Generalized network-based dimensionality reduction and analysis

nda constructs interpretable latent variables from communities in squared-association networks. It supports unsupervised dimensionality reduction with ndr() and supervised modelling with ndrlm().

Version 0.3.0 adds faster and safer association calculations, sparse and parallel workflows, fixed memberships, inference and bootstrap diagnostics, new centrality definitions, prediction methods, configurable graphics, and ordinary least squares, ridge, elastic-net, and LOESS response engines.

Installation

Install the development version from GitHub:

remotes::install_github("kzst/nda")
library(nda)

Main function 1: `ndr()`

ndr():

estimates or accepts an association matrix;
constructs a squared-association network;
detects indicator communities;
creates one standardized, centrality-weighted latent score per retained community.

fit <- ndr(
  swiss,
  cor_method = "spearman",
  cor_type = "full",
  centrality = "pagerank",
  min_comm = 2,
  seed = 42
)

fit
summary(fit)
round(fit$loadings, 3)

# Score new observations with the training transformation.
predict(fit, swiss[1:5, , drop = FALSE])

Network and biplot functions return printable objects. Assignment is silent; print the object to render it.

network <- plot(
  fit,
  interactive = FALSE,
  show_weights = TRUE,
  shape = "automatic"
)
network

# The same visual grammar as an embedded interactive HTML widget.
interactive_network <- plot(fit, interactive = TRUE, show_weights = TRUE)
interactive_network

score_plot <- biplot(fit, view_3D = FALSE, interactive = FALSE)
score_plot

Community membership is shown with a light qualitative node-colour palette. Indicators use one common rectangular shape by default; positive and negative associations use green and red edges, and absolute association controls edge width. Shapes, colours, line types, and the minimum/maximum width remain configurable.

covar = TRUE accepts an arbitrary square similarity matrix. For example, NDR can cluster observations from normalized Euclidean similarities:

z <- scale(swiss)
d <- as.matrix(dist(z))
similarity <- 1 - d / max(d)
cluster_fit <- ndr(similarity, covar = TRUE, seed = 42)
table(cluster_fit$membership)

Important options include:

full, partial, and semipartial associations;
Pearson, Spearman, Kendall, distance, Gaussian-rank, and biweight methods;
eigenvector, authority, hub, and PageRank centrality;
fixed memberships and custom expected-edge matrices;
FDR-adjusted association screening;
bootstrap loading intervals;
sparse networks and multicore distance calculations;
prediction using stored training transformations.

Main function 2: `ndrlm()`

ndrlm() connects original or NDA-reduced predictors to original or NDA-reduced responses. Set latents to "none", "in", "out", or "both".

The example below uses the two-community latent input structure from the published low-sample-size mtcars application:

X <- mtcars[, setdiff(names(mtcars), "mpg")]
Y <- mtcars["mpg"]
membership <- c(
  cyl = 1, disp = 2, hp = 1, drat = 2, wt = 2,
  qsec = 1, vs = 1, am = 2, gear = 2, carb = 1
)

model <- ndrlm(
  Y,
  X,
  latents = "in",
  optimize = FALSE,
  regression = "lm",
  membership.X = membership,
  seed = 42
)

summary(model)
predict(model, X[1:5, , drop = FALSE])
head(fitted(model))
head(residuals(model))

For regression = "lm", each model$fits[[i]] is a genuine lm object and can be used directly with summary(), AIC(), BIC(), and lm.beta::lm.beta().

inherits(model$fits[[1]], "lm")
summary(model$fits[[1]])

Ridge and elastic net use glmnet and retain NDRLM engine wrappers:

ridge <- ndrlm(
  Y,
  X,
  latents = "none",
  optimize = FALSE,
  regression = "ridge",
  lambda = 0.1
)
predict(ridge, X[1:5, , drop = FALSE])

The package methods also accept legacy direct lm fits and early 0.3.0 wrapped lm fits. For engine-independent code, use summary(), predict(), fitted(), residuals(), and plot() on the enclosing ndrlm object.

static_paths <- plot(
  model,
  interactive = FALSE,
  shape = "automatic"
)
static_paths

interactive_paths <- plot(
  model,
  interactive = TRUE,
  show_edge_labels = TRUE
)
interactive_paths

Observed variables are rectangles and latent variables are circles. Each latent variable and its indicators share one light community colour. Positive paths are green, negative paths are red, and absolute standardized coefficient or loading magnitude controls width within its edge role. The plot computes standardized structural coefficients for ordinary linear, ridge, and elastic-net regression from the complete training observations; use standardized.paths = FALSE only when raw coefficients are required. Measurement links between observed and latent variables are dashed; structural paths are solid. Latent nodes are centred on their indicator groups, absent layers leave no empty horizontal column, and the default node fills are slightly translucent and lightened. Edge labels use a light background offset from the path to avoid overwriting thick lines. For legacy unnamed fit lists, the path target is recovered from the fitted response formula before any original-variable fallback is considered; standardized coefficients are reconstructed from the legacy fitted model.

Static plots use the package’s layered grid renderer by default. The legacy igraph renderer remains available with static.engine = "igraph". Both static and interactive variants are printable objects, so the following pattern works with screen and file devices:

p <- plot(model, interactive = FALSE)  # no drawing during assignment
p                                      # draw now

pdf("ndrlm-paths.pdf", width = 10, height = 6)
p
dev.off()

Expensive NSGA-II optimization is optional:

optimized <- ndrlm(
  Y,
  X,
  latents = "in",
  regression = "lm",
  target = "adj.r.square",
  popsize = 20,
  generations = 50,
  restarts = 3,
  seed = 42
)

plot(
  optimized$NSGA_trace$evaluation,
  optimized$NSGA_trace$best_target,
  type = "s",
  xlab = "Objective evaluation",
  ylab = "Best target value"
)

Supporting functions

Purpose	Functions
Structured example data	`data_gen()`
Global, row, or column scaling	`normalize()`
Distance covariance and correlation	`dCov()`, `dCor()`
Partial and semipartial distance correlation	`pdCor()`, `spdCor()`
KMO-based feature selection	`fs.KMO()`
Loading-based feature selection and iterative refitting	`fs.dimred()`

Vignettes

The source package builds three official vignettes:

nda-function-tour: a visual tour of the package;
ndr-workflow: the generalized NDR workflow;
ndrlm-workflow: supervised latent-variable regression.

List or open them from an installed package with:

vignette(package = "nda")
vignette("nda-function-tour", package = "nda")
vignette("ndr-workflow", package = "nda")
vignette("ndrlm-workflow", package = "nda")

Reproducibility and performance

Use seed whenever community detection, bootstrap inference, or NSGA-II is involved.
Use small n_boot and optimization settings while developing, then increase them for final analysis.
Parallel workers mainly benefit distance-association and bootstrap tasks.
sparsecalc = TRUE is useful after meaningful edge thresholding.
Report association, community, centrality, filtering, optimization, and regression settings with fitted results.

References

Kosztyan, Z. T., Kurbucz, M. T., and Katona, A. I. (2022). Network-based dimensionality reduction of high-dimensional, low-sample-size datasets. Knowledge-Based Systems, 109180. https://doi.org/10.1016/j.knosys.2022.109180

Kosztyan, Z. T., Katona, A. I., Kurbucz, M. T., and Lantos, Z. (2024). Generalized network-based dimensionality analysis. Expert Systems with Applications, 238, 121779. https://doi.org/10.1016/j.eswa.2023.121779

Authors

Zsolt T. Kosztyan, Marcell T. Kurbucz, Attila I. Katona, and Zahid Khan.