# 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_cmeansFunction
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
Clustering.FuzzyCMeansResultType
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
Clustering.wcountsFunction
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)).

## 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}:
0.335455  0.331203  0.333343
0.33558   0.3301    0.33432
0.334436  0.329446  0.336118
0.330863  0.333282  0.335855
0.33164   0.335676  0.332684
0.333801  0.334683  0.331517
0.332706  0.332124  0.335171
0.332161  0.334503  0.333335
0.332186  0.335867  0.331948
0.333986  0.333386  0.332628
⋮
0.330116  0.336267  0.333617
0.334624  0.332446  0.332929
0.332683  0.331851  0.335467
0.332318  0.335346  0.332335
0.334245  0.331238  0.334517
0.333157  0.331976  0.334867
0.332984  0.332928  0.334088
0.330632  0.336557  0.332811
0.33306   0.334489  0.33245