libpysal.weights.W¶

class libpysal.weights.W(neighbors, weights=None, id_order=None, silence_warnings=False, ids=None)[source]¶

Spatial weights class. Class attributes are described by their docstrings. to view, use the help function.

Parameters:

neighborsdict: Key is region ID, value is a list of neighbor IDS. For example, {'a':['b'],'b':['a','c'],'c':['b']}.
weightsdict: Key is region ID, value is a list of edge weights. If not supplied all edge weights are assumed to have a weight of 1. For example, {'a':[0.5],'b':[0.5,1.5],'c':[1.5]}.
id_orderlist: An ordered list of ids, defines the order of observations when iterating over W if not set, lexicographical ordering is used to iterate and the id_order_set property will return False. This can be set after creation by setting the id_order property.
silence_warningsbool: By default libpysal will print a warning if the dataset contains any disconnected components or islands. To silence this warning set this parameter to True.
idslist: Values to use for keys of the neighbors and weights dict objects.

Attributes:

asymmetries: List of id pairs with asymmetric weights sorted in ascending index location order.
cardinalities: Number of neighbors for each observation.
component_labels: Store the graph component in which each observation falls.
diagW2: Diagonal of \(WW\).
diagWtW: Diagonal of \(W^{'}W\).
diagWtW_WW: Diagonal of \(W^{'}W + WW\).
histogram: Cardinality histogram as a dictionary where key is the id and value is the number of neighbors for that unit.
id2i: Dictionary where the key is an ID and the value is that ID’s index in W.id_order.
id_order: Returns the ids for the observations in the order in which they would be encountered if iterating over the weights.
id_order_set: Returns True if user has set id_order, False if not.
islands: List of ids without any neighbors.
max_neighbors: Largest number of neighbors.
mean_neighbors: Average number of neighbors.
min_neighbors: Minimum number of neighbors.
n: Number of units.
n_components: Store whether the adjacency matrix is fully connected.
neighbor_offsets: Given the current id_order, neighbor_offsets[id] is the offsets of the id’s neighbors in id_order.
nonzero: Number of nonzero weights.
pct_nonzero: Percentage of nonzero weights.
s0: s0 is defined as
s1: s1 is defined as
s2: s2 is defined as
s2array: Individual elements comprising s2.
sd: Standard deviation of number of neighbors.
sparse: Sparse matrix object.
trcW2: Trace of \(WW\).
trcWtW: Trace of \(W^{'}W\).
trcWtW_WW: Trace of \(W^{'}W + WW\).
transform: Getter for transform property.

Examples

>>> from libpysal.weights import W
>>> neighbors = {0: [3, 1], 1: [0, 4, 2], 2: [1, 5], 3: [0, 6, 4], 4: [1, 3, 7, 5], 5: [2, 4, 8], 6: [3, 7], 7: [4, 6, 8], 8: [5, 7]}
>>> weights = {0: [1, 1], 1: [1, 1, 1], 2: [1, 1], 3: [1, 1, 1], 4: [1, 1, 1, 1], 5: [1, 1, 1], 6: [1, 1], 7: [1, 1, 1], 8: [1, 1]}
>>> w = W(neighbors, weights)
>>> "%.3f"%w.pct_nonzero
'29.630'

Read from external .gal file.

>>> import libpysal
>>> w = libpysal.io.open(libpysal.examples.get_path("stl.gal")).read()
>>> w.n
78
>>> "%.3f"%w.pct_nonzero
'6.542'

Set weights implicitly.

>>> neighbors = {0: [3, 1], 1: [0, 4, 2], 2: [1, 5], 3: [0, 6, 4], 4: [1, 3, 7, 5], 5: [2, 4, 8], 6: [3, 7], 7: [4, 6, 8], 8: [5, 7]}
>>> w = W(neighbors)
>>> round(w.pct_nonzero,3)
29.63
>>> from libpysal.weights import lat2W
>>> w = lat2W(100, 100)
>>> w.trcW2
39600.0
>>> w.trcWtW
39600.0
>>> w.transform='r'
>>> round(w.trcW2, 3)
2530.722
>>> round(w.trcWtW, 3)
2533.667

Cardinality Histogram:

>>> w.histogram
[(2, 4), (3, 392), (4, 9604)]

Disconnected observations (islands):

>>> from libpysal.weights import W
>>> w = W({1:[0],0:[1],2:[], 3:[]})

UserWarning: The weights matrix is not fully connected: There are 3 disconnected components. There are 2 islands with ids: 2, 3.

__init__(neighbors, weights=None, id_order=None, silence_warnings=False, ids=None)[source]¶

Methods

`__init__`(neighbors[, weights, id_order, ...])
`asymmetry`([intrinsic])	Asymmetry check.
`from_WSP`(WSP[, silence_warnings])	Create a pysal W from a pysal WSP object (thin weights matrix).
`from_adjlist`(adjlist[, focal_col, ...])	Return an adjacency list representation of a weights object.
`from_file`([path, format])	Read a weights file into a W object.
`from_networkx`(graph[, weight_col])	Convert a `networkx` graph to a PySAL `W` object.
`from_shapefile`(args, *kwargs)
`from_sparse`(sparse)	Convert a `scipy.sparse` array to a PySAL `W` object.
`full`()	Generate a full `numpy.ndarray`.
`get_transform`()	Getter for transform property.
`plot`(gdf[, indexed_on, ax, color, node_kws, ...])	Plot spatial weights objects.
`remap_ids`(new_ids)	In place modification throughout `W` of id values from `w.id_order` to `new_ids` in all.
`set_shapefile`(shapefile[, idVariable, full])	Adding metadata for writing headers of `.gal` and `.gwt` files.
`set_transform`([value])	Transformations of weights.
`symmetrize`([inplace])	Construct a symmetric KNN weight.
`to_WSP`()	Generate a `WSP` object.
`to_adjlist`([remove_symmetric, drop_islands, ...])	Compute an adjacency list representation of a weights object.
`to_file`([path, format])	Write a weights to a file.
`to_networkx`()	Convert a weights object to a `networkx` graph.
`to_sparse`([fmt])	Generate a `scipy.sparse` array object from a pysal W.

Attributes

`asymmetries`	List of id pairs with asymmetric weights sorted in ascending index location order.
`cardinalities`	Number of neighbors for each observation.
`component_labels`	Store the graph component in which each observation falls.
`diagW2`	Diagonal of \(WW\).
`diagWtW`	Diagonal of \(W^{'}W\).
`diagWtW_WW`	Diagonal of \(W^{'}W + WW\).
`histogram`	Cardinality histogram as a dictionary where key is the id and value is the number of neighbors for that unit.
`id2i`	Dictionary where the key is an ID and the value is that ID's index in `W.id_order`.
`id_order`	Returns the ids for the observations in the order in which they would be encountered if iterating over the weights.
`id_order_set`	Returns `True` if user has set `id_order`, `False` if not.
`islands`	List of ids without any neighbors.
`max_neighbors`	Largest number of neighbors.
`mean_neighbors`	Average number of neighbors.
`min_neighbors`	Minimum number of neighbors.
`n`	Number of units.
`n_components`	Store whether the adjacency matrix is fully connected.
`neighbor_offsets`	Given the current `id_order`, `neighbor_offsets[id]` is the offsets of the id's neighbors in `id_order`.
`nonzero`	Number of nonzero weights.
`pct_nonzero`	Percentage of nonzero weights.
`s0`	`s0` is defined as
`s1`	`s1` is defined as
`s2`	`s2` is defined as
`s2array`	Individual elements comprising `s2`.
`sd`	Standard deviation of number of neighbors.
`sparse`	Sparse matrix object.
`transform`	Getter for transform property.
`trcW2`	Trace of \(WW\).
`trcWtW`	Trace of \(W^{'}W\).
`trcWtW_WW`	Trace of \(W^{'}W + WW\).

property asymmetries¶: List of id pairs with asymmetric weights sorted in ascending index location order.

asymmetry(intrinsic=True)[source]¶

Asymmetry check.

Parameters:

intrinsicbool

Default is True. Intrinsic symmetry is defined as:

\[w_{i,j} == w_{j,i}\]

If intrinsic is False symmetry is defined as:

\[i \in N_j \ \& \ j \in N_i\]

where \(N_j\) is the set of neighbors for \(j\).

Returns:

asymmetrieslist: Empty if no asymmetries are found. If there are asymmetries, then a list of (i,j) tuples is returned sorted in ascending index location order.

Examples

>>> from libpysal.weights import lat2W
>>> w=lat2W(3,3)
>>> w.asymmetry()
[]
>>> w.transform='r'
>>> w.asymmetry()
[(0, 1), (0, 3), (1, 0), (1, 2), (1, 4), (2, 1), (2, 5), (3, 0), (3, 4), (3, 6), (4, 1), (4, 3), (4, 5), (4, 7), (5, 2), (5, 4), (5, 8), (6, 3), (6, 7), (7, 4), (7, 6), (7, 8), (8, 5), (8, 7)]
>>> result = w.asymmetry(intrinsic=False)
>>> result
[]
>>> neighbors={0:[1,2,3], 1:[1,2,3], 2:[0,1], 3:[0,1]}
>>> weights={0:[1,1,1], 1:[1,1,1], 2:[1,1], 3:[1,1]}
>>> w=W(neighbors,weights)
>>> w.asymmetry()
[(0, 1), (1, 0)]

property cardinalities¶: Number of neighbors for each observation.

property component_labels¶: Store the graph component in which each observation falls.

property diagW2¶

Diagonal of \(WW\).

See also

trcW2

property diagWtW¶

Diagonal of \(W^{'}W\).

See also

trcWtW

property diagWtW_WW¶: Diagonal of \(W^{'}W + WW\).

classmethod from_WSP(WSP, silence_warnings=True)[source]¶

Create a pysal W from a pysal WSP object (thin weights matrix).

Parameters:

wspWSP: PySAL sparse weights object
silence_warningsbool: By default libpysal will print a warning if the dataset contains any disconnected components or islands. To silence this warning set this parameter to True.

Returns:

wW: PySAL weights object

Examples

>>> from libpysal.weights import lat2W, WSP, W

Build a 10x10 scipy.sparse matrix for a rectangular 2x5 region of cells (rook contiguity), then construct a PySAL sparse weights object (wsp).

>>> sp = lat2SW(2, 5)
>>> wsp = WSP(sp)
>>> wsp.n
10
>>> wsp.sparse[0].todense()
matrix([[0, 1, 0, 0, 0, 1, 0, 0, 0, 0]], dtype=int8)

Create a standard PySAL W from this sparse weights object.

>>> w = W.from_WSP(wsp)
>>> w.n
10
>>> print(w.full()[0][0])
[0 1 0 0 0 1 0 0 0 0]

classmethod from_adjlist(adjlist, focal_col='focal', neighbor_col='neighbor', weight_col=None)[source]¶

Return an adjacency list representation of a weights object.

Parameters:

adjlistpandas.DataFrame: Adjacency list with a minimum of two columns.
focal_colstr: Name of the column with the “source” node ids.
neighbor_colstr: Name of the column with the “destination” node ids.
weight_colstr: Name of the column with the weight information. If not provided and the dataframe has no column named “weight” then all weights are assumed to be 1.

classmethod from_file(path='', format=None)[source]¶

Read a weights file into a W object.

Parameters:

pathstr: location to save the file
formatstr: string denoting the format to write the weights to.

Returns:

W object

classmethod from_networkx(graph, weight_col='weight')[source]¶

Convert a networkx graph to a PySAL W object.

Parameters:

graphnetworkx.Graph: The graph to convert to a W.
weight_colstr: If the graph is labeled, this should be the name of the field to use as the weight for the W.

Returns:

wlibpysal.weights.W: A W object containing the same graph as the networkx graph.

classmethod from_shapefile(*args, **kwargs)[source]¶

classmethod from_sparse(sparse)[source]¶

Convert a scipy.sparse array to a PySAL W object.

Parameters:

sparsescipy.sparse array

Returns:

wlibpysal.weights.W: A W object containing the same graph as the scipy.sparse graph.

Notes

When the sparse array has a zero in its data attribute, and the corresponding row and column values are equal, the value for the pysal weight will be 0 for the “loop”.

full()[source]¶

Generate a full numpy.ndarray.

Parameters:

selflibpysal.weights.W: spatial weights object

Returns:

(fullw, keys)tuple: The first element being the full numpy.ndarray and second element keys being the ids associated with each row in the array.

Examples

>>> from libpysal.weights import W, full
>>> neighbors = {
...     'first':['second'],'second':['first','third'],'third':['second']
... }
>>> weights = {'first':[1],'second':[1,1],'third':[1]}
>>> w = W(neighbors, weights)
>>> wf, ids = full(w)
>>> wf
array([[0., 1., 0.],
       [1., 0., 1.],
       [0., 1., 0.]])
>>> ids
['first', 'second', 'third']

get_transform()[source]¶

Getter for transform property.

Returns:

transformationstr, None: Valid transformation value. See the transform parameters in set_transform() for a detailed description.

See also

set_transform

Examples

>>> from libpysal.weights import lat2W
>>> w=lat2W()
>>> w.weights[0]
[1.0, 1.0]
>>> w.transform
'O'
>>> w.transform='r'
>>> w.weights[0]
[0.5, 0.5]
>>> w.transform='b'
>>> w.weights[0]
[1.0, 1.0]

property histogram¶: Cardinality histogram as a dictionary where key is the id and value is the number of neighbors for that unit.

property id2i¶: Dictionary where the key is an ID and the value is that ID’s index in W.id_order.

property id_order¶: Returns the ids for the observations in the order in which they would be encountered if iterating over the weights.

property id_order_set¶

Returns True if user has set id_order, False if not.

Examples

>>> from libpysal.weights import lat2W
>>> w=lat2W()
>>> w.id_order_set
True

property islands¶: List of ids without any neighbors.

property max_neighbors¶: Largest number of neighbors.

property mean_neighbors¶: Average number of neighbors.

property min_neighbors¶: Minimum number of neighbors.

property n¶: Number of units.

property n_components¶: Store whether the adjacency matrix is fully connected.

property neighbor_offsets¶

Given the current id_order, neighbor_offsets[id] is the offsets of the id’s neighbors in id_order.

Returns:

neighbor_listlist: Offsets of the id’s neighbors in id_order.

Examples

>>> from libpysal.weights import W
>>> neighbors={'c': ['b'], 'b': ['c', 'a'], 'a': ['b']}
>>> weights ={'c': [1.0], 'b': [1.0, 1.0], 'a': [1.0]}
>>> w=W(neighbors,weights)
>>> w.id_order = ['a','b','c']
>>> w.neighbor_offsets['b']
[2, 0]
>>> w.id_order = ['b','a','c']
>>> w.neighbor_offsets['b']
[2, 1]

property nonzero¶: Number of nonzero weights.

property pct_nonzero¶: Percentage of nonzero weights.

plot(gdf, indexed_on=None, ax=None, color='k', node_kws=None, edge_kws=None)[source]¶

Plot spatial weights objects. Requires matplotlib, and implicitly requires a geopandas.GeoDataFrame as input.

Parameters:

gdfgeopandas.GeoDataFrame: The original shapes whose topological relations are modelled in W.
indexed_onstr: Column of geopandas.GeoDataFrame that the weights object uses as an index. Default is None, so the index of the geopandas.GeoDataFrame is used.
axmatplotlib.axes.Axes: Axis on which to plot the weights. Default is None, so plots on the current figure.
colorstr: matplotlib color string, will color both nodes and edges the same by default.
node_kwsdict: Keyword arguments dictionary to send to pyplot.scatter, which provides fine-grained control over the aesthetics of the nodes in the plot.
edge_kwsdict: Keyword arguments dictionary to send to pyplot.plot, which provides fine-grained control over the aesthetics of the edges in the plot.

Returns:

fmatplotlib.figure.Figure: Figure on which the plot is made.
axmatplotlib.axes.Axes: Axis on which the plot is made.

Notes

If you’d like to overlay the actual shapes from the geopandas.GeoDataFrame, call gdf.plot(ax=ax) after this. To plot underneath, adjust the z-order of the plot as follows: gdf.plot(ax=ax,zorder=0).

Examples

>>> from libpysal.weights import Queen
>>> import libpysal as lp
>>> import geopandas
>>> gdf = geopandas.read_file(lp.examples.get_path("columbus.shp"))
>>> weights = Queen.from_dataframe(gdf)
>>> tmp = weights.plot(
...     gdf, color='firebrickred', node_kws=dict(marker='*', color='k')
... )

remap_ids(new_ids)[source]¶

In place modification throughout W of id values from w.id_order to new_ids in all.

Parameters:

new_idslist, numpy.ndarray: Aligned list of new ids to be inserted. Note that first element of new_ids will replace first element of w.id_order, second element of new_ids replaces second element of w.id_order and so on.

Examples

>>> from libpysal.weights import lat2W
>>> w = lat2W(3, 3)
>>> w.id_order
[0, 1, 2, 3, 4, 5, 6, 7, 8]
>>> w.neighbors[0]
[3, 1]
>>> new_ids = ['id%i'%id for id in w.id_order]
>>> _ = w.remap_ids(new_ids)
>>> w.id_order
['id0', 'id1', 'id2', 'id3', 'id4', 'id5', 'id6', 'id7', 'id8']
>>> w.neighbors['id0']
['id3', 'id1']

property s0¶: s0 is defined as

\[s0=\sum_i \sum_j w_{i,j}\]

property s1¶: s1 is defined as

\[s1=1/2 \sum_i \sum_j \Big(w_{i,j} + w_{j,i}\Big)^2\]

property s2¶: s2 is defined as

\[s2=\sum_j \Big(\sum_i w_{i,j} + \sum_i w_{j,i}\Big)^2\]

property s2array¶

Individual elements comprising s2.

See also

s2

property sd¶: Standard deviation of number of neighbors.

set_shapefile(shapefile, idVariable=None, full=False)[source]¶

Adding metadata for writing headers of .gal and .gwt files.

Parameters:

shapefilestr: The shapefile name used to construct weights.
idVariablestr: The name of the attribute in the shapefile to associate with ids in the weights.
fullbool: Write out the entire path for a shapefile (True) or only the base of the shapefile without extension (False). Default is True.

set_transform(value='B')[source]¶

Transformations of weights.

Parameters:

transformstr

This parameter is not case sensitive. The following are valid transformations.

B – Binary
R – Row-standardization (global sum \(=n\))
D – Double-standardization (global sum \(=1\))
V – Variance stabilizing
O – Restore original transformation (from instantiation)

Notes

Transformations are applied only to the value of the weights at instantiation. Chaining of transformations cannot be done on a W instance.

Examples

>>> from libpysal.weights import lat2W
>>> w=lat2W()
>>> w.weights[0]
[1.0, 1.0]
>>> w.transform
'O'
>>> w.transform='r'
>>> w.weights[0]
[0.5, 0.5]
>>> w.transform='b'
>>> w.weights[0]
[1.0, 1.0]

property sparse¶: Sparse matrix object. For any matrix manipulations required for w, w.sparse should be used. This is based on scipy.sparse.

symmetrize(inplace=False)[source]¶: Construct a symmetric KNN weight. This ensures that the neighbors of each focal observation consider the focal observation itself as a neighbor. This returns a generic W object, since the object is no longer guaranteed to have k neighbors for each observation.

to_WSP()[source]¶

Generate a WSP object.

Returns:

implicitlibpysal.weights.WSP: Thin W class

See also

WSP

Examples

>>> from libpysal.weights import W, WSP
>>> neighbors={'first':['second'],'second':['first','third'],'third':['second']}
>>> weights={'first':[1],'second':[1,1],'third':[1]}
>>> w=W(neighbors,weights)
>>> wsp=w.to_WSP()
>>> isinstance(wsp, WSP)
True
>>> wsp.n
3
>>> wsp.s0
4

to_adjlist(remove_symmetric=False, drop_islands=None, focal_col='focal', neighbor_col='neighbor', weight_col='weight', sort_joins=False)[source]¶

Compute an adjacency list representation of a weights object.

Parameters:

remove_symmetricbool: Whether or not to remove symmetric entries. If the W is symmetric, a standard directed adjacency list will contain both the forward and backward links by default because adjacency lists are a directed graph representation. If this is True, a W created from this adjacency list MAY NOT BE THE SAME as the original W. If you would like to consider (1,2) and (2,1) as distinct links, leave this as False.
drop_islandsbool: Whether or not to preserve islands as entries in the adjacency list. By default, observations with no neighbors do not appear in the adjacency list. If islands are kept, they are coded as self-neighbors with zero weight.
focal_colstr: Name of the column in which to store “source” node ids.
neighbor_colstr: Name of the column in which to store “destination” node ids.
weight_colstr: Name of the column in which to store weight information.
sort_joinsbool: Whether or not to lexicographically sort the adjacency list by (focal_col, neighbor_col). Default is False.

to_file(path='', format=None)[source]¶

Write a weights to a file. The format is guessed automatically from the path, but can be overridden with the format argument.

See libpysal.io.FileIO for more information.

Parameters:

pathstr: location to save the file
formatstr: string denoting the format to write the weights to.

Returns:

None

to_networkx()[source]¶

Convert a weights object to a networkx graph.

Returns:

A networkx graph representation of the W object.

to_sparse(fmt='coo')[source]¶

Generate a scipy.sparse array object from a pysal W.

Parameters:

fmt{‘bsr’, ‘coo’, ‘csc’, ‘csr’}: scipy.sparse format

Returns:

scipy.sparse array: A scipy.sparse array with a format of fmt.

Notes

The keys of the w.neighbors are encoded to determine row,col in the sparse array.

property transform¶

Getter for transform property.

Returns:

transformationstr, None: Valid transformation value. See the transform parameters in set_transform() for a detailed description.

See also

set_transform

Examples

>>> from libpysal.weights import lat2W
>>> w=lat2W()
>>> w.weights[0]
[1.0, 1.0]
>>> w.transform
'O'
>>> w.transform='r'
>>> w.weights[0]
[0.5, 0.5]
>>> w.transform='b'
>>> w.weights[0]
[1.0, 1.0]

property trcW2¶

Trace of \(WW\).

See also

diagW2

property trcWtW¶

Trace of \(W^{'}W\).

See also

diagWtW

property trcWtW_WW¶: Trace of \(W^{'}W + WW\).