Pith. sign in

REVIEW 2 major objections 3 minor 76 references

Anchored Geodesic Analysis for Multivariate Extremes

T0 review · 2 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Anchored geodesic analysis turns multivariate-extreme dimension reduction into an exact eigenproblem: the optimal rank-p component space is the top-p eigenspace of the anchored second-moment matrix.

desk verdict Genuinely new spectral geodesic dimension reduction for extremal angular laws, with a clean population theory; the rank-Pareto consistency claim is narrower than the abstract says, but the paper itself flags it. read the letter →

arxiv 2607.13112 v1 pith:DYJX3JQG submitted 2026-07-14 stat.ME math.STq-fin.RMstat.TH

Alberto Quaini , Chen Zhou This is my paper
classification stat.MEmath.STq-fin.RMstat.TH MSC 62G3262H25
keywords multivariateextremesangularmeasuregeodesicdimensionreductionanchoredcomponentanalysisspectraldecompositiontailsimulationportfolioriskrank-Paretoestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

AGCA claims that dimension reduction for multivariate extreme angular laws can be solved exactly by a spectral decomposition, once the model space is restricted to great subspheres through a fixed reference anchor and the loss is sine-squared geodesic distance. The paper proves that the anchored geodesic reconstruction risk equals the Euclidean reconstruction risk of bounded tangent departures, so the optimal component directions are the leading eigenvectors of the anchored second-moment matrix and residual risk is the trailing eigensum. This matters because extremal dependence is naturally described by the angular law of tail observations; an exact, computable reduction makes tail structure interpretable and supports low-rank simulation of tail functionals such as portfolio capped excesses and value-at-risk with explicit error bounds. Consistency of rank-Pareto estimates and an oracle central limit theorem give the method inferential backing. In a 24-portfolio daily-loss panel, ten components explain about 91 percent of anchored variation and approximate capped-excess and normalized value-at-risk summaries within about 1.25 percent average relative error.

What carries the argument

The central object is the bounded anchored tangent departure u_mu(g) = (I - mu mu^T)g, which sends every positive-sphere direction into the tangent hyperplane at the anchor with norm at most one. The key identity is sin^2 d_g(g, M_mu(W)) = ||u_mu(g) - Pi_W u_mu(g)||^2, with M_mu(W) = span({mu} union W) intersected with the open hemisphere H_mu. Because of this identity, geodesic reconstruction through the anchor under the sine-squared loss is equivalent to Euclidean orthogonal projection of the departure U_mu, and the population and empirical fitting problems become eigendecompositions of the anchored second-moment matrix Sigma_mu = E[U_mu U_mu^T]. The eigenvectors are the AGCA loadings, the

What would settle it

Fix an anchor mu and a one- or two-dimensional tangent subspace W in R^3, take a dense grid of directions g on the positive sphere, and compute the geodesic distance to the anchored great subsphere M_mu(W) by brute-force optimization; the projection identity says sin^2 of that distance must equal ||u_mu(g) - Pi_W u_mu(g)||^2 to numerical precision. A material mismatch at any grid point would refute the identity that carries the spectral reduction.

Watch

Extended reading notes

Core claim

The paper's central discovery is that, for every tangent subspace W through a fixed interior anchor mu, the sine-squared geodesic distance from a positive-sphere direction g to the anchored great subsphere M_mu(W) equals the squared Euclidean distance between the bounded tangent departure u_mu(g) and its projection onto W. Under the limiting angular law G, the anchored risk R_mu(W)=E[sin^2 d_g(G, M_mu(W))] therefore equals tr((I - Pi_W) Sigma_mu), where Sigma_mu = E[U_mu U_mu^T] is the second-moment matrix of anchored departures, and the rank-p optimal component space is the span of the top p eigenvectors of Sigma_mu. This turns a nonlinear geodesic reconstruction problem into ordinary eigen

Load-bearing premise

The load-bearing premise is that the joint tail obeys Euclidean-polar regular variation, meaning the radial size is regularly varying and has a limiting angular law; without some radial anti-concentration near the top-k threshold, rank-Pareto selection can fail to be consistent even when directions converge.

Editorial extensions

If this is right

  • Rank-p AGCA summaries are fully determined by the leading p eigenvectors of Sigma_mu; residual risk is the sum of the remaining eigenvalues, so explained-variation curves, loadings, and scores are closed-form diagnostics.
  • Low-rank reconstructions simulate bounded Lipschitz tail functionals, including capped portfolio excesses, and homogeneous tail scores, including value-at-risk, with explicit error bounds: a threshold error plus a constant times rho^{beta/2}.
  • Face- and axis-supported angular laws, which arise under asymptotic independence, enter the same finite second-moment problem without singularities; a single-axis law yields one interpretable contrast loading away from balanced complete dependence.
  • For rank-Pareto margins, top-k AGCA estimates are consistent under Euclidean-polar regular variation, and the oracle estimator satisfies a CLT whose covariance is that of an independent sample from the limiting angular law, giving plug-in confidence intervals for oracle spectra and explained variation.
  • In the empirical panel, ten components explain about 91 percent of anchored variation and approximate capped-excess and normalized value-at-risk portfolio summaries with about 1.25 percent average relative error.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: because the AGCA risk is exactly a trace of a projected second moment, sparse or otherwise constrained eigenproblems could be applied directly to select a small set of interpretable loading contrasts without changing the residual-risk interpretation; the paper does not develop that sparsity direction.
  • Editorial inference: the oracle CLT relies only on angular convergence plus a second-order angular-bias condition, so the same inferential logic should transfer to any bounded, projection-friendly transform of angular directions, not only the u_mu coordinate used here.
  • Editorial inference: the total-variation obstruction means that for rare-event probabilities over arbitrary sets, the low-rank AGCA simulator needs an explicit residual or noise layer around the fitted subsphere; the paper derives the bounds but leaves the residual-augmented simulation scheme to future work.
  • Editorial inference: the near-axis loading structure suggests AGCA could serve as a diagnostic flag for asymptotically independent variables, since a variable-specific axis regime produces a one-dimensional contrast toward that axis; the paper presents this as a population phenomenon rather than as a testing procedure.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper introduces anchored geodesic component analysis (AGCA) for angular laws of multivariate extremes. Using a fixed interior anchor μ and the bounded loss sin² d_g, it shows that geodesic reconstruction by great subspheres through μ is exactly equivalent to Euclidean projection of the bounded tangent departure u_μ(G), so the population and empirical problems reduce to eigenanalysis of Σ_μ = E[u_μ(G)u_μ(G)ᵀ]. Theorem 1 gives the population eigensolution; Theorem 2 proves oracle top-k consistency under angular convergence; Theorem 3 proves rank-Pareto consistency under Euclidean-polar regular variation; Theorem 4 gives an oracle CLT with a covariance equal to that of an independent sample from the limiting angular law. Propositions 4–6 and Corollaries 2–3 provide explicit tail-simulation bounds for bounded Lipschitz functionals, homogeneous tail scores, capped portfolio excesses, and normalized VaR, together with a total-variation obstruction. Section 4 applies the method to daily Fama–French and OSAP portfolio-loss panels, reporting a concentrated anchored spectrum (ten components explain about 91% of anchored variation in the main panel) and small in-sample portfolio-functional errors. Supplementary material contains proofs, simulations, threshold/anchor sensitivity, and positive post-projection results.

Significance. The central theoretical contribution is strong and internally coherent. Proposition 1 and Theorem 1 give an exact, parameter-free spectral solution to a geodesic dimension-reduction problem on extremal angular laws, converting a nonlinear reconstruction problem into ordinary PCA of a bounded second-moment matrix. The oracle CLT (Theorem 4 and Corollaries 4–5) is a valuable inferential result, and the explicit simulation error bounds (Propositions 4–5) with the honest total-variation obstruction (Proposition 6) are commendable. The paper is also unusually transparent about its own limitations: Section 3.4 and Supplementary S2.9 state that no primitive rank-Pareto CLT is derived, and Supplementary S2.7 explains exactly where Assumption 2 enters for selected-set stability. The availability of R code and replication data strengthens the contribution. The main caveats are presentation-level: the abstract overstates the rank-Pareto consistency claim, and the empirical numbers are in-sample descriptive diagnostics rather than out-of-sample validation.

major comments (2)
  1. [Abstract; Section 3.2 (Theorem 3); Supplementary S2.7] The abstract and introduction state unqualified 'top-k consistency for oracle and rank-Pareto AGCA summaries.' The only rank-Pareto consistency result, Theorem 3, is proved under Assumption 2 (Euclidean-polar regular variation), not under the minimal angular Assumption 1. As Supplementary S2.7 makes explicit, Proposition S15 uses the vanishing band ratio F̄_R(r(1−δ))/F̄_R(r) → 1, a consequence of Assumption 2; under Assumption 1 alone the radial survival function only satisfies rF̄_R(r) ∈ [1, d^{3/2}], so the paper itself notes that a rank perturbation of vanishing relative size could in principle reshuffle a non-vanishing fraction of the top-k_n set. Since all Section 4 empirical claims use rank-Pareto margins, the abstract and introduction should carry the same qualification as the theorem, and the paper should state explicitly whether the rank-Pareto consistency claim under Assumption
  2. [Section 4; Figures 3–4] The headline empirical numbers (91% anchored variation explained at rank 10; about 1.25% average relative error for capped excess and normalized VaR) are computed on the same selected tail sample used to fit the AGCA model. The bootstrap diagnostic intervals are conditional on the selected directions and the rank-Pareto transform, and Section 4.1 explicitly states that they exclude marginal estimation, threshold selection, and serial dependence. As written, the abstract and Section 4.2 may be read as out-of-sample validation of the tail simulator. The paper should clearly label these as in-sample descriptive fit diagnostics in the abstract and Section 4, and, if predictive performance is intended, should either provide a genuinely out-of-sample or repeated-split validation protocol or explicitly defer such validation to future work.
minor comments (3)
  1. [Section 3.4] The sentence 'Weighted empirical-tail approximations indicate that both terms can enter at the same k^{-1/2} scale' is a heuristic assertion without proof or reference. Please mark it explicitly as heuristic or supply the supporting calculation/reference, since it motivates the important disclaimer about rank-based CLTs.
  2. [Supplementary S3.8, Table S2] The rank-margin row for AVE_{μ,2} shows coverage 0.878 with the oracle plug-in interval, demonstrating that the oracle formula is not automatically valid after rank-Pareto standardization. This is a useful caution; consider moving one sentence from the supplementary into the main text near Section 3.4, so that practitioners do not apply Corollary 5 directly to rank-Pareto estimates.
  3. [General] Minor language/typography issues: 'Fréchet' appears as 'Frechet' in several figure captions; the notation for the empirical explained variation [A VE uses a bracket that is hard to typeset; and the caption of Figure 1 says 'canonical AGC' where 'AGC1' would be clearer. These are cosmetic but should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: AGCA's core derivation is an exact geometric-spectral reduction under explicit assumptions, and its empirical summaries are in-sample descriptions rather than forced predictions.

full rationale

The central derivation chain is not circular. Proposition 1 proves an exact spherical-geometry identity—sin^2 of the geodesic distance to an anchored great subsphere equals the squared Euclidean residual of the bounded tangent departure—and Theorem 1 follows from this identity plus the Ky Fan spectral principle. The loss is deliberately chosen so that the reduction is exact, but the risk is defined independently as sin^2 d_g(G, M_mu(W)), not as tr((I-Pi_W)Sigma_mu); the equality is derived, not assumed. The anchor mu_0 is fixed by definition, not fitted from the data, and the Pareto index alpha=1 is a consequence of the rank-Pareto marginal standardization used in the empirical section, not a fitted parameter that is then relabeled as a prediction. The tail-simulation bounds in Propositions 4 and 5 are mathematical consequences of Assumption 2 and residual risk; they do not reduce to the object they are said to predict. The estimation results are stated under explicit conditions (Assumption 1 for the oracle consistency, Assumption 2 for rank-based consistency, Assumption 3 for the oracle CLT), and the paper itself flags the gap between oracle and rank-Pareto inference in Section 3.4 and S2.7. The abstract's phrase 'top-k consistency for oracle and rank-Pareto AGCA summaries' is broader than the theorem's Assumption 2 scope, but that is an overstatement of support, not circularity. There are no self-citations used as load-bearing evidence; the cited EVT results are external standard tools. The empirical section evaluates the fitted AGCA on the same selected extreme directions, but the paper presents these as descriptive summaries and reconstructions, not as out-of-sample predictions derived from the same data by tautology. No step in the paper exhibits the defining feature of circularity: a claimed prediction that is equivalent, by construction or by self-citation, to its own input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The method rests on standard EVT angular convergence and regular variation assumptions; the central derivation itself adds no fitted constants. The main subjectively chosen inputs are the tail threshold k and the reported rank p. No new physical or probabilistic entity is invented; the anchor μ_0 is a fixed benchmark, not an invented mechanism.

free parameters (3)
  • Tail threshold k (main fit) = 667 of n=13,340 (5%)
    The headline AVE/error numbers are computed at this hand-chosen threshold; sensitivity is shown over k=67–1334 but no data-driven selection is made.
  • Reported rank p = p=10 (also p=3,5,8,12)
    The effective-dimension claim selects a rank by inspection; rank is a model dimension, not estimated from data.
  • Portfolio evaluation class sizes = 1 equal + 4 block-equal + 250 long-only + 250 limited-leverage
    The 1.25% average relative error depends on this fixed evaluation set; numbers are in-sample reconstruction errors, not out-of-sample predictions.
assumptions (5)
  • domain assumption Assumption 1: L(X/||X||_2 | ||X||_2>r) converges to L(G) on S_+^{d-1}.
    The population target exists only if such a limiting angular law exists; invoked throughout Section 2.
  • domain assumption Assumption 2: Euclidean-polar regular variation with index α>0 and limiting law P_α G.
    Needed for rank-Pareto consistency (Theorem 3, proof S2.7) and for the tail-simulation bounds (Propositions 4–5).
  • domain assumption Assumption 3: second-order angular bias √k_n η_μ(2k_n/n)→0.
    Needed for the oracle CLT (Theorem 4); it assumes away asymptotic bias of the angular second moment at the chosen threshold.
  • domain assumption Continuous marginal distributions and continuous radius R_1.
    Used to make top-k sets unambiguous and rank transforms exact; stated in Theorems 2–3.
  • standard math Standard spectral facts: Ky Fan maximum principle and eigenvalue/projector perturbation bounds.
    Basis of Theorem 1 and Corollary 4; not reproved.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Anchored Geodesic Analysis for Multivariate Extremes." pith.science (2026). https://pith.science/paper/DYJX3JQG

@misc{pith2026260713112,
  author       = {Pith},
  title        = {Pith review of: Anchored Geodesic Analysis for Multivariate Extremes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DYJX3JQG}},
  note         = {Machine review of arXiv:2607.13112}
}
abstract

Extremal dependence is naturally described by the angular law of large multivariate observations. We introduce anchored geodesic component analysis (AGCA), a dimension-reduction method for extremal angular laws on the positive unit sphere. AGCA approximates angular variation by great subspheres constrained to pass through a chosen reference direction, with balanced complete dependence as the default anchor. Under a bounded sine-squared geodesic loss, the population and empirical problems reduce exactly to eigenanalysis of a second-moment matrix of anchored tangent departures. The resulting scores, loadings, residual risks and explained-variation summaries describe departures from the benchmark and remain well defined for face and near-axis extremes. Low-rank AGCA reconstructions also support tail simulation: bounded Lipschitz functionals and homogeneous tail scores, including portfolio capped excesses and value-at-risk, inherit explicit error bounds from the AGCA residual risk. We establish top-\(k\) consistency for oracle and rank-Pareto AGCA summaries and an oracle central limit theorem whose covariance is that of an independent sample from the limiting angular law. In daily equity-portfolio losses, AGCA finds concentrated benchmark-relative tail directions: ten components explain about \(91\%\) of anchored variation and approximate capped-excess and normalized value-at-risk summaries with about \(1.25\%\) average relative error.

Figures

Figures reproduced from arXiv: 2607.13112 by the authors.

Figure 1
Figure 1. Sphere-level geometry in two controlled simulations. The left panel shows a low-dimensional angular law; the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Geometry of anchored geodesic component analysis. In the left panel, the open hemisphere [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Fama–French empirical summaries. Left: cumulative anchored variation explained at the main [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Canonical-anchor AGCA loadings for the Fama–French panel. Columns are portfolios, grouped by bivariate [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

76 extracted references · 2 linked inside Pith

  1. [1]

    Avella Medina, M., R. A. Davis, and G. Samorodnitsky (2025). Insights into kernel pca with application to multivariate extremes. SIAM Journal on Mathematics of Data Science\/ 7\/ (2), 777--801

    2025

  2. [2]

    Butsch, L. and V. Fasen-Hartmann (2025). Estimation of the number of principal components in high-dimensional multivariate extremes. Scandinavian Journal of Statistics\/ 52\/ (4), 2270--2313

    2025

  3. [3]

    Chautru, E. (2015). Dimension reduction in multivariate extreme value analysis. Electronic Journal of Statistics\/ 9\/ (1), 383--418

    2015

  4. [4]

    Chen, A. Y. and T. Zimmermann (2026). Open Source Asset Pricing . https://www.openassetpricing.com/. Accessed July 7, 2026

    2026

  5. [6]

    Cooley, D. and E. Thibaud (2019). Decompositions of dependence for high-dimensional extremes. Biometrika\/ 106\/ (3), 587--604

    2019

  6. [7]

    and H.-G

    Dai, X. and H.-G. M \"u ller (2018). Principal component analysis for functional data on riemannian manifolds and spheres. The Annals of Statistics\/ 46\/ (6B), 3334--3361

    2018

  7. [8]

    De Haan, L. and A. Ferreira (2006). Extreme value theory: an introduction . Springer

    2006

  8. [10]

    Drees, H. and A. Sabourin (2021). Principal component analysis for multivariate extremes. Electronic Journal of Statistics\/ 15\/ (1), 908--943

    2021

Show all 76 references
  1. [11]

    Einmahl, J. H. J., L. De Haan, and V. I. Piterbarg (2001). Nonparametric estimation of the spectral measure of an extreme value distribution. The Annals of Statistics\/ 29\/ (5), 1401--1423

    2001

  2. [12]

    Einmahl, J. H. J., L. De Haan, and A. K. Sinha (1997). Estimating the spectral measure of an extreme value distribution. Stochastic Processes and their Applications\/ 70\/ (2), 143--171

    1997

  3. [13]

    Einmahl, J. H. J., A. Krajina, and J. Segers (2012). An M -estimator for tail dependence in arbitrary dimensions. The Annals of Statistics\/ 40\/ (3), 1764--1793

    2012

  4. [14]

    Einmahl, J. H. J. and J. Segers (2009). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. The Annals of Statistics\/ 37\/ (5B), 2953--2989

    2009

  5. [15]

    Engelke, S. and A. S. Hitz (2020). Graphical models for extremes. Journal of the Royal Statistical Society: Series B (Statistical Methodology)\/ 82\/ (4), 871--932

    2020

  6. [16]

    Fletcher, P. T., C. Lu, S. M. Pizer, and S. Joshi (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging\/ 23\/ (8), 995--1005

    2004

  7. [17]

    Fomichov, V. and J. Ivanovs (2023). Spherical clustering in detection of groups of concomitant extremes. Biometrika\/ 110\/ (1), 135--153

    2023

  8. [18]

    French, K. R. (2026). Kenneth R. French Data Library . https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. Accessed July 7, 2026

    2026

  9. [19]

    Sabourin, and S

    Goix, N., A. Sabourin, and S. Cl \'e men c on (2017). Sparse representation of multivariate extremes with applications to anomaly detection. Journal of Multivariate Analysis\/ 161 , 12--31

    2017

  10. [20]

    Huckemann, S. and H. Ziezold (2006). Principal component analysis for riemannian manifolds, with an application to triangular shape spaces. Advances in Applied Probability\/ 38\/ (2), 299--319

    2006

  11. [21]

    Jan en, A. and P. Wan (2020). k -means clustering of extremes. Electronic Journal of Statistics\/ 14\/ (1), 1211--1233

    2020

  12. [22]

    Jung, S., I. L. Dryden, and J. S. Marron (2012). Analysis of principal nested spheres. Biometrika\/ 99\/ (3), 551--568

    2012

  13. [23]

    Rootz \'e n, and J

    Lhaut, S., H. Rootz \'e n, and J. Segers (2026). Simulation of multivariate extremes: A wasserstein--aitchison gan approach. Extremes\/ 29\/ (2), 157--194

    2026

  14. [24]

    Meyer, N. and O. Wintenberger (2021). Sparse regular variation. Advances in Applied Probability\/ 53\/ (4), 1115--1148

    2021

  15. [25]

    Kiriliouk, and J

    Mourahib, A., A. Kiriliouk, and J. Segers (2025). Multivariate generalized pareto distributions along extreme directions. Extremes\/ 28\/ (2), 239--272

    2025

  16. [26]

    Resnick, S. I. (2007). Heavy-tail phenomena: probabilistic and statistical modeling . Springer

    2007

  17. [27]

    Segers, and J

    Rootz \'e n, H., J. Segers, and J. L. Wadsworth (2018). Multivariate generalized pareto distributions: Parametrizations, representations, and properties. Journal of Multivariate Analysis\/ 165 , 117--131

    2018

  18. [28]

    Rootz \'e n, H. and N. Tajvidi (2006). Multivariate generalized pareto distributions. Bernoulli\/ 12\/ (5), 917--930

    2006

  19. [29]

    Lauze, and M

    Sommer, S., F. Lauze, and M. Nielsen (2014). Optimization over geodesics for exact principal geodesic analysis. Advances in Computational Mathematics\/ 40\/ (2), 283--313

    2014

  20. [30]

    Khanzadeh, Y

    Tabaghi, P., M. Khanzadeh, Y. Wang, and S. Mirarab (2024). Principal component analysis in space forms. IEEE Transactions on Signal Processing\/ 72 , 4428--4443

    2024

  21. [31]

    Wan, P. (2026). Characterizing extremal dependence on a hyperplane. Biometrika\/ 113\/ (2), asag015

    2026

  22. [32]

    Biometrika , volume=

    Decompositions of dependence for high-dimensional extremes , author=. Biometrika , volume=. 2019 , publisher=

    2019

  23. [33]

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=

    Graphical models for extremes , author=. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=. 2020 , publisher=

    2020

  24. [34]

    Electronic Journal of Statistics , volume=

    Principal component analysis for multivariate extremes , author=. Electronic Journal of Statistics , volume=. 2021 , publisher=

    2021

  25. [35]

    2006 , publisher=

    Extreme value theory: an introduction , author=. 2006 , publisher=

    2006

  26. [36]

    IEEE Transactions on Medical Imaging , volume=

    Principal geodesic analysis for the study of nonlinear statistics of shape , author=. IEEE Transactions on Medical Imaging , volume=. 2004 , publisher=

    2004

  27. [37]

    Advances in Applied Probability , volume=

    Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces , author=. Advances in Applied Probability , volume=. 2006 , publisher=

    2006

  28. [38]

    2013 , publisher=

    Matrix Analysis , author=. 2013 , publisher=

    2013

  29. [39]

    Biometrika , volume=

    A useful variant of the Davis--Kahan theorem for statisticians , author=. Biometrika , volume=. 2015 , publisher=

    2015

  30. [40]

    Extremes , volume=

    Multivariate generalized Pareto distributions along extreme directions , author=. Extremes , volume=. 2025 , publisher=

    2025

  31. [41]

    2007 , publisher=

    Heavy-tail phenomena: probabilistic and statistical modeling , author=. 2007 , publisher=

    2007

  32. [42]

    Journal of Multivariate Analysis , volume=

    Multivariate generalized Pareto distributions: Parametrizations, representations, and properties , author=. Journal of Multivariate Analysis , volume=. 2018 , publisher=

    2018

  33. [43]

    Bernoulli , volume=

    Multivariate generalized Pareto distributions , author=. Bernoulli , volume=. 2006 , publisher=

    2006

  34. [44]

    Biometrika , volume=

    Characterizing extremal dependence on a hyperplane , author=. Biometrika , volume=. 2026 , publisher=

    2026

  35. [45]

    SIAM Journal on Mathematics of Data Science , volume=

    Insights into kernel PCA with application to multivariate extremes , author=. SIAM Journal on Mathematics of Data Science , volume=. 2025 , publisher=

    2025

  36. [46]

    Scandinavian Journal of Statistics , volume=

    Estimation of the number of principal components in high-dimensional multivariate extremes , author=. Scandinavian Journal of Statistics , volume=. 2025 , publisher=

    2025

  37. [47]

    Electronic Journal of Statistics , volume=

    Dimension reduction in multivariate extreme value analysis , author=. Electronic Journal of Statistics , volume=. 2015 , publisher=

    2015

  38. [48]

    arXiv preprint arXiv:2503.22296 , year=

    Asymptotic behavior of principal component projections for multivariate extremes , author=. arXiv preprint arXiv:2503.22296 , year=

    arXiv

  39. [49]

    Einmahl, John H. J. and Krajina, Andrea and Segers, Johan , journal=. An. 2012 , publisher=

    2012

  40. [50]

    The Annals of Statistics , volume=

    Rank-based inference for bivariate extreme-value copulas , author=. The Annals of Statistics , volume=. 2009 , publisher=

    2009

  41. [51]

    arXiv preprint arXiv:2606.07213 , year=

    Principal component analysis for multivariate extremes , author=. arXiv preprint arXiv:2606.07213 , year=

    arXiv

  42. [52]

    Journal of Multivariate Analysis , volume=

    Sparse representation of multivariate extremes with applications to anomaly detection , author=. Journal of Multivariate Analysis , volume=. 2017 , doi=

    2017

  43. [53]

    Advances in Applied Probability , volume=

    Sparse regular variation , author=. Advances in Applied Probability , volume=. 2021 , publisher=

    2021

  44. [54]

    Extremes , volume=

    Simulation of multivariate extremes: A Wasserstein--Aitchison GAN approach , author=. Extremes , volume=. 2026 , doi=

    2026

  45. [55]

    Biometrika , volume=

    Analysis of principal nested spheres , author=. Biometrika , volume=. 2012 , publisher=

    2012

  46. [56]

    Principal component analysis for functional data on Riemannian manifolds and spheres , journal=

    Dai, Xiongtao and M. Principal component analysis for functional data on Riemannian manifolds and spheres , journal=. 2018 , publisher=

    2018

  47. [57]

    Advances in Computational Mathematics , volume=

    Optimization over geodesics for exact principal geodesic analysis , author=. Advances in Computational Mathematics , volume=. 2014 , publisher=

    2014

  48. [58]

    IEEE Transactions on Signal Processing , volume=

    Principal component analysis in space forms , author=. IEEE Transactions on Signal Processing , volume=. 2024 , publisher=

    2024

  49. [59]

    2025 , howpublished=

    2025

  50. [60]

    French, Kenneth R. , year=

  51. [61]

    and Zimmermann, Tom , year=

    Chen, Andrew Y. and Zimmermann, Tom , year=

  52. [62]

    2023 , howpublished=

    2023

  53. [63]

    Proceedings of the National Academy of Sciences , volume=

    Global estimates of mortality associated with long-term exposure to outdoor fine particulate matter , author=. Proceedings of the National Academy of Sciences , volume=. 2018 , publisher=

    2018

  54. [64]

    The Journal of Clinical Investigation , volume=

    Particulate matter air pollution: effects on the respiratory system , author=. The Journal of Clinical Investigation , volume=. 2025 , publisher=

    2025

  55. [65]

    The lancet , volume=

    Global, regional, and national comparative risk assessment of 84 behavioural, environmental and occupational, and metabolic risks or clusters of risks for 195 countries and territories, 1990--2017: a systematic analysis for the Global Burden of Disease Study 2017 , author=. Th...

    1990

  56. [66]

    Stochastic Processes and their Applications , volume=

    Estimating the spectral measure of an extreme value distribution , author=. Stochastic Processes and their Applications , volume=. 1997 , publisher=

    1997

  57. [67]

    The Annals of Statistics , volume=

    Nonparametric estimation of the spectral measure of an extreme value distribution , author=. The Annals of Statistics , volume=. 2001 , publisher=

    2001

  58. [68]

    The Annals of Statistics , volume=

    Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution , author=. The Annals of Statistics , volume=. 2009 , publisher=

    2009

  59. [69]

    1986 , publisher=

    Empirical Processes with Applications to Statistics , author=. 1986 , publisher=

    1986

  60. [70]

    Zeitschrift f

    Limit theorems for the ratio of the empirical distribution function to the true distribution function , author=. Zeitschrift f. 1978 , publisher=

    1978

  61. [71]

    1987 , publisher=

    Regular Variation , author=. 1987 , publisher=

    1987

  62. [72]

    Electronic Journal of Statistics , volume=

    k -means clustering of extremes , author=. Electronic Journal of Statistics , volume=. 2020 , publisher=

    2020

  63. [73]

    Biometrika , volume=

    Spherical clustering in detection of groups of concomitant extremes , author=. Biometrika , volume=. 2023 , publisher=

    2023

  64. [74]

    Bingham, N. H., C. M. Goldie, and J. L. Teugels (1987). Regular Variation . Cambridge University Press

    1987

  65. [75]

    Genest, C. and J. Segers (2009). Rank-based inference for bivariate extreme-value copulas. The Annals of Statistics\/ 37\/ (5B), 2990--3022

    2009

  66. [76]

    Horn, R. A. and C. R. Johnson (2013). Matrix Analysis\/ (2 ed.). Cambridge University Press

    2013

  67. [77]

    Shorack, G. R. and J. A. Wellner (1986). Empirical Processes with Applications to Statistics . Wiley

    1986

  68. [78]

    Wellner, J. A. (1978). Limit theorems for the ratio of the empirical distribution function to the true distribution function. Zeitschrift f \"u r Wahrscheinlichkeitstheorie und verwandte Gebiete\/ 45\/ (1), 73--88

    1978

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.