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 exact sampling variance, diag(L Σ Lᵀ) — 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	UnivariateSpline (picks knots from the data)
Local polynomial	Penalized spline with GCV (penalty chosen from the data)
Penalized spline at fixed lam	LOESS with robust=True (the default)
LOESS with robust=False	L1 trend filter
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
Penalized spline, λ=5·10⁴	1.018	0.952	0.056
LOESS, frac 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, se=True, noise="ar1")

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.

Pointwise versus whole-curve

A 95% pointwise interval fails somewhere along a 130-point curve far more often than 5% of the time. 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.

The columns

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

derivative_value      the point estimate
derivative_se         NaN when unavailable
derivative_ci_lower   NaN when derivative_se is NaN
derivative_ci_upper
se_method             'operator' | 'native' | 'bootstrap' | None
significant_trend     False when no interval exists

derivative_se of NaN with se_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 se=True: the exact route costs one smoother evaluation per observation, and that should be a choice rather than a surprise.