guides/struct.md

In Albluca/ElPiGraph.R: Elastic principal graph construction

• Setup
• Obtaining notable structures in trees
• Obtaining notable structures in circles
• Looking at the nodes composing the structures
• Looking at the points associted with the structures

The ElPiGraph package contains a number of functions that that can be used to facilitate the analysis of the obtained graph. This tutorial explains how to extract notable substructures from the obtained graph and visualize them.

Setup

A a first step in the tutorial we need to generate graphs from data. We will use tree and circle structures. We begin by constructing the tree on the example data included in the package.

library(ElPiGraph.R)
library(igraph)

## 
## Attaching package: 'igraph'

## The following objects are masked from 'package:stats':
## 
##     decompose, spectrum

## The following object is masked from 'package:base':
## 
##     union

library(magrittr)

TreeEPG <- computeElasticPrincipalTree(X = tree_data, NumNodes = 60, Lambda = .03, Mu = .01,
                                       drawAccuracyComplexity = FALSE, drawEnergy = FALSE)

## [1] "Creating a chain in the 1st PC with 2 nodes"
## [1] "Constructing tree 1 of 1 / Subset 1 of 1"
## [1] "Performing PCA on the data"
## [1] "Using standard PCA"
## [1] "3 dimensions are being used"
## [1] "100% of the original variance has been retained"
## [1] "Computing EPG with 60 nodes on 492 points and 3 dimensions"
## [1] "Using a single core"
## Nodes = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 
## BARCODE  ENERGY  NNODES  NEDGES  NRIBS   NSTARS  NRAYS   NRAYS2  MSE MSEP    FVE FVEP    UE  UR  URN URN2    URSD
## 1|2||60  0.01848 60  59  51  2   0   0   0.00548 0.005053    0.9898  0.9906  0.01249 0.0005134   0.03081 1.848   0
## 29.213 sec elapsed
## [[1]]

CircleEPG <- computeElasticPrincipalCircle(X = circle_data, NumNodes = 40,
                                       drawAccuracyComplexity = FALSE, drawEnergy = FALSE)

## [1] "Using a single core"
## [1] "Creating a circle in the plane induced buy the 1st and 2nd PCs with 3 nodes"
## [1] "Constructing curve 1 of 1 / Subset 1 of 1"
## [1] "Performing PCA on the data"
## [1] "Using standard PCA"
## [1] "3 dimensions are being used"
## [1] "100% of the original variance has been retained"
## [1] "Computing EPG with 40 nodes on 200 points and 3 dimensions"
## [1] "Using a single core"
## Nodes = 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 
## BARCODE  ENERGY  NNODES  NEDGES  NRIBS   NSTARS  NRAYS   NRAYS2  MSE MSEP    FVE FVEP    UE  UR  URN URN2    URSD
## 0||40    0.04107 40  40  40  0   0   0   0.02892 0.02738 0.9729  0.9744  0.0105  0.00165 0.066   2.64    0
## 1.541 sec elapsed
## [[1]]

We then generate igraph networks from the ElPiGraph structure

Tree_Graph <- ConstructGraph(PrintGraph = TreeEPG[[1]])
Circle_Graph <- ConstructGraph(PrintGraph = CircleEPG[[1]])

Obtaining notable structures in trees

The first step in the analysis consists in selecting various substructures present in the tree. This can be done via the GetSubGraph function, by specified the appropriate value for the structure parameter. For trees the most relevant options include 'end2end' (that extracts the paths connecting all of the leaves of the tree), branches (that extracts all the different branches composing the tree), and branching, (that extracts all the branching sub-trees).

Tree_e2e <- GetSubGraph(Net = Tree_Graph, Structure = 'end2end')

Tree_Brches <- GetSubGraph(Net = Tree_Graph, Structure = 'branches')

Tree_BrBrPt <- GetSubGraph(Net = Tree_Graph, Structure = 'branches&bpoints')

Tree_SubTrees <- GetSubGraph(Net = Tree_Graph, Structure = 'branching')

The extracted structures can be visualized on the data using the PlotPG function, with minimal manipulation:

PlotPG(X = tree_data, TargetPG = TreeEPG[[1]], PGCol = V(Tree_Graph) %in% Tree_e2e[[1]], PointSize = NA)

## [[1]]

As we can see from the plot, the function will highlight the edges connecting the nodes selected (marked TRUE), the edges connecting the nodes not selected (marked FALSE), and the edges connecting the the groups (marked Multi).

Similarly, we have

PlotPG(X = tree_data, TargetPG = TreeEPG[[1]], PGCol = V(Tree_Graph) %in% Tree_Brches[[1]], PointSize = NA)

## [[1]]

and

PlotPG(X = tree_data, TargetPG = TreeEPG[[1]], PGCol = V(Tree_Graph) %in% Tree_SubTrees[[1]], PointSize = NA)

## [[1]]

It is also possible to visualize multiple structures. In this case it is better to use GetSubGraph by setting KeepEnds to FALSE

Tree_Brches_NoEnds <- GetSubGraph(Net = Tree_Graph, Structure = 'branches', KeepEnds = FALSE)

BrID <- sapply(1:length(Tree_Brches_NoEnds), function(i){
  rep(i, length(Tree_Brches_NoEnds[[i]]))}) %>%
  unlist()

NodesID <- rep(0, vcount(Tree_Graph))
NodesID[unlist(Tree_Brches_NoEnds)] <- BrID

PlotPG(X = tree_data, TargetPG = TreeEPG[[1]], PGCol = NodesID, PointSize = NA, p.alpha = .05)

## [[1]]

Obtaining notable structures in circles

Similarly to tree, GetSubGraph can be used to obtain circles by setting structure to circle. When looking for circles it is possible to specify the length of the circle via the Nodes parameter, if unspecified, the function will try to find the largest circle in the data (this is potentially time consuming for large structures). Furthermore, by setting Circular to TRUE we will get a path with coinciding initial and terminal node.

Circle_all <- GetSubGraph(Net = Circle_Graph, Structure = 'circle', Circular = TRUE)

## [1] "Looking for the largest cycle"
## [1] "A cycle of lenght 40 has been found"

Since the graph is a circle, all the edges will be part of any substructure selected:

PlotPG(X = circle_data, TargetPG = CircleEPG[[1]], PGCol = V(Tree_Graph) %in% Circle_all[[1]], PointSize = NA)

## [[1]]

Looking at the nodes composing the structures

To select the structure of interest it is necessary to understand in a more precise way how they are mapped to graph.

Let us consider the first subs-tree computed:

Tree_SubTrees[[1]]

##  [1]  1  7 21 57 20 52 34 44 30  9 55  5 47 29 16 28 37 60 12  6 40 19 26
## [24]  8 13  2  3  4

To look how the nodes maps to the graph, we can label the nodes in the PlotPG function. This can be done using the NodeLabels parameter.

PlotPG(X = tree_data, TargetPG = TreeEPG[[1]], PGCol = V(Tree_Graph) %in% Tree_SubTrees[[1]], PointSize = NA, NodeLabels = 1:nrow(TreeEPG[[1]]$NodePositions))

## [[1]]

If the labels are hard so see, is possible to adjust their size via that LabMult parameter.

PlotPG(X = tree_data, TargetPG = TreeEPG[[1]], PGCol = V(Tree_Graph) %in% Tree_SubTrees[[1]], PointSize = NA, NodeLabels = 1:nrow(TreeEPG[[1]]$NodePositions), LabMult = 3)

## [[1]]

A similar procedure can be used to look at one of the path between the leaves

Tree_e2e[[1]]

## + 26/60 vertices, named, from 12a263f:
##  [1] 2  8  40 60 16 5  30 52 21 1  57 34 9  47 28 12 19 3  15 18 27 31 35
## [24] 36 45 51

PlotPG(X = tree_data, TargetPG = TreeEPG[[1]], PGCol = V(Tree_Graph) %in% Tree_e2e[[1]], PointSize = NA, NodeLabels = 1:nrow(TreeEPG[[1]]$NodePositions), LabMult = 3)

## [[1]]

Note that GetSubGraph returns each structure only once. Therefore, the path from node 4 to node 2 will not be present in the list.

Looking at the points associted with the structures

With the previourly computed information, it is also easy to extract the points asscoited with each substructure via the association of points to nodes. We begin by associating the points to the nodes

PartStruct <- PartitionData(X = tree_data, NodePositions = TreeEPG[[1]]$NodePositions)

And then obtain the points associted with each substrucutre

PtInBr <- lapply(Tree_Brches, function(x){which(PartStruct$Partition %in% x)})

Again, we can visualize the inforlation with PlotPG

PlotPG(X = tree_data, TargetPG = TreeEPG[[1]], GroupsLab = 1:nrow(tree_data) %in% PtInBr[[1]])

## [[1]]

PlotPG(X = tree_data, TargetPG = TreeEPG[[1]], GroupsLab = 1:nrow(tree_data) %in% PtInBr[[3]])

## [[1]]

If we use the 'branches&bpoints' structure options, we obtain a complete partition of the tree into branches and branching points, that can looked at in different ways

PointLabel = rep("", length(PartStruct$Partition))

for(i in 1:length(Tree_BrBrPt)){
  PointLabel[PartStruct$Partition %in% Tree_BrBrPt[[i]]] <- names(Tree_BrBrPt)[i]
}

barplot(table(PointLabel), las = 2, ylab="Number of points")

PlotPG(X = tree_data, TargetPG = TreeEPG[[1]], GroupsLab = PointLabel)

## [[1]]

boxplot(tree_data[, 1] ~ PointLabel, ylab = "Coordinate on the 1st dimension", las = 2)

boxplot(tree_data[, 2] ~ PointLabel, ylab = "Coordinate on the 2nd dimension", las = 2)

boxplot(tree_data[, 3] ~ PointLabel, ylab = "Coordinate on the 3rd dimension", las = 2)

Albluca/ElPiGraph.R documentation built on May 28, 2019, 11:02 a.m.