Silhouette

class gtda.diagrams.Silhouette(power=1.0, n_bins=100, n_jobs=None)

Power-weighted silhouettes of persistence diagrams.

Based on ideas in 1. Given a persistence diagram consisting of birth-death-dimension triples [b, d, q], subdiagrams corresponding to distinct homology dimensions are considered separately, and their respective silhouettes are obtained by sampling the silhouette function over evenly spaced locations from appropriate ranges of the filtration parameter.

Important note:

• Input collections of persistence diagrams for this transformer must satisfy certain requirements, see e.g. fit.

Parameters

• power (float, optional, default: 1.) – The power to which persistence values are raised to define the power-weighted silhouettes.

• n_bins (int, optional, default: 100) – The number of filtration parameter values, per available homology dimension, to sample during fit.

• n_jobs (int or None, optional, default: None) – The number of jobs to use for the computation. None means 1 unless in a joblib.parallel_backend context. -1 means using all processors.

homology_dimensions_

Homology dimensions seen in fit, sorted in ascending order.

Type

tuple

samplings_

For each number in homology_dimensions_, a discrete sampling of filtration parameters, calculated during fit according to the minimum birth and maximum death values observed across all samples.

Type

dict

See also

PersistenceLandscape, PersistenceEntropy, HeatKernel, Amplitude, PairwiseDistance, BettiCurve, gtda.homology.VietorisRipsPersistence

Notes

The samplings in samplings_ are in general different between different homology dimensions. This means that the j-th entry of a silhouette in homology dimension q typically arises from a different parameter values to the j-th entry of a curve in dimension q’.

References

1

F. Chazal, B. T. Fasy, F. Lecci, A. Rinaldo, and L. Wasserman, “Stochastic Convergence of Persistence Landscapes and Silhouettes”; In Proceedings of the thirtieth annual symposium on Computational Geometry, Kyoto, Japan, 2014, pp. 474–483; DOI: 10.1145/2582112.2582128.

__init__(power=1.0, n_bins=100, n_jobs=None)

Initialize self. See help(type(self)) for accurate signature.

fit(X, y=None)

Store all observed homology dimensions in homology_dimensions_ and, for each dimension separately, store evenly sample filtration parameter values in samplings_. Then, return the estimator.

This method is here to implement the usual scikit-learn API and hence work in pipelines.

Parameters

• X (ndarray of shape (n_samples, n_features, 3)) – Input data. Array of persistence diagrams, each a collection of triples [b, d, q] representing persistent topological features through their birth (b), death (d) and homology dimension (q). It is important that, for each possible homology dimension, the number of triples for which q equals that homology dimension is constants across the entries of X.

• y (None) – There is no need for a target in a transformer, yet the pipeline API requires this parameter.

Returns

self

Return type

object

fit_transform(X, y=None, **fit_params)

Fit to data, then transform it.

Fits transformer to X and y with optional parameters fit_params and returns a transformed version of X.

Parameters

• X (ndarray of shape (n_samples, n_features, 3)) – Input data. Array of persistence diagrams, each a collection of triples [b, d, q] representing persistent topological features through their birth (b), death (d) and homology dimension (q). It is important that, for each possible homology dimension, the number of triples for which q equals that homology dimension is constants across the entries of X.

• y (None) – There is no need for a target in a transformer, yet the pipeline API requires this parameter.

Returns

Xt – One silhouette (represented as a one-dimensional array) per sample and per homology dimension seen in fit. Index i along axis 1 corresponds to the i-th homology dimension in homology_dimensions_.

Return type

ndarray of shape (n_samples, n_homology_dimensions, n_bins)

fit_transform_plot(X, y=None, sample=0, **plot_params)

Fit to data, then apply transform_plot.

Parameters

• X (ndarray of shape (n_samples, ..)) – Input data.

• y (ndarray of shape (n_samples,) or None) – Target values for supervised problems.

• sample (int) – Sample to be plotted.

• **plot_params – Optional plotting parameters.

Returns

Xt – Transformed one-sample slice from the input.

Return type

ndarray of shape (1, ..)

get_params(deep=True)

Get parameters for this estimator.

Parameters

deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns

params – Parameter names mapped to their values.

Return type

mapping of string to any

plot(Xt, sample=0, homology_dimensions=None, plotly_params=None)

Plot a sample from a collection of silhouettes arranged as in the output of transform. Include homology in multiple dimensions.

Parameters

• Xt (ndarray of shape (n_samples, n_homology_dimensions, n_bins)) – Collection of silhouettes, such as returned by transform.

• sample (int, optional, default: 0) – Index of the sample in Xt to be plotted.

• homology_dimensions (list, tuple or None, optional, default: None) – Which homology dimensions to include in the plot. None means plotting all dimensions present in homology_dimensions_.

• plotly_params (dict or None, optional, default: None) – Custom parameters to configure the plotly figure. Allowed keys are "traces" and "layout", and the corresponding values should be dictionaries containing keyword arguments as would be fed to the update_traces and update_layout methods of plotly.graph_objects.Figure.

Returns

fig – Plotly figure.

Return type

plotly.graph_objects.Figure object

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters

**params (dict) – Estimator parameters.

Returns

self – Estimator instance.

Return type

object

transform(X, y=None)

Compute silhouettes of diagrams in X.

Parameters

• X (ndarray of shape (n_samples, n_features, 3)) – Input data. Array of persistence diagrams, each a collection of triples [b, d, q] representing persistent topological features through their birth (b), death (d) and homology dimension (q). It is important that, for each possible homology dimension, the number of triples for which q equals that homology dimension is constants across the entries of X.

• y (None) – There is no need for a target in a transformer, yet the pipeline API requires this parameter.

Returns

Xt – One silhouette (represented as a one-dimensional array) per sample and per homology dimension seen in fit. Index i along axis 1 corresponds to the i-th homology dimension in homology_dimensions_.

Return type

ndarray of shape (n_samples, n_homology_dimensions, n_bins)

transform_plot(X, sample=0, **plot_params)

Take a one-sample slice from the input collection and transform it. Before returning the transformed object, plot the transformed sample.

Parameters

• X (ndarray of shape (n_samples, ..)) – Input data.

• sample (int) – Sample to be plotted.

• **plot_params – Optional plotting parameters.

Returns

Xt – Transformed one-sample slice from the input.

Return type

ndarray of shape (1, ..)