<!--
%\VignetteEngine{knitr}
%\VignetteIndexEntry{SOM for dissimilarity data}
%\VignettePackage{SOMbrero}
-->

# Self-Organizing Map for dissimilarity matrices

## Basic package description

To be able to run the SOM algorithm, you have to load the package called 
```SOMbrero```. The function used to run it is called ```trainSOM()``` and is 
detailed below.

__This documentation only considers the case of dissimilarity matrices.__

```{r loading, results='hide', echo=FALSE, warning=FALSE, message=FALSE}
library(SOMbrero)
```

### 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 matrix or a data
frame. This set must be a dissimilarity matrix, i.e., a symmetric matrix of
positive numbers, with zero entries on the diagonal.

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.

Type        |   Energy  |   Obs |   Prototypes  |   Add   |   Super Cluster
------------|:---------:|:-----:|:-------------:|:-------:|:----------------:
no type     |   x       |       |               |         |
hitmap      |           |   x   |               |         |         x
color       |           |       |               |   x     |         
lines       |           |       |       x       |   x     |         x2
barplot     |           |       |       x       |   x     |         x2
radar       |           |       |       x       |   x     |         x2
pie         |           |       |               |   x     |         x2
boxplot     |           |       |               |   x     |         
3d          |           |       |               |         |
poly.dist   |           |       |       x       |         |         x
umatrix     |           |       |       x       |         |
smooth.dist |           |       |       x       |         |
words       |           |       |               |   x     |
names       |           |   x   |               |   x     |
graph       |           |       |               |   x     |         x
mds         |           |       |       x       |         |         x
grid.dist   |           |       |       x       |         |         
grid        |           |       |               |         |         x
dendrogram  |           |       |               |         |         x
dendro3d    |           |       |               |         |         x

In the "Super Cluster" column, a plot marked by "x2" means it is available for 
both data set variables and additional variables.

## Case study: the ```lesmis``` data set

The ```lesmis``` data set provides 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 ```lesmis``` data contain two
objects: the first one ```lesmis``` is an ```igraph``` object (see [the igraph
web page](http://igraph.sourceforge.net/)),
with `r data(lesmis); vcount(lesmis)` nodes and `r ecount(lesmis)` edges.

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

```{r lesmisDescr, fig.width=12, fig.height=12}
data(lesmis)
lesmis
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 to the characters' names to ease the use of the 
graphical functions of ```SOMbrero```.

### Training the SOM

```{r lesmisTrain, cache=TRUE}
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.

### 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.
```{r lesmisClustering}
mis.som$clustering
table(mis.som$clustering)
plot(mis.som)
```

The clustering can be displayed using the ```plot``` function
with ```type=names```.
```{r lesmisPseudoNamesPlot, fig.height=12, fig.width=12, warning=FALSE}
plot(mis.som, what="obs", type="names")
```

or by sur-imposing the original igraph object on the map:
```{r lesmisProjGraph}
plot(mis.som, what="add", type="graph", var=lesmis)
```

Clusters profile overviews can be plotted either with *e.g.*, lines or radar.
```{r lesmisProtoProfiles}
plot(mis.som, what="prototypes", type="lines")
plot(mis.som, what="prototypes", type="radar")
```

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 behaviour as in the other SOM 
types.

* ```"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.

```{r lesmisProtoDist}
plot(mis.som, what="prototypes", type="poly.dist", print.title=TRUE)
plot(mis.som, what="prototypes", type="smooth.dist")
plot(mis.som, what="prototypes", type="umatrix", print.title=TRUE)
plot(mis.som, what="prototypes", type="mds")
plot(mis.som, what="prototypes", type="grid.dist")
```

Here we can see that the prototypes located in the top left corner of the map
(*e.g.*, clusters 4 and 5) are far from the others.

Finally, with a graphical overview of the clustering
```{r lesmisColorOverview, fig.height=12, fig.width=12}
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 cluster 5 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 5. The same kind of
conclusion holds for cluster 11, 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 quite important with the SOM algorithm, it is 
possible to perform a hierarchical clustering. First, let us have an overview of
the dendrogram:
```{r lesmisSCOverview}
plot(superClass(mis.som))
```

According to the proportion of variance explained by super clusters, 5 groups 
seem to be a good choice.
```{r lesmisSC}
sc.mis <- superClass(mis.som, k=5)
summary(sc.mis)
table(sc.mis$cluster)
plot(sc.mis)
plot(sc.mis, type="grid", plot.legend=TRUE)
plot(sc.mis, type="lines", print.title=TRUE)
plot(sc.mis, type="mds", plot.legend=TRUE)
plot(sc.mis, type="dendro3d")
```

```{r lesmisSCColorOverview, fig.width=12, fig.height=12}
library(RColorBrewer)
plot(lesmis, vertex.size=0, vertex.label.color=
       brewer.pal(6, "Set2")[sc.mis$cluster[mis.som$clustering]])
legend(x="left", legend=paste("SC",1:5), col=brewer.pal(5, "Set2"), pch=19)
```

* cluster 1 constains ```Valjean``` which has a central position in the MDS
visualization;

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

* cluster 3 contains ```Gavroche```, the abandonned child of the
```Thenardier```, and the characters of his sub-story;

* cluster 4 countains the Thenardier family: mister and misses ```Thenardier```,
their daughter ```Eponine``` and also the characters involved in their story. It
also contains ```Javert``` who is seeking to find the main character of the 
story, ```Valjean```, and ```Marius```. All these characters are strongly 
related to ```Cosette``` who is ```Marius```'s lover and has lived in the
```Thenardier``` family during her childhood;

* cluster 5 countains ```Fantine``` and the characters involved in her 
sub-story.

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

```{r lesmisSCProjGraph}
projectIGraph(sc.mis, lesmis)
par(mar=rep(0,4))
plot(sc.mis, type="projgraph", variable=lesmis, s.radius=2)
```
