# Source code for gudhi.point_cloud.dtm

```
# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
# Author(s): Marc Glisse
#
# Copyright (C) 2020 Inria
#
# Modification(s):
# - YYYY/MM Author: Description of the modification
from .knn import KNearestNeighbors
import numpy as np
__author__ = "Marc Glisse"
__copyright__ = "Copyright (C) 2020 Inria"
__license__ = "MIT"
[docs]class DistanceToMeasure:
"""
Class to compute the distance to the empirical measure defined by a point set, as introduced in :cite:`dtm`.
"""
[docs] def __init__(self, k, q=2, **kwargs):
"""
Args:
k (int): number of neighbors (possibly including the point itself).
q (float): order used to compute the distance to measure. Defaults to 2.
kwargs: same parameters as :class:`~gudhi.point_cloud.knn.KNearestNeighbors`, except that
metric="neighbors" means that :func:`transform` expects an array with the distances
to the k nearest neighbors.
"""
self.k = k
self.q = q
self.params = kwargs
[docs] def fit(self, X, y=None):
"""
Args:
X (numpy.array): coordinates for mass points.
"""
if self.params.setdefault("metric", "euclidean") != "neighbors":
self.knn = KNearestNeighbors(
self.k, return_index=False, return_distance=True, sort_results=False, **self.params
)
self.knn.fit(X)
return self
[docs] def transform(self, X):
"""
Args:
X (numpy.array): coordinates for query points, or distance matrix if metric is "precomputed",
or distances to the k nearest neighbors if metric is "neighbors" (if the array has more
than k columns, the remaining ones are ignored).
Returns:
numpy.array: a 1-d array with, for each point of X, its distance to the measure defined
by the argument of :func:`fit`.
"""
if self.params["metric"] == "neighbors":
distances = X[:, : self.k]
else:
distances = self.knn.transform(X)
distances = distances ** self.q
dtm = distances.sum(-1) / self.k
dtm = dtm ** (1.0 / self.q)
# We compute too many powers, 1/p in knn then q in dtm, 1/q in dtm then q or some log in the caller.
# Add option to skip the final root?
return dtm
[docs]class DTMDensity:
"""
Density estimator based on the distance to the empirical measure defined by a point set, as defined
in :cite:`dtmdensity`. Note that this implementation only renormalizes when asked, and the renormalization
only works for a Euclidean metric, so in other cases the total measure may not be 1.
.. note:: When the dimension is high, using it as an exponent can quickly lead to under- or overflows.
We recommend using a small fixed value instead in those cases, even if it won't have the same nice
theoretical properties as the dimension.
"""
[docs] def __init__(self, k=None, weights=None, q=None, dim=None, normalize=False, n_samples=None, **kwargs):
"""
Args:
k (int): number of neighbors (possibly including the point itself). Optional if it can be guessed
from weights or metric="neighbors".
weights (numpy.array): weights of each of the k neighbors, optional. They are supposed to sum to 1.
q (float): order used to compute the distance to measure. Defaults to dim.
dim (float): final exponent representing the dimension. Defaults to the dimension, and must be specified
when the dimension cannot be read from the input (metric is "neighbors" or "precomputed").
normalize (bool): normalize the density so it corresponds to a probability measure on ℝᵈ.
Only available for the Euclidean metric, defaults to False.
n_samples (int): number of sample points used for fitting. Only needed if `normalize` is True and
metric is "neighbors".
kwargs: same parameters as :class:`~gudhi.point_cloud.knn.KNearestNeighbors`, except that
metric="neighbors" means that :func:`transform` expects an array with the distances to
the k nearest neighbors.
"""
if weights is None:
self.k = k
if k is None:
assert kwargs.get("metric") == "neighbors", 'Must specify k or weights, unless metric is "neighbors"'
self.weights = None
else:
self.weights = np.full(k, 1.0 / k)
else:
self.weights = weights
self.k = len(weights)
assert k is None or k == self.k, "k differs from the length of weights"
self.q = q
self.dim = dim
self.params = kwargs
self.normalize = normalize
self.n_samples = n_samples
[docs] def fit(self, X, y=None):
"""
Args:
X (numpy.array): coordinates for mass points.
"""
if self.params.setdefault("metric", "euclidean") != "neighbors":
self.knn = KNearestNeighbors(
self.k, return_index=False, return_distance=True, sort_results=False, **self.params
)
self.knn.fit(X)
if self.params["metric"] != "precomputed":
self.n_samples = len(X)
return self
[docs] def transform(self, X):
"""
Args:
X (numpy.array): coordinates for query points, or distance matrix if metric is "precomputed",
or distances to the k nearest neighbors if metric is "neighbors" (if the array has more
than k columns, the remaining ones are ignored).
"""
q = self.q
dim = self.dim
if dim is None:
assert self.params["metric"] not in {
"neighbors",
"precomputed",
}, "dim not specified and cannot guess the dimension"
dim = len(X[0])
if q is None:
q = dim
k = self.k
weights = self.weights
if self.params["metric"] == "neighbors":
distances = np.asarray(X)
if weights is None:
k = distances.shape[1]
weights = np.full(k, 1.0 / k)
else:
distances = distances[:, :k]
else:
distances = self.knn.transform(X)
distances = distances ** q
dtm = (distances * weights).sum(-1)
if self.normalize:
dtm /= (np.arange(1, k + 1) ** (q / dim) * weights).sum()
density = dtm ** (-dim / q)
if self.normalize:
import math
if self.params["metric"] == "precomputed":
self.n_samples = len(X[0])
# Volume of d-ball
Vd = math.pi ** (dim / 2) / math.gamma(dim / 2 + 1)
density /= self.n_samples * Vd
return density
# We compute too many powers, 1/p in knn then q in dtm, d/q in dtm then whatever in the caller.
# Add option to skip the final root?
```