Concepts

EVI and EI workflows

EVI workflow

The core EVI path is:

1. Choose a block extraction scheme and summary target.
2. Build a log block-summary curve across a candidate block-size grid.
3. Select a plateau window on the log-log scale.
4. Fit a regression slope on the selected window to estimate xi.
5. Map the fitted scaling law to design-life levels.

flowchart LR
    raw["Observed raw series"] --> blocks["Block extraction over candidate block sizes"]
    blocks --> summaries["Block-summary curve<br/>Q_tau(M_b), mean(M_b), or mode(M_b)"]
    summaries --> plateau["Observed plateau selection<br/>on log(summary) vs log(block size)"]
    raw --> bootstrap["Super-block bootstrap draws<br/>from the original series"]
    bootstrap --> covariance["Cross-block covariance estimate<br/>for FGLS"]
    plateau --> fit["Observed scaling fit<br/>xi = slope on selected plateau"]
    covariance --> fit
    fit --> design["Design-life mapping<br/>evaluate the same fitted law at larger block sizes"]

In this package, the design-life-level step is not a separate annual-maxima fit. UniBM reuses the same block-quantile scaling law and simply evaluates it at larger block sizes that correspond to longer design-life spans. The fitting view therefore lives on the block size axis, while the planning-horizon interpretation lives on the design-life years axis through the chosen observation clock.

The relevant term for the current package output is a design-life level: a quantile of the maximum over a design-life span, or equivalently a T-year block-maximum tau-quantile on the chosen observation clock. The current package default is tau = 0.5, so the headline exported curve is best read as a median design-life level rather than as a classical return-period level. The severity workflow also exports companion design-life levels at tau = 0.90, 0.95, 0.99. Those higher curves reuse the same headline plateau and slope xi and only shift the intercept, so they should be read as shared-xi upper design-life quantiles rather than as separate headline EVI fits.

Main entrypoints:

• unibm.evi.generate_block_sizes
• unibm.evi.blocks.block_summary_curve
• unibm.evi.estimation.estimate_target_scaling
• unibm.estimate_evi_quantile
• unibm.estimate_design_life_level

EI workflow

1. Prepare block-size paths from the raw series.
2. Construct native block-maxima EI paths across the candidate block sizes.
3. Pool one preferred stable-window estimator, for example the BB-sliding-FGLS path.
4. Compare against reference estimators such as K-gaps.

flowchart LR
    raw["Observed raw series"] --> cdf["Full-sample empirical CDF<br/>Fhat(X_t) over the whole series"]
    cdf --> scores["Observed score sequence<br/>Northrop: -log(Fhat)<br/>BB: 1 - Fhat"]
    scores --> path["Observed BM path over block sizes<br/>statistics -> theta_b -> z_b"]
    path --> stable["Observed stable-window selection"]
    stable --> pooled["Observed pooled EI fit<br/>OLS or FGLS on the selected window"]
    pooled --> theta["Headline theta estimate"]
    raw --> bootraw["Circular raw-series bootstrap draws"]
    bootraw --> bootpath["Bootstrap z-path draws<br/>recomputed from each resample"]
    bootpath --> bootcov["Stable-window covariance<br/>estimated from bootstrap paths"]
    bootcov --> pooled

The important separation is that the observed path and the bootstrap paths play different roles:

• the observed path is what gets pooled and turned into the headline theta estimate
• the bootstrap paths are not averaged into a replacement path; they are used only to estimate the cross-block covariance for the FGLS weighting step

Main entrypoints:

• unibm.ei.preparation.prepare_ei_bundle
• unibm.ei.selection.select_stable_path_window
• unibm.ei.bm.estimate_pooled_bm_ei
• unibm.ei.threshold.estimate_k_gaps
• unibm.ei.threshold.estimate_ferro_segers

Interpreting EVI vs EI outputs

The EVI plateau and the EI stable window answer different questions and need not coincide.

• The EVI plateau is the block-size region where the log block-summary curve is sufficiently linear to support a stable xi estimate.
• The EI stable window is the block-size region where the extremal-index path is sufficiently stable to support a formal theta estimate.

Different block-maximum quantiles Q_tau(M_b) are expected to share the same asymptotic slope xi and to differ mainly in intercept. In the direct block-maxima framework used here, serial dependence is already internalized in the fitted block-maximum law, so the design-life-level curve should be read directly from the dependent-series fit rather than through a second BM-side theta adjustment. In the current package workflow, this is implemented explicitly by treating tau = 0.50 as the headline fit and deriving the 0.90 / 0.95 / 0.99 curves by holding the same plateau and slope fixed while re-estimating only tau-specific intercepts.

In practice:

• use the EVI/design-life-level outputs for severity on the original physical scale
• use the EI outputs for persistence, clustering, and recovery burden