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fuzzyCMeans

Executes one iteration of the fuzzy C-means clustering algorithm. Unlike the standard C-means algorithm, this uses fuzzy membership values that allow points to belong to multiple clusters with varying degrees of membership. 
export const fuzzyCMeans = (points: Point[], centroids: Point[])

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Parameters

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points
Point[]
required
Array of points to cluster. Each point has x and y coordinates.
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centroids
Point[]
required
Array of initial centroids. Each centroid has x and y coordinates.
The fuzzification parameter is hardcoded to m=2 in this implementation. This parameter controls the degree of fuzziness in the clustering.

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Returns

Returns an object containing the results of one fuzzy C-means iteration:
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distanceMatrix
number[][]
Matrix of distances where each element [i][j] is the distance between the i-th centroid and the j-th point.
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membershipMatrix
number[][]
Fuzzy membership matrix where each element [i][j] represents the degree of membership (0 to 1) of the j-th point to the i-th cluster.
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newCentroids
Point[]
New array of centroids calculated using fuzzy weighted means.
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costValues
number[]
Array of fuzzy cost values for each centroid.
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costFunction
number
Total cost as the sum of all individual fuzzy cost values.

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Example

import { fuzzyCMeans } from './utils/FuzzyCmeans';

const points = [
  { x: 1, y: 2 },
  { x: 3, y: 4 },
  { x: 5, y: 6 },
  { x: 7, y: 8 }
];

const centroids = [
  { x: 2, y: 3 },
  { x: 6, y: 7 }
];

const result = fuzzyCMeans(points, centroids);

console.log(result.membershipMatrix);
// Output: [[0.85, 0.72, 0.23, 0.15], [0.15, 0.28, 0.77, 0.85]]
console.log(result.newCentroids);
console.log(result.costFunction);

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Fuzzification Parameter

The fuzzy C-means algorithm uses a fuzzification parameter m that controls the degree of fuzziness in the clustering:
  • m = 1: Hard clustering (equivalent to standard C-means)
  • m = 2: Default fuzzy clustering (used in this implementation)
  • m > 2: Increasingly fuzzy clustering
The parameter affects how membership values are distributed across clusters. With m=2, points can have meaningful membership in multiple clusters, allowing for soft boundaries between clusters.

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Helper Functions

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getFuzzyMembershipMatrix

Calculates the fuzzy membership matrix given a distance matrix and a fuzzification parameter. 
export const getFuzzyMembershipMatrix = (
  distanceMatrix: number[][],
  fuzzyParameter: number
): number[][]
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distanceMatrix
number[][]
required
Matrix of distances between points and centroids.
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fuzzyParameter
number
required
Fuzzification parameter (typically set to 2).
Returns: A fuzzy membership matrix where each element [i][j] represents the degree of membership of the j-th point to the i-th cluster.

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getNewFuzzyCentroids

Calculates new centroids based on the fuzzy membership matrix. 
export const getNewFuzzyCentroids = (
  points: Point[],
  membershipMatrix: number[][],
  fuzzyParameter: number
): Point[]
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points
Point[]
required
Array of points.
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membershipMatrix
number[][]
required
Fuzzy membership matrix.
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fuzzyParameter
number
required
Fuzzification parameter (typically set to 2).
Returns: A new array of centroids calculated using fuzzy weighted means.

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getFuzzyCostValues

Calculates the cost values for each centroid based on the fuzzy membership and distance matrices. 
export const getFuzzyCostValues = (
  distanceMatrix: number[][],
  membershipMatrix: number[][],
  fuzzyParameter: number
): number[]
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distanceMatrix
number[][]
required
Matrix of distances between points and centroids.
​
membershipMatrix
number[][]
required
Fuzzy membership matrix.
​
fuzzyParameter
number
required
Fuzzification parameter (typically set to 2).
Returns: An array of cost values for each centroid, calculated using fuzzy membership weights.

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Comparison with Standard C-Means

The key difference between fuzzy C-means and standard C-means is that fuzzy C-means allows points to have partial membership in multiple clusters, while standard C-means assigns each point to exactly one cluster.
Standard C-Means:
  • Hard assignment: Each point belongs to exactly one cluster
  • Membership values: Binary (0 or 1)
  • Best for: Well-separated clusters
Fuzzy C-Means:
  • Soft assignment: Points can belong to multiple clusters
  • Membership values: Continuous (0 to 1)
  • Best for: Overlapping or ambiguous cluster boundaries

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