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## Summary Fixes elastic#2772, where an uncertain fitted trend could dominate a long forecast and produce implausible curvature, including values outside a percentage metric's natural range. ## Mathematical rationale The trend component returns a polynomial regression prediction `f(x)` and its covariance-derived forecast variance. Previously, a forecast evaluated the fitted polynomial directly at every future time. This means a small but poorly determined quadratic coefficient is multiplied by `x^2`; over a long horizon that term can become much larger than the local signal even when the model is highly uncertain about it. The existing prediction-horizon guard does not cover this case: the requested duration can be operationally reasonable while the selected trend is still statistically unreliable to extrapolate. The fix keeps the model's near-term prediction but attenuates its *change from the first forecast point* according to the fraction of variance explained by the fitted trend: ```text w(t) = V_trend / (V_trend + V_extrapolation(t)) y_fixed(t) = f(t0) + w(t) * (f(t) - f(t0)) ``` Here `t0` is the first forecast time, `V_trend` is the component's long-term variance, and `V_extrapolation(t)` is the additional uncertainty incurred by projecting the fitted polynomial from the regression origin to `t`. Thus `w(t)` is close to one when the trend is reliable, preserving the original prediction; it falls toward zero when extrapolation uncertainty dominates, smoothly anchoring the forecast at the boundary instead of allowing an uncertain high-order term to diverge. This uses no new persisted state, settings, metric bounds, or customer-wide runtime cost. The covariance calculation must use the regression basis. The trend model is parameterized at its regression origin, so the variance of the change `f(t) - f(t0)` uses the basis delta: ```text [0, t - t0, t^2 - t0^2] ``` Using `[1, t, t^2]` instead incorrectly includes the intercept and computes the variance of an absolute prediction rather than the extrapolated change. This PR corrects that basis. ## Behaviour and coverage - Exact zero extrapolation variance retains the original trend prediction (`w = 1`); this prevents a regression for deterministic affine trends. - Invalid or negative numerical variances take the conservative anchored fallback. - The forecast remains continuous at the first prediction point. - Added tests cover the restored elastic#2772 state, prefix continuity, affine trends, and exact zero-uncertainty trends. This targets the intended observability workload: ordinary trend forecasts remain unchanged when their extrapolation is well supported, while temporary changes and weakly identified curvature cannot dominate a long-horizon forecast. (cherry picked from commit aca6006) # Conflicts: # lib/maths/time_series/unittest/CTrendComponentTest.cc
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