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:

$\operatorname{cluster}(x) = \underset{i=1,\cdots,k}{\operatorname{argmax}} \ \mathcal{N}(x; \mu_i, \Sigma_i)$

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

$\mathbb{P}(x \in C_i) = \frac{\mathcal{N}(x; \mu_i, \Sigma_i)} {\sum_{j=1}^k \mathcal{N}(x; \mu_j, \Sigma_j)},$

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 expectation-maximization 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 scikit-learn library.

Classes:

 GaussianMixture(data[, transformer, ...]) 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

Dataset

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, the Transformer 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, 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 (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:

 is_trained Return whether the algorithm is trained. learning_samples_indices 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 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 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 (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 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 (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