Function linearRegression

Rolling linear regression — the whole family in one study, over a period-bar least-squares fit of the column against the bar index (kernel K7, corpus §6.7):

${prefix}Value      the fit at the window's LAST bar        TA-Lib LINEARREG
${prefix}Slope      change in the fit per bar               TA-Lib LINEARREG_SLOPE
${prefix}Intercept  the fit at the window's FIRST bar       TA-Lib LINEARREG_INTERCEPT
${prefix}Angle      atan(slope) in DEGREES                  TA-Lib LINEARREG_ANGLE
${prefix}R2         the fraction of variance explained      (no TA-Lib function)
Copy

Four of the five are exact against TA-Lib, mask and values (measured ≤ 1.3e-12 at period 14 and ≤ 1.9e-12 at period 5, the angle being the loosest of the four; warm-ups identical); the fifth has no vendor implementation and is a pandas replication. One study rather than five because they are five readings of one fit: splitting them would run the same O(N) regression up to five times to append columns that are two multiplications of each other — the macd / directionalMovement family rule.

One warm-up, unlike the other families

Every column lands on bar period − 1 and none later: they are all read off the same fit, so there is no column here whose definition needs more bars than another's (contrast macd's signal or the DMS' ADX). That is also TA-Lib's lookback for all four of its functions.

${prefix}Intercept is the window's FIRST bar, not its last

The regression's x runs 0 … period − 1 with x = 0 the oldest bar in the window, so the intercept is the fitted price period − 1 bars ago and ${prefix}Value is the fitted price now. TA-Lib publishes the same pair, and the identity is asserted in the oracle rather than assumed: Value = Intercept + Slope·(period − 1), and timeSeriesForecast is the same line one bar further on.

A reader who expects "intercept" to mean "the line's value at the current bar" will read it upside down on a trending series — at period 14 on the oracle input the two differ by up to 13.68 points, on a series whose entire range is 19.4 — which is why both are appended rather than one.

${prefix}Angle is scale-DEPENDENT, and that is TA-Lib's definition

atan(slope) treats a slope of "1 price unit per bar" as 45°, so the angle depends on the units of the price: the same instrument quoted in cents rather than dollars reads a different angle, and a $400 stock and a $4 stock moving the same percentage per bar do not. Nothing normalises it — not by price, not by σ, not by the chart's aspect ratio, which is what the "angle" metaphor is borrowed from.

TA-Lib's LINEARREG_ANGLE does exactly this and this study matches it bar-for-bar, so the column is here for parity. It is stated this plainly because the property is invisible in the name: the batch's property tests assert that scaling the input moves the angle (every other reading in the family is either invariant or equivariant), and a caller who wants a scale-free trend reading wants ${prefix}R2 or a slope divided by price.

${prefix}R2 — how well the line fits, 0 … 1

The coefficient of determination, corr(x, y)²: 1 is a perfect straight line over the window and 0 is a slope that explains nothing. It is invariant to both scale and shift (a correlation is), which makes it the one column in the family a caller can compare across instruments. TA-Lib has no equivalent — its CORREL is between two series, not against the bar index — so the oracle is a pandas replication with the analytic first-valid bar asserted and a measured separation from the un-squared |corr|.

Edges

• period must be at least 2. One point does not determine a line — the regression denominator n²(n²−1)/12 is exactly 0 at n = 1 — so the study throws rather than emitting 0/0 on every bar (the choppinessIndex log10(1) = 0 precedent).
• A flat window: slope 0, R2 undefined. The slope's numerator is forced to zero by the same condition that empties it, so 0 is a real reading (the accumulationDistribution flat-bar case) and Value, Intercept and the forecast all read the flat price back. R2 is a genuine 0/0 — a line explaining all of zero variance — and is undefined, the stochastic flat-window case. Both are exact, not approximated; see the kernel's note on why that costs a counter.
• Scale and shift: Value and Intercept are equivariant to both (f(a·y + b) = a·f(y) + b), Slope is equivariant to scale and invariant to shift, R2 is invariant to both, and Angle is invariant to shift only. Each is pinned by its own property test rather than by one loop, because the five answers genuinely differ.
• The strict window: a bar is emitted only when all period cells are finite, because x names a position and dropping a cell would fit the line against the wrong abscissa (the wma rule). So a leading gap shifts the start and an interior gap blanks period bars and then recovers.
• A misnamed column reads all-missing rather than throwing — columnValues' door, the atr behaviour.