Geometric Analysis of Configurations in High-Dimensional Spaces

Computes Average Membership Degree (AMD) from soft-membership matrices and constructs AMD curves across candidate configuration counts. The current curve backend uses repeated fuzzy c-means fits while keeping the AMD definition, membership estimation, curve construction, and final assignment as separate operations. Fitted memberships, convergence diagnostics, preprocessing transformations, seeds, row maps, and process-level reproducibility links are retained. The package reports numerical curve features but leaves their scientific interpretation to an explicitly justified analysis. The method was introduced by Mendoza and Araújo (2022) .

AMDconfigurations

Average Membership Degree and AMD curves

AMDconfigurations implements the computational core of Average Membership Degree (AMD). The package separates four operations that should not be confused:

1. amd_score() evaluates AMD from an existing soft-membership matrix.
2. compute_amd_curve() estimates one membership matrix for each candidate number of configurations and evaluates AMD on it.
3. The user inspects and justifies the relevant resolution from the complete curve.
4. assign_configurations() extracts the exact stored fit for that explicitly selected resolution.

AMD is defined independently of fuzzy c-means. Fuzzy c-means is the current membership-estimation backend used to construct curves. At a specified resolution and data representation, its groups are empirical configurations / clusters of the fitted FCM partition. They need not coincide with generator components, density modes or modal basins. Count correctness does not validate assignments or the substantive meaning of those configurations.

Users moving from public version 0.1.0 should read inst/MIGRATION.md. Saved 0.1.0 curve objects remain inspectable, but they do not contain enough information to reconstruct the new minimum-objective primary curve.

Installation

Version 0.4.0 includes the drop-37 reader. Download the checked source archive AMDconfigurations_0.4.0.tar.gz from the GitHub releases and install it explicitly:

install.packages("AMDconfigurations_0.4.0.tar.gz", repos = NULL, type = "source")

Core example

The example below uses three generating components. We extract a fitted FCM partition at c=3 to illustrate the API; this does not equate components, density modes and empirical configurations. Real analyses must justify c from the observed curve and supporting diagnostics.

library(AMDconfigurations)

set.seed(8301)
X <- rbind(
  matrix(rnorm(120, mean = 0, sd = 0.7), ncol = 3),
  matrix(rnorm(120, mean = 4, sd = 0.7), ncol = 3),
  matrix(rnorm(120, mean = 8, sd = 0.7), ncol = 3)
)

candidate_c <- 2:6
starts <- 5L
seed_matrix <- outer(
  seq_len(starts),
  candidate_c,
  function(start, c) 100000L + 1000L * c + start
)
colnames(seed_matrix) <- as.character(candidate_c)

curve <- compute_amd_curve(
  X,
  its = starts,
  nin = min(candidate_c),
  nsp = max(candidate_c),
  seeds = seed_matrix,
  iter_max = 1000,
  verbose = FALSE
)

plot(curve)

# Explicit illustrative resolution; not an independently verified modal truth.
chosen_c <- 3L
assigned <- assign_configurations(curve, c = chosen_c)

curve$c_peak and its compatibility alias curve$c_opt are numerical summaries, not automatic scientific decisions. A maximum at the right boundary does not establish a number of groups. An internal peak must be interpreted in the context of the subsequent curve, possible plateaus, stability and the examined range.

Curve and stored-partition diagnostics

Version 0.2.0 adds a descriptive operation:

diagnostics <- diagnose_amd(curve)
diagnostics$candidates    # Curve neighbours and membership/size summaries
diagnostics$groups        # Per-group hard counts and soft membership masses
diagnostics$transitions   # Label-invariant comparisons of successive partitions
diagnostics$dictionary    # Definitions and scales
diagnostics$unavailable   # What this operation does not measure

This operation reads exactly the fits already stored in curve; it does not fit them again or choose K. It needs no original data and constructs no sample distance matrix. Winning membership, entropy and ambiguity describe memberships, not density or geometric compactness. ARI and NMI describe agreement between successive partitions, not stability under resampling. The default top-two gap of 0.1 is an explicit descriptive convention, not a universal threshold for structure. A strict local-maximum flag is likewise only a numerical summary, not a rule for interpreting an entire curve.

Optional drop-37 model-assisted reader

The base workflow does not require a model, ranger or digest. The optional API accepts the external amd_predictor_bundle_v3_drop37.rds and its SHA-256. The model and matching manifest.json are separate assets of release v0.4.0, alongside the checked source archive. Download them explicitly from the GitHub releases page linked above. The model is about 49 MiB and is not part of the CRAN source package.

The five discovery-trained forests use 194 raw and 231 processed predictors, excluding the 37 dimensional descriptors responsible for the tested common-unit sensitivity. Their own exact-K and one/several correctness calibrators were fitted on selection populations only.

# Optional dependencies; install them only if using the reader.
# install.packages(c("ranger", "digest"))
bundle_path <- "path/to/amd_predictor_bundle_v3_drop37.rds"
bundle_sha256 <- "3407a5a55c30a70e150afaa16a52ee4f2c8d3ed265941f3af8c4cb9d0c99035f"
reader <- load_amd_predictor(bundle_path, sha256 = bundle_sha256)
result <- predict_amd(X, reader, seed = 123, verbose = FALSE)
plot(result)
result$proposed_k
result$reference_reliability  # conditional on reference simulations
result$agreement              # forest votes, not correctness probability
result$reliability            # applicability to this distribution is unestablished

# A compatible saved curve avoids repeating the principal fits:
# c=2:14, 20 starts, m=2, iter_max=2000, no scaling or preselection.
# result <- predict_amd(X, reader, curve = compatible_curve, seed = 123)

The reader still performs five computational restarts per c required by the feature contract. These are not sample resampling. Numerical curve maximum, assisted proposal and scientific choice remain separate. K targets the numerically resolved density-mode count of the simulation reference, not necessarily generator components; AMD itself does not estimate density.

The reader estimates the population density-mode count. proposed_k is its proposal; agreement describes the five forest votes. Neither consensus nor the numerical AMD maximum is an automatic scientific choice of resolution.

reliability is NA with status not_established_for_analysed_distribution. Continuous own-calibrator values remain in reference_reliability, separately for exact K and one/several. These are estimates conditional on the reference simulations, not certified probabilities for a new distribution. Matching N and D does not establish distributional applicability. A later prospective bank showed an exact-count confidence excess of 2.40 percentage points; a controlled distribution challenge showed an excess of 6.18 points. Unanimous forests can share the same error.

Reference values are suppressed by default outside N={150,600}, D={2,6,15}, assessed output support K=1:8, or assessed missingness and training-constant conditions. allow_extrapolation=TRUE exposes off-grid reference values only; it never certifies applicability or overrides feature/output guards. No confidence cutoff chooses K. Do not alter data to match the numerical grid. See system.file("READER_EVIDENCE.txt", package="AMDconfigurations").

Historical v1/v2 bundles remain reproducible with their matching historical package versions and are rejected by the active drop-37 interface. Family- exclusion readers are archived diagnostics and are not included in this bundle.

A working-memory estimate is checked before new fits; the default 512 MiB descriptor limit excludes loaded forests and is not a total-memory guarantee. Partition distances use all rows. Frozen global-distance summaries use a seeded sample of 2000 rows when N exceeds 2000, with sample count and sampling flags reported. No additional approximation bypasses the guard.

Reproducibility

Every curve records its seeds, convergence diagnostics, retained variables, scaling transformation and row map. It also contains a manifest_id linking it to the process-level environment returned by:

manifest <- amd_reproducibility_manifest()
identical(curve$manifest_id, manifest$manifest_id)

Immediately after each curve, retrieve amd_reproducibility_manifest() and save one copy of every distinct manifest_id; curves sharing an identifier can share that copy. This also covers the uncommon case in which RNGkind() changes within one R process. Named seed matrices are recommended because they preserve the seeds associated with each (start, c) pair when the candidate range is extended.

Interpretation limits

The computational core computes AMD and preserves fitted objects and diagnostics. It does not provide a universal automatic rule for reading peaks, plateaus or sustained declines. Whether AMD height or peak morphology measures compactness, separation or configuration definition remains a research question.

Reference

Mendoza, M. & Araujo, M. B. (2022). Biogeography of bird and mammal trophic structures. Ecography, 2022(5), e06289. doi:10.1111/ecog.06289.

License

Package code: MIT License. The new drop-37 model bundle: CC BY 4.0, attributed to Manuel Mendoza. Historical bundles retain their existing licences. The transcriptomic example dataset retains its separately attributed CC BY 4.0 licence; see inst/COPYRIGHTS and the dataset help.