R package seriation - Infrastructure for Ordering Objects Using Seriation

Introduction

Seriation arranges a set of objects into a linear order given available data with the goal of revealing structural information. This package provides the infrastructure for ordering objects with an implementation of many seriation/ordination techniques to reorder data matrices, dissimilarity matrices, correlation matrices, and dendrograms (see below for a complete list). The package provides several visualizations (grid and ggplot2) to reveal structural information, including permuted image plots, reordered heatmaps, Bertin plots, clustering visualizations like dissimilarity plots, and visual assessment of cluster tendency plots (VAT and iVAT).

Here are some quick guides on applications of seriation:

• Introduction the R package seriation
• How to reorder heatmaps
• How to reorder correlation matrices
• How to evaluate clusters using dissimilarity plots

Implemented seriation methods and criteria:

• Documentation of the implemented seriation methods
• Documentation of the implemented seriation criteria
• A visual comparison between seriation methods

The following R packages use seriation: adepro, arulesViz, baizer, ChemoSpec, ClusteredMutations, corrgram, corrplot, corrr, dendextend, DendSer, dendsort, disclapmix, elaborator, flexclust, ggraph, heatmaply, MEDseq, ockc, protti, qlcVisualize, RMaCzek, SFS, tidygraph, treeheatr, vcdExtra

To cite package ‘seriation’ in publications use:

Hahsler M, Hornik K, Buchta C (2008). “Getting things in order: An introduction to the R package seriation.” Journal of Statistical Software, 25(3), 1-34. ISSN 1548-7660, doi:10.18637/jss.v025.i03 https://doi.org/10.18637/jss.v025.i03.

@Article{,
  title = {Getting things in order:  An introduction to the R package seriation},
  author = {Michael Hahsler and Kurt Hornik and Christian Buchta},
  year = {2008},
  journal = {Journal of Statistical Software},
  volume = {25},
  number = {3},
  pages = {1--34},
  doi = {10.18637/jss.v025.i03},
  month = {March},
  issn = {1548-7660},
}

Available seriation methods to reorder dissimilarity data

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 heuristic to optimize various criteria
• GW - Hierarchical clustering reordered by the Gruvaeus and Wainer heuristic
• HC - Hierarchical clustering (single link, avg. link, complete link)
• OLO - Hierarchical clustering with optimal leaf ordering

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 - classical metric multidimensional scaling
• MDS_angle - order by the angular order in the 2D MDS projection space split by the larges gap
• isoMDS - 1D Krusakl’s non-metric multidimensional scaling
• isomap - 1D isometric feature mapping ordination
• monoMDS - order along 1D global and local non-metric multidimensional scaling using monotone regression (NMDS)
• metaMDS - 1D non-metric multidimensional scaling (NMDS) with stable solution from random starts
• Sammon - Order along the 1D Sammon’s non-linear mapping
• smacof - 1D MDS using majorization (ratio MDS, interval MDS, ordinal MDS)
• TSNE - Order along the 1D t-distributed stochastic neighbor embedding (t-SNE)
• UMAP - Order along the 1D embedding produced by uniform manifold approximation and projection

Optimization

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

• ARSA - optimize the linear seriation criterion using simulated annealing
• Branch-and-bound to minimize the unweighted/weighted column gradient
• GA - Genetic algorithm with warm start to optimize any seriation criteria
• GSA - General simulated annealing to optimize any seriation criteria
• SGD - stochastic gradient descent to find a local optimum given an initial order and a seriation criterion.
• QAP - Quadratic assignment problem heuristic (optimizes 2-SUM, linear seriation, inertia, banded anti-Robinson form)
• Spectral seriation to optimize the 2-SUM criterion (unnormalized, normalized)
• TSP - Traveling salesperson solver to minimize the Hamiltonian path length

Other Methods

• Identity permutation
• OPTICS - Order of ordering points to identify the clustering structure
• R2E - Rank-two ellipse seriation
• Random permutation
• Reverse order
• SPIN - Sorting points into neighborhoods (neighborhood algorithm, side-to-site algorithm)
• VAT - Order of the visual assessment of clustering tendency

A detailed comparison of the most popular methods is available in the paper An experimental comparison of seriation methods for one-mode two-way data. (read the preprint).

Available seriation methods to reorder data matrices, count tables, and data.frames

For matrices, rows and columns are reordered.

Seriating rows and columns simultaneously

Row and column order influence each other.

• BEA - Bond Energy Algorithm to maximize the measure of effectiveness (ME)
• BEA_TSP - TSP to optimize the measure of effectiveness
• BK_unconstrained - Algorithm by Brower and Kyle (1988) to arrange binary matrices.
• CA - calculates a correspondence analysis of a matrix of frequencies (count table) and reorders according to the scores on a correspondence analysis dimension

Seriating rows and columns separately using dissimilarities

• Heatmap - reorders rows and columns independently by calculating row/column distances and then applying a seriation method for dissimilarities (see above)

Seriate rows in a 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.

• PCA_angle - order by the angular order in the 2D PCA projection space split by the larges gap
• LLE reorder along a 1D locally linear embedding
• Means - reorders using row means
• PCA - orders along the first principal component
• TSNE - Order along the 1D t-distributed stochastic neighbor embedding (t-SNE)
• UMAP - Order along the 1D embedding produced by uniform manifold approximation and projection

Other methods

• AOE - order by the angular order of the first two eigenvectors for correlation matrices.
• Identity permutation
• Random permutation
• Reverse order

Installation

Stable CRAN version: Install from within R with

install.packages("seriation")

Current development version: Install from r-universe.

install.packages("seriation",
    repos = c("https://mhahsler.r-universe.dev",
              "https://cloud.r-project.org/"))

Usage

The used example dataset contains the joint probability of disagreement between Supreme Court Judges from 1995 to 2002. The goal is to reveal structural information in this data. We load the library, read the data, convert the data to a distance matrix, and then use the default seriation method to reorder the objects.

library(seriation)
data("SupremeCourt")

d <- as.dist(SupremeCourt)
d

##           Breyer Ginsburg Kennedy OConnor Rehnquist Scalia Souter Stevens
## Ginsburg   0.120                                                         
## Kennedy    0.250    0.267                                                
## OConnor    0.209    0.252   0.156                                        
## Rehnquist  0.299    0.308   0.122   0.162                                
## Scalia     0.353    0.370   0.188   0.207     0.143                      
## Souter     0.118    0.096   0.248   0.220     0.293  0.338               
## Stevens    0.162    0.145   0.327   0.329     0.402  0.438  0.169        
## Thomas     0.359    0.368   0.177   0.205     0.137  0.066  0.331   0.436

order <- seriate(d)
order

## object of class 'ser_permutation', 'list'
## contains permutation vectors for 1-mode data
## 
##   vector length seriation method
## 1             9         Spectral

Here is the resulting permutation vector.

get_order(order)

##    Scalia    Thomas Rehnquist   Kennedy   OConnor    Souter    Breyer  Ginsburg 
##         6         9         5         3         4         7         1         2 
##   Stevens 
##         8

Next, we visualize the original and permuted distance matrix.

pimage(d, main = "Judges (original alphabetical order)")
pimage(d, order, main = "Judges (reordered by seriation)")

Darker squares around the main diagonal indicate groups of similar objects. After seriation, two groups are visible.

We can compare the available seriation criteria. Seriation improves all measures. Note that some measures are merit measures while others represent cost. See the manual page for details.

rbind(alphabetical = criterion(d), seriated = criterion(d, order))

##              2SUM AR_deviations AR_events BAR Gradient_raw Gradient_weighted
## alphabetical  872        10.304        80 1.8            8              0.54
## seriated      811         0.064         5 1.1          158             19.76
##              Inertia Lazy_path_length Least_squares LS MDS_stress  ME
## alphabetical     267              6.9           967 99       0.62  99
## seriated         364              4.6           942 86       0.17 101
##              Moore_stress Neumann_stress Path_length RGAR   Rho
## alphabetical          7.0            3.9         1.8 0.48 0.028
## seriated              2.5            1.3         1.1 0.03 0.913

Some seriation methods also return a linear configuration where more similar objects are located closer to each other.

get_config(order)

##    Breyer  Ginsburg   Kennedy   OConnor Rehnquist    Scalia    Souter   Stevens 
##      0.24      0.28     -0.15     -0.11     -0.27     -0.42      0.21      0.61 
##    Thomas 
##     -0.41

plot_config(order)

We can see a clear divide between the two groups in the configuration.

References

• Michael Hahsler, Kurt Hornik and Christian Buchta, Getting Things in Order: An Introduction to the R Package seriation, Journal of Statistical Software, 25(3), 2008. DOI: 10.18637/jss.v025.i03
• Michael Hahsler. An experimental comparison of seriation methods for one-mode two-way data. European Journal of Operational Research, 257:133-143, 2017. DOI: 10.1016/j.ejor.2016.08.066 (read the preprint)
• Hahsler, M. and Hornik, K. (2011): Dissimilarity plots: A visual exploration tool for partitional clustering. Journal of Computational and Graphical Statistics, 10(2):335–354. doi:10.1198/jcgs.2010.09139 (read the preprint; code examples)
• Reference manual for package seriation.