PairwiseDistance¶

class
gtda.diagrams.
PairwiseDistance
(metric='landscape', metric_params=None, order=2.0, n_jobs=None)[source]¶ Distances between pairs of persistence diagrams.
Given two collections of persistence diagrams consisting of birthdeathdimension triples [b, d, q], a collection of distance matrices or a single distance matrix between pairs of diagrams is calculated according to the following steps:
All diagrams are partitioned into subdiagrams corresponding to distinct homology dimensions.
Pairwise distances between subdiagrams of equal homology dimension are calculated according to the parameters metric and metric_params. This gives a collection of distance matrices, \(\mathbf{D} = (D_{q_1}, \ldots, D_{q_n})\).
The final result is either \(\mathbf{D}\) itself as a threedimensional array, or a single distance matrix constructed by taking norms of the vectors of distances between diagram pairs.
Important notes:
 Parameters
metric (
'bottleneck'
'wasserstein'
'betti'
'landscape'
'silhouette'
'heat'
'persistence_image'
, optional, default:'landscape'
) –Distance or dissimilarity function between subdiagrams:
'bottleneck'
and'wasserstein'
refer to the identically named perfectmatching–based notions of distance.'betti'
refers to the \(L^p\) distance between Betti curves.'landscape'
refers to the \(L^p\) distance between persistence landscapes.'silhouette'
refers to the \(L^p\) distance between silhouettes.'heat'
refers to the \(L^p\) distance between Gaussiansmoothed diagrams.'persistence_image'
refers to the \(L^p\) distance between Gaussiansmoothed diagrams represented on birthpersistence axes.
metric_params (dict or None, optional, default:
None
) –Additional keyword arguments for the metric function (passing
None
is equivalent to passing the defaults described below):If
metric == 'bottleneck'
the only argument is delta (float, default:0.01
). When equal to0.
, an exact algorithm is used; otherwise, a faster approximate algorithm is used and symmetry is not guaranteed.If
metric == 'wasserstein'
the available arguments are p (float, default:2.
) and delta (float, default:0.01
). Unlike the case of'bottleneck'
, delta cannot be set to0.
and an exact algorithm is not available.If
metric == 'betti'
the available arguments are p (float, default:2.
) and n_bins (int, default:100
).If
metric == 'landscape'
the available arguments are p (float, default:2.
), n_bins (int, default:100
) and n_layers (int, default:1
).If
metric == 'silhouette'
the available arguments are p (float, default:2.
), power (float, default:1.
) and n_bins (int, default:100
).If
metric == 'heat'
the available arguments are p (float, default:2.
), sigma (float, default:0.1
) and n_bins (int, default:100
).If
metric == 'persistence_image'
the available arguments are p (float, default:2.
), sigma (float, default:0.1
), n_bins (int, default:100
) and weight_function (callable or None, default:None
).
order (float or None, optional, default:
2.
) – IfNone
,transform
returns for each pair of diagrams a vector of distances corresponding to the dimensions inhomology_dimensions_
. Otherwise, the \(p\)norm of these vectors with \(p\) equal to order is taken.n_jobs (int or None, optional, default:
None
) – The number of jobs to use for the computation.None
means 1 unless in ajoblib.parallel_backend
context.1
means using all processors.

effective_metric_params_
¶ Dictionary containing all information present in metric_params as well as relevant quantities computed in
fit
. Type
dict
See also
Amplitude
,Scaler
,Filtering
,BettiCurve
,PersistenceLandscape
,PersistenceImage
,HeatKernel
,Silhouette
,gtda.homology.VietorisRipsPersistence
Notes
To compute distances without first splitting the computation between different homology dimensions, data should be first transformed by an instance of
ForgetDimension
.Hera is used as a C++ backend for computing bottleneck and Wasserstein distances between persistence diagrams. Python bindings were modified for performance from the Dyonisus 2 package.

__init__
(metric='landscape', metric_params=None, order=2.0, n_jobs=None)[source]¶ Initialize self. See help(type(self)) for accurate signature.

fit
(X, y=None)[source]¶ Store all observed homology dimensions in
homology_dimensions_
and computeeffective_metric_params
. Then, return the estimator.This method is here to implement the usual scikitlearn API and hence work in pipelines.
 Parameters
X (ndarray of shape (n_samples_fit, 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_fit, 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 – Distance matrix or collection of distance matrices between diagrams in X and diagrams seen in
fit
. In the second case, index i along axis 2 corresponds to the ith homology dimension inhomology_dimensions_
. Return type
ndarray of shape (n_samples, n_samples_fit, n_homology_dimensions) if order is
None
, else (n_samples, n_samples_fit)

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

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)[source]¶ Computes a distance or vector of distances between the diagrams in X and the diagrams seen in
fit
. 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 – Distance matrix or collection of distance matrices between diagrams in X and diagrams seen in
fit
. In the second case, index i along axis 2 corresponds to the ith homology dimension inhomology_dimensions_
. Return type
ndarray of shape (n_samples, n_samples_fit, n_homology_dimensions) if order is
None
, else (n_samples, n_samples_fit)