In Albluca/rpgraph: r interface to VDAOEngine to build elastic principal graphs

This document describes how to combine different graph grammars to produce advanced topologies

An hairy circle

The simple_circle dataset included in the package describes points placed on a three dimensional circle. To produce an circle with additional brances we stard by producing a circle

library(rpgraph)
PCircle <- computeElasticPrincipalGraph(Data = simple_circle, NumNodes = 20, Method = 'CircleConfiguration')

plotData2D(Data = simple_circle, PrintGraph = PCircle[[1]] ,
           GroupsLab = rep(1, nrow(simple_tree)),
           Main = "Pincipal circle",
           Xlab = "Dimension 1", Ylab = "Dimension 2")

It is then possible to add branches by contructing a tree starting from the circle obtained

HairyCircle <- computeElasticPrincipalGraph(Data = simple_circle, NumNodes = 40,
                                            Method = 'DefaultPrincipalTreeConfiguration',
                                            NodesPositions = PCircle[[1]]$Nodes, Edges = PCircle[[1]]$Edges)

plotData2D(Data = simple_circle, PrintGraph = HairyCircle[[1]] ,
           GroupsLab = rep(1, nrow(simple_tree)),
           Main = "An hairy circle",
           Xlab = "Dimension 1", Ylab = "Dimension 2")

An lollipop

Is it also possible to construct a circle with only one branching point. However, this requires a more advanced approach. We will start by adding a strong branch to the simple_circle dataset and by getting the initial principal circle.

library(rpgraph)

Data <- rbind(simple_circle,
      cbind(seq(1,2,.1) + rnorm(11, sd=0.1),
      seq(1,2,.1) + rnorm(11, sd=0.1),
      rnorm(11, sd=0.1))
)


PCircle <- computeElasticPrincipalGraph(Data = Data, NumNodes = 20, Method = 'CircleConfiguration')

we will now apply the tree grammar une step at the time untill we get a single branching points

NBranches <- 0

tGraph <- PCircle

while(NBranches != 1){
  tGraph <- computeElasticPrincipalGraph(Data = Data, NumNodes = nrow(tGraph[[1]]$Nodes) + 1,
                                         Method = 'DefaultPrincipalTreeConfiguration',
                                         NodesPositions = tGraph[[1]]$Nodes, Edges = tGraph[[1]]$Edges)

  Br <- strsplit(as.character(tGraph[[1]]$Report$BARCODE), "||", fixed = TRUE)[[1]][1]
  NBranches <- sum(as.integer(strsplit(Br, "|", fixed = TRUE)[[1]]))
}

We now has a circle with a single branching point.

plotData2D(Data = Data, PrintGraph = tGraph[[1]] ,
           GroupsLab = rep(1, nrow(simple_circle)),
           Main = "A basic lollipop",
           Xlab = "Dimension 1", Ylab = "Dimension 2")

To avoid adding new branching points, we need to use the principal curve grammar from now on

Lollipop <- computeElasticPrincipalGraph(Data = Data, NumNodes = 40,
                                         Method = 'CurveConfiguration',
                                         NodesPositions = tGraph[[1]]$Nodes, Edges = tGraph[[1]]$Edges)

plotData2D(Data = Data, PrintGraph = Lollipop[[1]] ,
           GroupsLab = rep(1, nrow(simple_circle)),
           Main = "A lollipop",
           Xlab = "Dimension 1", Ylab = "Dimension 2")

An tree with a maximum number of branches

Using an approch similar to whas has been done before, it is possible to create a principal tre with a fixed number of branches. Assumme for example that we wat to fit a principal tree to the simple_tree dataset with only 3 branches. To do that we start by constructing a tree with only three brances

tGraph <- computeElasticPrincipalGraph(Data = simple_tree, NumNodes = 3,
                                       Method = 'DefaultPrincipalTreeConfiguration')

NBranches <- unlist(strsplit(as.character(tGraph[[1]]$Report$BARCODE), "||", fixed = TRUE))[1]

while(NBranches < 3){
  tGraph <- computeElasticPrincipalGraph(Data = simple_tree, NumNodes = nrow(tGraph[[1]]$Nodes) + 1,
                                         Method = 'DefaultPrincipalTreeConfiguration',
                                         NodesPositions = tGraph[[1]]$Nodes, Edges = tGraph[[1]]$Edges)

  Br <- strsplit(as.character(tGraph[[1]]$Report$BARCODE), "||", fixed = TRUE)[[1]][1]
  NBranches <- sum(as.integer(strsplit(Br, "|", fixed = TRUE)[[1]]))
}

plotData2D(Data = simple_tree, PrintGraph = tGraph[[1]] ,
           GroupsLab = rep(1, nrow(simple_tree)),
           Main = "A minimal tree with 3 branches",
           Xlab = "Dimension 1", Ylab = "Dimension 2")

Now the tree can be extended by using the curve topology

tGraph <- computeElasticPrincipalGraph(Data = simple_tree, NumNodes = 50,
                                         Method = 'CurveConfiguration',
                                         NodesPositions = tGraph[[1]]$Nodes, Edges = tGraph[[1]]$Edges)

plotData2D(Data = simple_tree, PrintGraph = tGraph[[1]] ,
           GroupsLab = rep(1, nrow(simple_tree)),
           Main = "A tree with 3 branches",
           Xlab = "Dimension 1", Ylab = "Dimension 2")