Bootstrap Confidence Intervals

Use bootstrap resampling when you need uncertainty bounds for Sobol indices, not just point estimates.

Scalar-output bootstrap

import jax
import gsax
from gsax.benchmarks.ishigami import PROBLEM, evaluate

sampling_result = gsax.sample(PROBLEM, n_samples=4096, seed=42)
Y = evaluate(sampling_result.samples)

result = gsax.analyze(
    sampling_result,
    Y,
    prenormalize=True,
    num_resamples=200,
    conf_level=0.95,
    ci_method="quantile",
    key=jax.random.key(0),
)

print("S1:", result.S1)
print("ST:", result.ST)
print("S1 lower:", result.S1_conf[0])
print("S1 upper:", result.S1_conf[1])
print("ST lower:", result.ST_conf[0])
print("ST upper:", result.ST_conf[1])
print("S2 lower:", result.S2_conf[0])
print("S2 upper:", result.S2_conf[1])

Confidence-interval shapes

The bootstrap adds a leading dimension of 2 for [lower, upper]:

• scalar output: S1_conf.shape == (2, D)
• multi-output: S1_conf.shape == (2, K, D)
• time-series multi-output: S1_conf.shape == (2, T, K, D)

S2_conf follows the same rule with two trailing parameter axes.

ci_method only changes how those two endpoints are summarized from the bootstrap draws:

• quantile uses percentile bootstrap endpoints.
• gaussian uses symmetric gaussian endpoints from the bootstrap standard deviation around the point estimate.

Practical caveats

• A jax.random.key(...) is required when num_resamples > 0.
• prenormalize=True applies SALib-style output standardization once over the sample axis before the bootstrap starts. The resamples reuse that transformed output array; they are not re-standardized per resample.
• Set num_resamples=0 to skip bootstrap entirely when you only need point estimates.
• If calc_second_order=False during sampling, then result.S2 and result.S2_conf are both None.
• Bootstrap intervals follow the same output-shape rules as the point estimates, so the page on Multi-Output & Time-Series is the right companion when your model is not scalar.
• Confidence intervals always remain lower/upper endpoint arrays even when prenormalize=True. ci_method="gaussian" is closer to SALib's CI construction, but gsax still returns endpoints rather than SALib-style confidence half-widths.

See also

• Save and Reload Samples if you want to bootstrap a stored design.
• Multi-Output & Time-Series for concrete shape examples on (N, K) and (N, T, K) outputs.
• xarray Labeled Output for exporting confidence intervals as _lower and _upper dataset variables.