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

`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.**

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).

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 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 |

`lesmis`

data setThe `lesmis`

data set is based on the co-appearance graph of the characters
of the novel Les MisÃ©rables (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 co-appearances. 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)
```

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`

.

```
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")
```

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.

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)
```

The clustering can be displayed using the `plot`

function with
`type = names`

.

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

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 ThÃ©nardier 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)
```

In the latter plot (which is still messy at this stage of the analysis), nodes correspond to clusters and are positioned 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(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())
```

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`

.

`"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 encode the number of observations in the neuron;`"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 the \(x\) coordinates representing the distance between the prototypes in the input space, and \(y\) coordinates representing the distance between the corresponding neurons on the grid.

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

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)
```

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.

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.
```

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(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)
```

cluster 1 contains

`Myriel`

and the characters involved in his sub-story;cluster 2 contains

`Valjean`

which has a central position in the graph visualization, and most of the important character of the novel (including Javert, Fantine and Cosette);cluster 3 contains people almost only connected to

`Fantine`

who links them to the rest of the novel;cluster 4 contains

`Gavroche`

, the abandoned child of the`Thenardier`

, and the characters of his sub-story (including Mr ThÃ©nardier and Gavroche's two brothers and his sister,`Eponine`

);cluster 5 is a bit harder to interpret, with secondary characters related to

`Thenardier`

and to the main characters of the novel.

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

```
projectIGraph(sc.mis, lesmis)
```

```
## IGRAPH ae6b199 UNW- 5 7 --
## + attr: layout (g/n), name (v/c), size (v/n), weight (e/n)
## + edges from ae6b199 (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)
```