Uncertainty

A trend estimate without a standard error is a number you cannot act on. The interactive explorer below is the fastest way to see what incline’s intervals do and where they stop being trustworthy.

What the uncertainty actually looks like

Every estimator here reports a standard error, and how it gets one depends on what the smoother is. Linear smoothers have an exact operator, so the variance is closed-form; the rest are bootstrapped; Gaussian processes are asked for their own posterior. Move the smoothing slider and watch the band trade width for bias.

Series

Method

Smoothing

Show

Uncertainty AR(1) noise Bias-corrected Whole-curve band Table

Observed series and the smooth

Estimated slope, per unit time

A trend is credible where the band clears the zero line. The band flares at both ends because a smoother has data on one side only there.

How a standard error gets computed

Which machinery applies is decided by what the smoother is, never by its name.

Route	When	What you get
operator	The derivative is a fixed linear map of the data	The sampling variance diag(L Σ Lᵀ), conditional on the fitted or supplied noise covariance — no asymptotics, no resampling
native	The smoother is a probability model (Gaussian process, state space)	Its own posterior variance, which it already knows
bootstrap	Everything else	A simulated sampling distribution

Whether a smoother is linear is settled by probing it, not by assumption:

Linear — exact variance	Nonlinear — bootstrapped
Savitzky-Golay	Smoothing spline with GCV (penalty chosen from the data)
Local polynomial	LOESS with robust=True (the default)
Smoothing spline at fixed penalty	L1 trend filter
LOESS with robust=False
Naive differencing

The declaration is enforced. A smoother that claims to be linear has its operator checked against its own output before any exact standard error is issued, so a wrong claim raises rather than quietly producing wrong inference.

Two things a standard error does not tell you

It is about the smooth, not the truth. Every smoother estimates the derivative of its own smoothed curve. The gap between that and the true derivative is smoothing bias, and it is governed by the bandwidth you chose. Measured over 120 replicates on a known trend:

Method	reported SE ÷ actual spread	coverage of its own estimand	coverage of the true derivative
Savitzky-Golay, window 21	1.010	0.950	0.950
Naive differencing	1.001	0.949	0.949
Local polynomial, bw 0.15	1.008	0.947	0.043
Smoothing spline, λ=5·10⁴	1.018	0.952	0.056
LOESS, span 0.3	1.026	0.952	0.089

The variance is right in every row. The last column collapses only where the bandwidth oversmooths — the interval is correctly sized and centered in the wrong place. bias_correct=True re-centers it, at roughly five times the width; on the LOESS row that moves coverage from 0.089 to 0.941.

It assumes independent noise unless told otherwise. Under AR(1) errors with φ=0.7 the independence assumption reports standard errors 29% of their true size. Pass noise='ar1':

from incline import sgolay_trend

result = sgolay_trend(df, with_uncertainty=True, noise="ar1")

noise is an estimation option, not a label. Without with_uncertainty=True, it is accepted only by an adaptive smoothing spline, where covariance changes penalty selection and the fitted curve. Fixed smoothers reject it because it cannot affect their point estimate. Gaussian process and state-space smoothers model noise internally and reject the external option in both modes.

The autocorrelation is estimated from second differences of the raw series, never from the smoother’s residuals — smoothing strips the low-frequency noise along with the trend, and residual-based estimates of φ come out around 0.21 when the truth is 0.7.

For a nonlinear smoother, Given(covariance) uses a Gaussian parametric bootstrap: each replicate draws an error vector from the complete supplied covariance and refits the smoother. This preserves arbitrary off-diagonal dependence that residual or block resampling cannot reconstruct from one series. Supplying a covariance does not specify higher moments, so Gaussian errors are the explicit distributional assumption on this route.

For an adaptive smoothing spline, the fitted covariance also enters the point fit: the roughness penalty is selected by covariance-aware generalized maximum likelihood and the curve is fit by penalized generalized least squares. Its bootstrap draws Gaussian errors from that covariance and repeats covariance and penalty estimation in every replicate. This follows the correlated-spline framework of Diggle and Hutchinson (1989) and Wang (1998).

The GML fit reports generalized_penalty in its provenance and output frame. It is the coefficient on roughness in the covariance-weighted objective $ (y-f)^T \Sigma^{-1}(y-f) + \lambda f^T Kf $. Its scale therefore depends on the fitted covariance. It is a diagnostic, not a value to pass back through the public penalty argument, which configures SciPy’s independent-error spline.

Pointwise versus whole-curve

A 95% pointwise interval fails somewhere along a 130-point curve far more often than 5% of the time. For fixed linear smoothers, simultaneous=True widens to a band that covers the whole curve at once — for the explorer’s default series that multiplier is 3.46 rather than 1.96. Bootstrap and native-posterior smoothers reject this option because they do not provide a validated whole-curve band.

The columns

Every estimator returns the same schema, whether or not it can support a standard error:

derivative_value      the point estimate
derivative_standard_error         NaN when unavailable
derivative_ci_lower   NaN when derivative_standard_error is NaN
derivative_ci_upper
uncertainty_method             'operator' | 'native' | 'bootstrap' | None
confidence_level                interval level, or NaN
simultaneous                    whether this is a whole-curve band
bias_corrected                  whether pilot-fit correction was applied
significant_trend     False when no interval exists

derivative_standard_error of NaN with uncertainty_method of None is a deliberate, documented state. It is never a missing column, so downstream code can always index it.

Standard errors are opt-in via with_uncertainty=True: the exact route costs one smoother evaluation per observation, and that should be a choice rather than a surprise.