Reproducibility and validation

The 1.0.3 maintenance candidate starts from the CRAN 1.0.2 source archive, published on 2020-02-29. It repairs compatibility and interface errors while retaining the historical local quadratic approximation solver, initialization, lambda grid, contiguous CV folds, elimination threshold and iteration cap.

The article is Yuzbasi, Arashi and Akdeniz (2021). The published algorithm is described in Section 3.1. Package examples demonstrate the interfaces; they do not claim to reproduce the article’s tables or figures.

Input dictionary

Input	Meaning
X	Numeric design matrix: rows are observations, columns are predictors
y	Numeric response vector with one entry per observation
q	Positive bridge-penalty exponent, at most two
lambda	Nonnegative fitted penalty grid
R	Restriction matrix with one column per standardized predictor
r	Target vector or one-column matrix with one entry per restriction
eta	Historical elimination and squared coefficient-change threshold
converge	Initial convergence-distance value
nfolds	Number of contiguous validation folds
num_threads	Requested fold-level thread count when supported by the build

Missing or nonfinite data, constant predictors and incompatible dimensions are rejected. The package does not impute missing values or silently remove constant predictors. Supply linearly independent restriction rows. The historical solver does not return iteration counts or a convergence certificate; validate a fitted path before substantive interpretation. Zero penalties are accepted only when the resulting linear systems can be resolved. Singular or numerically unresolved systems fail explicitly; no automatic pseudoinverse or approximate estimate is substituted.

Checking restriction scales

The restriction defines linear combinations of standardized predictor slopes. The fitted intercept is excluded. To inspect the restriction residual, use the same column deviations as the solver. In this example path is a fitted restricted model; X, R and r are its supplied inputs.

center <- colMeans(X)
scale <- sqrt(colMeans(sweep(X, 2, center, "-")^2))
standardized_slopes <- path$betas[-1, , drop = FALSE] * scale
restriction_residual <- sweep(R %*% standardized_slopes, 1, as.numeric(r), "-")
max(abs(restriction_residual))

The calculated residual measures departure from each supplied standardized-scale restriction over the fitted path. Thresholding can affect residuals. This check does not certify convergence or global optimality.

Local validation

The maintenance tests exercise an independent ridge reference for q = 2, coefficient and prediction methods, restriction scales, small CV fits and invalid inputs. Use a single BLAS/OpenMP thread for reproducible small checks. The second-stage tests include 36 frozen synthetic cases from the first maintenance candidate, with q = 0.5, 1, 2, ordinary, wide and correlated designs, and all four fitting interfaces. They compare coefficient paths, CV errors, selected penalties and support at the historical threshold. Independent constrained ridge and uneven-fold CV references provide additional checks. One- and two-thread requests are compared when supported by the installed build. These tests are regression checks, not a full simulation study or a proof of global optimality. Their frozen reference records its source-archive checksum, seed and R version without requiring the historical package at test time.

From the directory containing a built source archive:

OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 VECLIB_MAXIMUM_THREADS=1 \
  R CMD check --as-cran rbridge_1.0.3.tar.gz

Record sessionInfo() and package versions with any study outputs. Examples use seed 2019. The repository’s published version and software environment should be recorded separately from any data-analysis configuration.

Performance and parallel builds

Fixed cross-products are cached, active-set products are reused until the set changes, and the restricted solver shares a linear-system factorization across coefficient and restriction right-hand sides. CV reuses its training slices and calculates predictions for the path in one matrix multiplication. Roundoff can differ from the historical order of operations, so compatibility is checked numerically rather than asserted to be bit-for-bit on every platform.

rbridge_info() reports the actual compiled OpenMP capability. The build uses R’s portable OpenMP flags and falls back to sequential CV when those flags are empty. No additional OpenMP runtime is installed by the package. Worker errors are collected and reported after the workers finish. Set BLAS threads to one before starting R to avoid nested parallelism.

The development workspace supplies a separate timing driver with frozen seeds, isolated engine processes, warmups, timing shards, resume checks and a SHA-256 manifest. This driver is development tooling and is not a runtime dependency of the package. Full timings require their numerical gates to pass before any speed claim. Small smoke timings establish executability only. Record the actual hardware and engine hashes and report the tested designs with any benchmark.

Completed timing evidence

The completed timing protocol contains four synthetic designs: 80 observations and 16 predictors; 300 and 60; 500 and 180; and a wide design with 60 observations and 120 predictors. The bridge-penalty exponent q takes values 0.5, 1 and 2. Each of the four interfaces uses the same seed 2019, 24 penalties and, where applicable, five contiguous CV folds. Each engine has two warmups and 15 timed fits per case, in an isolated R process. Startup and loading are excluded.

The reference is the first repaired 1.0.3 candidate, identified by its engine hash. Both engines share the same unreleased version number. CV and BLAS use one thread, so these measurements assess serial engine optimization. All 96 shards and 48 comparisons pass the SHA-256, dimension and numerical gates. Coefficient/CV relative differences are at most 5.51e-14 and prediction differences at most 3.31e-15. Selected penalties, support at 1e-7 and reported variable counts are identical. A relative error divides the largest absolute difference by the greater of one and the largest reference magnitude.

Interface	Median of case speed ratios	Range
bridge()	3.14x	1.52-14.62x
rbridge()	2.59x	1.18-3.93x
cv.bridge()	2.99x	1.31-13.00x
cv.rbridge()	2.57x	1.41-3.88x

Each ratio divides the median baseline time by the median optimized time. Results apply to the recorded Apple M3 Pro, R 4.6.0 environment. They use one data seed and a fixed baseline-then-candidate order; engine order was not randomized across sessions. They do not measure accuracy, certify convergence, or compare directly with the original CRAN 1.0.2 engine. The timer’s precision notice is retained in the provenance; solver warnings were treated as errors.

Inspect the report, case comparisons, raw timing repetitions and protocol and environment. The GitHub preparation includes the frozen historical driver and reference source for this recorded comparison. Its engine hashes depend on installed build contents; rebuilding on another platform establishes a new signature. The current development driver uses byte ordering for file hashes, making new signatures independent of the operating system’s collation setting. The completed historical run remains verifiable with its frozen driver and LC_ALL=en_US.UTF-8; changing that driver invalidates its recorded signature.

Reproducing the published study

The package source provides estimators and demonstrations. The original complete simulation and empirical reproduction drivers are not included in this source archive. A full reproduction requires the original datasets, configuration, seeds and scripts; neither the README demonstration nor a successful package check establishes agreement with the published results.