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}:
 0.332687  0.332566  0.334747
 0.333311  0.333275  0.333414
 0.333581  0.333197  0.333223
 0.331212  0.334853  0.333935
 0.332916  0.33288   0.334204
 0.334561  0.332931  0.332508
 0.33173   0.336071  0.332199
 0.335626  0.332506  0.331868
 0.331009  0.334966  0.334025
 0.332777  0.334055  0.333169
 ⋮                           
 0.334764  0.33347   0.331766
 0.331685  0.334359  0.333956
 0.331176  0.333994  0.33483 
 0.333982  0.333628  0.33239 
 0.333607  0.331976  0.334417
 0.33087   0.335357  0.333774
 0.33393   0.333991  0.332079
 0.333438  0.333452  0.33311 
 0.333656  0.333123  0.333221