Fuzzy C-means

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:

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

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 $m > 1$ is a user-defined fuzziness parameter.

Clustering.fuzzy_cmeans — Function.
fuzzy_cmeans(data::AbstractMatrix, C::Int, fuzziness::Real,
             [...]) -> 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::Int: the number of fuzzy clusters, $2 ≤ C < n$.
  • fuzziness::Real: clusters fuzziness (see $m$ in the mathematical formulation), $\mathrm{fuzziness} > 1$.

Optional keyword arguments:

  • dist_metric::Metric (defaults to Euclidean): the Metric object that defines the distance between the data points
  • maxiter, tol, display: 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
Missing docstring.

Missing docstring for wcounts(::FuzzyCMeansResult). Check Documenter's build log for details.

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 Array{Float64,2}:
 0.332426  0.335326  0.332249
 0.338547  0.330885  0.330569
 0.336168  0.332667  0.331166
 0.335341  0.333433  0.331227
 0.334174  0.332594  0.333232
 0.331969  0.33368   0.334351
 0.33327   0.333149  0.333581
 0.333658  0.335752  0.33059
 0.332174  0.335772  0.332054
 0.334529  0.334403  0.331068
 ⋮                   
 0.336722  0.330313  0.332965
 0.33687   0.332734  0.330396
 0.329968  0.333983  0.336049
 0.331693  0.333535  0.334772
 0.335914  0.333517  0.330568
 0.330851  0.335326  0.333822
 0.334354  0.333439  0.332206
 0.33422   0.333913  0.331868
 0.332509  0.334279  0.333212