Anchored Geodesic Component Analysis for Extremes

Implements anchored geodesic component analysis for multivariate extremes. The package provides Pareto and rank-Pareto marginal standardization, top-k angular extraction, anchored eigensolutions, reconstruction, diagnostics, nonparametric bootstrap tools, and plotting methods for benchmark-relative angular variation.

AGCA4extremes

Author: Alberto Quaini

AGCA4extremes implements anchored geodesic component analysis (AGCA) for multivariate extremes. AGCA summarizes how extreme angular profiles vary around a benchmark direction, most often the canonical balanced-dependence anchor.

The package provides:

• rank-Pareto marginal standardization;
• Pareto standardization from user-supplied CDFs;
• top-k angular extraction and optional runs declustering;
• canonical, spherical Frechet, and principal anchors;
• AGCA scores, loadings, eigenvalues, explained variation, reconstruction, and residual risk;
• bootstrap, threshold, and anchor-sensitivity diagnostics;
• base R plotting methods and functional approximation summaries.

The numerical core is implemented with Rcpp and RcppArmadillo; the R layer provides validation, S3 methods, diagnostics, plotting, and documentation.

Installation

During development, install from the package directory:

install.packages(c("Rcpp", "RcppArmadillo"))
devtools::install("AGCA4extremes")

After public release:

install.packages("AGCA4extremes")

Basic Use

library(AGCA4extremes)

data(agca_10d_simulation)

x <- agca_10d_simulation[paste0("X", 1:10)]
fit <- agca(x, k = 250, p = 5)

fit
summary(fit)
plot(fit, type = "variation")

The default marginal transformation is rank-Pareto. Larger observations in each margin are treated as more extreme, so financial return data should be converted to losses before calling agca().

Example Diagnostics

The bundled agca_10d_simulation data set is generated from a 10-dimensional heavy-tailed design. Variables X1--X8 share a low-dimensional extremal mechanism, while X9 and X10 contain independent Pareto sources that create near-axis extreme regimes.

data(agca_10d_simulation)

x <- agca_10d_simulation[paste0("X", 1:10)]
fit <- agca(x, k = 500, p = 4, seed = 1)

agca_rank_summary(fit)

Explained variation:

plot(fit, type = "variation")

Scores for the first two anchored geodesic components:

cols <- c(shared_low_rank = "#1B9E77", axis_9 = "#5B3A29", axis_10 = "#7570B3")
plot(
  fit$scores[, 1], fit$scores[, 2],
  col = cols[agca_10d_simulation$regime[fit$tail$index]],
  pch = 16,
  xlab = "AGC1 score",
  ylab = "AGC2 score"
)
legend("topright", legend = names(cols), col = cols, pch = 16, bty = "n")

Loadings:

plot(fit, type = "loadings", p = 1)
plot(fit, type = "loadings", p = 2)

Threshold and anchor diagnostics:

threshold_stability(x, k = c(250, 350, 500, 750), p = 4)
anchor_sensitivity(x, k = 500, p = 4)

Bootstrap uncertainty for rank summaries:

boot <- bootstrap_agca(fit, B = 99, ranks = c(1, 2, 4), seed = 1)
summary(boot)
plot(boot, statistic = "variation_explained")

Main Functions

• agca() fits the full workflow from data to an AGCA object.
• rank_pareto() and pareto_from_cdf() perform marginal standardization.
• tail_directions() extracts large-radius angular observations.
• decluster_runs() performs simple runs declustering for radial extremes.
• agca_fit_directions() fits AGCA directly to angular directions.
• agca_reconstruct() maps leading anchored coordinates back to the sphere.
• agca_rank_summary() reports residual risk and anchored variation explained.
• bootstrap_agca() resamples angular directions to quantify sampling uncertainty.
• threshold_stability() and anchor_sensitivity() provide diagnostics.

Paper Replication

The CRAN package is intentionally lean. Large raw data, generated results, figures, and full paper workflows should live in a separate replicateAGCApaper repository that depends on AGCA4extremes.

Development Assistance

OpenAI Codex was used as a programming assistant during development, mainly for code scaffolding, refactoring, documentation, and tests. All methodological choices, validation, final code, and responsibility for the package remain with the author.