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/e-doc-relationalSOM.html

Basic package description

SOMbrero implements different variants of the Self-Organizing Map algorithm (also called Kohonen's algorithm). To process a given dataset with the SOM algorithm, you can use the function trainSOM().

This documentation only considers the case of dissimilarity matrices.

Arguments

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 square matrix or data frame, which entries are dissimilarity measures between pairs of observations. The diagonal of this matrix must contain only zeros.

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).

Outputs

The trainSOM function returns an object of class somRes (see help(trainSOM) for further details on this class).

Graphics

The following table indicates which graphics are available for a relational 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
lines x x x x
meanline x x
barplot x x x x
pie x x
boxplot x x
poly.dist x x
umatrix x
smooth.dist x
mds x x
grid.dist x
words x
names x x x
graph x x
projgraph x x
grid x
dendrogram x
dendro3d x

Case study: the lesmis data set

The lesmis data set is based on the coappearance graph of the characters of the novel Les Miserables (Victor Hugo). Each vertex stands for a character whose name is given by the vertex label. One edge means that the corresponding two characters appear in a common chapter in the book. Each edge also has a value indicating the number of coappearances. The co-appearance network has been extracted by D.E. Knuth (1993).

The lesmis data contain two objects: the first one,lesmis, is an igraph object (see the igraph web page), with 77 nodes and 254 edges.

Further information on this data set is provided with help(lesmis).

data(lesmis)
lesmis
## IGRAPH 3babff7 U--- 77 254 -- 
## + attr: layout (g/n), id (v/n), label (v/c), value (e/n)
## + edges from 3babff7:
##  [1]  1-- 2  1-- 3  1-- 4  3-- 4  1-- 5  1-- 6  1-- 7  1-- 8  1-- 9  1--10
## [11] 11--12  4--12  3--12  1--12 12--13 12--14 12--15 12--16 17--18 17--19
## [21] 18--19 17--20 18--20 19--20 17--21 18--21 19--21 20--21 17--22 18--22
## [31] 19--22 20--22 21--22 17--23 18--23 19--23 20--23 21--23 22--23 17--24
## [41] 18--24 19--24 20--24 21--24 22--24 23--24 13--24 12--24 24--25 12--25
## [51] 25--26 24--26 12--26 25--27 12--27 17--27 26--27 12--28 24--28 26--28
## [61] 25--28 27--28 12--29 28--29 24--30 28--30 12--30 24--31 31--32 12--32
## [71] 24--32 28--32 12--33 12--34 28--34 12--35 30--35 12--36 35--36 30--36
## + ... omitted several edges
plot(lesmis, vertex.size = 0)

plot of chunk lesmisDescr

The dissim.lesmis object is a matrix with entries equal to the length of the shortest path between two characters (obtained with the function shortest.paths of package igraph). Note that its row and column names have been initialized with the characters' names to ease the use of the graphical functions of SOMbrero.

Training the SOM

set.seed(622)
mis.som <- trainSOM(x.data=dissim.lesmis, type = "relational", nb.save = 10,
                   init.proto = "random", radius.type = "letremy")
plot(mis.som, what="energy")

plot of chunk lesmisTrain

The dissimilarity matrix dissim.lesmis is passed to the trainSOM function as input. As the SOM intermediate backups have been registered (nb.save = 10), the energy evolution can be plotted: it stabilized in the last 100 iterations.

Resulting clustering

The clustering component provides the classification of each of the 77 characters. The table function is a simple way to view data distribution on the map.

mis.som$clustering
##           Myriel         Napoleon   MlleBaptistine      MmeMagloire 
##               25               25               19               19 
##     CountessDeLo         Geborand     Champtercier         Cravatte 
##               25               25               25               25 
##            Count           OldMan          Labarre          Valjean 
##               25               25               22               22 
##       Marguerite           MmeDeR          Isabeau          Gervais 
##               16               22               23               23 
##        Tholomyes        Listolier          Fameuil      Blacheville 
##               11               11               11               11 
##        Favourite           Dahlia          Zephine          Fantine 
##               11               11               11               11 
##    MmeThenardier       Thenardier          Cosette           Javert 
##                2                6                7               17 
##     Fauchelevent       Bamatabois         Perpetue         Simplice 
##               18               21               11               17 
##      Scaufflaire           Woman1            Judge     Champmathieu 
##               22               22               21               21 
##           Brevet       Chenildieu      Cochepaille        Pontmercy 
##               21               21               21                9 
##     Boulatruelle          Eponine          Anzelma           Woman2 
##                6                1                2               17 
##   MotherInnocent          Gribier        Jondrette        MmeBurgon 
##               18               18               15               15 
##         Gavroche     Gillenormand           Magnon MlleGillenormand 
##               15                3                3               13 
##     MmePontmercy      MlleVaubois   LtGillenormand           Marius 
##                8               13                8                4 
##        BaronessT           Mabeuf         Enjolras       Combeferre 
##                3                5               10                5 
##        Prouvaire          Feuilly       Courfeyrac          Bahorel 
##               10                5                5               10 
##          Bossuet             Joly        Grantaire   MotherPlutarch 
##               10               10               10                5 
##        Gueulemer            Babet       Claquesous     Montparnasse 
##                1                1                1                1 
##        Toussaint           Child1           Child2           Brujon 
##               17               15               15                1 
##     MmeHucheloup 
##               10
table(mis.som$clustering)
## 
##  1  2  3  4  5  6  7  8  9 10 11 13 15 16 17 18 19 21 22 23 25 
##  6  2  3  1  5  2  1  2  1  7  9  2  5  1  4  3  2  6  5  2  8
plot(mis.som)

plot of chunk lesmisClustering

The clustering can be displayed using the plot function with type = names.

plot(mis.som, what = "obs", type = "names")

plot of chunk lesmisPseudoNamesPlot

In this clustering, the main character, Valjean, is in a central position (in cluster 8) and some clusters are easily identified as sub-stories around Javert. For instance, clusters 10, 15 and 20 are related to the Thénardier family, with (for instance), cluster 20 being the cluster of Gavroche and his two brothers (named children 1 and 2).

The original graph can also be superimposed on the map:

plot(mis.som, what = "add", type = "graph", var = lesmis)

plot of chunk lesmisProjGraph

In the latter plot (which is still messy at this stage of the analysis), nodes correspond to clusters and are positionned at the cluster location on the map. The size of the nodes is proportional to the number of characters classified in this cluster and edges between nodes have a width proportional to the total weight between any two characters from the two linked clusters.

Clusters profile overviews can be plotted either with e.g., lines or barplot, that both provide an information similar to that given by "names".

plot(mis.som, what = "prototypes", type = "lines")  +
  guides(color = guide_legend(keyheight = 0.5, ncol = 2, label.theme = element_text(size = 6))) + 
  theme(axis.text.x = element_blank(), axis.ticks.x = element_blank())

plot of chunk lesmisProtoProfiles

plot(mis.som, what = "prototypes", type = "barplot")  +
  guides(fill = guide_legend(keyheight = 0.5, ncol = 2, label.theme = element_text(size = 6))) + 
  theme(axis.text.x = element_blank(), axis.ticks.x = element_blank())

plot of chunk lesmisProtoProfiles

On these graphics, one variable is represented respectively with a point or a slice. It is therefore easy to see which variable affects which cluster.

To see how different the clusters are, some graphics show the distances between prototypes. These graphics have exactly the same interpretation as for the other data types processed by SOMbrero.

plot(mis.som, what = "prototypes", type = "poly.dist")

plot of chunk lesmisProtoDist

Here we can see that the prototypes located in the top left and top right corners of the map (e.g., clusters 5 and clusters 19-20 and 24-25) are further from the other neurons than in average.

Finally, with a graphical overview of the clustering

plot(lesmis, vertex.label.color = rainbow(25)[mis.som$clustering], 
     vertex.size = 0)
legend(x = "left", legend = 1:25, col = rainbow(25), pch = 19)

plot of chunk lesmisColorOverview

We can see that (for instance) cluster 25 is very relevant to the story: as the characters of this cluster appear only in the sub-story of the Bishop Myriel, he is the only connection for all other characters of cluster 25. The same kind of conclusion holds for cluster 20 (with Gavroche), among others. Most of the other clusters have a small number of observations: it thus seems relevant to compute super clusters.

Compute super clusters

As the number of clusters is rather large with the SOM algorithm, it is possible to perform a hierarchical clustering on top of SOM results. First, let us have an overview of the dendrogram:

plot(superClass(mis.som))
## Warning in plot.somSC(superClass(mis.som)): Impossible to plot the rectangles: no super clusters.

plot of chunk lesmisSCOverview

According to the proportion of variance explained by super clusters, 5 groups seem to be a good choice (4 groups would have been relevant also. The clustering with 5 groups creates a group with only one cluster in it).

sc.mis <- superClass(mis.som, k = 5)
summary(sc.mis)
## 
##    SOM Super Classes
##      Initial number of clusters :  25 
##      Number of super clusters   :  5 
## 
## 
##   Frequency table
## 1 2 3 4 5 
## 3 6 6 6 4 
## 
##   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  1  2  3  3  1  2  2  3  3  2  2  2  3  3  4  4  4  5  5  4  4  4  5  5 
## 
## 
##   ANOVA
##          F                       :  9.13755 
##          Degrees of freedom      :  4 
##          p-value                 :  5.00329e-06 
##                  significativity :  ***
table(sc.mis$cluster)
## 
## 1 2 3 4 5 
## 3 6 6 6 4
plot(sc.mis)

plot of chunk lesmisSC

plot(lesmis, vertex.size = 0, 
     vertex.label.color = rainbow(5)[sc.mis$cluster[mis.som$clustering]])
legend(x = "left", legend = paste("SC", 1:5), col = rainbow(5), pch = 19)

plot of chunk lesmisSCColorOverview

SOMbrero also contains functions to compute a projected graph based on the super-clusters and to display it:

projectIGraph(sc.mis, lesmis)
## IGRAPH ff42ac0 UNW- 5 7 -- 
## + attr: layout (g/n), name (v/c), size (v/n), weight (e/n)
## + edges from ff42ac0 (vertex names):
## [1] 1--2 1--3 1--4 2--3 2--4 3--4 4--5
par(mar = rep(0,4))
plot(sc.mis, what = "add", type = "projgraph", variable = lesmis, s.radius = 2)