The kmeans algorithm for clustering.
The kmeans 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
A probability measure may also be provided, using the distances from the point to each of the centroids:
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 scikitlearn library.
Classes:

The kmeans clustering algorithm. 
 class gemseo.mlearning.cluster.kmeans.KMeans(data, transformer=None, var_names=None, n_clusters=5, random_state=0, **parameters)[source]
The kmeans clustering algorithm.
 Parameters
n_clusters (int) – The number of clusters of the Kmeans algorithm.
random_state (Optional[int]) – If None, use a random generation of the initial centroids. If not None, the integer is used to make the initialization deterministic.
data (Dataset) –
transformer (Optional[TransformerType]) –
var_names (Optional[Iterable[str]]) –
parameters (Optional[Union[int,float,bool,str]]) –
 Return type
None
Classes:
Decorators for the internal MLAlgo methods.
Attributes:
Return whether the algorithm is trained.
Methods:
learn
([samples])Train the machine learning algorithm from the learning dataset.
load_algo
(directory)Load a machine learning algorithm from a directory.
predict
(data)Predict the clusters from the input data.
predict_proba
(data[, hard])Predict the probability of belonging to each cluster from input data.
save
([directory, path, save_learning_set])Save the machine learning algorithm.
 class DataFormatters
Decorators for the internal MLAlgo methods.
 property is_trained
Return whether the algorithm is trained.
 learn(samples=None)
Train the machine learning algorithm from the learning dataset.
 Parameters
samples (Optional[List[int]]) – The indices of the learning samples. If None, use the whole learning dataset.
 Return type
None
 load_algo(directory)
Load a machine learning algorithm from a directory.
 Parameters
directory (str) – 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 ith rows represent the input data of the ith 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 (Union[numpy.ndarray, Dict[str, numpy.ndarray]]) – The input data.
 Returns
The predicted cluster for each input data sample.
 Return type
Union[int, numpy.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 ith rows represent the input data of the ith 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, Dict[str, numpy.ndarray]]) – The input data.
hard (bool) – Whether clustering should be hard (True) or soft (False).
 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 (Optional[str]) – The name of the directory to save the algorithm.
path (str) – The path to parent directory where to create the directory.
save_learning_set (bool) – If False, do not save the learning set to lighten the saved files.
 Returns
The path to the directory where the algorithm is saved.
 Return type
str