ANalysis Of Cluster VAriability

Description

The ANOCVA (ANalysis Of Cluster VAriability) is a non-parametric statistical test to compare clusters with applications in functional magnetic resonance imaging data. The ANOCVA allows us to compare the clustering structure of multiple groups simultaneously and also to identify features that contribute to the differential clustering.

Usage

anocva(dataDist, id, replicates = 1000, r = NULL,
  clusteringFunction = NULL, p = 1, maxClust = 20,
  criterion = c("slope", "silhouette"), showElapTime = TRUE)

Arguments

Details

The test statistic used is the one proposed by Caetano de Jesus (2017).

Value

ANOCVA p-values

References

Fujita A, Takahashi DY, Patriota AG, Sato JR (2014a) A non-parametric statistical test to compare clusters with applications in functional magnetic resonance imaging data. Statistics in Medicine 33: 4949–4962

Vidal MC, Sato JR, Balardin JB, Takahashi DY, Fujita A (2017) ANOCVA in R: a software to compare clusters between groups and its application to the study of autism spectrum disorder. Frontiers in Neuroscience 11:1–8

Caetano de Jesus DA. (2017) Evaluation of ANOCVA test for cluster comparison through simulations. Master Dissertation. Institute of Mathematics and Statistics, University of São Paulo.

Examples

# Install packages if necessary
# install.packages('MASS')
# install.packages('cluster')

library(anocva)
library(MASS)
library(cluster)

set.seed(5000)

# Defines a k-means function that returns cluster labels directly
myKmeans = function(dist, k){
  return(kmeans(dist, k, iter.max = 50, nstart = 5)$cluster)
}

# Number of subjects in each population
nsub = 20
# Number of items in each subject
nitem = 30

# Generate simulated data
data = array(NA, c(nsub*2, nitem*2, 2))
dataDist = array(NA, c(nsub*2, nitem*2, nitem*2))
meanx = 2
delta = 0.5
# Covariance matrix
sigma = matrix(c(0.03, 0, 0, 0.03), 2)
for (i in seq(nsub*2)){
  sub = rbind(mvrnorm(nitem, mu = c(0, 0), Sigma = sigma ),
              mvrnorm(nitem, mu = c(meanx,0), Sigma = sigma))
  data[i,,] = sub
  # If it's a sample of population 2.
  if (i > nsub){
    data[i,10,1] = data[i,10,1] + delta
  }
  # Euclidian distance
  dataDist[i,,] = as.matrix(dist(data[i,,]))
}

# Population 1 subject
plot(data[5,,], asp = 1, xlab = '', ylab = '', main = 'Population 1 - subject example')

# Population 2 subject
plot(data[35,,], asp = 1, xlab = '', ylab = '', main = 'Population 2 - subject example')

# The first nsub subjects belong to population 1 while the next nsub subjects belong to population 2
id = c(rep(1, nsub), rep(2, nsub))

## Not run: 
# ANOCVA call with different clustering function (myKmeans) and inside estimation of
# the number of clusters (r)
res1 = anocva(dataDist, id, replicates=500, r = NULL,
              clusteringFunction = myKmeans,
              p = 1, criterion = "slope")

## End(Not run)

# Estimate the number of clusters previously by using Spectral Clustering and Slope criterion
r = nClustMulti(dataDist, clusteringFunction = spectralClustering, criterion = 'slope')

# Calls ANOCVA statistical test
res = anocva(dataDist, id, replicates=500, r = r,
             clusteringFunction = spectralClustering,
             p = 1, criterion = "slope")

# DeltaS p-value
res$pValueDeltaS

# DeltaSq p-values
res$pValueDeltaSq

# Identifies which items have significant p-values with a significance level of 0.05.
which(res$pValueDeltaSq < 0.05)

# Identifies which items have significant FDR adjusted p-values (q-values)
# with a significance level of 0.05.
qValue = p.adjust(res$pValueDeltaSq, "fdr")
which(qValue < 0.05)

Calculates ANOVA statistics for original data and bootstrap replicates.

Description

Calculates ANOVA statistics for original data and bootstrap replicates.

Usage

anocvaStats(idx, dataDist, id, k, N, r, clusteringFunction)

Arguments

Value

ANOCVA deltaS and deltaSq statistics

References

Fujita A, Takahashi DY, Patriota AG, Sato JR (2014a) A non-parametric statistical test to compare clusters with applications in functional magnetic resonance imaging data. Statistics in Medicine 33: 4949–4962

Caetano de Jesus DA. (2017) Evaluation of ANOCVA test for cluster comparison through simulations. Master Dissertation. Institute of Mathematics and Statistics, University of São Paulo.

Verifies if a clustering function is set. If not, uses the spectral clustering as default clustering method.

Description

Verifies if a clustering function is set. If not, uses the spectral clustering as default clustering method.

Usage

checkClusteringFunction(clusteringFunction)

Arguments

Value

A clustering function

If the number of clusters has not been set, estimates it by using Slope criterion in the range 2, 20.

Description

If the number of clusters has not been set, estimates it by using Slope criterion in the range 2, 20.

Usage

checkNClust(dataDist, r = NULL, p = 1, maxClust = 20, clusteringFunction,
  criterion = c("slope", "silhouette"))

Arguments

Check for 0,1 Interval Normalization.

Description

Verifies if the data is normalized in the range 0,1. If they are not, the normalization is performed and a warning issued.

Usage

checkRange01(data)

Arguments

Value

The data matrix normalized in the range 0,1.

Examples

set.seed(2000)

simuData = runif(100, min = 0.5, max=7)
sprintf("The minimum value is %.2f and the maximum is %.2f.", min(simuData), max(simuData))

simuData = checkRange01(simuData)
sprintf("Now the minimum value is %.2f and the maximum is %.2f.", min(simuData), max(simuData))

Optimal Number of Clusters Estimation

Description

Estimates the optimal number of clusters using either Slope or Silhouette criterion. The optimal number of clusters will be verified in the range 2,..., maxClust.

Usage

nClust(meanDist, p = 1, maxClust = 20, clusteringFunction,
  criterion = c("slope", "silhouette"))

Arguments

Value

The optimal number of clusters.

References

Fujita A, Takahashi DY, Patriota AG (2014b) A non-parametric method to estimate the number of clusters. Computational Statistics & Data Analysis 73:27–39

Rousseeuw PJ (1987) Sihouettes: a graphical aid to the interpretation and validation of cluster analysis. Journal of Computational and Applied Mathematics 20:53–65

Examples

# Install packages if necessary
# install.packages('MASS')
# install.packages('cluster')

library(MASS)
library(cluster)
library(anocva)

set.seed(2000)

# Defines a k-means function that returns cluster labels directly
myKmeans = function(dist, k){
  return(kmeans(dist, k, iter.max = 50, nstart = 5)$cluster)
}

# Generate simulated data
nitem = 70
sigma = matrix(c(0.04, 0, 0, 0.04), 2)
simuData = rbind(mvrnorm(nitem, mu = c(0, 0), Sigma = sigma ),
             mvrnorm(nitem, mu = c(3,0), Sigma = sigma),
             mvrnorm(nitem, mu = c(2.5,2), Sigma = sigma))

plot(simuData, asp = 1, xlab = '', ylab = '', main = 'Data for clustering')

# Calculate distances and perform {0,1} normalization
distMatrix = as.matrix(dist(simuData))
distMatrix = checkRange01(distMatrix)

# Estimate the optimal number of clusters
r = nClust(meanDist = distMatrix, p = 1, maxClust = 10,
           clusteringFunction = myKmeans, criterion = "silhouette")
sprintf("The optimal number of clusters found was %d.", r)

# K-means Clustering
labels = myKmeans(distMatrix, r)

plot(simuData, col = labels, asp = 1, xlab = '', ylab = '', main = 'K-means clustered data')

Multiple Samples Optimal Number of Clusters Estimation

Description

Estimates the optimal number of clusters for multiple samples using either Slope or Silhouette criterion. The optimal number of clusters will be verified in the range 2,..., maxClust. Takes the mean of all samples in order to perform the estimation.

Usage

nClustMulti(dataDist, p = 1, maxClust = 20, clusteringFunction,
  criterion = c("slope", "silhouette"))

Arguments

Value

The optimal number of clusters.

References

Fujita A, Takahashi DY, Patriota AG (2014b) A non-parametric method to estimate the number of clusters. Computational Statistics & Data Analysis 73:27–39

Rousseeuw PJ (1987) Sihouettes: a graphical aid to the interpretation and validation of cluster analysis. Journal of Computational and Applied Mathematics 20:53–65

Examples

# Install packages if necessary
# install.packages('MASS')
# install.packages('cluster')

library(anocva)
library(MASS)
library(cluster)

set.seed(5000)

# A k-means function that returns cluster labels directly.
myKmeans = function(dist, k){
  return(kmeans(dist, k, iter.max = 50, nstart = 5)$cluster)
}

# Number of subjects in each population
nsub = 25
# Number of items in each subject
nitem = 60

# Generate simulated data
data = array(NA, c(nsub, nitem*2, 2))
data.dist = array(NA, c(nsub, nitem*2, nitem*2))
meanx = 2
delta = 0.5
# Covariance matrix
sigma = matrix(c(0.03, 0, 0, 0.03), 2)
for (i in seq(nsub)){
  sub = rbind(mvrnorm(nitem, mu = c(0, 0), Sigma = sigma ),
              mvrnorm(nitem, mu = c(meanx,0), Sigma = sigma))
  data[i,,] = sub
  data.dist[i,,] = as.matrix(dist(data[i,,]))
}

# Estimate the optimal number of clusters
r = nClustMulti(dataDist = data.dist, p = 1, maxClust = 20,
                clusteringFunction = myKmeans, criterion = "slope")
sprintf("The optimal number of clusters found was %d.", r)

Estimates the optimal number of clusters using the Silhouette criterion when silhouette statistics already exists.

Description

Estimates the optimal number of clusters using the Silhouette criterion when silhouette statistics already exists.

Usage

optimalSilhouette(silStats)

Arguments

Value

The optimal number of clusters.

References

Rousseeuw PJ (1987) Sihouettes: a graphical aid to the interpretation and validation of cluster analysis. Journal of Computational and Applied Mathematics 20:53–65

Estimates the optimal number of clusters using the Slope criterion when silhouette statistics already exists. The optimal number of clusters will be verified in the range 2,..., maxClust.

Description

Estimates the optimal number of clusters using the Slope criterion when silhouette statistics already exists. The optimal number of clusters will be verified in the range 2,..., maxClust.

Usage

optimalSlope(silStats, p = 1, maxClust)

Arguments

Value

The optimal number of clusters.

References

Fujita A, Takahashi DY, Patriota AG (2014b) A non-parametric method to estimate the number of clusters. Computational Statistics & Data Analysis 73:27–39

Spectral clustering

Description

Unnormalized spectral clustering function. Uses Partitioning Around Medoids clustering instead of K-means.

Usage

spectralClustering(W, k)

Arguments

Value

Cluster labels

References

Von Luxburg, U (2007) A tutorial on spectral clustering. Statistics and computing 17:395–416.

Ng A, Jordan M, Weiss Y (2002) On spectral clustering: analysis and an algorithm. In: Advances in Neural Information Processing Systems. Dietterich T, Becker S, Ghahramani Z (Eds.), vol. 14. MIT Press, (pp. 849–856).

Examples

# Install igraph if necessary
# install.packages('igraph')
# install.packages('cluster')

library(anocva)

set.seed(2000)

if (requireNamespace("igraph", quietly = TRUE)) {

  # Create a tree graph
  treeGraph = igraph::make_tree(80, children = 4, mode = "undirected")

  # Visualize the tree graph
  plot(treeGraph, vertex.size = 10, vertex.label = NA)

  # Get the adjacency matrix of the tree graph
  adj = as.matrix(igraph::get.adjacency(treeGraph))

  # Cluster the tree graph in to four clusters
  cluster = spectralClustering(adj, 4)

  # See the result clustering
  plot(treeGraph, vertex.size=10, vertex.color = cluster, vertex.label = NA)
}