High-Dimensional Methods for Elliptically Symmetric Distributions

Fast, documented implementations of robust estimation, testing, dimension reduction, classification, and clustering methods for high-dimensional elliptically symmetric data. Computational kernels use 'Rcpp' and 'RcppArmadillo'. The package follows methods reviewed in Feng (2026), "High-Dimensional Data Analysis for Elliptically Symmetric Distributions" < https://github.com/flnankai/HDElliptical/releases>.

HDElliptical

HDElliptical is the companion R package for High-Dimensional Data Analysis for Elliptically Symmetric Distributions. Version 0.1.2 implements Chapters 1--7, with Rcpp/RcppArmadillo kernels for the computationally intensive parts and direct links between the book, the source papers, the public API, and regression tests.

Current status

The Chapter 1 foundation layer is implemented and tested:

• spatial signs, the modified-Weiszfeld spatial median, and empirical spatial ranks;
• spatial sign covariance, multivariate spatial Kendall, and spatial-rank covariance matrices;
• Tyler's trace-normalized shape estimator and the angular central Gaussian log-likelihood;
• the joint Hettmansperger-Randles location-shape estimator; and
• simulation from spherical and elliptically symmetric distributions.

The first Chapter 2 block is also implemented: one- and two-sample Hotelling tests, fixed-dimensional spatial sign/rank tests, and the high-dimensional Srivastava--Du, Park--Ayyala, Bai--Saranadasa, Chen--Qin, and Srivastava--Katayama--Kano mean tests, plus the Cai--Liu--Xia precision-adjusted maximum test, the Feng--Zou--Wang--Zhu scale-invariant Behrens--Fisher test, the common-covariance Composite T2 test, and the Wang--Peng--Li high-dimensional raw spatial-sign test and Feng--Sun scalar-invariant one-sample sign test, together with the Feng--Liu--Ma one-sample INST, the Feng--Zou--Wang and Li--Wang--Zou two-sample sign tests, the Feng--Zhang--Liu high-dimensional two-sample spatial-rank test, and the Huang--Liu--Zhou--Feng two-sample inverse-norm sign test (tINST), together with the Xu--Lin--Wei--Pan analytical adaptive sum-of-powers test (aSPU), the scaled-spatial-median spatial-sign MAX/MAXSUM procedures, the Yan--Zhao--Feng general-(m) weighted MAX/MAXSUM procedures, and the Zhang--Feng adaptive rank max/sum/Cauchy tests. The late Chapter 2 block is now implemented as well: the Zhang--Zhou--Guo one-sample normal-reference test with paired and same-unit linear-hypothesis wrappers, the Zhang--Zhu--Zhang scale-invariant two-sample normal-reference test, the Wang--Xu approximate-randomization test, the Feng--Wang PDQ two-sample spatial-sign test, the Zhao--Feng strong-correlation one-sample spatial-sign test, and the Feng--Zhou--Wang fixed-ridge and Cauchy-aggregated elliptical regularized Hotelling tests. The formula-complete generic weighted layer is also public through generic_weighted_hr_location() and oracle_weighted_sign_sum_test(). The oracle statistic requires an explicit null_sd, or supplied nu2 and trace_R2, before reporting a calibrated p-value; no feasible plug-in is invented from incomplete prose. The aSPU interface distinguishes the primary paper's raw mean-difference powers from the book's scale-invariant studentized-coordinate variant. CLX exposes distinct oracle, supplied feasible, and adaptive-threshold precision paths. The Zhang--Feng sum and combined rank paths require either a supplied long-run variance or an explicitly requested Ouyang--Parzen lag; the package does not invent an automatic bandwidth absent from the source paper. The compiled kernels expose the raw formula components and reject singular or invalid variance estimates instead of silently applying a ridge or pseudoinverse.

The only Chapter 2 item not implemented is the structured-correlation review at ch2_location.tex:2438-2451. It is explicitly blocked pending source: a legally accessible primary full text has not been obtained, and the book gives no executable statistic or calibration formulas. No public API is invented from that prose.

Chapter 3 is now implemented as 38 callable APIs. It covers classical and high-dimensional covariance/sphericity tests, covariance thresholding and POET, the convex-l1 EC2 covariance program, Gaussian off-diagonal graphical lasso and Gaussian CLIME, elliptical sign/rank/adaptive sphericity, SSCM proportionality, certificate-checked SCLIME/SGLASSO precision estimation, tensor elliptical graphs, the Ollila shrinkage/BASIC/BASICS/linear-pooling line, high-dimensional HR, and elliptical factor reconstruction/precision. Solver functions return feasibility, positive-definiteness, KKT, and convergence diagnostics and do not silently add a ridge, eigenvalue floor, pseudoinverse, or nearest-PD repair. The distinct SCIO and scaled-lasso precision reviews, and the adaptive/MC+ extensions of EC2, remain review-only because the short book passages do not close their additional solver and tuning contracts.

Chapter 4 is now implemented as 33 callable APIs. It covers unconditional and conditional alpha tests, mutual-fund FDR, classical/dense-max/spatial-sign/ERHT change-point tests and ERHT-WBS segmentation, Gaussian and robust white-noise tests, the radial-directional diagnostic, weighted-sign alpha completion, book Gaussian-alpha and conditional-Wald benchmarks, and Gaussian/panel/vector/exact rank-U independence tests. The separate mutual-independence max--sum studentization review remains source-blocked; the available sources do not uniquely determine its calibration, and the exact rank-U vector method is not presented as a substitute. Randomization, wild bootstrap, permutation, and Gaussian-process draws appear only where they are intrinsic to a method's calibration; the package contains no paper-specific simulation grids, size/power replications, tables, scripts, or data.

Chapter 5 is now implemented as 23 callable APIs on a common certified binary-classifier fit and prediction contract. It covers oracle and classical Gaussian/elliptical LDA and QDA, independence/FAIR/thresholded and direct sparse linear rules, sparse Gaussian and spatial-sign QDA, certified precision plug-ins, and robust GQDA/HR-GQDA. Fits record score orientation and scale, priors, tuning, feasibility/KKT and determinant/SPD certificates; invalid fits cannot be used for prediction and receive no hidden numerical repair.

Chapter 6 is now implemented as 16 callable APIs across four modules: classical PCA/CCA/Bartlett testing with robust factor subspaces, RTS factors and factor-number selection; spatial-sign, Kendall and generalized-sign PCA; TPM/Fantope/PMD sparse PCA and CCA; and the primary metric-lasso sscca() interface, kept distinct from the book's sign_whitened_sparse_cca(). Centers, divisors, ranks, eigengaps, ties, supports, l1 boundaries, ridge operators, KKT certificates and failure states are explicit; there is no hidden floor, pseudoinverse or numerical repair.

Chapter 7 is now implemented as 14 exported APIs backed by 19 registered compiled kernels. It covers Lloyd K-means, full-covariance Gaussian-mixture EM, sparse K-means and K-medians, CHIME, IF-PCA, K-spatial medians, SM-SSCM, Sparse-SM and its tau/K selectors, plus semc_fit(), its prediction method, and semc_select_k_gap(). Initialization, ties, empty clusters, cycles, seeds, solver certificates and returned final states are explicit. CHIME canonicalizes component labels so omega <= 1/2 after initialization and every update, and the spatial kernels preserve finite close-large, subnormal and opposite-extreme geometry.

SEMC is no longer source-blocked. It is an independent rewrite against the primary source and the pinned flnankai/GEMcluster@10fce04fe690fe274dd5d237cfcd3d5c6a4139f6 MIT release. Primary-paper and official-software-only controls are distinguished explicitly; formula and pinned-software fixtures are tested without copying author-code expressions or reproducing a paper simulation.

The fresh Chapter 1--7 gate contains 176 exports, 3 S3 registrations, and 140 Rcpp attributes matched one-for-one by 140 native registrations. Roxygen generated 223 Rd files covering 169 public topics. Across 56 test files there are 688 named test_that blocks and 5,781 static expect_* calls; the isolated installed-package run reported 6,267 passes and zero failures, errors, warnings, or skips. Two independent generation passes produced the same 226-file hash manifest, and all 226 generated files matched the synchronized release tree. The final 0.1.0 source tarball built from this release state completed both release checks. Standard R CMD check --no-manual reported 0 ERROR, 0 WARNING, and 1 NOTE; remote-enabled R CMD check --as-cran --no-manual reported 0 ERROR, 0 WARNING, and 2 NOTEs. Both ran examples, tests, and vignettes; the as-CRAN run also executed donttest examples. Both checks share the conservative Windows DLL scan for linked _exit, abort, and exit symbols; an exact direct-call scan of the R and src/ sources found zero calls. The additional as-CRAN NOTE identifies this as a new CRAN submission.

The public repository and issue tracker were bootstrapped on 2026-08-16 and now resolve at https://github.com/flnankai/HDElliptical. The final as-CRAN run kept incoming remote checks enabled and completed without errors or warnings. The annotated v0.1.0 tag and GitHub release were created only after the Windows, macOS, and Linux Actions jobs passed and the post-bootstrap tarball, check logs, and SHA-256 manifest were regenerated.

The normative method ledger has 87 rows: 83 implemented, 2 review-only, and 2 source-blocked. The two blocked rows are the Chapter 2 structured-correlation review and Chapter 4 mutual-independence studentization; SEMC is implemented.

The dated method coverage, machine-readable traceability, book errata, and validation snapshot keep the book formula, primary-paper correction, public API, compiled kernel, and test evidence traceable for release and manuscript review.

The installed-package deterministic benchmark harness runs one representative callable workflow for each of Chapters 1--7. Its machine-local report and CSV output record timings and deterministic fingerprints. They are software benchmarks, not paper-specific simulation, size/power, or empirical-reproduction studies.

Installation

# Current GitHub version:
remotes::install_github("flnankai/HDElliptical")

For a local checkout:

install.packages(".", repos = NULL, type = "source")

Quick start

library(HDElliptical)

set.seed(20260814)
x <- relliptical(
  n = 200,
  location = c(0, 0, 0),
  shape = matrix(c(2, .5, 0, .5, 1, .2, 0, .2, .7), 3),
  radial = function(n) abs(rt(n, df = 3))
)

spatial_median(x)
sscm(x)
spatial_kendall(x)
tyler_shape(x)
hr_estimator(x)

# Chapter 2 examples
srivastava_du_one_sample_test(x)
park_ayyala_one_sample_test(x)

y <- relliptical(
  n = 200,
  location = c(0.1, 0, 0),
  shape = matrix(c(2, .5, 0, .5, 1, .2, 0, .2, .7), 3),
  radial = function(n) abs(rt(n, df = 3))
)
paired_y <- x + matrix(rnorm(length(x), sd = 0.2), nrow(x))
cai_liu_xia_two_sample_test(x, y)
feng_zou_wang_zhu_two_sample_test(x[1:10, ], y[1:11, ])
composite_t2_two_sample_test(x[1:8, 1:3], y[1:8, 1:3])
wang_peng_li_one_sample_test(x)
feng_sun_one_sample_test(x[1:10, ], tol = 1e-6)
inst_one_sample_test(x[1:10, ], tol = 1e-6)
feng_zou_wang_two_sample_sign_test(x[1:10, ], y[1:11, ], tol = 1e-6)
li_wang_zou_two_sample_sign_test(x[1:10, ], y[1:11, ], tol = 1e-6)
feng_zhang_liu_spatial_rank_test(x[1:10, ], y[1:11, ], tol = 1e-6)
tinst_two_sample_test(x, y)
xu_lin_wei_pan_aspu_test(x, y)
scaled_spatial_median(x)
spatial_sign_max_test(x)
spatial_sign_maxsum_test(x[1:10, ], tol = 1e-6)
weighted_scaled_spatial_median(x, m = 0)
yan_zhao_feng_weighted_max_test(x, m = -1)
yan_zhao_feng_weighted_maxsum_test(x[1:10, ], m = -1, tol = 1e-6)
zhang_feng_rank_one_sample_test(x, component = "max")
zhang_feng_rank_two_sample_test(x, y, component = "max")
zhang_zhou_guo_one_sample_test(x)
zhang_zhou_guo_paired_test(x, paired_y)
zhang_zhou_guo_linear_hypothesis_test(x, rbind(c(1, -1, 0)))
zhang_zhu_zhang_two_sample_test(x, y)
wang_xu_approx_randomization_test(
  x[1:8, ], y[1:8, ], calibration = "exact"
)
feng_wang_pdq_two_sample_test(
  x[1:10, ], y[1:11, ], B = 19, seed = 2605
)
zhao_feng_strongcorr_sign_test(x[1:12, ], B = 19, seed = 2601)
elliptical_regularized_hotelling_test(x, rho = 0.5)
elliptical_regularized_hotelling_cauchy_test(
  x, rho = c(0.2, 0.5, 1)
)

# Chapter 3 examples
john_sphericity_test(x)
zou_peng_feng_wang_sphericity_test(x)
ollila_raninen_shrinkage_covariance(x, sphericity = "ell3")
elliptical_factor_number(x, max_factors = 1)
spatial_sign_poet(x, factors = 1, threshold = 0.1)

# Chapter 5 examples
training <- rbind(x, y)
group <- factor(rep(c("class1", "class2"), each = nrow(x)))
lda_fit <- classical_lda_classifier(training, group)
predict(lda_fit, training[1:4, ], type = "class")

Every function treats rows as observations and columns as variables. Exact zero directions follow the convention U(0) = 0; estimators expose convergence and zero-residual diagnostics instead of silently discarding observations.

Reproducibility policy

Each method is accepted into the package only after it has:

1. a source location and bibliographic key;
2. a documented callable API;
3. numerical or invariance tests;
4. a compiled implementation when it materially reduces runtime; and
5. a check against the original paper whenever the book omits an implementable quantity or contains a suspected typographical error.

The package is under active development and is not yet ready for scientific production use.

The package targets reusable method implementations. It does not ship the paper-specific Monte Carlo grids, simulation tables, or power-study scripts from the many source articles; compact deterministic and fixed-seed checks are kept only to verify formulas, invariances, numerical stability, and API contracts.

Randomization and wild-bootstrap draws inside wang_xu_approx_randomization_test(), feng_wang_pdq_two_sample_test(), and zhao_feng_strongcorr_sign_test() are intrinsic parts of those tests' null calibrations. They are callable-method computations, not replications of the papers' simulation studies. No paper size/power simulation scenarios or result tables are included.

Chapter 7 selector permutations and IF-PCA empirical-null generation are also intrinsic method calibrations. They use supplied plans or explicit local seeds and restore the caller's RNG state; they are not paper simulation replications.