Pointwise confidence intervals cover the true function at each individual point with the stated probability, but they say nothing about whether the entire curve lies within the band. A 95% pointwise band means each point has 95% coverage independently (across many points, you would expect some to miss). Simultaneous confidence bands guarantee (approximately) that the whole function lies within the band with the stated probability.
Whittaker provides two ways to get simultaneous bands:
predict(..., interval="simultaneous") — overall simultaneous bands for the full linear predictor or response, covered in the prediction guide.- simultaneous_ci() — term-level simultaneous bands for individual smooth terms, useful for inference about where a specific smooth is significantly different from zero.
This article focuses on simultaneous_ci().
Why simultaneous bands matter
When you look at a pointwise confidence band for a smooth and ask “where does this band exclude zero?”, you are performing multiple implicit tests (one at every evaluation point). Pointwise bands do not adjust for this multiplicity, so some apparent “significant” regions may be false positives.
Simultaneous bands solve this by computing a wider critical value that accounts for the supremum of the standardized deviation across the entire curve. If the simultaneous band excludes zero at a region, you can be confident the smooth is truly non-zero there at the stated level.
Setup
import numpy as np
import whittaker as wk
rng = np.random.default_rng(23)
n = 300
x = np.linspace(0, 2 * np.pi, n)
y = np.sin(x) + rng.normal(0, 0.3, n)
data = {"x": x, "y": y}
model = wk.GAM("y ~ s(x, k=10)").fit(data, method="REML")
model.summary()
GAM fit summary
============================================================
Formula: y ~ s(x, k=10)
Family: Gaussian(link='identity')
Inference: REML
Observations: 300
Coefficients: 10
Parametric coefficients:
Term Estimate Std.Err t value p-value
------------------------ ---------- ---------- ---------- ----------
(Intercept) -0.0014 0.0180 -0.076 0.9395
Approximate significance of smooth terms:
Term EDF Ref.df Chi.sq p-value
------------------------ ------ ------ ---------- ----------
s(x, k=10) 7.47 8 1457.725 < 1e-16
Total EDF: 8.47
Scale est: 0.097016
Deviance: 28.2828
Null dev: 170.3338
Dev. expl: 83.4%
GCV score: 0.099835
AIC: 159.97
BIC: 191.35
Computing simultaneous bands for a smooth
simultaneous_ci() returns a SimultaneousCIResult dataclass with the smooth’s estimate, standard errors, lower and upper bounds, the term label, and the critical value used. All fields are available as attributes (e.g., sci.estimate), and dict-style access (e.g., sci["estimate"]) still works for backward compatibility:
grid = {"x": np.linspace(0, 2 * np.pi, 200)}
sci = model.simultaneous_ci(grid, term=0)
print(f"Term: {sci['term_label']}")
print(f"Critical value: {sci['crit_value']:.3f}")
print(f"Compare to z_{0.975} = 1.96 for pointwise")
Term: s(x, k=10)
Critical value: 3.023
Compare to z_0.975 = 1.96 for pointwise
The critical value is always larger than the pointwise z-value (1.96 for 95%), which is why the simultaneous band is wider.
Pointwise vs. simultaneous comparison
Let’s visualize both bands side by side to see the difference:
import altair as alt
preds = model.predict(grid, type="terms", se=True)
term_label = sci["term_label"]
pw_estimate = preds.terms[term_label]
pw_se = preds.se[term_label]
pw_lower = pw_estimate - 1.96 * pw_se
pw_upper = pw_estimate + 1.96 * pw_se
x_vals = grid["x"]
plot_data = []
for i in range(len(x_vals)):
plot_data.append({
"x": float(x_vals[i]),
"estimate": float(sci["estimate"][i]),
"pw_lower": float(pw_lower[i]),
"pw_upper": float(pw_upper[i]),
"sim_lower": float(sci["lower"][i]),
"sim_upper": float(sci["upper"][i]),
})
pw_band = alt.Chart({"values": plot_data}).mark_area(
opacity=0.3, color="steelblue"
).encode(x=alt.X("x:Q"), y="pw_lower:Q", y2="pw_upper:Q")
sim_band = alt.Chart({"values": plot_data}).mark_area(
opacity=0.15, color="darkorange"
).encode(x=alt.X("x:Q"), y="sim_lower:Q", y2="sim_upper:Q")
line = alt.Chart({"values": plot_data}).mark_line(
color="black"
).encode(x="x:Q", y=alt.Y("estimate:Q", title="s(x)"))
zero = alt.Chart({"values": [{"y": 0}]}).mark_rule(
color="firebrick", strokeDash=[4, 4]
).encode(y="y:Q")
(sim_band + pw_band + line + zero).properties(
width=500, height=300,
title="Pointwise (blue) vs. simultaneous (orange) 95% bands"
)
The simultaneous band (orange) is wider everywhere. Where the simultaneous band excludes zero, you have strong evidence that the smooth is non-zero, adjusted for the fact that you are making this claim about the entire curve.
Selecting a specific term
For models with multiple smooths, specify the term by index or name:
x2 = rng.uniform(0, 5, n)
y2 = np.sin(x) + 0.5 * x2 + rng.normal(0, 0.3, n)
data2 = {"x": x, "x2": x2, "y": y2}
model2 = wk.GAM("y ~ s(x) + s(x2)").fit(data2, method="REML")
grid2 = {"x": np.linspace(0, 2 * np.pi, 100), "x2": np.linspace(0, 5, 100)}
sci_x = model2.simultaneous_ci(grid2, term="x")
sci_x2 = model2.simultaneous_ci(grid2, term="x2")
print(f"s(x) critical value: {sci_x['crit_value']:.3f}")
print(f"s(x2) critical value: {sci_x2['crit_value']:.3f}")
s(x) critical value: 3.054
s(x2) critical value: 2.600
Each term gets its own critical value because the multiplicity correction depends on the term’s basis dimension and the correlation structure of its basis functions.
Unconditional bands
By default, the bands are conditional on the estimated smoothing parameters. Setting unconditional=True additionally accounts for the uncertainty in the smoothing parameters themselves, producing even wider bands:
sci_cond = model.simultaneous_ci(grid, term=0)
sci_uncond = model.simultaneous_ci(grid, term=0, unconditional=True)
mean_width_cond = (sci_cond["upper"] - sci_cond["lower"]).mean()
mean_width_uncond = (sci_uncond["upper"] - sci_uncond["lower"]).mean()
print(f"Mean band width (conditional): {mean_width_cond:.4f}")
print(f"Mean band width (unconditional): {mean_width_uncond:.4f}")
print(f"Ratio: {mean_width_uncond / mean_width_cond:.2f}x")
Mean band width (conditional): 0.2933
Mean band width (unconditional): 0.2951
Ratio: 1.01x
Use unconditional bands when the smoothing parameter is uncertain (e.g., when the REML criterion surface is flat as shown by smoothing parameter sensitivity).
Controlling the simulation
The critical value is computed via posterior simulation (drawing from the Bayesian posterior of the coefficients and computing the maximum standardized deviation). You can control the number of simulations and the random seed:
sci_1k = model.simultaneous_ci(grid, term=0, n_sim=1_000, seed=0)
sci_50k = model.simultaneous_ci(grid, term=0, n_sim=50_000, seed=0)
print(f"Critical value (1,000 sims): {sci_1k['crit_value']:.4f}")
print(f"Critical value (50,000 sims): {sci_50k['crit_value']:.4f}")
Critical value (1,000 sims): 2.9133
Critical value (50,000 sims): 3.0407
The default of 10,000 simulations is usually sufficient. Increase it if you need very precise critical values (e.g., for publication).
When to use simultaneous vs. pointwise
| “Is the smooth significantly non-zero at this specific x?” |
Pointwise CI |
| “Over what range of x is the smooth significantly non-zero?” |
Simultaneous band |
| “Can I claim the entire fitted curve lies within this band?” |
Simultaneous band |
| Exploratory analysis, quick checks |
Pointwise CI |
| Publication-quality inference |
Simultaneous band |