Benchmarks

Benchmarks in VineCopulas.jl serve three different purposes:

• the small Julia-only BenchmarkTools.SUITE in benchmark/benchmarks.jl tracks representative evaluation, fitting, selection, transforms and CDF workloads; Tachometer compares it on pull requests;

• correctness gates show that the Julia implementation agrees with an external reference on deterministic fixtures;

• performance benchmarks explain where the Julia implementation is fast, where it is near parity, and where optimization work remains.

Correctness before timing

A faster result is not meaningful until both implementations are known to be evaluating or fitting the same model. Read correctness reports before interpreting speed tables.

The continuous suite is intentionally distinct from the deeper R-backed campaigns below: the latter remain useful for parity and diagnosis, but are not required for ordinary pull-request benchmark comparisons.

Correctness against rvinecopulib

The external reference is R rvinecopulib, which wraps the C++ vinecopulib library. The correctness gate checks:

• pair-copula log densities, h-functions, and inverse h-functions;

• fixed general R-vine evaluation and transforms;

• fixed-structure fitting;

• automatic Dissmann-style structure selection.

Equivalent R-vine matrices can encode the same conditioned and conditioning sets, so the gate compares represented edges and numerical behavior rather than raw matrix text.

Run:

PARITY_N=800 PARITY_FIT_MODE=common \
  bash benchmark/correctness/run_correctness_gate.sh

PARITY_N=800 PARITY_FIT_MODE=default \
  bash benchmark/correctness/run_correctness_gate.sh

The v0.1.1 release-candidate gate passed both supported parity modes:

Mode	Fit		Parameters	Structure	Families	Fitted parameters
common	automatic		10 / 10	PASS	PASS	PASS
common	fixed		11 / 11	PASS	PASS	PASS
default	automatic		11 / 11	PASS	PASS	PASS
default	fixed		11 / 11	PASS	PASS	PASS

The fixed-model stage also passed all pair fixtures and both general R-vine fixtures. Representative fixed-vine discrepancies were around for log density and below   for inverse transforms.

Performance overview

The standard evaluation battery measures log density, Rosenblatt transform, inverse Rosenblatt transform, simulation, and numerical CDF behavior. It keeps evaluation separate from fitting because they stress different parts of the implementation.

Run:

bash benchmark/run_main.sh

Fitting benchmarks use a separate entry point:

bash benchmark/fitting/run_fit.sh

Tip

Use the benchmark reports to choose optimization targets. For example, standalone pair primitives and fused vine traversal can have different bottlenecks.

The reference evaluation campaign used   observations on an Apple Silicon macOS system with Julia 1.12.6, R 4.5.3, and rvinecopulib 0.7.3.1.0. These numbers are machine-specific, but the profile is useful:

Family		Julia	rvinecopulib	Interpretation
Gaussian	5	14.0 ms	21.9 ms	Julia 1.56× faster
Gaussian	10	16.8 ms	33.0 ms	Julia 1.96× faster
Gaussian	20	37.9 ms	72.0 ms	Julia 1.90× faster
Clayton	5	13.7 ms	18.8 ms	Julia 1.38× faster
Clayton	10	17.1 ms	27.6 ms	Julia 1.62× faster
Clayton	20	39.3 ms	61.8 ms	Julia 1.57× faster
Gumbel	5	31.0 ms	30.1 ms	near parity
Gumbel	10	38.4 ms	45.3 ms	Julia 1.18× faster
Gumbel	20	85.3 ms	100.1 ms	Julia 1.17× faster
Frank	5	20.8 ms	15.6 ms	R 1.33× faster
Frank	10	24.0 ms	21.9 ms	R 1.10× faster
Frank	20	54.8 ms	49.0 ms	R 1.12× faster

Gaussian and Clayton are consistently faster in this battery. Gumbel is close to parity, while Frank log density is modestly faster in rvinecopulib.

The fitting benchmark is currently the main performance gap:

Task	Common model space	Default model space
Gaussian pair selection	R 3.40× faster	R 7.25× faster
Clayton pair selection	R 3.71× faster	R 8.45× faster
Fixed-structure R-vine	R 3.16× faster	R 6.74× faster
Automatic R-vine	R 3.31× faster	R 6.93× faster

This is not a correctness issue: the corresponding common and default correctness gates pass. It is the clearest performance target for future work.

Generic vs specialized pair conditionals

The architectural question is not simply whether hfunc1(C, u, v) is fast. It is whether the specialized vine spelling still adds value over the canonical Copulas.jl conditioning interface:

hfunc1(C, u, v)
cdf(condition(C, 2, v), u)

hinv1(C, q, v)
quantile(condition(C, 2, v), q)

Focused diagnostics live in:

julia --project=benchmark benchmark/diagnostics/condition_fallback.jl
julia --project=benchmark benchmark/diagnostics/fused_pair_kernels.jl
julia --project=benchmark benchmark/diagnostics/vine_engine_allocations.jl

The goal is to decide which optimized paths remain justified. Generic compatibility comes first; family-specific or fused methods should be retained when they are measurably faster, more stable, or avoid important allocations.

Reproducing results

Instantiate the benchmark environment:

julia --project=benchmark -e '
using Pkg
Pkg.develop(path=".")
Pkg.resolve()
Pkg.instantiate()
'

Install the R reference dependency once:

Rscript -e 'install.packages("rvinecopulib", repos="https://cloud.r-project.org")'

Then run the correctness and performance commands above from the repository root. Generated reports are written under benchmark/reports/ and benchmark/correctness/results/.

Note

Benchmark numbers are local measurements. They are useful for tracking regressions and prioritizing engineering work, not as hardware-independent guarantees.