Function centerOfGravity

Center of Gravity (John F. Ehlers, Stocks & Commodities, May 2002 — "The CG Oscillator") — where the window's price mass balances, measured in bars back from the current one:

${output}[i] = − Σ_{k=0}^{period−1} (k + 1) · column[i − k]
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                    Σ_{k=0}^{period−1} column[i − k]
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The current bar carries weight 1 and the oldest carries period, so on positive prices the reading is a negative number between −period and −1: a flat window balances in the middle at exactly −(period + 1)/2, and it moves up (toward −1) as recent bars get heavier, i.e. as the price rises. Ehlers' point was that this is a moment rather than a momentum — it has essentially no lag, because a change in the newest bar moves the balance point immediately.

It is not a regression, despite arriving in the same batch: it takes one position-weighted moment of the price rather than fitting a line, so nothing here has a slope or a residual.

The sign and the zero line — TradingView's convention, not Ehlers'

Ehlers' published EasyLanguage adds (Length + 1)/2 at the end, which re-centres a flat window on 0; TradingView's built-in ta.cog leaves it off, so a flat window reads −(period + 1)/2 (−5.5 at the default period 10). This ships the uncentred form, the one more consumers plot. The two differ by a constant that depends only on period, so the shape is identical and a caller who wants Ehlers' zero line adds (period + 1)/2. The oracle pins the choice with a separation assert rather than leaving it to be discovered from a chart, and the sign convention is pinned the same way (the ascending-weight reading — oldest bar lightest — is 0.17 away at period 10, a full window's worth of the reading's own range on the fixture).

TA-Lib has no Center of Gravity, so the oracle is a pandas replication with the analytic first-valid bar (period − 1) asserted.

No kernel: it is wma and sma, exactly

The weights run down from the oldest bar, which neither rollingWeightedMeanValues (per-row weights, not positional) nor symmetricWeightedValues (a fixed 4-bar 1,2,2,1) can express — but no new kernel is needed either, because the descending weights are the ascending ones subtracted from a constant. Writing u for the position in the window (0 oldest, n − 1 newest), the numerator weight k + 1 is n − u, so

Σ (n − u)·p = (n + 1)·Σp − Σ (u + 1)·p = (n + 1)·n·SMA − WMA·n(n+1)/2
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and dividing by Σp = n·SMA collapses the whole study to

${output} = (period + 1) · ( WMA / (2 · SMA) − 1 )
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— the K2 engine's wma over rollingMeanValues' sma, both O(N) and flat in period, both already carrying the package's rebuild-every- period numerics and its strict-window mask. The identity is exact algebra rather than an approximation, and it is pinned by a test against the naive O(N·period) definition (agreement ≤ 1.1e-14 at period 20 on the oracle input; measured, not pinned: 8.0e-15 over 50k bars at a price of 1e12 on one series and 2.5e-14 on another — a few ulps either way), so a future editor can check the shortcut rather than trust it.

Edges

• Bounded −period … −1 on a positive source column, and both ends are reachable only in the limit (all the weight on the oldest bar, or on the newest). The bound is a property of positive weights over positive prices, not of the formula: a column that crosses zero (an oscillator, a spread) can read outside it — measured −0.667 at period 2 on [5, 1, −1, 4], the zero-crossing case the bounds test does not cover because it runs on closes only.
• A zero-sum window reads undefined, and the guard is live. The division is the study's output, and the numerator is not forced to zero with the denominator — [1, −1] at period 2 sums to 0 with a weighted sum of −1 — so an unguarded x / 0 would reach withColumn as −Infinity and throw. This is the choppinessIndex live-guard case, not the ulcerIndex dead one. Unreachable on prices (they are positive, so a zero sum means an all-zero window and 0/0 is already NaN); reachable, and unit-tested, over another study's output that crosses zero.
• Scale-invariant, not shift-invariant. Every term of both sums scales with the price and the ratio divides it out; adding a constant does not cancel — it adds (n+1)/2·c to the numerator and c to the mean, dragging the balance point toward the middle. Both halves are pinned, the second as a direction.
• The strict window — a bar is emitted only when all period cells are finite, because a positional weight cannot skip a cell without silently re-weighting the rest (wma's rule, and rollingMeanValues masks the same way, so the two halves agree on where a reading exists). A leading gap shifts the start; an interior gap blanks period bars and then recovers.
• A misnamed column reads all-missing rather than throwing.