Bootstrap Confidence Intervals¶

Bootstrap methods provide confidence intervals and p-values for trend estimates. The implementation follows:

Hirsch, R.M., Archfield, S.A., and De Cicco, L.A. (2015), A bootstrap method for estimating uncertainty of water quality trends. Environmental Modelling & Software, 73, 148-166.

Bootstrap Pairs¶

Provides confidence intervals for pairwise trend comparisons:

boot = w.bootstrap_pairs(
    year1=1985,
    year2=2010,
    n_boot=100,
    block_length=200,
    seed=42,
)

What It Does¶

For each bootstrap replicate:

Block resample the Sample DataFrame (preserving temporal structure)
Estimate two one-year surfaces (one for each target year)
Run the trend decomposition (same as run_pairs)
Apply bias correction to account for systematic differences between the bootstrap and original surfaces

Results¶

The returned dictionary contains:

Key	Description
`observed`	Original `run_pairs` result
`boot_conc`	Array of bootstrap concentration trend replicates
`boot_flux`	Array of bootstrap flux trend replicates
`p_conc`	Two-sided p-value for concentration trend
`p_flux`	Two-sided p-value for flux trend
`ci_conc`	90% confidence interval for concentration change
`ci_flux`	90% confidence interval for flux change
`likelihood_conc_up`	Probability that concentration increased
`likelihood_flux_up`	Probability that flux increased
`like_conc_up`	Likelihood descriptor for concentration increase
`like_conc_down`	Likelihood descriptor for concentration decrease
`like_flux_up`	Likelihood descriptor for flux increase
`like_flux_down`	Likelihood descriptor for flux decrease

Bootstrap Groups¶

Same approach applied to group comparisons:

boot = w.bootstrap_groups(
    group1_years=(1985, 1996),
    group2_years=(1997, 2010),
    n_boot=100,
    seed=42,
)

For group bootstrap, each replicate re-estimates the full surface (not just per-year surfaces), then runs the group trend decomposition.

Block Resampling¶

The block_length parameter (default 200 Julian days) defines the size of temporal blocks drawn with replacement. Block resampling preserves the temporal correlation structure of the sample data while allowing statistical inference.

Likelihood Descriptors¶

The likelihood_descriptor function maps a probability to a qualitative description using EGRETci thresholds:

Probability Range	Descriptor
0.0 -- 0.05	Highly unlikely
0.05 -- 0.1	Very unlikely
0.1 -- 0.33	Unlikely
0.33 -- 0.67	About as likely as not
0.67 -- 0.9	Likely
0.9 -- 0.95	Very likely
0.95 -- 1.0	Highly likely

Number of Replicates¶

The n_boot parameter controls how many bootstrap replicates are drawn. More replicates give more stable p-values and confidence intervals. Typical values:

100 — quick exploratory analysis
500--1000 — publication-quality results

Each replicate requires re-fitting the surfaces, so computation time scales linearly with n_boot.