Fit a Thin Plate Regression Spline

Use the default TPRS basis to fit a smooth non-linear curve.

Thin plate regression splines (TPRS) are the default smooth basis in Whittaker. They minimise a penalty on integrated squared second derivatives, producing a smooth curve that adapts to the data without requiring you to place knots manually. Use s(x) or s(x, bs='tp') as both are identical. TPRS works well as a general-purpose smooth for any continuous predictor.

Load data

Load the mcycle dataset, which records head acceleration during a simulated motorcycle crash. It is a classic benchmark for smoothing because the signal has strong curvature.

import whittaker as wk

data = wk.load_dataset("mcycle", as_frame=True)
data.head()

	times	accel
0	0.0	-11.075042
1	0.2	2.887586
2	0.4	3.909896
3	0.6	14.610615
4	2.2	-0.172051

The two columns are times (milliseconds after impact) and accel (head acceleration in g).

Fit

Fit a Gaussian GAM with a TPRS smooth on times. The bs='tp' argument is shown explicitly here but is the default (both formulas produce the same model).

# Fit TPRS smooth on times
model = wk.GAM("accel ~ s(times, bs='tp')").fit(data)

model.summary()

GAM fit summary
============================================================
Formula:    accel ~ s(times)
Family:     Gaussian(link='identity')
Observations: 133
Coefficients: 10

Parametric coefficients:
  Term                       Estimate    Std.Err    t value    p-value
  ------------------------ ---------- ---------- ---------- ----------
  (Intercept)                -45.6924     1.8364    -24.882    < 1e-16

Approximate significance of smooth terms:
  Term                        EDF Ref.df     Chi.sq    p-value
  ------------------------ ------ ------ ---------- ----------
  s(times)                   7.92      8    663.396    < 1e-16

Total EDF:  8.92
Deviance:   55653.7152
Null dev:   357878.4929
Dev. expl:  84.4%
GCV score:  480.746119
Scale est:  448.517253
AIC:        1198.44
BIC:        1224.22

Check the estimated degrees of freedom (EDF) for the smooth. An EDF near 1 would indicate a nearly linear effect; a higher EDF reflects more curvature. The k-index should be above 1 and the p-value for the smooth should be small, confirming that the non-linear term is necessary.

Partial effects

Plot the fitted smooth to see the shape of the relationship.

wk.partial_effects(model)

The partial effects plot shows the TPRS smooth with a pointwise confidence band. Notice how the curve captures the sharp dip around 15–20 ms and the rebound afterward (a pattern that a straight line would completely miss).