# 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.

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'


>>> import libpysal
>>> 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.

Attributes:
asymmetries

List of id pairs with asymmetric weights.

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.

__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]) 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) Generate a full numpy.ndarray. 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. 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. Convert a weights object to a networkx graph.

Attributes

 asymmetries List of id pairs with asymmetric weights. 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.

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 asymmetries, then a list of (i,j) tuples is returned.

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 $$WW$$.

property diagWtW

Diagonal of $$W^{'}W$$.

property diagWtW_WW

Diagonal of $$W^{'}W + WW$$.

classmethod from_WSP(WSP, silence_warnings=True)[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]
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:
transformation

Valid transformation value. See the transform parameters in set_transform() for a detailed description.

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
>>> 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_ids

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.

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

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


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.

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.

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.
property transform

Getter for transform property.

Returns:
transformation

Valid transformation value. See the transform parameters in set_transform() for a detailed description.

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 $$WW$$.

property trcWtW

Trace of $$W^{'}W$$.

property trcWtW_WW

Trace of $$W^{'}W + WW$$.