gemseo / mlearning / clustering

# kmeans module¶

The k-means algorithm for clustering.

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

$\begin{split}\\operatorname{cluster}(x) = \\underset{i=1,\\cdots,k}{\\operatorname{argmin}} \\|x-c_i\\|.\end{split}$

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 BaseMLClusteringAlgo class.

## Dependence¶

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

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

The k-means clustering algorithm.

Parameters:
• data (Dataset) – The learning dataset.

• transformer (TransformerType) –

The strategies to transform the variables. The values are instances of BaseTransformer while the keys are the names of either the variables or the groups of variables, e.g. "inputs" or "outputs" in the case of the regression algorithms. If a group is specified, the BaseTransformer will be applied to all the variables of this group. If IDENTITY, do not transform the variables.

By default it is set to {}.

• var_names (Iterable[str] | None) – The names of the variables. If None, consider all variables mentioned in the learning dataset.

• n_clusters (int) –

The number of clusters of the K-means algorithm.

By default it is set to 5.

• random_state (int | None) –

The random state passed to the method generating the initial centroids Use an integer for reproducible results.

By default it is set to 0.

• **parameters (int | float | bool | str | None) – The parameters of the machine learning algorithm.

Raises:

ValueError – When both the variable and the group it belongs to have a transformer.

DataFormatters: ClassVar[type[BaseDataFormatters]]

The data formatters for the learning and prediction methods.

EPS = 2.220446049250313e-16
LIBRARY: ClassVar[str] = 'scikit-learn'

The name of the library of the wrapped machine learning algorithm.

SHORT_ALGO_NAME: ClassVar[str] = 'KMeans'

The short name of the machine learning algorithm, often an acronym.

Typically used for composite names, e.g. f"{algo.SHORT_ALGO_NAME}_{dataset.name}" or f"{algo.SHORT_ALGO_NAME}_{discipline.name}".

algo: Any

The interfaced machine learning algorithm.

input_names: list[str]

The names of the variables.

labels: list[int]

The indices of the clusters for the different samples.

learning_set: Dataset

The learning dataset.

n_clusters: int

The number of clusters.

parameters: dict[str, MLAlgoParameterType]

The parameters of the machine learning algorithm.

resampling_results: dict[str, tuple[BaseResampler, list[BaseMLAlgo], list[ndarray] | ndarray]]

The resampler class names bound to the resampling results.

A resampling result is formatted as (resampler, ml_algos, predictions) where resampler is a BaseResampler, ml_algos is the list of the associated machine learning algorithms built during the resampling stage and predictions are the predictions obtained with the latter.

resampling_results stores only one resampling result per resampler type (e.g., "CrossValidation", "LeaveOneOut" and "Boostrap").

transformer: dict[str, BaseTransformer]

The strategies to transform the variables, if any.

The values are instances of BaseTransformer while the keys are the names of either the variables or the groups of variables, e.g. “inputs” or “outputs” in the case of the regression algorithms. If a group is specified, the BaseTransformer will be applied to all the variables of this group.

API

API

K-means

K-means