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

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

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]

The k-means clustering algorithm.

Parameters
• data (Dataset) – The learning dataset.

• transformer (Mapping[str, TransformerType] | None) –

The strategies to transform the variables. The values are instances of Transformer 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 Transformer will be applied to all the variables of this group. If None, do not transform the variables.

By default it is set to None.

• var_names (Iterable[str] | None) –

The names of the variables. If None, consider all variables mentioned in the learning dataset.

By default it is set to None.

• n_clusters (int) –

The number of clusters of the K-means algorithm.

By default it is set to 5.

• random_state (int | None) –

If None, use a random generation of the initial centroids. If not None, the integer is used to make the initialization deterministic.

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.

class DataFormatters

Decorators for the internal MLAlgo methods.

Noindex

learn(samples=None, fit_transformers=True)

Train the machine learning algorithm from the learning dataset.

Parameters
• samples (Sequence[int] | None) –

The indices of the learning samples. If None, use the whole learning dataset.

By default it is set to None.

• fit_transformers (bool) –

Whether to fit the variable transformers.

By default it is set to True.

Return type

None

Load a machine learning algorithm from a directory.

Parameters

directory (str | Path) – The path to the directory where the machine learning algorithm is saved.

Return type

None

predict(data)

Predict the clusters from the input data.

The user can specify these input data either as a NumPy array, e.g. array([1., 2., 3.]) or as a dictionary, e.g. {'a': array([1.]), 'b': array([2., 3.])}.

If the numpy arrays are of dimension 2, their i-th rows represent the input data of the i-th sample; while if the numpy arrays are of dimension 1, there is a single sample.

The type of the output data and the dimension of the output arrays will be consistent with the type of the input data and the dimension of the input arrays.

Parameters

data (DataType) – The input data.

Returns

The predicted cluster for each input data sample.

Return type

int | ndarray

predict_proba(data, hard=True)

Predict the probability of belonging to each cluster from input data.

The user can specified these input data either as a numpy array, e.g. array([1., 2., 3.]) or as a dictionary, e.g. {'a': array([1.]), 'b': array([2., 3.])}.

If the numpy arrays are of dimension 2, their i-th rows represent the input data of the i-th sample; while if the numpy arrays are of dimension 1, there is a single sample.

The dimension of the output array will be consistent with the dimension of the input arrays.

Parameters
• data (Union[numpy.ndarray, Mapping[str, numpy.ndarray]]) – The input data.

• hard (bool) –

Whether clustering should be hard (True) or soft (False).

By default it is set to True.

Returns

The probability of belonging to each cluster, with shape (n_samples, n_clusters) or (n_clusters,).

Return type

numpy.ndarray

save(directory=None, path='.', save_learning_set=False)

Save the machine learning algorithm.

Parameters
• directory (str | None) –

The name of the directory to save the algorithm.

By default it is set to None.

• path (str | Path) –

The path to parent directory where to create the directory.

By default it is set to ..

• save_learning_set (bool) –

Whether to save the learning set or get rid of it to lighten the saved files.

By default it is set to False.

Returns

The path to the directory where the algorithm is saved.

Return type

str

property is_trained: bool

Return whether the algorithm is trained.

property learning_samples_indices: Sequence[int]

The indices of the learning samples used for the training.
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