QuanTAlib/lib/cycles/cg/cg.md

CG: Ehlers Center of Gravity

Property	Value
Category	Cycle
Inputs	Source (close)
Parameters	period (default 10)
Outputs	Single series (Cg)
Output range	Varies (see docs)
Warmup	period bars

TL;DR

• CG identifies potential turning points using the physics concept of weighted center of mass applied to a price window.
• Parameterized by period (default 10).
• Output range: Varies (see docs).
• Requires period bars of warmup before first valid output (IsHot = true).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

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

Performance Profile

Operation Count (Streaming Mode)

Operation	Count	Cost (cycles)	Subtotal
ADD/SUB	2×N	1	2N
MUL	N	3	3N
DIV	1	15	15
Total	~3N+1	—	~5N+15

The RecalculateSums() loop iterates over the full buffer each bar, making this O(N) per bar. For default N = 10: ~65 cycles. A periodic resync every 1000 bars maintains numerical stability.

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	Exact weighted center-of-mass calculation
Timeliness	9/10	Leads price movement by construction
Smoothness	7/10	Raw oscillator; no internal smoothing
Memory	9/10	O(N) ring buffer + 2 running sums

Resources

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