The Gaussian mixture algorithm for clustering.
The Gaussian mixture algorithm groups the data into clusters. The number of clusters is fixed. Each cluster \(i=1, \cdots, k\) is defined by a mean \(\mu_i\) and a covariance matrix \(\Sigma_i\).
The prediction of the cluster value of a point is simply the cluster where the probability density of the Gaussian distribution defined by the given mean and covariance matrix is the highest:
where \(\mathcal{N}(x; \mu_i, \Sigma_i)\) is the value of the probability density function of a Gaussian random variable \(X \sim \mathcal{N}(\mu_i, \Sigma_i)\) at the point \(x\) and \(\x\mu_i\_{\Sigma_i^{1}} = \sqrt{(x\mu_i)^T \Sigma_i^{1} (x\mu_i)}\) is the Mahalanobis distance between \(x\) and \(\mu_i\) weighted by \(\Sigma_i\). Likewise, the probability of belonging to a cluster \(i=1, \cdots, k\) may be determined through
where \(C_i = \{x\,  \, \operatorname{cluster}(x) = i \}\).
When fitting the algorithm, the cluster centers \(\mu_i\) and the covariance matrices \(\Sigma_i\) are computed using the expectationmaximization algorithm.
This concept is implemented through the GaussianMixture
class
which inherits from the MLClusteringAlgo
class.
Dependence¶
This clustering algorithm relies on the GaussianMixture class of the scikitlearn library.
Classes:

The Gaussian mixture clustering algorithm. 
 class gemseo.mlearning.cluster.gaussian_mixture.GaussianMixture(data, transformer=None, var_names=None, n_components=5, **parameters)[source]
The Gaussian mixture clustering algorithm.
 learning_set
The learning dataset.
 Type
 parameters
The parameters of the machine learning algorithm.
 Type
Dict[str,MLAlgoParameterType]
 transformer
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, theTransformer
will be applied to all the variables of this group. If None, do not transform the variables. Type
Dict[str,Transformer]
 algo
The interfaced machine learning algorithm.
 Type
Any
 input_names
The names of the variables.
 Type
List[str]
 labels
The indices of the clusters for the different samples.
 Type
List(int)
 n_clusters
The number of clusters.
 Type
int
Initialize self. See help(type(self)) for accurate signature.
 Parameters
data (Dataset) – The learning dataset.
transformer (Optional[TransformerType]) –
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, theTransformer
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 (Optional[Iterable[str]]) –
The names of the variables. If None, consider all variables mentioned in the learning dataset.
By default it is set to None.
n_components (int) –
The number of components of the Gaussian mixture.
By default it is set to 5.
**parameters (Optional[Union[int,float,str,bool]]) – The parameters of the machine learning algorithm.
 Return type
None
Classes:
Decorators for the internal MLAlgo methods.
Attributes:
Return whether the algorithm is trained.
The indices of the learning samples used for the training.
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[Sequence[int]]) –
The indices of the learning samples. If None, use the whole learning dataset.
By default it is set to None.
 Return type
None
 property learning_samples_indices
The indices of the learning samples used for the training.
 load_algo(directory)
Load a machine learning algorithm from a directory.
 Parameters
directory (Union[str, pathlib.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 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, Mapping[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, 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 (Optional[str]) –
The name of the directory to save the algorithm.
By default it is set to None.
path (Union[str, pathlib.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