Linear Manifold Clustering¶

Linear manifold clustering algorithm (LMCLUS) discovers clusters which are described by a following model:

$x = \mu^{N \times 1} + B^{N \times K} \phi^{K \times 1} + \bar{B}^{N \times N-K} \epsilon^{N-K \times 1}$

where $N$ is a dimension of the dataset, $K$ is dimension of the manifold, $\mu \in \mathbb{R}^N$ is a linear manifold translation vector, $B$ is a matrix whose columns are orthonormal vectors that span $\mathbb{R}^K$ , $\bar{B}$ is a matrix whose columns span subspace orthogonal to spanned by columns of $B$ , $\phi$ is a zero-mean random vector whose entries are i.i.d. from a support of linear manifold, $\epsilon$ is a zero-mean random vector with small variance independent of $\phi$ .

Clustering¶

This package implements the LMCLUS algorithm in the lmclus function:

lmclus(X, p)¶

Performs linear manifold clustering over the given dataset.

Parameters:	X – The given sample matrix. Each column of `X` is a sample. p – The clustering parameters as instance of LMCLUSParameters.

This function returns an LMCLUSResult instance.

Results¶

Let M be an instance of Manifold, n be the number of observations, and d be the dimension of the linear manifold cluster.

indim(M)¶: Returns a dimension of the observation space.

outdim(M)¶: Returns a dimension of the linear manifold cluster which is the dimension of the subspace.

size(M)¶: Returns the number of points in the cluster which is the size of the cluster.

points(M)¶: Returns indexes of points assigned to the cluster.

mean(M)¶: Returns the translation vector $\mu$ which contains coordinates of the linear manifold origin.

projection(M)¶: Returns the basis matrix with columns corresponding to orthonormal vectors that span the linear manifold.”

separation(M)¶: Returns the instance of Separation object.

Example¶

using LMCLUS

# Load test data, remove label column and flip
X = readdlm(Pkg.dir("LMCLUS", "test", "testData"), ',')[:,1:end-1]'

# Initialize clustering parameters with
# maximum dimensionality for clusters.
# I should be less then original space dimension.
params = LMCLUSParameters(5)

# perform clustering and returns a collection of clusters
clust = lmclus(X, params)

# pick the first cluster
M = manifold(clust, 1)

# obtain indexes of points assigned to the cluster
l = points(M)

# obtain the linear manifold cluster translation vector
mu = mean(M)

# get basis vectors that span manifold as columns of the returned matrix
B = projection(M)

# get separation properties
S = separation(M)