Fuzzy C-means

Fuzzy C-means is a clustering method that provides cluster membership weights instead of "hard" classification (e.g. K-means).

From a mathematical standpoint, fuzzy C-means solves the following optimization problem:

\[\arg\min_\mathcal{C} \ \sum_{i=1}^n \sum_{j=1}^C w_{ij}^\mu \| \mathbf{x}_i - \mathbf{c}_j \|^2, \ \text{where}\ w_{ij} = \left(\sum_{k=1}^{C} \left(\frac{\left\|\mathbf{x}_i - \mathbf{c}_j \right\|}{\left\|\mathbf{x}_i - \mathbf{c}_k \right\|}\right)^{\frac{2}{\mu-1}}\right)^{-1}\]

Here, $\mathbf{c}_j$ is the center of the $j$-th cluster, $w_{ij}$ is the membership weight of the $i$-th point in the $j$-th cluster, and $\mu > 1$ is a user-defined fuzziness parameter.

Clustering.fuzzy_cmeans — Function

fuzzy_cmeans(data::AbstractMatrix, C::Integer, fuzziness::Real;
             [dist_metric::SemiMetric], [...]) -> FuzzyCMeansResult

Perform Fuzzy C-means clustering over the given data.

Arguments

data::AbstractMatrix: $d×n$ data matrix. Each column represents one $d$-dimensional data point.
C::Integer: the number of fuzzy clusters, $2 ≤ C < n$.
fuzziness::Real: clusters fuzziness ($μ$ in the mathematical formulation), $μ > 1$.

Optional keyword arguments:

dist_metric::SemiMetric (defaults to Euclidean): the SemiMetric object that defines the distance between the data points
maxiter, tol, display, rng: see common options

source

Clustering.FuzzyCMeansResult — Type

FuzzyCMeansResult{T<:AbstractFloat}

The output of fuzzy_cmeans function.

Fields

centers::Matrix{T}: the $d×C$ matrix with columns being the centers of resulting fuzzy clusters
weights::Matrix{Float64}: the $n×C$ matrix of assignment weights ($\mathrm{weights}_{ij}$ is the weight (probability) of assigning $i$-th point to the $j$-th cluster)
iterations::Int: the number of executed algorithm iterations
converged::Bool: whether the procedure converged

source

Clustering.wcounts — Function

wcounts(R::ClusteringResult) -> Vector{Float64}
wcounts(R::FuzzyCMeansResult) -> Vector{Float64}

Get the weighted cluster sizes as the sum of weights of points assigned to each cluster.

For non-weighted clusterings assumes the weight of every data point is 1.0, so the result is equivalent to convert(Vector{Float64}, counts(R)).

source

Examples

using Clustering

# make a random dataset with 1000 points
# each point is a 5-dimensional vector
X = rand(5, 1000)

# performs Fuzzy C-means over X, trying to group them into 3 clusters
# with a fuzziness factor of 2. Set maximum number of iterations to 200
# set display to :iter, so it shows progressive info at each iteration
R = fuzzy_cmeans(X, 3, 2, maxiter=200, display=:iter)

# get the centers (i.e. weighted mean vectors)
# M is a 5x3 matrix
# M[:, k] is the center of the k-th cluster
M = R.centers

# get the point memberships over all the clusters
# memberships is a 20x3 matrix
memberships = R.weights

1000×3 Matrix{Float64}:
 0.335587  0.331904  0.332509
 0.333195  0.334676  0.332129
 0.330883  0.337093  0.332024
 0.333009  0.333025  0.333965
 0.333156  0.334844  0.332
 0.333366  0.33151   0.335123
 0.334077  0.333166  0.332756
 0.331714  0.334724  0.333562
 0.330663  0.336738  0.332598
 0.33229   0.335949  0.331762
 ⋮                   
 0.334832  0.330876  0.334292
 0.333322  0.333547  0.333131
 0.331661  0.333502  0.334837
 0.331433  0.336429  0.332138
 0.33399   0.333018  0.332993
 0.331101  0.335586  0.333314
 0.334731  0.332668  0.3326
 0.334471  0.332054  0.333475
 0.332104  0.33377   0.334126