# CG: Ehlers Center of Gravity

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.

## Historical Context

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.

## Architecture & Physics

### 1. Weighted Sum (Numerator)

Position-weighted accumulation over the lookback window:

$$Num = \sum_{i=1}^{n} i \cdot P_{t-n+i}$$

where $i$ ranges from 1 (oldest) to $n$ (newest), giving linearly increasing weight to more recent data.

### 2. Simple Sum (Denominator)

$$Den = \sum_{i=1}^{n} P_{t-n+i}$$

### 3. Center of Gravity

$$CG_t = \frac{Num}{Den} - \frac{n + 1}{2}$$

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).

### 4. Complexity

Streaming uses running sums for both numerator and denominator: $O(1)$ per bar with $O(n)$ memory for the ring buffer.

## Mathematical Foundation

### Parameters

| Parameter | Description | Default | Constraint |
|-----------|-------------|---------|------------|
| `period` | Lookback window length | 10 | $> 0$ |

### Pseudo-code

```
function CG(source, period):
    buffer ← RingBuffer(period)
    runNum ← 0       // weighted sum
    runDen ← 0       // simple sum

    for each price in source:
        buffer.Add(price)
        if buffer.Count < period: continue

        // Compute from buffer (or maintain running sums)
        num = 0
        den = 0
        for i = 0 to period-1:
            w = i + 1
            num += w * buffer[i]
            den += buffer[i]

        cg = (den ≠ 0) ? (num / den) - (period + 1) / 2.0 : 0

        emit cg
```

### Output Interpretation

| Condition | Meaning |
|-----------|---------|
| $CG > 0$ | Weight concentrated in recent prices (bullish momentum) |
| $CG < 0$ | Weight concentrated in older prices (bearish momentum) |
| Zero crossing up | Momentum shifting bullish |
| Zero crossing down | Momentum shifting bearish |
| Hanging at extremes | Strong trend in progress |

## Resources

- **Ehlers, J.F.** *Cybernetic Analysis for Stocks and Futures*. Wiley, 2002.
- **Ehlers, J.F.** *Rocket Science for Traders*. Wiley, 2001.