silhouette module¶

The silhouette coefficient to measure the quality of a clustering algorithm.

The silhouette module implements the concept of silhouette coefficient measure for machine learning algorithms.

This concept is implemented through the SilhouetteMeasure class and overloads the MLClusteringMeasure._compute_measure() method.

The silhouette coefficient is defined for each point as the difference between the average distance from the point to each of the other points in its cluster and the average distance from the point to each of the points in the nearest cluster different from its own.

More formally, the silhouette coefficient \(s_i\) of a point \(x_i\) is given by

\[\begin{split}a_i = \\frac{1}{|C_{k_i}| - 1} \\sum_{j\\in C_{k_i}\setminus\{i\} } \\|x_i-x_j\\|\\\\ b_i = \\underset{\\ell=1,\\cdots,K\\atop{\\ell\\neq k_i}}{\\min}\\ \\frac{1}{|C_\\ell|} \\sum_{j\\in C_\\ell} \\|x_i-x_j\\|\\\\ s_i = \\frac{b_i-a_i}{\\max(b_i,a_i)}\end{split}\]

where \(k_i\) is the index of the cluster to which \(x_i\) belongs, \(K\) is the number of clusters, \(C_k\) is the set of indices of points belonging to the cluster \(k\) (\(k=1,\\cdots,K\)), and \(|C_k| = \\sum_{j\\in C_k} 1\) is the number of points in the cluster \(k\), \(k=1,\\cdots,K\).

Classes:

SilhouetteMeasure(algo)

The silhouette coefficient measure for machine learning.

class gemseo.mlearning.qual_measure.silhouette.SilhouetteMeasure(algo)[source]¶

Bases: gemseo.mlearning.qual_measure.cluster_measure.MLPredictiveClusteringMeasure

The silhouette coefficient measure for machine learning.

Attributes

algo (MLAlgo) – The machine learning algorithm.
algo (MLAlgo) – The machine learning algorithm.
algo (MLAlgo) – The machine learning algorithm.
algo (MLAlgo) – The machine learning algorithm.

Parameters

algo (MLPredictiveClusteringAlgo) – A machine learning algorithm for clustering.

Return type

None

Attributes:

`BOOTSTRAP`
`KFOLDS`
`LEARN`
`LOO`
`SMALLER_IS_BETTER`
`TEST`

Methods:

`evaluate`([method, samples])	Evaluate the quality measure.
`evaluate_bootstrap`([n_replicates, samples, …])	Evaluate the quality measure using the bootstrap technique.
`evaluate_kfolds`([n_folds, samples, multioutput])	Evaluate the quality measure using the k-folds technique.
`evaluate_learn`([samples, multioutput])	Evaluate the quality measure using the learning dataset.
`evaluate_loo`([samples, multioutput])	Evaluate the quality measure using the leave-one-out technique.
`evaluate_test`(test_data[, samples, multioutput])	Evaluate the quality measure using a test dataset.
`is_better`(val1, val2)	Compare the quality between two values.

BOOTSTRAP = 'bootstrap'¶

KFOLDS = 'kfolds'¶

LEARN = 'learn'¶

LOO = 'loo'¶

SMALLER_IS_BETTER = False¶

TEST = 'test'¶

evaluate(method='learn', samples=None, **options)¶

Evaluate the quality measure.

Parameters

method (str) – The name of the method to evaluate the quality measure.
samples (Optional[List[int]]) – The indices of the learning samples. If None, use the whole learning dataset.
**options – The options of the estimation method (e.g. ‘test_data’ for
'test' method (the) –
for the bootstrap one ('n_replicates') –
...) –
options (Optional[Union[List[int], bool, int, gemseo.core.dataset.Dataset]]) –

Returns

The value of the quality measure.

Raises

ValueError – If the name of the method is unknown.

Return type

Union[float, numpy.ndarray]

evaluate_bootstrap(n_replicates=100, samples=None, multioutput=True)[source]¶

Evaluate the quality measure using the bootstrap technique.

Parameters

n_replicates (int) – The number of bootstrap replicates.
samples (Optional[List[int]]) – The indices of the learning samples. If None, use the whole learning dataset.
multioutput (bool) – If True, return the quality measure for each output component. Otherwise, average these measures.

Returns

The value of the quality measure.

Return type

Union[float, numpy.ndarray]

evaluate_kfolds(n_folds=5, samples=None, multioutput=True)[source]¶

Evaluate the quality measure using the k-folds technique.

Parameters

n_folds (int) – The number of folds.
samples (Optional[List[int]]) – The indices of the learning samples. If None, use the whole learning dataset.
multioutput (bool) – If True, return the quality measure for each output component. Otherwise, average these measures.

Returns

The value of the quality measure.

Return type

Union[float, numpy.ndarray]

evaluate_learn(samples=None, multioutput=True)¶

Evaluate the quality measure using the learning dataset.

Parameters

samples (Optional[List[int]]) – The indices of the learning samples. If None, use the whole learning dataset.
multioutput (bool) – If True, return the quality measure for each output component. Otherwise, average these measures.

Returns

The value of the quality measure.

Return type

Union[float, numpy.ndarray]

evaluate_loo(samples=None, multioutput=True)¶

Evaluate the quality measure using the leave-one-out technique.

Parameters

samples (Optional[List[int]]) – The indices of the learning samples. If None, use the whole learning dataset.
multioutput (bool) – If True, return the quality measure for each output component. Otherwise, average these measures.

Returns

The value of the quality measure.

Return type

Union[float, numpy.ndarray]

evaluate_test(test_data, samples=None, multioutput=True)[source]¶

Evaluate the quality measure using a test dataset.

Parameters

dataset – The test dataset.
samples (Optional[List[int]]) – The indices of the learning samples. If None, use the whole learning dataset.
multioutput (bool) – If True, return the quality measure for each output component. Otherwise, average these measures.
test_data (gemseo.core.dataset.Dataset) –

Returns

The value of the quality measure.

Return type

Union[float, numpy.ndarray]

classmethod is_better(val1, val2)¶

Compare the quality between two values.

This methods returns True if the first one is better than the second one.

For most measures, a smaller value is “better” than a larger one (MSE etc.). But for some, like an R2-measure, higher values are better than smaller ones. This comparison method correctly handles this, regardless of the type of measure.

Parameters

val1 (float) – The value of the first quality measure.
val2 (float) – The value of the second quality measure.

Returns

Whether val1 is of better quality than val2.

Return type

bool