ElPiGraph.R: Elastic principal graph construction

The alpha parameter
Excessinve branching due to noise

The alpha parameter

Under certain circumstances, it may be necessary to tune the form of the elastic energy to obtain a better results. This is the case, for example, when an excessive number of branches is introduced by the tree grammar.

To tune the elastic energy, it is necessary to set FinalEnergy = "Penalized" and specify values for the alpha parameter. This parameter can be used to give more weight to the components of the energy associted with the lenght of the edges, hence penalizing graphs that are longer. Increasing alpha will penalize stars with a larger number of edges, and will preferentially introduce 3-stars.

Excessinve branching due to noise

To exemplify the effect of alpha, we will increase the level of noise around the &st and 2nd dimension of the tree example dataset. This will generate spurious clusters of points that are likely to be identified as branches.

library(ElPiGraph.R)
set.seed(42)

nExp <- 1

NewPoints <- lapply(1:nExp, function(x){
  tree_data[,1:2] + rnorm(n = length(tree_data[,1:2]), sd = .15)
})

NewPoints <- do.call(rbind, NewPoints)

NoisyTree <- rbind(tree_data[,1:2], NewPoints)
NoisyTree_Cat <- c(rep("Real", nrow(tree_data)), rep("Noise", nrow(NewPoints)))

TreeEPG <- computeElasticPrincipalTree(X = NoisyTree, NumNodes = 60,
                                       Lambda = .01, Mu = .01,
                                       drawAccuracyComplexity = FALSE,
                                       drawEnergy = FALSE,
                                       drawPCAView = FALSE,
                                       n.cores = 1)

## [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] "2 dimensions are being used"
## [1] "100% of the original variance has been retained"
## [1] "Computing EPG with 60 nodes on 984 points and 2 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|7||60  0.0178  60  59  41  7   0   0   0.01075 0.01005 0.981   0.9822  0.006854    0.000189    0.01134 0.6805  0
## 19.149 sec elapsed

PlotPG(X = NoisyTree, TargetPG = TreeEPG[[1]], GroupsLab = NoisyTree_Cat,
       Do_PCA = FALSE)

## [[1]]

TreeEPG <- computeElasticPrincipalTree(X = NoisyTree, NumNodes = 60,
                                       Lambda = .01, Mu = .01,
                                       drawAccuracyComplexity = FALSE,
                                       drawEnergy = FALSE, drawPCAView = FALSE,
                                       n.cores = 1,
                                       FinalEnergy = "Penalized",
                                       alpha = 0.005)

## [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] "2 dimensions are being used"
## [1] "100% of the original variance has been retained"
## [1] "Computing EPG with 60 nodes on 984 points and 2 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
## 5||60    0.01834 60  59  48  5   0   0   0.0122  0.01153 0.9784  0.9796  0.0058  0.00034 0.0204  1.224   0
## 20.602 sec elapsed

PlotPG(X = NoisyTree, TargetPG = TreeEPG[[1]], GroupsLab = NoisyTree_Cat,
       Do_PCA = FALSE)

## [[1]]

Note that when using elastic energy controlled graph embeddment (Mode = 2) the standard energy funtion will be used.

TreeEPG <- computeElasticPrincipalTree(X = NoisyTree, NumNodes = 60,
                                       Lambda = .01, Mu = .01,
                                       drawAccuracyComplexity = FALSE,
                                       drawEnergy = FALSE, drawPCAView = FALSE,
                                       n.cores = 1, Mode = 2,
                                       FinalEnergy = "Penalized",
                                       alpha = 0.005)

## [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] "2 dimensions are being used"
## [1] "100% of the original variance has been retained"
## [1] "Computing EPG with 60 nodes on 984 points and 2 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
## 4||60    0.01834 60  59  50  4   0   0   0.01222 0.0116  0.9784  0.9795  0.005891    0.0002311   0.01387 0.8319  0
## 19.886 sec elapsed

PlotPG(X = NoisyTree, TargetPG = TreeEPG[[1]], GroupsLab = NoisyTree_Cat,
       Do_PCA = FALSE)

## [[1]]