K-means clustering algorithm

The k-means algorithm groups the data into clusters, where the number of clusters \(k\) is fixed. This is done by initializing \(k\) centroids in the design space. The points are grouped into clusters according to their nearest centroid.

When fitting the algorithm, each centroid is successively moved to the mean of its corresponding cluster, and the cluster value of each point is then reset to the cluster value of the closest centroid. This process is repeated until convergence.

Cluster values of new points may be predicted by returning the value of the closest centroid. Denoting \((c_1, \cdots, c_k) \in \mathbb{R}^{n \times k}\) the centroids, and assuming no overlap between the centroids, we may compute the prediction

\[\operatorname{cluster}(x) = \underset{i=1,\cdots,k}{\operatorname{argmin}} \|x-c_i\|.\]

A probability measure may also be provided, using the distances from the point to each of the centroids:

\[\begin{split}\mathbb{P}(x \in C_i) = \begin{cases} 1 & \operatorname{if} x = c_i\\ 0 & \operatorname{if} x = c_j,\ j \neq i\\ \frac{\frac{1}{\|x-c_i\|}}{\sum_{j=1}^k \frac{1}{\|x-c_j\|}} & \operatorname{if} x \neq c_j\, \forall j=1,\cdots,k \end{cases},\end{split}\]

where \(C_i = \{x\, | \, \operatorname{cluster}(x) = i \}\). Here, \(\mathbb{P}(x \in C_i)\) represents the probability of cluster \(i\) given the point \(x\).

This concept is implemented through the KMeans class which inherits from the MLClusteringAlgo class.

Dependence

This clustering algorithm relies on the KMeans class of the scikit-learn library.

class gemseo.mlearning.cluster.kmeans.KMeans(data, transformer=None, var_names=None, n_clusters=5, random_state=0, **parameters)[source]

KMeans clustering algorithm.

Constructor.

Parameters

  • data (Dataset) – learning dataset.

  • transformer (dict(str)) – transformation strategy for data groups. If None, do not transform data. Default: None.

  • var_names (list(str)) – names of the variables to consider.

  • n_clusters (int) – number of clusters. Default: 5.

  • random_state (int) – If None, use a random generation of the initial centroids. Use an int to make the randomness deterministic. Default: 0.

  • parameters – Scikit-learn algorithm parameters.

Example