Seriation Methods implemented in the R package seriation • seriation

This document contains the seriation methods currently implemented in the R package seriation. The methods are organized by methods for distance matrices and data matrices.

This list was created for the following version of seriation:

library("seriation")
packageVersion('seriation')

## [1] '1.6.0'

Some additional methods need to be registered.

register_DendSer()
register_optics()
register_smacof()
register_GA()
register_tsne()
register_umap()
register_vegan()

Seriation methods on the data x can be used in

seriate(x, method = "Spectral", control = NULL, rep = 1L)

The list below shows what seriation criterion (if any) is optimized by the method.

If the method uses randomization (see randomized below), then rep can be used to run the algorithms several times and return the best permutation.

Available control parameters can be passed on as a list using the control argument.

Methods for dissimilarity data (dist)

Seriation methods for dissimilarity data reorder the set of objects in the data. The methods fall into several groups based on the criteria they try to optimize, constraints (like dendrograms), and the algorithmic approach.

Dendrogram leaf order

These methods create a dendrogram using hierarchical clustering and then derive the seriation order from the leaf order in the dendrogram. Leaf reordering may be applied.

DendSer

Dendrogram seriation (Earle and Hurley, 2015).

optimizes: N/A (specified criterion restricted by dendrogram)
randomized: FALSE
registered by: register_DendSer()
control parameters:

	default	help
h	NULL	an hclust object (optional)
method	“complete”	hclust linkage method
criterion	NULL	criterion to optimize the dendrogram for
DendSer_args	NULL	more arguments are passed on to DendSer (? DendSer)
verbose	FALSE	N/A

DendSer_ARc

Dendrogram seriation for Anti-Robinson form cost (Earle and Hurley, 2015).

optimizes: ARc (Anti-Robinson form cost restricted by dendrogram)
randomized: FALSE
registered by: register_DendSer()
control parameters:

	default	help
h	NULL	an hclust object (optional)
method	“complete”	hclust linkage method
criterion	NULL	criterion to optimize the dendrogram for
DendSer_args	NULL	more arguments are passed on to DendSer (? DendSer)
verbose	FALSE	N/A

DendSer_BAR

Dendrogram seriation with BAR (Earle and Hurley, 2015).

optimizes: BAR (banded anti-Robinson form restricted by dendrogram)
randomized: FALSE
registered by: register_DendSer()
control parameters:

	default	help
h	NULL	an hclust object (optional)
method	“complete”	hclust linkage method
criterion	NULL	criterion to optimize the dendrogram for
DendSer_args	NULL	more arguments are passed on to DendSer (? DendSer)
verbose	FALSE	N/A

DendSer_LPL

Dendrogram seriation for Lazy path length (Earle and Hurley, 2015).

optimizes: Lazy_path_length (restricted by dendrogram)
randomized: FALSE
registered by: register_DendSer()
control parameters:

	default	help
h	NULL	an hclust object (optional)
method	“complete”	hclust linkage method
criterion	NULL	criterion to optimize the dendrogram for
DendSer_args	NULL	more arguments are passed on to DendSer (? DendSer)
verbose	FALSE	N/A

DendSer_PL

Dendrogram seriation for Path length (Earle and Hurley, 2015).

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
registered by: register_DendSer()
control parameters:

	default	help
h	NULL	an hclust object (optional)
method	“complete”	hclust linkage method
criterion	NULL	criterion to optimize the dendrogram for
DendSer_args	NULL	more arguments are passed on to DendSer (? DendSer)
verbose	FALSE	N/A

GW

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering and reordered by the Gruvaeus and Wainer (1972) heuristic

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
control parameters:

	default	help
hclust	NULL	a precomputed hclust object (optional)
linkage	“complete”	hclust method

GW_average

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering and reordered by the Gruvaeus and Wainer (1972) heuristic (avg.link)

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
control parameters: no parameters

GW_complete

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering and reordered by the Gruvaeus and Wainer (1972) heuristic (complete link)

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
control parameters: no parameters

GW_single

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering and reordered by the Gruvaeus and Wainer (1972) heuristic (single link)

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
control parameters: no parameters

GW_ward

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering and reordered by the Gruvaeus and Wainer (1972) heuristic (Ward’s method)

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
control parameters: no parameters

HC

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering

optimizes: N/A
randomized: FALSE
control parameters:

	default	help
hclust	NULL	a precomputed hclust object (optional)
linkage	“complete”	hclust method

HC_average

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering (avg. link).

optimizes: N/A
randomized: FALSE
control parameters: no parameters

HC_complete

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering (complete link).

optimizes: N/A
randomized: FALSE
control parameters: no parameters

HC_single

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering (single link)

optimizes: N/A
randomized: FALSE
control parameters: no parameters

HC_ward

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering (Ward’s method).

optimizes: N/A
randomized: FALSE
control parameters: no parameters

OLO

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering and reordered with optimal leaf ordering (Bar-Joseph et al., 2001)

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
control parameters:

	default	help
hclust	NULL	a precomputed hclust object (optional)
linkage	“complete”	hclust method

OLO_average

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering and reordered with optimal leaf ordering (Bar-Joseph et al., 2001) (avg. link)

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
control parameters: no parameters

OLO_complete

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering and reordered with optimal leaf ordering (Bar-Joseph et al., 2001) (complete link)

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
control parameters: no parameters

OLO_single

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering and reordered with optimal leaf ordering (Bar-Joseph et al., 2001) (single link)

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
control parameters: no parameters

OLO_ward

Using the order of the leaf nodes in a dendrogram obtained by hierarchical clustering and reordered with optimal leaf ordering (Bar-Joseph et al., 2001) (Ward’s method)

optimizes: Path_length (restricted by dendrogram)
randomized: FALSE
control parameters: no parameters

Dimensionality reduction

Find a seriation order by reducing the dimensionality to 1 dimension. This is typically done by minimizing a stress measure or the reconstruction error.

MDS

Order along the 1D classical metric multidimensional scaling

optimizes: MDS_stress (Euclidean distances)
randomized: FALSE
control parameters:

	default	help
add	FALSE	make the distances Euclidean using an additive constant (see ? cmdscale)

MDS_angle

Order by the angular order in the 2D MDS projection space split by the largest gap

optimizes: N/A
randomized: FALSE
control parameters:

	default	help
add	FALSE	make the distances Euclidean using an additive constant (see ? cmdscale)

isoMDS

Order along the 1D Kruskal’s non-metric multidimensional scaling

optimizes: MDS_stress (with monotonic transformation)
randomized: FALSE
control parameters:

	default	help
add	1e-09	small constant to avoid 0 distances
maxit	50	maximum number of iterations
trace	FALSE	trace optimization
tol	0.001	convergence tolerance
p	2	power for Minkowski distance in the configuration space

isomap

Isometric feature mapping ordination (Tenenbaum, 2000)

optimizes: N/A (Stress on shortest path distances)
randomized: FALSE
registered by: register_vegan()
control parameters:

	default	help
k	30	number of shortest dissimilarities retained for a point
path	“shortest”	method used to estimate the shortest path (“shortest”/“extended”)

monoMDS

Kruskal’s (1964a,b) non-metric multidimensional scaling (NMDS) using monotone regression.

optimizes: MDS_stress (Kruskal’s monotone regression stress)
randomized: TRUE
registered by: register_vegan()
control parameters:

	default	help
y		See ? monoMDS for help
model	“global”	N/A
threshold	0.8	N/A
maxit	200	N/A
weakties	TRUE	N/A
stress	1	N/A
scaling	TRUE	N/A
pc	TRUE	N/A
smin	1e-04	N/A
sfgrmin	1e-07	N/A
sratmax	0.999999	N/A

metaMDS

Nonmetric Multidimensional Scaling with Stable Solution from Random Starts.

optimizes: MDS_stress (Kruskal’s monotone regression stress)
randomized: FALSE
registered by: register_vegan()
control parameters:

	default	help
distance	“bray”	see ? metaMDS for help
try	20	N/A
trymax	20	N/A
engine	“monoMDS”	N/A
autotransform	TRUE	N/A
noshare	FALSE	N/A
wascores	TRUE	N/A
expand	TRUE	N/A
trace	0	N/A
plot	FALSE	N/A
previous.best		N/A
verbose	FALSE	N/A

Sammon_mapping

Order along the 1D Sammon’s non-linear mapping

optimizes: MDS_stress (scale free, weighted stress called Sammon’s error)
randomized: FALSE
control parameters:

	default	help
add	1e-09	small constant to avoid 0 distances
niter	100	maximum number of iterations
trace	FALSE	trace optimization
magic	0.2	initial value of the step size constant in diagonal Newton method
tol	1e-04	tolerance for stopping in units of stress

MDS_smacof

Seriation based on multidimensional scaling using stress majorization (de Leeuw & Mair, 2009).

optimizes: smacof_stress0 (MDS stress)
randomized: FALSE
registered by: register_smacof()
control parameters:

	default	help
type	“ratio”	MDS type: “interval”, “ratio”, “ordinal” (nonmetric MDS)
init	“torgerson”	start configuration method (“torgerson”/“random”)
relax	FALSE	use block relaxation for majorization?
modulus	1	number of smacof iterations per monotone regression call
itmax	1000	maximum number of iterations
eps	1e-06	convergence criterion
verbose	FALSE	N/A

tsne

Use 1D t-distributed stochastic neighbor embedding (t-SNE) of a distance matrix to create an order (van der Maaten and Hinton, 2008).

optimizes: N/A
randomized: TRUE
registered by: register_tsne()
control parameters:

	default	help
max_iter	1000	number of iterations
theta	0.5	speed/accuracy trade-off (increase for less accuracy)
perplexity	NULL	perplexity parameter (calculated as n - 1 / 3)
eta	100	learning rate
mds	TRUE	start from a classical MDS solution
verbose	FALSE	N/A

umap

Use 1D Uniform manifold approximation and projection (UMAP) embedding of the distances to create an order (McInnes and Healy, 2018)

optimizes: N/A
randomized: TRUE
registered by: register_umap()
control parameters:

	default	help
n_neighbors	NA	see ? umap::umap for help
n_components	1	N/A
metric	“euclidean”	N/A
n_epochs	1000	N/A
input	NA	N/A
init	“spectral”	N/A
min_dist	0.1	N/A
set_op_mix_ratio	1	N/A
local_connectivity	1	N/A
bandwidth	1	N/A
alpha	0.001	N/A
gamma	1	N/A
negative_sample_rate	5	N/A
a	NA	N/A
b	NA	N/A
spread	1	N/A
random_state	NA	N/A
transform_state	NA	N/A
knn	NA	N/A
knn_repeats	1	N/A
verbose	FALSE	N/A
umap_learn_args	NA	N/A

Optimization

These methods try to optimize a seriation criterion directly, typically using a heuristic approach.

ARSA

Minimize the linear seriation criterion using simulated annealing (Brusco et al, 2008).

optimizes: LS (Linear seriation criterion)
randomized: TRUE
control parameters:

	default	help
cool	0.5	cooling factor (smaller means faster cooling)
tmin	1e-04	stopping temperature
swap_to_inversion	0.5	probability for swap vs inversion local move
try_multiplier	100	number of local move tries per object
verbose	FALSE	N/A

Enumerate

Enumerate all permutations

optimizes: N/A (set via control criterion))
randomized: FALSE
control parameters:

	default	help
criterion	“Gradient_weighted”	Criterion measure to optimize
verbose	FALSE	N/A

BBURCG

Minimize the unweighted row/column gradient by branch-and-bound (Brusco and Stahl 2005). This is only feasible for a relatively small number of objects.

optimizes: Gradient_raw (Unweighted gradient condition)
randomized: FALSE
control parameters:

	default	help
eps	0	Distances need to be at least eps to count as violations
verbose	FALSE	N/A

BBWRCG

Minimize the weighted row/column gradient by branch-and-bound (Brusco and Stahl 2005). This is only feasible for a relatively small number of objects.

optimizes: Gradient_weighted (Weighted gradient condition)
randomized: FALSE
control parameters:

	default	help
eps	0	Distances need to be at least eps to count as violations
verbose	FALSE	N/A

GA

Use a genetic algorithm to optimize for various criteria.

optimizes: N/A (specified as parameter criterion)
randomized: TRUE
registered by: register_GA()
control parameters:

	default	help
criterion	“BAR”	criterion to be optimized
suggestions	c(“TSP”, “QAP_LS”, “Spectral”)	seed the population with these seriation methods
selection	function (object, r = 2/(object@popSize * (object@popSize - 1)), q = 2/object@popSize, …) {	selection operator function
crossover	function (object, parents, …) { if (gaControl(“useRcpp”)) gaperm_oxCrossover_Rcpp(objec	crossover operator function
mutation	function (object, parent, …) { if (runif(1) > ismProb) GA::gaperm_simMutation(object, p	mutation operator function
pcrossover	0.8	probability for crossover
pmutation	0.1	probability of mutations
popSize	100	population size
maxiter	1000	maximum number of generations
run	50	stop after run generations without improvement
parallel	FALSE	use multiple cores?
verbose	FALSE	N/A

GSA

Minimize a specified seriation measure (criterion) using simulated annealing. (Hahsler, Hornik, and Buchta, 2023)

optimizes: N/A (set via control criterion)
randomized: TRUE
control parameters:

	default	help
criterion	“Gradient_raw”	Criterion measure to optimize
cool	0.5	cooling factor (smaller means faster cooling)
t_min	1e-07	stopping temperature
localsearch	“LS_insert”	used local search move function
try_multiplier	5	number of local move tries per object
t0	NA	initial temperature (if NA then it is estimated)
p_initial_accept	0.01	Probability to accept a bad move at time 0 (used for t0 estimation)
warmstart	“Random”	permutation or seriation method for warmstart
verbose	FALSE	N/A

SGLS

Improve an existing solution using stochastic greedy local search. (Hahsler, Hornik, and Buchta, 2023)

optimizes: N/A (set via control criterion)
randomized: TRUE
control parameters:

	default	help
criterion	“Gradient_raw”	Criterion measure to optimize
init	“Spectral”	Start permutation or name of a seriation method
max_iter	NULL	number of iterations
localsearch	“LS_insert”	used local search move function
verbose	FALSE	N/A

QAP_LS

Quadratic assignment problem formulation for seriation solved using a simulated annealing solver to minimize the Linear Seriation Problem (LS) criterion (Hubert and Schultz 1976).

optimizes: LS (Linear seriation criterion)
randomized: TRUE
control parameters: no parameters

QAP_2SUM

Quadratic assignment problem formulation for seriation solved using a simulated annealing solver to minimize the 2-Sum Problem criterion (Barnard, Pothen, and Simon 1993).

optimizes: 2SUM (2-sum criterion)
randomized: TRUE
control parameters: no parameters

QAP_BAR

Quadratic assignment problem formulation for seriation solved using a simulated annealing solver to minimize the banded anti-Robinson form (Hahsler, 2017).

optimizes: BAR (Banded anti-robinson form)
randomized: TRUE
control parameters:

	default	help
b	function (n) max(1, floor(n * 0.2))	bandwidth (default is 20%)

QAP_Inertia

Quadratic assignment problem formulation for seriation solved using a simulated annealing solver to minimize the Inertia criterion (Hahsler, 2017).

optimizes: Inertia
randomized: TRUE
control parameters: no parameters

Spectral

Spectral seriation (Ding and He 2004) uses a relaxation to minimize the 2-Sum Problem (Barnard, Pothen, and Simon 1993). It uses the order of the Fiedler vector of the similarity matrix’s Laplacian.

optimizes: 2SUM (2-sum criterion)
randomized: FALSE
control parameters: no parameters

Spectral_norm

optimizes: 2SUM (2-sum criterion)
randomized: FALSE
control parameters: no parameters

TSP

Minimize Hamiltonian path length with a TSP solver.

optimizes: Path_length
randomized: TRUE
control parameters:

	default	help
method	“arbitrary insertion”	used TSP method (see ? solve_TSP)
rep	10	number of random restarts
two_opt	TRUE	use the 2-opt improvement heuristic?

Other methods

Identity

Identity permutation

optimizes: N/A
randomized: FALSE
control parameters: no parameters

optics

Use ordering points to identify the clustering structure (OPTICS) to create an order

optimizes: N/A
randomized: FALSE
registered by: register_optics()
control parameters:

	default	help
eps	Inf	upper limit of the size of the epsilon neighborhood (see ? optics)
minPts	5	minimum density for dense neighborhoods

R2E

Rank-two ellipse seriation (Chen 2002)

optimizes: N/A
randomized: FALSE
control parameters: no parameters

Random

Random permutation

optimizes: N/A
randomized: TRUE
control parameters: no parameters

Reverse

Reversed identity permutation

optimizes: N/A
randomized: FALSE
control parameters: no parameters

SPIN_NH

Sorting Points Into Neighborhoods (SPIN) (Tsafrir 2005). Neighborhood algorithm to concentrate low distance values around the diagonal.

optimizes: N/A (Energy)
randomized: FALSE
control parameters:

	default	help
sigma	c(20, 17, 15, 13, 11, 9, 7, 5, 3, 1)	emphasize local (small alpha) or global (large alpha) structure.
step	5	iterations to run for each sigma value.
W_function	NULL	custom function to create the weight matrix W
verbose	FALSE	N/A

SPIN_STS

Sorting Points Into Neighborhoods (SPIN) (Tsafrir 2005). Side-to-Side algorithm which tries to push out large distance values.

optimizes: N/A (Energy)
randomized: FALSE
control parameters:

	default	help
step	25L	iterations to run
nstart	10L	number of random restarts
X	function (n) seq(n) - (n + 1)/2	matrix to calculate the W matrix
verbose	FALSE	N/A

VAT

Visual assessment of clustering tendency (Bezdek and Hathaway (2002). Creates an order based on Prim’s algorithm for finding a minimum spanning tree (MST) in a weighted connected graph representing the distance matrix. The order is given by the order in which the nodes (objects) are added to the MST.

optimizes: N/A
randomized: FALSE
control parameters: no parameters

Methods for data matrices, tables and data.frames

Seriating rows and columns simultaneously

Row and column order influence each other.

BEA

Bond Energy Algorithm (BEA; McCormick 1972) to maximize the Measure of Effectiveness of a non-negative matrix.

optimizes: ME (Measure of effectiveness)
randomized: TRUE
control parameters: no parameters

BEA_TSP

Use a TSP to optimize the Measure of Effectiveness (Lenstra 1974).

optimizes: ME (Measure of effectiveness)
randomized: TRUE
control parameters:

	default	help
method	“arbitrary insertion”	used TSP method (see ? solve_TSP)
rep	10	number of random restarts
two_opt	TRUE	use the 2-opt improvement heuristic?

BK_unconstrained

Implements the method for binary matrices described in Brower and Kile (1988). Reorders using the mean row and column position of presences (1s).

optimizes: N/A
randomized: FALSE
control parameters: no parameters

CA

This method calculates a correspondence analysis of the matrix and computes an order according to the scores on a correspondence analysis dimension (Friendly 2023).

optimizes: N/A
randomized: FALSE
control parameters:

	default	help
dim	1L	CA dimension used for reordering
ca_param	NULL	List with parameters for the call to ca::ca()

Seriating rows and columns separately using dissimilarities

Heatmap

Calculates distances for rows and columns and then independently applies the specified seriation method for distances.

optimizes: N/A
randomized: FALSE
control parameters:

	default	help
dist_fun	list(row = function (x, method = “euclidean”, diag = FALSE, upper = FALSE, p = 2) { if (!is.n	A named list with functions to calculate row and column distances
seriation_method	list(row = “OLO_complete”, col = “OLO_complete”)	A named list with row and column seriation methods
seriation_control	list(row = NULL, col = NULL)	named list with control parameters for the seriation methods
scale	“none”	Scale “rows”, “cols”, or “none”
verbose	FALSE	N/A

Seriate rows using the data matrix

These methods need access to the data matrix instead of dissimilarities to reorder objects (rows). The same approach can be applied to columns.

LLE

Find an order using 1D locally linear embedding.

optimizes: N/A
randomized: FALSE
control parameters:

	default	help
k	30	used number of neighbors
reg	2	regularization method (see ? lle)

PCA

Uses the projection of the data on its first principal component to determine the order.

optimizes: N/A (Least squares for each dimension (for Euclidean distances).)
randomized: FALSE
control parameters:

	default	help
center	TRUE	center the data (mean = 0)?
scale	FALSE	scale to unit variance?
verbose	FALSE	FALSE

PCA_angle

Uses the angular order in the 2D PCA projection space split by the largest gap.

optimizes: N/A
randomized: FALSE
control parameters:

	default	help
center	TRUE	center the data (mean = 0)?
scale	FALSE	scale to unit variance?
verbose	FALSE	FALSE

AOE

Order by the angle of the first two eigenvectors for correlation matrices (Friendly, 2002)

optimizes: N/A
randomized: FALSE
control parameters: no parameters

Mean

Reorders rows and columns by row and column means.

optimizes: N/A
randomized: FALSE
control parameters:

	default	help
transformation	NULL	transformation function applied before calculating means (e.g., scale)

Identity

Identity permutation

optimizes: N/A
randomized: FALSE
control parameters: no parameters

Reverse

Reversed identity permutation

optimizes: N/A
randomized: FALSE
control parameters: no parameters

Random

Random permutation

optimizes: N/A
randomized: TRUE
control parameters: no parameters

References

Arabie, P. and L.J. Hubert (1990): The bond energy algorithm revisited, IEEE Transactions on Systems, Man, and Cybernetics, 20(1), 268–274. https://doi.org/10.1109/21.47829

Bar-Joseph, Z., E. D. Demaine, D. K. Gifford, and T. Jaakkola. (2001): Fast Optimal Leaf Ordering for Hierarchical Clustering. Bioinformatics, 17(1), 22–29. https://doi.org/10.1093/bioinformatics/17.suppl_1.S22

Barnard, S. T., A. Pothen, and H. D. Simon (1993): A Spectral Algorithm for Envelope Reduction of Sparse Matrices. In Proceedings of the 1993 ACM/IEEE Conference on Supercomputing, 493–502. Supercomputing ’93. New York, NY, USA: ACM. https://doi.org/10.1145/169627.169790

Bezdek, J.C. and Hathaway, R.J. (2002): VAT: a tool for visual assessment of (cluster) tendency. Proceedings of the 2002 International Joint Conference on Neural Networks (IJCNN ’02), Volume: 3, 2225–2230. https://doi.org/10.1109/IJCNN.2002.1007487

Brower, J.C. and Kile, K.M. (1988): Sedation of an original data matrix as applied to paleoecology. Lethaia, 21, 79–93. https://doi.org/10.1111/j.1502-3931.1988.tb01756.x

Brusco, M., Koehn, H.F., and Stahl, S. (2008): Heuristic Implementation of Dynamic Programming for Matrix Permutation Problems in Combinatorial Data Analysis. Psychometrika, 73(3), 503–522. https://doi.org/10.1007/s11336-007-9049-5

Brusco, M., and Stahl, S. (2005): Branch-and-Bound Applications in Combinatorial Data Analysis. New York: Springer. https://doi.org/10.1007/0-387-28810-4

Chen, C. H. (2002): Generalized Association Plots: Information Visualization via Iteratively Generated Correlation Matrices. Statistica Sinica, 12(1), 7–29.

Ding, C. and Xiaofeng He (2004): Linearized cluster assignment via spectral ordering. Proceedings of the Twenty-first International Conference on Machine learning (ICML ’04). https://doi.org/10.1145/1015330.1015407

Climer, S. and Xiongnu Zhang (2006): Rearrangement Clustering: Pitfalls, Remedies, and Applications, Journal of Machine Learning Research, 7(Jun), 919–943.

D. Earle, C. B. Hurley (2015): Advances in dendrogram seriation for application to visualization. Journal of Computational and Graphical Statistics, 24(1), 1–25.

Friendly, M. (2002): Corrgrams: Exploratory Displays for Correlation Matrices. The American Statistician, 56(4), 316–324. https://doi.org/10.1198/000313002533

Friendly, M. (2023). vcdExtra: ‘vcd’ Extensions and Additions. R package version 0.8-5, https://CRAN.R-project.org/package=vcdExtra.

Gruvaeus, G. and Wainer, H. (1972): Two Additions to Hierarchical Cluster Analysis, British Journal of Mathematical and Statistical Psychology, 25, 200–206. https://doi.org/10.1111/j.2044-8317.1972.tb00491.x

Hahsler, M. (2017): An experimental comparison of seriation methods for one-mode two-way data. European Journal of Operational Research, 257, 133–143. https://doi.org/10.1016/j.ejor.2016.08.066

Hahsler M, Buchta C, Hornik K (2023): seriation: Infrastructure for Ordering Objects Using Seriation. R package version 1.5.0. https://doi.org/10.32614/CRAN.package.seriation

Hahsler, M., Hornik, K., and Buchta, C. (2008): Getting things in order: An introduction to ithe R package seriation. Journal of Statistical Software, 25(3), 1–34. https://doi.org/10.18637/jss.v025.i03

Hubert, Lawrence, and James Schultz (1976): Quadratic Assignment as a General Data Analysis Strategy. British Journal of Mathematical and Statistical Psychology, 29(2). Blackwell Publishing Ltd. 190–241. https://doi.org/10.1111/j.2044-8317.1976.tb00714.x

Hurley, Catherine B. (2004): Clustering Visualizations of Multidimensional Data. Journal of Computational and Graphical Statistics, 13(4), 788–806. https://doi.org/10.1198/106186004X12425

Kruskal, J.B. (1964). Nonmetric multidimensional scaling: a numerical method. Psychometrika, 29, 115–129.

Lenstra, J.K (1974): Clustering a Data Array and the Traveling-Salesman Problem, Operations Research, 22(2) 413–414. https://doi.org/10.1287/opre.22.2.413

Mair P., De Leeuw J. (2015). Unidimensional scaling. In Wiley StatsRef: Statistics Reference Online, Wiley, New York. https://doi.org/10.1002/9781118445112.stat06462.pub2

McCormick, W.T., P.J. Schweitzer and T.W. White (1972): Problem decomposition and data reorganization by a clustering technique, Operations Research, 20(5), 993–1009. https://doi.org/10.1287/opre.20.5.993

McInnes, L and Healy, J, UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction, ArXiv e-prints 1802.03426, 2018.

Sammon, J. W. (1969) A non-linear mapping for data structure analysis. IEEE Trans. Comput., C-18 401–409.

Tenenbaum, J.B., de Silva, V. & Langford, J.C. (2000) A global network framework for nonlinear dimensionality reduction. Science 290, 2319-2323.

Tsafrir, D., Tsafrir, I., Ein-Dor, L., Zuk, O., Notterman, D.A. and Domany, E. (2005): Sorting points into neighborhoods (SPIN): data analysis and visualization by ordering distance matrices, Bioinformatics, 21(10) 2301–8. https://doi.org/10.1093/bioinformatics/bti329

van der Maaten, L.J.P.; Hinton, G.E. (Nov 2008). Visualizing Data Using t-SNE. Journal of Machine Learning Research. 9: 2579–2605.