import math
from functools import cached_property
import numpy as np
import pandas as pd
from scipy import sparse
from libpysal.weights import W
from ._contiguity import (
_block_contiguity,
_fuzzy_contiguity,
_queen,
_rook,
_vertex_set_intersection,
)
from ._kernel import _distance_band, _kernel
from ._matching import _spatial_matching
from ._parquet import _read_parquet, _to_parquet
from ._plotting import _explore_graph, _plot
from ._set_ops import SetOpsMixin
from ._spatial_lag import _lag_spatial
from ._triangulation import _delaunay, _gabriel, _relative_neighborhood, _voronoi
from ._utils import (
_evaluate_index,
_neighbor_dict_to_edges,
_resolve_islands,
_sparse_to_arrays,
)
ALLOWED_TRANSFORMATIONS = ("O", "B", "R", "D", "V", "C")
# listed alphabetically
__author__ = """"
Martin Fleischmann (martin@martinfleischmann.net)
Eli Knaap (ek@knaaptime.com)
Serge Rey (sjsrey@gmail.com)
Levi John Wolf (levi.john.wolf@gmail.com)
"""
[docs]
class Graph(SetOpsMixin):
"""Graph class encoding spatial weights matrices
The :class:`Graph` is currently experimental
and its API is incomplete and unstable.
"""
[docs]
def __init__(self, adjacency, transformation="O", is_sorted=False):
"""Weights base class based on adjacency list
It is recommenced to use one of the ``from_*`` or ``build_*`` constructors
rather than invoking ``__init__`` directly.
Each observation needs to be present in the focal,
at least as a self-loop with a weight 0.
Parameters
----------
adjacency : pandas.Series
A MultiIndexed pandas.Series with ``"focal"`` and ``"neigbor"`` levels
encoding adjacency, and values encoding weights. By convention,
isolates are encoded as self-loops with a weight 0.
transformation : str, default "O"
weights transformation used to produce the table.
- **O** -- Original
- **B** -- Binary
- **R** -- Row-standardization (global sum :math:`=n`)
- **D** -- Double-standardization (global sum :math:`=1`)
- **V** -- Variance stabilizing
- **C** -- Custom
is_sorted : bool, default False
``adjacency`` capturing the graph needs to be canonically sorted to
initialize the class. The MultiIndex needs to be ordered i-->j
on both focal and neighbor levels according to the order of ids in the
original data from which the Graph is created. Sorting is performed by
default based on the order of unique values in the focal level. Set
``is_sorted=True`` to skip this step if the adjacency is already canonically
sorted.
"""
if not isinstance(adjacency, pd.Series):
raise TypeError(
f"The adjacency table needs to be a pandas.Series. {type(adjacency)}"
)
if not adjacency.index.names == ["focal", "neighbor"]:
raise ValueError(
"The index of the adjacency table needs to be a MultiIndex named "
"['focal', 'neighbor']."
)
if not adjacency.name == "weight":
raise ValueError(
"The adjacency needs to be named 'weight'. "
f"'{adjacency.name}' was given instead."
)
if not pd.api.types.is_numeric_dtype(adjacency):
raise ValueError(
"The 'weight' needs to be of a numeric dtype. "
f"'{adjacency.dtype}' dtype was given instead."
)
if adjacency.isna().any():
raise ValueError("The adjacency table cannot contain missing values.")
if transformation.upper() not in ALLOWED_TRANSFORMATIONS:
raise ValueError(
f"'transformation' needs to be one of {ALLOWED_TRANSFORMATIONS}. "
f"'{transformation}' was given instead."
)
if not is_sorted:
# adjacency always ordered i-->j on both levels
ids = adjacency.index.get_level_values(0).unique().values
adjacency = adjacency.reindex(ids, level=0).reindex(ids, level=1)
self._adjacency = adjacency
self.transformation = transformation
def __getitem__(self, item):
"""Easy lookup based on focal index
Parameters
----------
item : hashable
hashable represting an index value
Returns
-------
pandas.Series
subset of the adjacency table for `item`
"""
if item in self.isolates:
return pd.Series(
[],
index=pd.Index([], name="neighbor"),
name="weight",
)
return self._adjacency.loc[item]
[docs]
def copy(self, deep=True):
"""Make a copy of this Graph's adjacency table and transformation
Parameters
----------
deep : bool, optional
Make a deep copy of the adjacency table, by default True
Returns
-------
Graph
libpysal.graph.Graph as a copy of the original
"""
return Graph(
self._adjacency.copy(deep=deep),
transformation=self.transformation,
is_sorted=True,
)
@cached_property
def adjacency(self):
"""Return a copy of the adjacency list
Returns
-------
pandas.Series
Underlying adjacency list
"""
return self._adjacency.copy()
[docs]
@classmethod
def from_W(cls, w): # noqa: N802
"""Create an experimental Graph from libpysal.weights.W object
Parameters
----------
w : libpysal.weights.W
Returns
-------
Graph
libpysal.graph.Graph from W
"""
return cls.from_weights_dict(dict(w))
[docs]
def to_W(self): # noqa: N802
"""Convert Graph to a libpysal.weights.W object
Returns
-------
libpysal.weights.W
representation of graph as a weights.W object
"""
ids, labels = pd.factorize(
self._adjacency.index.get_level_values("focal"), sort=False
)
neighbors = (
self._adjacency.reset_index(level=1)
.groupby(ids)
.apply(
lambda group: list(
group[
~((group.index == group.neighbor) & (group.weight == 0))
].neighbor
)
)
)
neighbors.index = labels[neighbors.index]
weights = (
self._adjacency.reset_index(level=1)
.groupby(ids)
.apply(
lambda group: list(
group[
~((group.index == group.neighbor) & (group.weight == 0))
].weight
)
)
)
weights.index = labels[weights.index]
return W(neighbors.to_dict(), weights.to_dict(), id_order=labels.tolist())
[docs]
@classmethod
def from_adjacency(
cls, adjacency, focal_col="focal", neighbor_col="neighbor", weight_col="weight"
):
"""Create a Graph from a pandas DataFrame formatted as an adjacency list
Parameters
----------
adjacency : pandas.DataFrame
a dataframe formatted as an ajacency list. Should have columns
"focal", "neighbor", and "weight", or columns that can be mapped
to these (e.g. origin, destination, cost)
focal : str, optional
name of column holding focal/origin index, by default 'focal'
neighbor : str, optional
name of column holding neighbor/destination index, by default 'neighbor'
weight : str, optional
name of column holding weight values, by default 'weight'
Returns
-------
Graph
libpysal.graph.Graph
"""
cols = dict(
zip(
[focal_col, neighbor_col, weight_col],
["focal_col", "neighbor_col", "weight_col"],
strict=True,
)
)
for col in cols:
assert col in adjacency.columns.tolist(), (
f'"{col}" was given for `{cols[col]}`, but the '
f"columns available in `adjacency` are: {adjacency.columns.tolist()}."
)
return cls.from_arrays(
adjacency[focal_col].values,
adjacency[neighbor_col].values,
adjacency[weight_col].values,
)
[docs]
@classmethod
def from_sparse(cls, sparse, ids=None):
"""Convert a ``scipy.sparse`` array to a PySAL ``Graph`` object.
Parameters
----------
sparse : scipy.sparse array
sparse representation of a graph
ids : list-like, default None
list-like of ids for geometries that is mappable to
positions from sparse. If None, the positions are used as labels.
Returns
-------
Graph
libpysal.graph.Graph based on sparse
"""
return cls.from_arrays(*_sparse_to_arrays(sparse, ids))
[docs]
@classmethod
def from_arrays(cls, focal_ids, neighbor_ids, weight, **kwargs):
"""Generate Graph from arrays of indices and weights of the same length
The arrays needs to be sorted in a way ensuring that focal_ids.unique() is
equal to the index of original observations from which the Graph is being built
Parameters
----------
focal_index : array-like
focal indices
neighbor_index : array-like
neighbor indices
weight : array-like
weights
**kwargs
keyword arguments passed to the class constructor
Returns
-------
Graph
libpysal.graph.Graph based on arrays
"""
w = cls(
pd.Series(
weight,
name="weight",
index=pd.MultiIndex.from_arrays(
[focal_ids, neighbor_ids], names=["focal", "neighbor"]
),
),
**kwargs,
)
return w
[docs]
@classmethod
def from_weights_dict(cls, weights_dict):
"""Generate Graph from a dict of dicts
Parameters
----------
weights_dict : dictionary of dictionaries
weights dictionary with the ``{focal: {neighbor: weight}}`` structure.
Returns
-------
Graph
libpysal.graph.Graph based on weights dictionary of dictionaries
"""
idx = {f: list(neighbors) for f, neighbors in weights_dict.items()}
data = {f: list(neighbors.values()) for f, neighbors in weights_dict.items()}
return cls.from_dicts(idx, data)
[docs]
@classmethod
def from_dicts(cls, neighbors, weights=None):
"""Generate Graph from dictionaries of neighbors and weights
Parameters
----------
neighbors : dict
dictionary of neighbors with the ``{focal: [neighbor1, neighbor2]}``
structure
weights : dict, optional
dictionary of neighbors with the ``{focal: [weight1, weight2]}``
structure. If None, assumes binary weights.
Returns
-------
Graph
libpysal.graph.Graph based on dictionaries
"""
head, tail, weight = _neighbor_dict_to_edges(neighbors, weights=weights)
return cls.from_arrays(head, tail, weight)
[docs]
@classmethod
def build_block_contiguity(cls, regimes):
"""Generate Graph from block contiguity (regime neighbors)
Block contiguity structures are relevant when defining neighbor relations
based on membership in a regime. For example, all counties belonging to
the same state could be defined as neighbors, in an analysis of all
counties in the US.
Parameters
----------
regimes : list-like
list-like of regimes. If pandas.Series, its index is used to encode Graph.
Otherwise a default RangeIndex is used.
Returns
-------
Graph
libpysal.graph.Graph encoding block contiguity
"""
ids = _evaluate_index(regimes)
return cls.from_dicts(_block_contiguity(regimes, ids=ids))
[docs]
@classmethod
def build_contiguity(cls, geometry, rook=True, by_perimeter=False, strict=False):
"""Generate Graph from geometry based on contiguity
Contiguity builder assumes that all geometries are forming a coverage, i.e.
a non-overlapping mesh and neighbouring geometries share only points or
segments of their exterior boundaries. In practice, ``build_contiguity`` is
capable of creating a Graph of partially overlapping geometries when
``strict=False, by_perimeter=False``, but that would not strictly follow the
definition of queen or rook contiguity.
Parameters
----------
geometry : array-like of shapely.Geometry objects
Could be geopandas.GeoSeries or geopandas.GeoDataFrame, in which case the
resulting Graph is indexed by the original index. If an array of
shapely.Geometry objects is passed, Graph will assume a RangeIndex.
rook : bool, optional
Contiguity method. If True, two geometries are considered neighbours if
they share at least one edge. If False, two geometries are considered
neighbours if they share at least one vertex. By default True
by_perimeter : bool, optional
If True, ``weight`` represents the length of the shared boundary between
adjacent units, by default False. For row-standardized version of perimeter
weights, use
``Graph.build_contiguity(gdf, by_perimeter=True).transform("r")``.
strict : bool, optional
Use the strict topological method. If False, the contiguity is determined
based on shared coordinates or coordinate sequences representing edges.
This assumes geometry coverage that is topologically correct. This method
is faster but can miss some relations. If True, the contiguity is
determined based on geometric relations that do not require precise
topology. This method is slower but will result in correct contiguity
even if the topology of geometries is not optimal. By default False.
Returns
-------
Graph
libpysal.graph.Graph encoding contiguity weights
"""
ids = _evaluate_index(geometry)
if hasattr(geometry, "geometry"):
# potentially cast GeoDataFrame to GeoSeries
geometry = geometry.geometry
if strict:
# use shapely-based constructors
if rook:
return cls.from_arrays(
*_rook(geometry, ids=ids, by_perimeter=by_perimeter)
)
return cls.from_arrays(
*_queen(geometry, ids=ids, by_perimeter=by_perimeter)
)
# use vertex-based constructor
return cls.from_arrays(
*_vertex_set_intersection(
geometry, rook=rook, ids=ids, by_perimeter=by_perimeter
)
)
[docs]
@classmethod
def build_distance_band(
cls, data, threshold, binary=True, alpha=-1.0, kernel=None, bandwidth=None
):
"""Generate Graph from geometry based on a distance band
Parameters
----------
data : numpy.ndarray, geopandas.GeoSeries, geopandas.GeoDataFrame
geometries containing locations to compute the
delaunay triangulation. If a geopandas object with Point
geometry is provided, the .geometry attribute is used. If a numpy.ndarray
with shapely geometry is used, then the coordinates are extracted and used.
If a numpy.ndarray of a shape (2,n) is used, it is assumed to contain x, y
coordinates.
threshold : float
distance band
binary : bool, optional
If True :math:`w_{ij}=1` if :math:`d_{i,j}<=threshold`, otherwise
:math:`w_{i,j}=0`.
If False :math:`wij=dij^{alpha}`, by default True.
alpha : float, optional
distance decay parameter for weight (default -1.0)
if alpha is positive the weights will not decline with
distance. Ignored if ``binary=True`` or ``kernel`` is not None.
kernel : str, optional
kernel function to use in order to weight the output graph. See
:meth:`Graph.build_kernel` for details. Ignored if ``binary=True``.
bandwidth : float (default: None)
distance to use in the kernel computation. Should be on the same scale as
the input coordinates. Ignored if ``binary=True`` or ``kernel=None``.
Returns
-------
Graph
libpysal.graph.Graph encoding distance band weights
"""
ids = _evaluate_index(data)
dist = _distance_band(data, threshold)
if binary:
head, tail, weight = _kernel(
dist,
kernel="boxcar",
metric="precomputed",
ids=ids,
bandwidth=np.inf,
)
elif kernel is not None:
head, tail, weight = _kernel(
dist,
kernel=kernel,
metric="precomputed",
ids=ids,
bandwidth=bandwidth,
)
else:
head, tail, weight = _kernel(
dist,
kernel=lambda distances, alpha: np.power(distances, alpha),
metric="precomputed",
ids=ids,
bandwidth=alpha,
)
adjacency = pd.DataFrame.from_dict(
{"focal": head, "neighbor": tail, "weight": weight}
).set_index("focal")
# drop diagonal
counts = adjacency.index.value_counts()
no_isolates = counts[counts > 1]
adjacency = adjacency[
~(
adjacency.index.isin(no_isolates.index)
& (adjacency.index == adjacency.neighbor)
)
]
# set isolates to 0 - distance band should never contain self-weight
adjacency.loc[~adjacency.index.isin(no_isolates.index), "weight"] = 0
return cls.from_arrays(
adjacency.index.values, adjacency.neighbor.values, adjacency.weight.values
)
[docs]
@classmethod
def build_fuzzy_contiguity(
cls,
geometry,
tolerance=None,
buffer=None,
predicate="intersects",
):
"""Generate Graph from fuzzy contiguity
Fuzzy contiguity relaxes the notion of contiguity neighbors
for the case of geometry collections that violate the condition
of planar enforcement. It handles three types of conditions present
in such collections that would result in missing links when using
the regular contiguity methods.
The first are edges for nearby polygons that should be shared, but are
digitized separately for the individual polygons and the resulting edges
do not coincide, but instead the edges intersect. This case can also be
covered by ``build_contiguty`` with the ``strict=False`` parameter.
The second case is similar to the first, only the resultant edges do not
intersect but are "close". The optional buffering of geometry then closes the
gaps between the polygons and a resulting intersection is encoded as a link.
The final case arises when one polygon is "inside" a second polygon but is not
encoded to represent a hole in the containing polygon.
It is also possible to create a contiguity based on a custom spatial predicate.
Parameters
----------
geoms : array-like of shapely.Geometry objects
Could be geopandas.GeoSeries or geopandas.GeoDataFrame, in which case the
resulting Graph is indexed by the original index. If an array of
shapely.Geometry objects is passed, Graph will assume a RangeIndex.
tolerance : float, optional
The percentage of the length of the minimum side of the bounding rectangle
for the ``geoms`` to use in determining the buffering distance. Either
``tolerance`` or ``buffer`` may be specified but not both.
By default None.
buffer : float, optional
Exact buffering distance in the units of ``geoms.crs``. Either
``tolerance`` or ``buffer`` may be specified but not both.
By default None.
predicate : str, optional
The predicate to use for determination of neighbors. Default is
'intersects'. If None is passed, neighbours are determined based
on the intersection of bounding boxes. See the documentation of
``geopandas.GeoSeries.sindex.query`` for allowed predicates.
Returns
-------
Graph
libpysal.graph.Graph encoding fuzzy contiguity
"""
ids = _evaluate_index(geometry)
heads, tails, weights = _fuzzy_contiguity(
geometry, ids, tolerance=tolerance, buffer=buffer, predicate=predicate
)
return cls.from_arrays(heads, tails, weights)
[docs]
@classmethod
def build_kernel(
cls,
data,
kernel="gaussian",
k=None,
bandwidth=None,
metric="euclidean",
p=2,
coincident="raise",
):
"""Generate Graph from geometry data based on a kernel function
Parameters
----------
data : numpy.ndarray, geopandas.GeoSeries, geopandas.GeoDataFrame
geometries over which to compute a kernel. If a geopandas object with Point
geoemtry is provided, the .geometry attribute is used. If a numpy.ndarray
with shapely geoemtry is used, then the coordinates are extracted and used.
If a numpy.ndarray of a shape (2,n) is used, it is assumed to contain x, y
coordinates. If metric="precomputed", data is assumed to contain a
precomputed distance metric.
kernel : string or callable (default: 'gaussian')
kernel function to apply over the distance matrix computed by `metric`.
The following kernels are supported:
- ``"triangular"``:
- ``"parabolic"``:
- ``"gaussian"``:
- ``"bisquare"``:
- ``"cosine"``:
- ``'boxcar'``/discrete: all distances less than `bandwidth` are 1, and all
other distances are 0
- ``"identity"``/None : do nothing, weight similarity based on raw distance
- ``callable`` : a user-defined function that takes the distance vector and
the bandwidth and returns the kernel: kernel(distances, bandwidth)
k : int (default: None)
number of nearest neighbors used to truncate the kernel. This is assumed
to be constant across samples. If None, no truncation is conduted.
bandwidth : float (default: None)
distance to use in the kernel computation. Should be on the same scale as
the input coordinates.
metric : string or callable (default: 'euclidean')
distance function to apply over the input coordinates. Supported options
depend on whether or not scikit-learn is installed. If so, then any
distance function supported by scikit-learn is supported here. Otherwise,
only euclidean, minkowski, and manhattan/cityblock distances are admitted.
p : int (default: 2)
parameter for minkowski metric, ignored if metric != "minkowski".
coincident: str, optional (default "raise")
Method for handling coincident points when ``k`` is not None. Options are
``'raise'`` (raising an exception when coincident points are present),
``'jitter'`` (randomly displace coincident points to produce uniqueness), &
``'clique'`` (induce fully-connected sub cliques for coincident points).
Returns
-------
Graph
libpysal.graph.Graph encoding kernel weights
"""
ids = _evaluate_index(data)
head, tail, weight = _kernel(
data,
bandwidth=bandwidth,
metric=metric,
kernel=kernel,
k=k,
p=p,
ids=ids,
coincident=coincident,
)
return cls.from_arrays(head, tail, weight)
[docs]
@classmethod
def build_knn(cls, data, k, metric="euclidean", p=2, coincident="raise"):
"""Generate Graph from geometry data based on k-nearest neighbors search
Parameters
----------
data : numpy.ndarray, geopandas.GeoSeries, geopandas.GeoDataFrame
geometries over which to compute a kernel. If a geopandas object with Point
geoemtry is provided, the .geometry attribute is used. If a numpy.ndarray
with shapely geoemtry is used, then the coordinates are extracted and used.
If a numpy.ndarray of a shape (2,n) is used, it is assumed to contain x, y
coordinates.
k : int
number of nearest neighbors.
metric : string or callable (default: 'euclidean')
distance function to apply over the input coordinates. Supported options
depend on whether or not scikit-learn is installed. If so, then any
distance function supported by scikit-learn is supported here. Otherwise,
only euclidean, minkowski, and manhattan/cityblock distances are admitted.
p : int (default: 2)
parameter for minkowski metric, ignored if metric != "minkowski".
coincident: str, optional (default "raise")
Method for handling coincident points. Options include
``'raise'`` (raising an exception when coincident points are present),
``'jitter'`` (randomly displace coincident points to produce uniqueness), &
``'clique'`` (induce fully-connected sub cliques for coincident points).
Returns
-------
Graph
libpysal.graph.Graph encoding KNN weights
"""
ids = _evaluate_index(data)
head, tail, weight = _kernel(
data,
bandwidth=np.inf,
metric=metric,
kernel="boxcar",
k=k,
p=p,
ids=ids,
coincident=coincident,
)
return cls.from_arrays(head, tail, weight)
[docs]
@classmethod
def build_spatial_matches(
cls,
data,
k,
metric="euclidean",
solver=None,
allow_partial_match=False,
**metric_kwargs,
):
"""
Match locations in one dataset to at least `n_matches`
locations in another (possibly identical) dataset
by minimizing the total distance between matched locations.
Letting :math:`d_{ij}` be
.. math::
\\text{minimize} \\sum_i^n \\sum_j^n d_{ij}m_{ij}
\\text{subject to}
\\sum_j^n m_{ij} >= k \\forall i
m_{ij} \\in {0,1} \\forall ij
Parameters
----------
x : numpy.ndarray, geopandas.GeoSeries, geopandas.GeoDataFrame
geometries that need matches. If a geopandas.Geo* object
is provided, the .geometry attribute is used. If a numpy.ndarray with
a geometry dtype is used, then the coordinates are extracted and used.
y : numpy.ndarray, geopandas.GeoSeries, geopandas.GeoDataFrame (default: None)
geometries that are used as a source for matching. If a geopandas object
is provided, the .geometry attribute is used. If a numpy.ndarray with
a geometry dtype is used, then the coordinates are extracted and
used. If none, matches are made within `x`.
n_matches : int (default: None)
number of matches
metric : string or callable (default: 'euclidean')
distance function to apply over the input coordinates. Supported options
depend on whether or not scikit-learn is installed. If so, then any
distance function supported by scikit-learn is supported here. Otherwise,
only euclidean, minkowski, and manhattan/cityblock distances are admitted.
solver : solver from pulp (default: None)
a solver defined by the pulp optimization library. If no solver is
provided, pulp's default solver will be used. This is generally
pulp.COIN(), but this may vary depending on your configuration.
return_mip : bool (default: False)
whether or not to return the instance of the pulp.LpProblem. By
default, the problem is not returned to the user.
allow_partial_match : bool (default: False)
whether to allow for partial matching. A partial match may have
a weight between zero and one, while a "full" match (by default)
must have a weight of either zero or one. A partial matching may
have a shorter total distance, but will result in a weighted
graph.
"""
head, tail, weight = _spatial_matching(
x=data,
metric=metric,
n_matches=k,
solver=solver,
allow_partial_match=allow_partial_match,
**metric_kwargs,
)
# ids need to be addressed here, rather than in the matching
# because x and y can have different id sets. It's only
# in W where we *know* we can just use one id vector.
return cls.from_arrays(head, tail, weight)
[docs]
@classmethod
def build_triangulation(
cls,
data,
method="delaunay",
bandwidth=np.inf,
kernel="boxcar",
clip="bounding_box",
rook=True,
coincident="raise",
):
"""Generate Graph from geometry based on triangulation
Parameters
----------
data : numpy.ndarray, geopandas.GeoSeries, geopandas.GeoDataFrame
geometries containing locations to compute the
delaunay triangulation. If a geopandas object with Point
geoemtry is provided, the .geometry attribute is used. If a numpy.ndarray
with shapely geoemtry is used, then the coordinates are extracted and used.
If a numpy.ndarray of a shape (2,n) is used, it is assumed to contain x, y
coordinates.
method : str, (default "delaunay")
method of extracting the weights from triangulation. Supports:
- ``"delaunay"``
- ``"gabriel"``
- ``"relative_neighborhood"``
- ``"voronoi"``
bandwidth : float, optional
distance to use in the kernel computation. Should be on the same scale as
the input coordinates, by default numpy.inf
kernel : str, optional
kernel function to use in order to weight the output graph. See
:meth:`Graph.build_kernel` for details. By default "boxcar"
clip : str (default: 'bbox')
Clipping method when ``method="voronoi"``. Ignored otherwise.
Default is ``'bounding_box'``. Options are as follows.
* ``None`` -- No clip is applied. Voronoi cells may be arbitrarily
larger that the source map. Note that this may lead to cells that are many
orders of magnitude larger in extent than the original map. Not recommended.
* ``'bounding_box'`` -- Clip the voronoi cells to the
bounding box of the input points.
* ``'convex_hull'`` -- Clip the voronoi cells to the convex hull of
the input points.
* ``'alpha_shape'`` -- Clip the voronoi cells to the tightest hull that
contains all points (e.g. the smallest alpha shape, using
:func:`libpysal.cg.alpha_shape_auto`).
* ``shapely.Polygon`` -- Clip to an arbitrary Polygon.
rook : bool, optional
Contiguity method when ``method="voronoi"``. Ignored otherwise.
If True, two geometries are considered neighbours if they
share at least one edge. If False, two geometries are considered neighbours
if they share at least one vertex. By default True
coincident: str, optional (default "raise")
Method for handling coincident points. Options include
``'raise'`` (raising an exception when coincident points are present),
``'jitter'`` (randomly displace coincident points to produce uniqueness), &
``'clique'`` (induce fully-connected sub cliques for coincident points).
Returns
-------
Graph
libpysal.graph.Graph encoding triangulation weights
"""
ids = _evaluate_index(data)
if method == "delaunay":
head, tail, weights = _delaunay(
data, ids=ids, bandwidth=bandwidth, kernel=kernel, coincident=coincident
)
elif method == "gabriel":
head, tail, weights = _gabriel(
data, ids=ids, bandwidth=bandwidth, kernel=kernel, coincident=coincident
)
elif method == "relative_neighborhood":
head, tail, weights = _relative_neighborhood(
data, ids=ids, bandwidth=bandwidth, kernel=kernel, coincident=coincident
)
elif method == "voronoi":
head, tail, weights = _voronoi(
data, ids=ids, clip=clip, rook=rook, coincident=coincident
)
else:
raise ValueError(
f"Method '{method}' is not supported. Use one of ['delaunay', "
"'gabriel', 'relative_neighborhood', 'voronoi']."
)
return cls.from_arrays(head, tail, weights)
@cached_property
def neighbors(self):
"""Get neighbors dictionary
Notes
-----
It is recommended to work directly with :meth:`Graph.adjacency` rather than
using the :meth:`Graph.neighbors`.
Returns
-------
dict
dict of tuples representing neighbors
"""
return (
self._adjacency.reset_index(level=1)
.groupby(level=0)
.apply(
lambda group: tuple(
group[
~((group.index == group.neighbor) & (group.weight == 0))
].neighbor
)
)
.to_dict()
)
@cached_property
def weights(self):
"""Get weights dictionary
Notes
-----
It is recommended to work directly with :meth:`Graph.adjacency` rather than
using the :meth:`Graph.weights`.
Returns
-------
dict
dict of tuples representing weights
"""
return (
self._adjacency.reset_index(level=1)
.groupby(level=0)
.apply(
lambda group: tuple(
group[
~((group.index == group.neighbor) & (group.weight == 0))
].weight
)
)
.to_dict()
)
@cached_property
def sparse(self):
"""Return a scipy.sparse array (COO)
Returns
-------
scipy.sparse.COO
sparse representation of the adjacency
"""
# pivot to COO sparse matrix and cast to array
return sparse.coo_array(
self._adjacency.astype("Sparse[float]").sparse.to_coo(sort_labels=True)[0]
)
@cached_property
def _components(self):
"""helper for n_components and component_labels"""
# TODO: remove casting to matrix once scipy supports arrays here
return sparse.csgraph.connected_components(sparse.coo_matrix(self.sparse))
@cached_property
def n_components(self):
"""Get a number of connected components
Returns
-------
int
number of components
"""
return self._components[0]
@cached_property
def component_labels(self):
"""Get component labels per observation
Returns
-------
numpy.array
Array of component labels
"""
return pd.Series(
self._components[1], index=self.unique_ids, name="component labels"
)
@cached_property
def cardinalities(self):
"""Number of neighbors for each observation
Returns
-------
pandas.Series
Series with a number of neighbors per each observation
"""
cardinalities = self._adjacency.astype(bool).groupby(level=0).sum()
cardinalities.name = "cardinalities"
return cardinalities
@cached_property
def isolates(self):
"""Index of observations with no neighbors
Isolates are encoded as a self-loop with
the weight == 0 in the adjacency table.
Returns
-------
pandas.Index
Index with a subset of observations that do not have any neighbor
"""
nulls = self._adjacency[self._adjacency == 0]
# since not all zeros are necessarily isolates, do the focal == neighbor check
return (
nulls[nulls.index.codes[0] == nulls.index.codes[1]]
.index.get_level_values(0)
.unique()
)
@cached_property
def unique_ids(self):
"""Unique IDs used in the Graph"""
return self._adjacency.index.get_level_values("focal").unique()
@cached_property
def n(self):
"""Number of observations."""
return self.unique_ids.shape[0]
@cached_property
def n_nodes(self):
"""Number of observations."""
return self.unique_ids.shape[0]
@cached_property
def n_edges(self):
"""Number of observations."""
return self._adjacency.shape[0] - self.isolates.shape[0]
@cached_property
def pct_nonzero(self):
"""Percentage of nonzero weights."""
p = 100.0 * self.sparse.nnz / (1.0 * self.n**2)
return p
@cached_property
def nonzero(self):
"""Number of nonzero weights."""
return (self._adjacency.drop(self.isolates) > 0).sum()
[docs]
def asymmetry(self, intrinsic=True):
r"""Asymmetry check.
Parameters
----------
intrinsic : bool, optional
Default is ``True``. Intrinsic symmetry is defined as:
.. math::
w_{i,j} == w_{j,i}
If ``intrinsic`` is ``False`` symmetry is defined as:
.. math::
i \in N_j \ \& \ j \in N_i
where :math:`N_j` is the set of neighbors for :math:`j`,
e.g., ``True`` requires equality of the weight to consider
two links equal, ``False`` requires only a presence of a link
with a non-zero weight.
Returns
-------
pandas.Series
A ``Series`` of ``(i,j)`` pairs of asymmetries sorted
ascending by the focal observation (index value),
where ``i`` is the focal and ``j`` is the neighbor.
An empty ``Series`` is returned if no asymmetries are found.
"""
if intrinsic:
wd = self.sparse.transpose() - self.sparse
else:
transformed = self.transform("b")
wd = transformed.sparse.transpose() - transformed.sparse
ids = np.nonzero(wd)
if len(ids[0]) == 0:
return pd.Series(
index=pd.Index([], name="focal"),
name="neighbor",
dtype=self._adjacency.index.dtypes["focal"],
)
else:
i2id = dict(
zip(np.arange(self.unique_ids.shape[0]), self.unique_ids, strict=True)
)
focal, neighbor = np.nonzero(wd)
focal = focal.astype(self._adjacency.index.dtypes["focal"])
neighbor = neighbor.astype(self._adjacency.index.dtypes["focal"])
for i in i2id:
focal[focal == i] = i2id[i]
neighbor[neighbor == i] = i2id[i]
ijs = pd.Series(
neighbor, index=pd.Index(focal, name="focal"), name="neighbor"
).sort_index()
return ijs
[docs]
def higher_order(self, k=2, shortest_path=True, diagonal=False, lower_order=False):
"""Contiguity weights object of order :math:`k`.
Proper higher order neighbors are returned such that :math:`i` and :math:`j`
are :math:`k`-order neighbors if the shortest path from :math:`i-j` is of
length :math:`k`.
Parameters
----------
k : int, optional
Order of contiguity. By default 2.
shortest_path : bool, optional
If True, :math:`i,j` and :math:`k`-order neighbors if the shortest
path for :math:`i,j` is :math:`k`. If False, :math:`i,j` are
`k`-order neighbors if there is a path from :math:`i,j` of length
:math:`k`. By default True.
diagonal : bool, optional
If True, keep :math:`k`-order (:math:`i,j`) joins when :math:`i==j`.
If False, remove :math:`k`-order (:math:`i,j`) joins when
:math:`i==j`. By default False.
lower_order : bool, optional
If True, include lower order contiguities. If False return only weights of
order :math:`k`. By default False.
Returns
-------
Graph
higher order weights
"""
# TODO: remove casting to matrix once scipy implements matrix_power on array.
binary = self.transform("B")
sp = sparse.csr_matrix(binary.sparse)
if lower_order:
wk = sum(sp**x for x in range(2, k + 1))
shortest_path = False
else:
wk = sp**k
rk, ck = wk.nonzero()
sk = set(zip(rk, ck, strict=True))
if shortest_path:
for j in range(1, k):
wj = sp**j
rj, cj = wj.nonzero()
sj = set(zip(rj, cj, strict=True))
sk.difference_update(sj)
if not diagonal:
sk = {(i, j) for i, j in sk if i != j}
return Graph.from_sparse(
sparse.coo_array(
(
np.ones(len(sk), dtype=np.int8),
([s[0] for s in sk], [s[1] for s in sk]),
),
shape=sp.shape,
),
ids=self.unique_ids,
)
[docs]
def lag(self, y):
"""Spatial lag operator
If weights are row standardized, returns the mean of each
observation's neighbors; if not, returns the weighted sum
of each observation's neighbors.
Parameters
----------
y : array-like
array-like (N,) shape where N is equal to number of observations in self.
Returns
-------
numpy.ndarray
array of numeric values for the spatial lag
"""
return _lag_spatial(self, y)
[docs]
def to_parquet(self, path, **kwargs):
"""Save Graph to a Apache Parquet
Graph is serialized to the Apache Parquet using the underlying adjacency
object stored as a Parquet table and custom metadata containing transformation.
Requires pyarrow package.
Parameters
----------
path : str | pyarrow.NativeFile
path or any stream supported by pyarrow
**kwargs
additional keyword arguments passed to pyarrow.parquet.write_table
See also
--------
read_parquet
"""
_to_parquet(self, path, **kwargs)
[docs]
def to_networkx(self):
"""Convert Graph to a ``networkx`` graph.
If Graph is symmetric, returns ``nx.Graph``, otherwise returns a ``nx.DiGraph``.
Returns
-------
networkx.Graph | networkx.DiGraph
Representation of libpysal Graph as networkx graph
"""
try:
import networkx as nx
except ImportError:
raise ImportError("NetworkX is required.") from None
graph_type = nx.Graph if self.asymmetry().empty else nx.DiGraph
return nx.from_pandas_edgelist(
self._adjacency.reset_index(),
source="focal",
target="neighbor",
edge_attr="weight",
create_using=graph_type,
)
[docs]
def plot(
self,
gdf,
focal=None,
nodes=True,
color="k",
edge_kws=None,
node_kws=None,
focal_kws=None,
ax=None,
figsize=None,
limit_extent=False,
):
"""Plot edges and nodes of the Graph
Creates a ``maptlotlib`` plot based on the topology stored in the
Graph and spatial location defined in ``gdf``.
Parameters
----------
gdf : geopandas.GeoDataFrame
Geometries indexed using the same index as Graph. Geometry types other than
points are converted to centroids encoding start and end point of Graph
edges.
focal : hashable | array-like[hashable] | None, optional
ID or an array-like of IDs of focal geometries whose weights shall be
plotted. If None, all weights from all focal geometries are plotted.
By default None
nodes : bool, optional
Plot nodes as points, by default True
color : str, optional
The color of all objects, by default "k"
edge_kws : dict, optional
Keyword arguments dictionary to send to ``LineCollection``,
which provides fine-grained control over the aesthetics
of the edges in the plot. By default None
node_kws : dict, optional
Keyword arguments dictionary to send to ``ax.scatter``,
which provides fine-grained control over the aesthetics
of the nodes in the plot. By default None
focal_kws : dict, optional
Keyword arguments dictionary to send to ``ax.scatter``,
which provides fine-grained control over the aesthetics
of the focal nodes in the plot on top of generic ``node_kws``.
Values of ``node_kws`` are updated from ``focal_kws``.
Ignored if ``focal=None``. By default None
ax : matplotlib.axes.Axes, optional
Axis on which to plot the weights. If None, a new figure and axis are
created. By default None
figsize : tuple, optional
figsize used to create a new axis. By default None
limit_extent : bool, optional
limit the extent of the axis to the extent of the plotted graph, by default
False
Returns
-------
matplotlib.axes.Axes
Axis with the resulting plot
Notes
-----
If you'd like to overlay the actual geometries from the
``geopandas.GeoDataFrame``, create an axis by plotting the ``GeoDataFrame``
and plot the Graph on top.
ax = gdf.plot()
gdf_graph.plot(gdf, ax=ax)
"""
return _plot(
self,
gdf,
focal=focal,
nodes=nodes,
color=color,
node_kws=node_kws,
edge_kws=edge_kws,
focal_kws=focal_kws,
ax=ax,
figsize=figsize,
limit_extent=limit_extent,
)
[docs]
def explore(
self,
gdf,
focal=None,
nodes=True,
color="black",
edge_kws=None,
node_kws=None,
focal_kws=None,
m=None,
**kwargs,
):
"""Plot graph as an interactive Folium Map
Parameters
----------
gdf : geopandas.GeoDataFrame
geodataframe used to instantiate to Graph
focal : list, optional
subset of focal observations to plot in the map, by default None.
If none, all relationships are plotted
nodes : bool, optional
whether to display observations as nodes in the map, by default True
color : str, optional
color applied to nodes and edges, by default "black"
edge_kws : dict, optional
additional keyword arguments passed to geopandas explore function
when plotting edges, by default None
node_kws : dict, optional
additional keyword arguments passed to geopandas explore function
when plotting nodes, by default None
focal_kws : dict, optional
additional keyword arguments passed to geopandas explore function
when plotting focal observations, by default None. Only applicable when
passing a subset of nodes with the `focal` argument
m : Folilum.Map, optional
folium map objecto to plot on top of, by default None
**kwargs : dict, optional
additional keyword arguments are passed directly to geopandas.explore, when
``m=None`` by default None
Returns
-------
folium.Map
folium map
"""
return _explore_graph(
self,
gdf,
focal=focal,
nodes=nodes,
color=color,
edge_kws=edge_kws,
node_kws=node_kws,
focal_kws=focal_kws,
m=m,
**kwargs,
)
[docs]
def subgraph(self, ids):
"""Returns a subset of Graph containing only nodes specified in ids
The resulting subgraph contains only the nodes in ``ids`` and the edges
between them or zero-weight self-loops in case of isolates.
The order of ``ids`` reflects a new canonical order of the resulting
subgraph. This means ``ids`` should be equal to the index of the DataFrame
containing data linked to the graph to ensure alignment of sparse representation
of subgraph.
Parameters
----------
ids : array-like
An array of node IDs to be retained
Returns
-------
Graph
A new Graph that is a subset of the original
Notes
-----
Unlike the implementation in ``networkx``, this creates a copy since
Graphs in ``libpysal`` are immutable.
"""
masked_adj = self._adjacency[ids]
filtered_adj = masked_adj[
masked_adj.index.get_level_values("neighbor").isin(ids)
]
return Graph.from_arrays(
*_resolve_islands(
filtered_adj.index.get_level_values("focal"),
filtered_adj.index.get_level_values("neighbor"),
ids,
filtered_adj.values,
)
)
[docs]
def eliminate_zeros(self):
"""Remove graph edges with zero weight
Eliminates edges with weight == 0 that do not encode an
isolate. This is useful to clean-up edges that will make
no effect in operations like :meth:`lag`.
Returns
-------
Graph
subset of Graph with zero-weight edges eliminated
"""
# get a mask for isolates
isolates = self._adjacency.index.codes[0] == self._adjacency.index.codes[1]
# substract isolates from mask of zeros
zeros = (self._adjacency == 0) != isolates
return Graph(self._adjacency[~zeros], is_sorted=True)
[docs]
def assign_self_weight(self, weight=1):
"""Assign values to edges representing self-weight.
The value for each ``focal == neighbor`` location in
the graph is set to ``weight``.
Parameters
----------
weight : float | array-like
Defines the value(s) to which the weight representing the relationship with
itself should be set. If a constant is passed then each self-weight will get
this value (default is 1). An array of length ``Graph.n`` can be passed to
set explicit values to each self-weight (assumed to be in the same order as
original data).
Returns
-------
Graph
A new ``Graph`` with added self-weights.
"""
addition = pd.Series(
weight,
index=pd.MultiIndex.from_arrays(
[self.unique_ids, self.unique_ids], names=["focal", "neighbor"]
),
name="weight",
)
adj = (
pd.concat([self.adjacency.drop(self.isolates), addition])
.reindex(self.unique_ids, level=0)
.reindex(self.unique_ids, level=1)
)
return Graph(adj, is_sorted=True)
[docs]
def apply(self, y, func, **kwargs):
"""Apply a reduction across the neighbor sets
Applies ``func`` over groups of ``y`` defined by neighbors for each focal.
Parameters
----------
y : array_like
array of values to be grouped. Can be 1-D or 2-D and will be coerced to a
pandas object
func : function, str, list, dict or None
Function to use for aggregating the data passed to pandas ``GroupBy.apply``.
Returns
-------
Series | DataFrame
pandas object indexed by unique_ids
"""
if not isinstance(y, pd.Series | pd.DataFrame):
y = pd.DataFrame(y) if hasattr(y, "ndim") and y.ndim == 2 else pd.Series(y)
grouper = y.take(self._adjacency.index.codes[1]).groupby(
self._adjacency.index.codes[0]
)
result = grouper.apply(func, **kwargs)
result.index = self.unique_ids
if isinstance(result, pd.Series):
result.name = None
return result
[docs]
def aggregate(self, func):
"""Aggregate weights within a neighbor set
Apply a custom aggregation function to a group of weights of the same focal
geometry.
Parameters
----------
func : callable
A callable accepted by pandas ``groupby.agg`` method
Returns
-------
pd.Series
Aggregated weights
"""
return self._adjacency.groupby(level=0).agg(func)
def _arrange_arrays(heads, tails, weights, ids=None):
"""
Rearrange input arrays so that observation indices
are well-ordered with respect to the input ids. That is,
an "early" identifier should always preceed a "later" identifier
in the heads, but the tails should be sorted with respect
to heads *first*, then sorted within the tails.
"""
if ids is None:
ids = np.unique(np.hstack((heads, tails)))
lookup = list(ids).index
input_df = pd.DataFrame.from_dict(
{"focal": heads, "neighbor": tails, "weight": weights}
)
return (
input_df.set_index(["focal", "neighbor"])
.assign(
focal_loc=input_df.focal.apply(lookup).values,
neighbor_loc=input_df.neighbor.apply(lookup).values,
)
.sort_values(["focal_loc", "neighbor_loc"])
.reset_index()
.drop(["focal_loc", "neighbor_loc"], axis=1)
.values.T
)
def read_parquet(path, **kwargs):
"""Read Graph from a Apache Parquet
Read Graph serialized using `Graph.to_parquet()` back into the `Graph` object.
The Parquet file needs to contain adjacency table with a structure required
by the `Graph` constructor and optional metadata with the type of transformation.
Parameters
----------
path : str | pyarrow.NativeFile | file-like object
path or any stream supported by pyarrow
**kwargs
additional keyword arguments passed to pyarrow.parquet.read_table
Returns
-------
Graph
deserialized Graph
"""
adjacency, transformation = _read_parquet(path, **kwargs)
return Graph(adjacency, transformation, is_sorted=True)