *Cybernetic Analysis for Stocks and Futures*. Wiley, 2002.
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- **Ehlers, J.F.** *Rocket Science for Traders*. Wiley, 2001.
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