To limit this documentation size, most figures are not displayed in the version
of the vignette included in the package. To see all figures, you can rerun
the vignettes, changing the following option to TRUE:
knitr::opts_chunk$set(include = FALSE)
Alternatively, this compilation is also available at: http://sombrero.nathalievialaneix.eu/articles/c-doc-numericSOM.html
SOMbrero implements different variants of the Self-Organizing Map algorithm
(also called Kohonen's algorithm). To run the standard version of the algorithm,
use the function trainSOM() on a data frame or matrix with numerical columns.
The standard numeric SOM and its use in SOMbrero are illustrated below.
This documentation only considers the case of numerical data.
The trainSOM function has several arguments, but only the first one is
required. This argument is x.data which is the dataset used to train the
SOM. In this documentation, it is passed to the function as a matrix or a data
frame with numerical variables in columns and observations of these variables in
rows.
The other arguments are the same as the arguments passed to the initSOM
function (they are parameters defining the algorithm, see help(initSOM)
for further details).
The trainSOM function returns an object of class somRes (see
help(trainSOM) for further details on this class).
The following table indicates which graphics are available for a numeric SOM.
| What SOM or SC Type |
SOM Energy |
Obs |
Prototypes |
Add |
SuperCluster (no what) |
Obs |
Prototypes |
Add |
|---|---|---|---|---|---|---|---|---|
| (no type) | x | |||||||
| hitmap | x | x | ||||||
| color | x | x | x | x | x | |||
| lines | x | x | x | x | x | x | ||
| meanline | x | x | x | x | ||||
| barplot | x | x | x | x | x | x | ||
| pie | x | x | ||||||
| boxplot | x | x | x | x | ||||
| 3d | x | |||||||
| poly.dist | x | x | ||||||
| umatrix | x | |||||||
| smooth.dist | x | |||||||
| mds | x | x | ||||||
| grid.dist | x | |||||||
| words | x | |||||||
| names | x | x | ||||||
| grid | x | |||||||
| dendrogram | x | |||||||
| dendro3d | x |
The first case study shows the clustering of points randomly distributed in the square \([0,1]^2\). The data are generated by:
The numeric SOM algorithm is used to cluster the data:
The energy evolves as described in the following graphic:
plot(my.som, what="energy")
The resulting clustering distribution can be visualized by the hitmap:
The observations are almost uniformly distributed on the map.
The clustering component allows us to plot the initial data according to the final clustering.
# prepare a vector of colors
my.colors <- rainbow(prod(my.som$parameters$the.grid$dim))[my.som$clustering]
# points depicted with the same color are in the same final cluster
plot(my.som$data[,1], my.som$data[,2], col=my.colors, pch=19, xlab="x1",
ylab="x2", main="Data according to final clustering")
The values of the prototypes can be represented with the plot function and help interpret the clusters:
par(mfrow=c(1,2))
plot(my.som, what="prototypes", type="color", var=1)
plot(my.som, what="prototypes", type="color", var=2)
Here, the interpretation is simple enough: high values of the first variables x1 are located at the top of the map and small values at the bottom of the map. Large values of x2 are located at the right hand side of the map, whereas, small values are located at the left hand side.
We obtain the same results with a similar plot on the observation mean values:
The prototypes coordinates are also registered for each intermediate backup so
they can be displayed on different graphics to see the evolution in the
prototype organization.
At the beginning of the algorithm, the prototypes are randomly distributed in [0,1]2 and then, they organize as a regular rectangular grid in \([0,1]^2\).
This second case study is performed on the famous (Fisher's or Anderson's) iris data set that gives the measurements in centimeters of the variables sepal length and width and petal length and width, respectively, for 50 flowers from each of 3 species of iris (setosa, versicolor, and virginica).
NB: In the following analysis, variables are centered and scaled to unit variance, which is the default behavior of the algorithm.
The first four variables of the data set (that are the numeric variables) are used to map each flower on the SOM grid.
set.seed(255)
# run the SOM algorithm with verbose set to TRUE
iris.som <- trainSOM(x.data = iris[,1:4], dimension = c(5,5), verbose = TRUE,
nb.save = 5, topo = "hexagonal")
## Self-Organizing Map algorithm...
##
## Parameters of the SOM
##
## SOM mode : online
## SOM type : numeric
## Affectation type : standard
## Grid :
## Self-Organizing Map structure
##
## Features :
## topology : hexagonal
## x dimension : 5
## y dimension : 5
## distance type: euclidean
##
## Number of iterations : 750
## Number of intermediate backups : 5
## Initializing prototypes method : random
## Data pre-processing type : unitvar
## Neighbourhood type : gaussian
##
## 0 % done
## 10 % done
## 20 % done
## 30 % done
## 40 % done
## 50 % done
## 60 % done
## 70 % done
## 80 % done
## 90 % done
## 100 % done
iris.som
## Self-Organizing Map object...
## online learning, type: numeric
## 5 x 5 grid with hexagonal topology
## neighbourhood type: gaussian
## distance type: euclidean
As the energy is registered during the intermediate backups, we can have a look at its evolution.
plot(iris.som, what="energy")
Here the energy does not stabilize as in the case of dist.type="letremy"
because the Gaussian annealing of the neighborhood is continuous and not
stepwise.
The clustering component contains the final classification of the dataset. It is a vector with length equal to the number of rows of the input dataset.
iris.som$clustering
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
## 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
## 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
## 5 5 5 5 5 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5
## 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
## 5 2 5 5 5 5 5 5 5 5 23 23 21 6 16 6 23 1 23 1
## 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
## 1 18 6 17 1 23 17 1 11 1 23 11 16 11 17 23 21 21 17 1
## 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
## 1 1 6 16 12 25 23 11 12 6 6 18 6 1 6 12 6 17 1 6
## 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
## 21 16 21 21 21 21 6 21 21 21 21 21 21 16 21 21 21 21 21 11
## 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
## 21 16 21 21 21 21 16 22 21 21 21 21 21 16 16 21 21 21 22 21
## 141 142 143 144 145 146 147 148 149 150
## 21 21 16 21 21 21 21 21 21 22
table(iris.som$clustering)
##
## 1 2 4 5 6 11 12 16 17 18 21 22 23 25
## 11 1 1 48 11 5 3 10 5 2 41 3 8 1
which can also be visualized by a hitmap plot:
plot(iris.som, what="obs", type="hitmap")
To assess the relevance of each explanatory variable in the definition of the
clusters, the function summary includes an ANOVA with the predictor being
the clustering, for each (numeric) input variable.
summary(iris.som)
##
## Summary
##
## Class : somRes
##
## Self-Organizing Map object...
## online learning, type: numeric
## 5 x 5 grid with hexagonal topology
## neighbourhood type: gaussian
## distance type: euclidean
##
## Final energy : 1.091265
## Topographic error: 0.1466667
##
## ANOVA :
##
## Degrees of freedom : 13
##
## F pvalue significativity
## Sepal.Length 43.463 0 ***
## Sepal.Width 17.240 0 ***
## Petal.Length 298.228 0 ***
## Petal.Width 173.062 0 ***
Here, all variables have significantly different means among the different clusters and can thus be considered to be relevant for the clustering definition.
Another useful function is predict.somRes. This function predicts the
neuron to which a new observation would be assigned. The first argument must be
a somRes object and the second one the new observation. Let us have a try
on the first observation of the iris data set:
# call predict.somRes
predict(iris.som, iris[1,1:4])
## 5
## 5
# check the result of the final clustering with the SOM algorithm
iris.som$clustering[1]
## 1
## 5
Some graphics are shared between observations and prototypes and can be used to display the prototypes' or the observations' values for the different variables in the neurons of the map. For observations, the mean values are sometimes displayed instead of the individual values.
par(mfrow = c(2,2))
plot(iris.som, what = "obs", type = "color", variable = 1)
plot(iris.som, what = "obs", type = "color", variable = 2)
plot(iris.som, what = "obs", type = "color", variable = 3)
plot(iris.som, what = "obs", type = "color", variable = 4)
Some neurons are empty (no observations affected to them): neurons 3, 7, 8, 9, 10, 13, 14, 15, 19, 20, 24. They are the illustration of a large difference between observations classified in clusters 4-5 with the rest of the observations (to some extend, the same can be said about observations in cluster 25).
Clusters 4-5 are characterized by the following facts (visible on "color"
plots): larger values for Sepal.Width, smaller values for Petal.Length and
Petal.Width and average values for Sepal.Length. Similar conclusions are
obtained from the "lines" and "barplot" plots. Please note that, for these
two last plots, it is advised to display the neuron numbers (option
show.names = TRUE) because the hexagonal display is not properly rendered in
them.
Individual information on observations in clustered can be obtained with the following plots:
They display either the observation distribution within the cluster for all
the variables in the dataset (for "boxplot" and "lines") or the names (row
numbers by default) of the observations classified within the cluster.
Some more graphics handling prototypes have been implemented:
"3d" provides the same results as "color" but on a 3 dimensional plot:
x is the x dimension of the grid, y is the y dimension of the grid and z is the
value of the prototype for the variable variable (by name or number in the
dataset) of the corresponding neuron. For the hexagonal topology, the plot is
obtained using a linear interpolation on a regular square grid.Also, some graphics are provided to visualize the distance between prototypes on the grid:
plot(iris.som, what = "prototypes", type = "poly.dist", show.names = FALSE)
"poly.dist" represents the distances between neighboring prototypes with
polygons plotted for each cell of the grid. The smaller the distance between
a polygon's vertex and a cell border, the closer the pair of prototypes.
The colors indicates the number of observations in the neuron (white is used
for empty neurons);
"umatrix" fills the neurons of the grid using colors that represent
the average distance between the current prototype and its neighbors;
"smooth.dist" plots the mean distance between the current prototype and
its neighbors with a color gradation;
mds plots the number of the neuron on a map according to a Multi
Dimensional Scaling (MDS) projection;
grid.dist plots a point for each pair of prototypes, with x coordinates
representing the distance between the prototypes in the input space, and y
coordinates representing the distance between the corresponding neurons on the
grid.
These graphics show that there is a big gap (large distances) between the top left corner and the rest of the map and between the top right corner and the rest of the map (which is consistent with what was already observed in the previous plots). In addition, the bottom right corner of the maps has a few neurons (21, 22, 23) that are very closed to each others.
The clustering can be analyzed together with an additional variable (here, the
flower species) using what = "add":
class(iris$Species)
## [1] "factor"
levels(iris$Species)
## [1] "setosa" "versicolor" "virginica"
plot(iris.som, what = "add", type = "pie", variable = iris$Species) +
scale_fill_brewer(type = "qual") +
guides(fill = guide_legend(title = "Species"))
This plot shows that the clustering produced by the SOM is indeed relevant to identify the three different species of iris: they are well separated on the map and almost all clusters only contain one species of iris. The Setosa species is the most distinct from the other two, isolated in the top left corner of the map.
The "color" plot available for "add" is similar to the "obs"
or "prototypes" cases. Here we choose the first variable of the iris data
set as an additional variable to illustrate its use. We thus obtain the same
plot as above (see section Graphics common to observations and prototypes).
The "lines", "barplot", "radar" and "boxplot" plots
available for "add" are similar to the "obs" or "prototypes"
cases.
"words" is only implemented for an additional variable. In this case, the
additional variable must be a contingency matrix: the words used on the plot
are the names of the columns and the presence or lack of the word is expressed
by respectively 1 or 0. The size of the words on the grid depends on the rate
of presence in the observations of the current neuron. To illustrate its use,
we define a contingency table my.cont.mat that corresponds to the flower
Species:
"names" is similar to the "names" case implemented for "obs".
Here we choose to give the argument variable the row names of the iris
data set: so we obtain the same plot as above (see
More graphics on observations).
Similarly, this plot can be used with the variable iris$Species:
which gives exactly the same plot as before for type "words" with the
contingency matrix corresponding to the variable iris$Species.
quality(iris.som)
## $topographic
## [1] 0.1466667
##
## $quantization
## [1] 1.22527
By default, the quality function calculates both quantization and topographic
errors. It is also possible to specify which one you want using the
argument quality.type.
The topographic error value varies between 0 (good projection quality) and 1 (poor projection quality). Here, the topographic quality of the mapping is equal to 0.15, which means that around 14.7% of the observations have a second best unit in the neighborhood of the best matching unit.
The quantization error is an unbounded positive number. The closer it is to 0, the better the projection quality.
In the SOM algorithm, the number of clusters is necessarily close to the number of neurons on the grid (not necessarily equal as some neurons may have no observations assigned to them). This - quite large - number may not suit the original data for a clustering purpose.
A usual way to address clustering with SOM is to perform a hierarchical
clustering on the prototypes. This clustering is directly available in the
package SOMbrero using the function superClass. To do so, you can
first have a quick overview to decide on the number of super clusters which
suits your data.
plot(superClass(iris.som))
## Warning in plot.somSC(superClass(iris.som)): Impossible to plot the rectangles: no super clusters.
By default, the function plots both a dendrogram and the evolution of the
percentage of explained variance. Here, 3 super clusters seem to be the best
choice. The output of superClass is a somSC class object.
Basic functions have been defined for this class:
my.sc <- superClass(iris.som, k = 3)
summary(my.sc)
##
## SOM Super Classes
## Initial number of clusters : 25
## Number of super clusters : 3
##
##
## Frequency table
## 1 2 3
## 7 9 9
##
## Clustering
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## 1 2 2 2 2 1 1 2 2 2 3 1 1 2 2 3 3 3 1 1 3 3 3 3 3
##
##
## ANOVA
##
## Degrees of freedom : 2
##
## F pvalue significativity
## Sepal.Length 178.319 0 ***
## Sepal.Width 60.416 0 ***
## Petal.Length 847.100 0 ***
## Petal.Width 505.974 0 ***
plot(my.sc, plot.var = FALSE)
Like plot.somRes, the function plot.somSC has an argument type
which offers many different plots and can thus be combined with most of the
graphics produced by plot.somSC:
plot(my.sc, type = "grid")
Case "grid" fills the grid with colors according to the super clustering
(and can provide a legend).
Case "dendro3d" plots a 3d dendrogram.
A couple of plots from plot.somRes are also available for the super
clustering. Some identify the super clusters with colors:
And some others identify the super clusters with titles:
plot(my.sc, what = "prototypes", type = "color", variable = "Sepal.Length")
plot(my.sc, what = "prototypes", type = "poly.dist")
It is also possible to consider an additional variable using the
argument what='add':
Super cluster number 2 is located at the top left hand corner of the map and associated with the Setosa species. SC 3 is in the opposite corner and associated mainly with Virginica whereas SC 1 is in the diagonal between these two corners and associated with Versicolor.
This vignette has been computed with the following environment:
sessionInfo()
## R version 4.1.2 (2021-11-01)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 20.04.3 LTS
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.9.0
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.9.0
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=fr_FR.UTF-8 LC_COLLATE=C
## [5] LC_MONETARY=fr_FR.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=fr_FR.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=fr_FR.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] ggplot2_3.3.5 SOMbrero_1.4-1 markdown_1.1 igraph_1.2.10
##
## loaded via a namespace (and not attached):
## [1] tidyselect_1.1.1 xfun_0.24 purrr_0.3.4
## [4] lattice_0.20-45 colorspace_2.0-2 vctrs_0.3.8
## [7] generics_0.1.0 utf8_1.2.2 rlang_0.4.11
## [10] isoband_0.2.5 hexbin_1.28.2 pillar_1.6.2
## [13] glue_1.4.2 withr_2.4.2 DBI_1.1.1
## [16] RColorBrewer_1.1-2 lifecycle_1.0.0 plyr_1.8.6
## [19] stringr_1.4.0 metR_0.11.0 munsell_0.5.0
## [22] gtable_0.3.0 codetools_0.2-18 evaluate_0.14
## [25] labeling_0.4.2 ggwordcloud_0.5.0 knitr_1.33
## [28] fansi_0.5.0 highr_0.9 Rcpp_1.0.7
## [31] scales_1.1.1 backports_1.2.1 checkmate_2.0.0
## [34] scatterplot3d_0.3-41 mime_0.10 deldir_1.0-6
## [37] farver_2.1.0 interp_1.0-33 png_0.1-7
## [40] digest_0.6.27 stringi_1.6.2 dplyr_1.0.7
## [43] grid_4.1.2 tools_4.1.2 magrittr_2.0.1
## [46] tibble_3.1.3 crayon_1.4.1 pkgconfig_2.0.3
## [49] ellipsis_0.3.2 data.table_1.14.0 lubridate_1.7.10
## [52] assertthat_0.2.1 R6_2.5.1 compiler_4.1.2