Scope

Average Membership Degree (AMD) is a summary of a soft-membership matrix. It is not a clustering algorithm. The package currently uses fuzzy c-means to estimate memberships when constructing an AMD curve, but this backend is conceptually separate from the definition of the index.

For a membership matrix \(U=(u_{ij})\) with \(n\) observations and \(c\) columns,

\[ \operatorname{AMD}(U)=\frac{1}{n}\sum_{i=1}^{n}\max_j(u_{ij})-\frac{1}{c}. \]

Therefore \(0 \leq \operatorname{AMD}(U) \leq 1-1/c\). AMD is zero when every row is uniform and reaches its upper bound for crisp memberships. The formula also accepts \(c=1\), where AMD is zero, but operational curves compare alternative resolutions and begin at \(c=2\).

Constructing a curve

For each candidate \(c\), compute_amd_curve() runs several initialisations of the current fuzzy c-means backend. Random starts are computational attempts to estimate the same solution, not statistical replicates. The converged fit with the smallest fuzzy c-means objective supplies the primary curve point.

The following example contains three groups by construction.

set.seed(8301)
X <- rbind(
  matrix(rnorm(90, mean = 0, sd = 0.7), ncol = 3),
  matrix(rnorm(90, mean = 4, sd = 0.7), ncol = 3),
  matrix(rnorm(90, mean = 8, sd = 0.7), ncol = 3)
)

candidate_c <- 2:5
starts <- 3L
seed_matrix <- outer(
  seq_len(starts),
  candidate_c,
  function(start, c) 100000L + 1000L * c + start
)
colnames(seed_matrix) <- as.character(candidate_c)

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

data.frame(
  c = candidate_c,
  AMD = unname(amd_curve$curve),
  converged_starts = colSums(amd_curve$converged)
)
#>   c       AMD converged_starts
#> 2 2 0.3466353                3
#> 3 3 0.6309742                3
#> 4 4 0.6310310                3
#> 5 5 0.5942833                3

abs(amd_curve$curve[["3"]] - amd_curve$curve[["4"]])
#> [1] 5.679318e-05

plot(amd_curve, open_device = FALSE)

Named seed matrices make the mapping from (start, c) explicit. If a wider candidate range uses the same master matrix, all common fits remain identical.

New curve objects carry schema_version = 2L. Saved public 0.1.0 objects can still be printed, summarised and plotted from their stored mean and max values. They emit a warning because they do not contain objectives and fitted memberships from which the current primary curve could be reconstructed.

This deliberately retained example also illustrates why the complete curve is the primary object of interpretation. The data generator contains three groups, yet the AMD values at c=3 and c=4 are nearly tied and the numerical winner can be c=4. The subsequent decline supports a well-defined resolution around that short plateau, but the isolated arithmetic maximum does not by itself distinguish three generating groups from a four-part refinement. The known truth is used below to extract c=3; it is not inferred by which.max().

Reading before assigning

c_peak and its compatibility alias c_opt report the first numerical maximum of the evaluated primary curve. They do not encode a complete scientific rule. In particular:

• a maximum at the right boundary has no observed post-peak behaviour;
• nearly equal adjacent values may represent a resolution plateau;
• exact ties are reported at the smaller c;
• an internal maximum should be considered with its subsequent decline, stability and the range examined.

The package therefore requires an explicit c when extracting assignments. Here c=3 is used because it is known from the synthetic generator, not because the API selected it automatically.

chosen_c <- 3L
assignment <- assign_configurations(amd_curve, c = chosen_c)

identical(
  assignment$membership,
  amd_curve$selected_fits[[as.character(chosen_c)]]$membership
)
#> [1] TRUE

Describing stored partitions

The diagnostic interface reads the fits already stored in the curve. It does not refit them, construct sample distances or select a number of groups.

diagnostics <- diagnose_amd(amd_curve)
diagnostics$candidates[, c("c", "amd", "mean_winning_membership")]
#>   c       amd mean_winning_membership
#> 1 2 0.3466353               0.8466353
#> 2 3 0.6309742               0.9643075
#> 3 4 0.6310310               0.8810310
#> 4 5 0.5942833               0.7942833
diagnostics$transitions
#>   from_c to_c c_step    amd_change       ari       nmi from_fragmentation
#> 1      2    3      1  2.843389e-01 0.4365947 0.5302015          0.3333333
#> 2      3    4      1  5.679318e-05 0.8661654 0.9089745          0.1666667
#> 3      4    5      1 -3.674771e-02 0.8502425 0.9238963          0.1444444
#>   to_fragmentation
#> 1        0.1555556
#> 2        0.0000000
#> 3        0.0000000

The dictionary in diagnostics$dictionary defines each summary. Membership entropy and top-two gaps describe membership definition, not density or geometric compactness. Agreement between adjacent partitions is not stability under sample resampling. The default ambiguity gap is a descriptive convention, not a universal cutoff. Diagnostics not measured are listed explicitly in diagnostics$unavailable.

Optional model-assisted interpretation

Version 0.2.0 includes an optional reader. It uses five fitted random forests in an external corrected-v2 bundle; no models are downloaded or trained by the package. ranger and digest are needed only for this workflow. Obtain the model file and its expected SHA-256 from the same trusted release.

reader <- load_amd_predictor(bundle_path, sha256 = bundle_sha256)
result <- predict_amd(X, reader, seed = 123, verbose = FALSE)
plot(result)
result$proposed_k
result$reliability

The reader requires c=2:14, 20 principal starts, m=2, iter_max=2000, and no scaling or variable selection. A curve already computed with those settings can be passed as curve; the shorter illustrative curve above cannot. There are also five computational refits per c for the frozen descriptors, even when the main curve is reused. These refits do not resample observations.

The proposed K targets the numerically resolved density-mode count used as the simulation reference, not necessarily the number of generating components. AMD itself does not estimate a density. The proposal is separate from the numerical AMD maximum and does not establish the true number of configurations. Off-grid proposals themselves are labelled exploratory. Assignments still require the user’s explicit c. Reliability estimates correctness of the emitted one/several-modes decision and correctness of the exact proposed K, separately; it is not directly the probability that groups exist.

The corrected forests’ historical calibrators were reassessed on the previously examined final simulation bank, not in a new prospective validation. Reliability is suppressed by default unless N is 150 or 600, D is 2, 6 or 15, the output is within the observed support K=1:8 and no unassessed missingness occurs. Matching N and D does not certify new-data geometry; do not subsample or pad data merely to obtain a reported certainty. allow_extrapolation=TRUE can expose unvalidated numerical values away from the grid, but cannot validate them.

The frozen models omitted value columns for the zero-variance fraction (always zero) and minimum covariance-rank fraction (always one). Departures suppress certainty even on the grid and cannot be overridden with allow_extrapolation.

The descriptors include quadratic distance and covariance operations. A working-memory risk guard runs before new fits; it excludes the loaded forests and is neither a guaranteed total-memory bound nor a runtime limit. Partition distances use all observations; global distance summaries use a seeded sample of 2000 rows when N > 2000, as required by the frozen feature definition. The actual count and exact/subsampled flags are recorded. No additional approximation bypasses the guard. The manual workflow and stored-partition diagnostics remain usable without these operations.

Reproducibility record

The curve stores seeds, objectives, convergence, transformations, retained variables and the map of complete and omitted rows. It also links to a process-level software manifest.

manifest <- amd_reproducibility_manifest()
identical(amd_curve$manifest_id, manifest$manifest_id)
#> [1] TRUE
manifest[c("r_version", "amdconfigurations_version", "e1071_version", "rng_kind")]
#> $r_version
#> [1] "R version 4.6.1 (2026-06-24)"
#> 
#> $amdconfigurations_version
#> [1] "0.2.1"
#> 
#> $e1071_version
#> [1] "1.7.17"
#> 
#> $rng_kind
#> [1] "Mersenne-Twister" "Inversion"        "Rejection"

In a production analysis, retrieve the manifest immediately after every curve and save one copy of each distinct manifest_id. Curves sharing an identifier can share that copy. Each parallel worker, and each different RNGkind() used within a worker, receives its own identifier.

Current limits

The mean and max components are preserved for compatibility with earlier versions. They summarise all starts, including nonconverged starts, and are not the primary curve or a basis for scientific selection.

The package does not claim that AMD height, peak sharpness or curve prominence measures compactness, separation or definition in a universal sense.