Estimate the sigma-equivalent compactness of a dataset

This function calibrates the observed AMD peak of real data against synthetic datasets generated with varying levels of isotropic Gaussian noise (\(\sigma\)). For each candidate \(\sigma\), synthetic data are generated with the same number of samples, dimensionality, and number of clusters as the real data. The AMD peak of each synthetic dataset is computed, and the sigma-equivalent value is defined as the \(\sigma\) whose synthetic AMD peak best matches the real AMD peak (either by interpolation or nearest match).

Usage

estimate_sigma_equivalent(
  real_data,
  its,
  nin,
  nsp,
  k_opt = NULL,
  sigmas,
  iter_max = 20,
  make_plot = FALSE,
  return_plot = TRUE,
  quiet = TRUE,
  open_device_each = FALSE,
  device_width = 7,
  device_height = 5,
  cube_size = 100,
  method = c("interpolate", "nearest"),
  standardize = FALSE,
  seed_base = 7,
  plot_sigma_curves = FALSE
)

Arguments

real_data

A numeric matrix or data frame of samples × features.

its

Number of random initialisations per AMD computation.

nin

Minimum number of clusters to evaluate.

nsp

Maximum number of clusters to evaluate.

k_opt

Optional; the optimal number of clusters for the real data. If NULL, it is estimated internally.

sigmas

Numeric vector of candidate \(\sigma\) values to evaluate.

iter_max

Maximum number of iterations for fuzzy c-means.

make_plot

Logical; if TRUE, produce a comparative plot of \(\sigma\) vs synthetic AMD peaks.

return_plot

Logical; if TRUE, return the comparative plot object.

quiet

Logical; suppress console output from synthetic data generation.

open_device_each

Logical; if TRUE, open a new graphics device for each sigma-curve plot (when plot_sigma_curves = TRUE).

device_width, device_height

Size of graphics device for sigma-curve plots.

cube_size

Side length of the hypercube used to place synthetic centroids.

method

Method for estimating sigma-equivalent: "interpolate" or "nearest".

standardize

Logical; if TRUE, standardise synthetic data.

seed_base

Base seed for reproducibility.

plot_sigma_curves

Logical; if TRUE, plot the AMD curve for each candidate \(\sigma\).

Value

A list containing:

amd_real_peak

AMD peak of the real dataset.

k_opt

Optimal number of clusters for the real data.

table_sigma_amd

Data frame of \(\sigma\) vs synthetic AMD peaks.

sigma_equivalent

Interpolated sigma-equivalent value.

sigma_eq

Nearest-match sigma on the explored grid.

extrapolated

Logical; whether interpolation required extrapolation.

plot_comparative

Comparative plot object (if requested).

best_i

Index of best-matching sigma.

best_sigma

Best-matching sigma value.

best_res_syn

Full AMD results for the best synthetic dataset.

best_df_curve

Data frame of the AMD curve for the best sigma.

Examples

if (FALSE) { # \dontrun{
set.seed(1)
X <- matrix(rnorm(1000), nrow = 100, ncol = 10)
out <- estimate_sigma_equivalent(
  real_data = X,
  its = 5,
  nin = 2,
  nsp = 6,
  sigmas = seq(1, 10, by = 2)
)
out$sigma_equivalent
} # }