# Source code for gtda.diagrams.features

"""Feature extraction from persistence diagrams."""

from numbers import Real

import numpy as np
from joblib import Parallel, delayed, effective_n_jobs
from scipy.stats import entropy
from sklearn.base import BaseEstimator, TransformerMixin
from sklearn.utils import gen_even_slices
from sklearn.utils.validation import check_is_fitted

from ._metrics import _AVAILABLE_AMPLITUDE_METRICS, _parallel_amplitude
from ._features import _AVAILABLE_POLYNOMIALS, _implemented_polynomial_recipes
from ._utils import _subdiagrams, _bin, _homology_dimensions_to_sorted_ints
from ..utils.intervals import Interval
from ..utils.validation import validate_params, check_diagrams

[docs]@adapt_fit_transform_docs class PersistenceEntropy(BaseEstimator, TransformerMixin): """:ref:Persistence entropies <persistence_entropy> of persistence diagrams. 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 persistence entropies are calculated as the (base 2) Shannon entropies of the collections of differences d - b ("lifetimes"), normalized by the sum of all such differences. Optionally, these entropies can be normalized according to a simple heuristic, see normalize. **Important notes**: - Input collections of persistence diagrams for this transformer must satisfy certain requirements, see e.g. :meth:fit. - By default, persistence subdiagrams containing only triples with zero lifetime will have corresponding (normalized) entropies computed as numpy.nan. To avoid this, set a value of nan_fill_value different from None. Parameters ---------- normalize : bool, optional, default: False When True, the persistence entropy of each diagram is normalized by the logarithm of the sum of lifetimes of all points in the diagram. Can aid comparison between diagrams in an input collection when these have different numbers of (non-trivial) points. [1]_ nan_fill_value : float or None, optional, default: -1. If a float, (normalized) persistence entropies initially computed as numpy.nan are replaced with this value. If None, these values are left as numpy.nan. n_jobs : int or None, optional, default: None The number of jobs to use for the computation. None means 1 unless in a :obj:joblib.parallel_backend context. -1 means using all processors. Attributes ---------- homology_dimensions_ : tuple Homology dimensions seen in :meth:fit, sorted in ascending order. See also -------- NumberOfPoints, Amplitude, BettiCurve, PersistenceLandscape, HeatKernel, \ Silhouette, PersistenceImage References ---------- .. [1] A. Myers, E. Munch, and F. A. Khasawneh, "Persistent Homology of Complex Networks for Dynamic State Detection"; *Phys. Rev. E* **100**, 022314, 2019; DOI: 10.1103/PhysRevE.100.022314 <https://doi.org/10.1103/PhysRevE.100.022314>_. """ _hyperparameters = { 'normalize': {'type': bool}, 'nan_fill_value': {'type': (Real, type(None))} }
[docs] def __init__(self, normalize=False, nan_fill_value=-1., n_jobs=None): self.normalize = normalize self.nan_fill_value = nan_fill_value self.n_jobs = n_jobs
@staticmethod def _persistence_entropy(X, normalize=False, nan_fill_value=None): X_lifespan = X[:, :, 1] - X[:, :, 0] X_entropy = entropy(X_lifespan, base=2, axis=1) if normalize: lifespan_sums = np.sum(X_lifespan, axis=1) X_entropy /= np.log2(lifespan_sums) if nan_fill_value is not None: np.nan_to_num(X_entropy, nan=nan_fill_value, copy=False) X_entropy = X_entropy[:, None] return X_entropy
[docs] def fit(self, X, y=None): """Store all observed homology dimensions in :attr:homology_dimensions_. 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 : object """ X = check_diagrams(X) validate_params( self.get_params(), self._hyperparameters, exclude=['n_jobs']) # Find the unique homology dimensions in the 3D array X passed to fit # assuming that they can all be found in its zero-th entry homology_dimensions_fit = np.unique(X[0, :, 2]) self.homology_dimensions_ = \ _homology_dimensions_to_sorted_ints(homology_dimensions_fit) self._n_dimensions = len(self.homology_dimensions_) return self
[docs] def transform(self, X, y=None): """Compute the persistence entropies 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 : ndarray of shape (n_samples, n_homology_dimensions) Persistence entropies: one value per sample and per homology dimension seen in :meth:fit. Index i along axis 1 corresponds to the i-th homology dimension in :attr:homology_dimensions_. """ check_is_fitted(self) X = check_diagrams(X) with np.errstate(divide='ignore', invalid='ignore'): Xt = Parallel(n_jobs=self.n_jobs)( delayed(self._persistence_entropy)( _subdiagrams(X[s], [dim]), normalize=self.normalize, nan_fill_value=self.nan_fill_value ) for dim in self.homology_dimensions_ for s in gen_even_slices(len(X), effective_n_jobs(self.n_jobs)) ) Xt = np.concatenate(Xt).reshape(self._n_dimensions, len(X)).T return Xt
[docs]@adapt_fit_transform_docs class Amplitude(BaseEstimator, TransformerMixin): """:ref:Amplitudes <vectorization_amplitude_and_kernel> of persistence diagrams. For each persistence diagram in a collection, a vector of amplitudes or a single scalar amplitude is calculated according to the following steps: 1. The diagram is partitioned into subdiagrams according to homology dimension. 2. The amplitude of each subdiagram is calculated according to the parameters metric and metric_params. This gives a vector of amplitudes, :math:\\mathbf{a} = (a_{q_1}, \\ldots, a_{q_n}) where the :math:q_i range over the available homology dimensions. 3. The final result is either :math:\\mathbf{a} itself or a norm of :math:\\mathbf{a}, specified by the parameter order. **Important notes**: - Input collections of persistence diagrams for this transformer must satisfy certain requirements, see e.g. :meth:fit. - The shape of outputs of :meth:transform depends on the value of the order parameter. Parameters ---------- metric : 'bottleneck' | 'wasserstein' | 'betti' | \ 'landscape' | 'silhouette' | 'heat' | \ 'persistence_image', optional, default: 'landscape' Distance or dissimilarity function used to define the amplitude of a subdiagram as its distance from the (trivial) diagonal diagram: - 'bottleneck' and 'wasserstein' refer to the identically named perfect-matching--based notions of distance. - 'betti' refers to the :math:L^p distance between Betti curves. - 'landscape' refers to the :math:L^p distance between persistence landscapes. - 'silhouette' refers to the :math:L^p distance between silhouettes. - 'heat' refers to the :math:L^p distance between Gaussian-smoothed diagrams. - 'persistence_image' refers to the :math:L^p distance between Gaussian-smoothed diagrams represented on birth-persistence 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' there are no available arguments. - If metric == 'wasserstein' the only argument is p (float, default: 2.). - 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: None If None, :meth:transform returns for each diagram a vector of amplitudes corresponding to the dimensions in :attr:homology_dimensions_. Otherwise, the :math:p-norm of these vectors with :math: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 a :obj:joblib.parallel_backend context. -1 means using all processors. Attributes ---------- effective_metric_params_ : dict Dictionary containing all information present in metric_params as well as relevant quantities computed in :meth:fit. homology_dimensions_ : tuple Homology dimensions seen in :meth:fit, sorted in ascending order. See also -------- NumberOfPoints, PersistenceEntropy, PairwiseDistance, Scaler, Filtering, \ BettiCurve, PersistenceLandscape, HeatKernel, Silhouette, PersistenceImage Notes ----- To compute amplitudes without first splitting the computation between different homology dimensions, data should be first transformed by an instance of :class:ForgetDimension. """ _hyperparameters = { 'metric': {'type': str, 'in': _AVAILABLE_AMPLITUDE_METRICS.keys()}, 'order': {'type': (Real, type(None)), 'in': Interval(0, np.inf, closed='right')}, 'metric_params': {'type': (dict, type(None))} }
[docs] def __init__(self, metric='landscape', metric_params=None, order=None, n_jobs=None): self.metric = metric self.metric_params = metric_params self.order = order self.n_jobs = n_jobs
[docs] def fit(self, X, y=None): """Store all observed homology dimensions in :attr:homology_dimensions_ and compute :attr:effective_metric_params. 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 : object """ X = check_diagrams(X) validate_params( self.get_params(), self._hyperparameters, exclude=['n_jobs']) if self.metric_params is None: self.effective_metric_params_ = {} else: self.effective_metric_params_ = self.metric_params.copy() validate_params(self.effective_metric_params_, _AVAILABLE_AMPLITUDE_METRICS[self.metric]) # Find the unique homology dimensions in the 3D array X passed to fit # assuming that they can all be found in its zero-th entry homology_dimensions_fit = np.unique(X[0, :, 2]) self.homology_dimensions_ = \ _homology_dimensions_to_sorted_ints(homology_dimensions_fit) self.effective_metric_params_['samplings'], \ self.effective_metric_params_['step_sizes'] = \ _bin(X, self.metric, **self.effective_metric_params_) if self.metric == 'persistence_image': weight_function = self.effective_metric_params_.get( 'weight_function', None ) weight_function = \ np.ones_like if weight_function is None else weight_function self.effective_metric_params_['weight_function'] = weight_function return self
[docs] def transform(self, X, y=None): """Compute the amplitudes or amplitude vectors 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 : ndarray of shape (n_samples, n_homology_dimensions) if order \ is None, else (n_samples, 1) Amplitudes or amplitude vectors of the diagrams in X. In the second case, index i along axis 1 corresponds to the i-th homology dimension in :attr:homology_dimensions_. """ check_is_fitted(self) Xt = check_diagrams(X, copy=True) Xt = _parallel_amplitude(Xt, self.metric, self.effective_metric_params_, self.homology_dimensions_, self.n_jobs) if self.order is not None: Xt = np.linalg.norm(Xt, axis=1, ord=self.order).reshape(-1, 1) return Xt
[docs]@adapt_fit_transform_docs class NumberOfPoints(BaseEstimator, TransformerMixin): """Number of off-diagonal points in persistence diagrams, per homology dimension. 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 numbers of off-diagonal points are calculated. **Important note**: - Input collections of persistence diagrams for this transformer must satisfy certain requirements, see e.g. :meth:fit. Parameters ---------- n_jobs : int or None, optional, default: None The number of jobs to use for the computation. None means 1 unless in a :obj:joblib.parallel_backend context. -1 means using all processors. Attributes ---------- homology_dimensions_ : list Homology dimensions seen in :meth:fit, sorted in ascending order. See also -------- PersistenceEntropy, Amplitude, BettiCurve, PersistenceLandscape, HeatKernel, Silhouette, PersistenceImage """
[docs] def __init__(self, n_jobs=None): self.n_jobs = n_jobs
@staticmethod def _number_points(X): return np.count_nonzero(X[:, :, 1] - X[:, :, 0], axis=1)
[docs] def fit(self, X, y=None): """Store all observed homology dimensions in :attr:homology_dimensions_. 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 : object """ X = check_diagrams(X) # Find the unique homology dimensions in the 3D array X passed to fit # assuming that they can all be found in its zero-th entry homology_dimensions_fit = np.unique(X[0, :, 2]) self.homology_dimensions_ = \ _homology_dimensions_to_sorted_ints(homology_dimensions_fit) self._n_dimensions = len(self.homology_dimensions_) return self
[docs] def transform(self, X, y=None): """Compute a vector of numbers of off-diagonal points for each diagram 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 : ndarray of shape (n_samples, n_homology_dimensions) Number of points: one value per sample and per homology dimension seen in :meth:fit. Index i along axis 1 corresponds to the i-th homology dimension in :attr:homology_dimensions_. """ check_is_fitted(self) X = check_diagrams(X) Xt = Parallel(n_jobs=self.n_jobs)( delayed(self._number_points)(_subdiagrams(X, [dim])[s]) for dim in self.homology_dimensions_ for s in gen_even_slices(len(X), effective_n_jobs(self.n_jobs)) ) Xt = np.concatenate(Xt).reshape(self._n_dimensions, len(X)).T return Xt
[docs]@adapt_fit_transform_docs class ComplexPolynomial(BaseEstimator, TransformerMixin): """Coefficients of complex polynomials whose roots are obtained from points in persistence diagrams. Given a persistence diagram consisting of birth-death-dimension triples [b, d, q], subdiagrams corresponding to distinct homology dimensions are first considered separately. For each subdiagram, the polynomial whose roots are complex numbers obtained from its birth-death pairs is computed, and its :attr:n_coefficients_ highest-degree complex coefficients excluding the top one are stored into a single real vector by concatenating the vector of all real parts with the vector of all imaginary parts [1]_ (if not enough coefficients are available to form a vector of the required length, padding with zeros is performed). Finally, all such vectors coming from different subdiagrams are concatenated to yield a single vector for the diagram. There are three possibilities for mapping birth-death pairs :math:(b, d) to complex polynomial roots. They are: .. math:: :nowrap: \\begin{gather*} R(b, d) = b + \\mathrm{i} d, \\\\ S(b, d) = \\frac{d - b}{\\sqrt{2} r} (b + \\mathrm{i} d), \\\\ T(b, d) = \\frac{d - b}{2} [\\cos{r} - \\sin{r} + \ \\mathrm{i}(\\cos{r} + \\sin{r})], \\end{gather*} where :math:r = \\sqrt{b^2 + d^2}. **Important note**: - Input collections of persistence diagrams for this transformer must satisfy certain requirements, see e.g. :meth:fit. Parameters ---------- polynomial_type : 'R' | 'S' | 'T', optional, default: 'R' Type of complex polynomial to compute. n_coefficients : list, int or None, optional, default: 10 Number of complex coefficients per homology dimension. If an int then the number of coefficients will be equal to that value for each homology dimension. If None then, for each homology dimension in the collection of persistence diagrams seen in :meth:fit, the number of complex coefficients is defined to be the largest number of off-diagonal points seen among all subdiagrams in that homology dimension, minus one. n_jobs : int or None, optional, default: None The number of jobs to use for the computation. None means 1 unless in a :obj:joblib.parallel_backend context. -1 means using all processors. Attributes ---------- homology_dimensions_ : list Homology dimensions seen in :meth:fit, sorted in ascending order. n_coefficients_ : list Effective number of complex coefficients per homology dimension. Set in :meth:fit. See also -------- Amplitude, PersistenceEntropy References ---------- .. [1] B. Di Fabio and M. Ferri, "Comparing Persistence Diagrams Through Complex Vectors"; in *Image Analysis and Processing — ICIAP 2015*, 2015; DOI: 10.1007/978-3-319-23231-7_27 <https://doi.org/10.1007/978-3-319-23231-7_27>_. """ _hyperparameters = { 'n_coefficients': {'type': (int, type(None), list), 'in': Interval(1, np.inf, closed='left'), 'of': {'type': int, 'in': Interval(1, np.inf, closed='left')}}, 'polynomial_type': {'type': str, 'in': _AVAILABLE_POLYNOMIALS.keys()} }
[docs] def __init__(self, n_coefficients=10, polynomial_type='R', n_jobs=None): self.n_coefficients = n_coefficients self.polynomial_type = polynomial_type self.n_jobs = n_jobs
[docs] def fit(self, X, y=None): """Store all observed homology dimensions in :attr:homology_dimensions_ and compute :attr:n_coefficients_. 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 : object """ validate_params( self.get_params(), self._hyperparameters, exclude=['n_jobs']) X = check_diagrams(X) # Find the unique homology dimensions in the 3D array X passed to fit # assuming that they can all be found in its zero-th entry homology_dimensions_fit, counts = np.unique(X[0, :, 2], return_counts=True) self.homology_dimensions_ = \ _homology_dimensions_to_sorted_ints(homology_dimensions_fit) _n_homology_dimensions = len(self.homology_dimensions_) _homology_dimensions_counts = dict(zip(homology_dimensions_fit, counts)) if self.n_coefficients is None: self.n_coefficients_ = [_homology_dimensions_counts[dim] for dim in self.homology_dimensions_] elif type(self.n_coefficients) == list: if len(self.n_coefficients) != _n_homology_dimensions: raise ValueError( f'n_coefficients has been passed as a list of length ' f'{len(self.n_coefficients)} while diagrams in X have ' f'{_n_homology_dimensions} homology dimensions.' ) self.n_coefficients_ = self.n_coefficients else: self.n_coefficients_ = \ [self.n_coefficients] * _n_homology_dimensions self._polynomial_function = \ _implemented_polynomial_recipes[self.polynomial_type] return self
def _complex_polynomial(self, X, n_coefficients): Xt = np.zeros(2 * n_coefficients,) X = X[X[:, 0] != X[:, 1]] roots = self._polynomial_function(X) coefficients = np.poly(roots) coefficients = np.array(coefficients[1:]) dimension = min(n_coefficients, coefficients.shape[0]) Xt[:dimension] = coefficients[:dimension].real Xt[n_coefficients:n_coefficients + dimension] = \ coefficients[:dimension].imag return Xt
[docs] def transform(self, X, y=None): """Compute vectors of real and imaginary parts of coefficients of complex polynomials obtained from each diagram 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 : ndarray of shape (n_samples, n_homology_dimensions * 2 \ * n_coefficients_) Polynomial coefficients: real and imaginary parts of the complex polynomials obtained in each homology dimension from each diagram in X`. """ check_is_fitted(self) Xt = check_diagrams(X, copy=True) Xt = Parallel(n_jobs=self.n_jobs)( delayed(self._complex_polynomial)( _subdiagrams(Xt[[s]], [dim], remove_dim=True)[0], self.n_coefficients_[d]) for s in range(len(X)) for d, dim in enumerate(self.homology_dimensions_) ) Xt = np.concatenate(Xt).reshape(len(X), -1) return Xt