def __init__(self, neighbors, weights=None, id_order=None, silent_island_warning=False):
self.silent_island_warning = silent_island_warning
self.transformations = {}
self.neighbors = ROD(neighbors)
if not weights:
weights = {}
for key in neighbors:
weights[key] = [1.] * len(neighbors[key])
self.weights = ROD(weights)
self.transformations['O'] = self.weights # original weights
self.transform = 'O'
if id_order is None:
self._id_order = self.neighbors.keys()
self._id_order.sort()
self._id_order_set = False
else:
self._id_order = id_order
self._id_order_set = True
self._reset()
self._n = len(self.weights)
if self.islands and not self.silent_island_warning:
ni = len(self.islands)
if ni == 1:
print "WARNING: there is one disconnected observation (no neighbors)"
print "Island id: ",self.islands
else:
print "WARNING: there are %d disconnected observations"%ni
print "Island ids: ",self.islands
def __init__(self, neighbors, weights=None, id_order=None):
self.transformations = {}
self.neighbors = ROD(neighbors)
if not weights:
weights = {}
for key in neighbors:
weights[key] = [1.] * len(neighbors[key])
self.weights = ROD(weights)
self.transformations['O'] = self.weights # original weights
self.transform = 'O'
if id_order is None:
self._id_order = self.neighbors.keys()
self._id_order.sort()
self._id_order_set = False
else:
self._id_order = id_order
self._id_order_set = True
self._reset()
self._n = len(self.weights)
def id2i(self):
"""
Dictionary where the key is an ID and the value is that ID's
index in W.id_order.
"""
if 'id2i' not in self._cache:
self._id2i = {}
for i, id in enumerate(self._id_order):
self._id2i[id] = i
self._id2i = ROD(self._id2i)
self._cache['id2i'] = self._id2i
return self._id2i
def __init__(self, neighbors, weights=None, id_order=None):
self.transformations = {}
self.neighbors = ROD(neighbors)
if not weights:
weights = {}
for key in neighbors:
weights[key] = [1.] * len(neighbors[key])
self.weights = ROD(weights)
self.transformations['O'] = self.weights # original weights
self.transform = 'O'
if id_order is None:
self._id_order = self.neighbors.keys()
self._id_order.sort()
self._id_order_set = False
else:
self._id_order = id_order
self._id_order_set = True
self._reset()
self._n = len(self.weights)
class W(object):
"""
Spatial weights
Parameters
----------
neighbors : dictionary
key is region ID, value is a list of neighbor IDS
Example: {'a':['b'],'b':['a','c'],'c':['b']}
weights = None : dictionary
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.
Example: {'a':[0.5],'b':[0.5,1.5],'c':[1.5]}
id_order = None : list
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.
Attributes
----------
asymmetries
cardinalities
diagW2
diagWtW
diagWtW_WW
histogram
id2i
id_order
id_order_set
islands
max_neighbors
mean_neighbors
min_neighbors
n
neighbor_offsets
nonzero
pct_nonzero
s0
s1
s2
s2array
sd
sparse
trcW2
trcWtW
trcWtW_WW
transform
Examples
--------
>>> from pysal import W, lat2W
>>> 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)
>>> w.pct_nonzero
0.29629629629629628
Read from external gal file
>>> import pysal
>>> w = pysal.open(pysal.examples.get_path("stl.gal")).read()
>>> w.n
78
>>> w.pct_nonzero
0.065417488494411577
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)
>>> w.pct_nonzero
0.29629629629629628
>>> w = lat2W(100, 100)
>>> w.trcW2
39600.0
>>> w.trcWtW
39600.0
>>> w.transform='r'
>>> w.trcW2
2530.7222222222586
>>> w.trcWtW
2533.6666666666774
Cardinality Histogram
>>> w=pysal.rook_from_shapefile(pysal.examples.get_path("sacramentot2.shp"))
>>> w.histogram
[(1, 1), (2, 6), (3, 33), (4, 103), (5, 114), (6, 73), (7, 35), (8, 17), (9, 9), (10, 4), (11, 4), (12, 3), (13, 0), (14, 1)]
"""
def __init__(self, neighbors, weights=None, id_order=None):
self.transformations = {}
self.neighbors = ROD(neighbors)
if not weights:
weights = {}
for key in neighbors:
weights[key] = [1.] * len(neighbors[key])
self.weights = ROD(weights)
self.transformations['O'] = self.weights # original weights
self.transform = 'O'
if id_order is None:
self._id_order = self.neighbors.keys()
self._id_order.sort()
self._id_order_set = False
else:
self._id_order = id_order
self._id_order_set = True
self._reset()
self._n = len(self.weights)
def _reset(self):
"""
Reset properties
"""
self._cache = {}
@property
def sparse(self):
"""
Sparse matrix object
For any matrix manipulations required for w, w.sparse should be
used. This is based on scipy.sparse.
"""
if 'sparse' not in self._cache:
self._sparse = self._build_sparse()
self._cache['sparse'] = self._sparse
return self._sparse
def _build_sparse(self):
"""
construct the sparse attribute
"""
row = []
col = []
data = []
gc.disable()
id2i = self.id2i
for id, neigh_list in self.neighbor_offsets.iteritems():
card = self.cardinalities[id]
row.extend([id2i[id]] * card)
col.extend(neigh_list)
data.extend(self.weights[id])
gc.enable()
row = np.array(row)
col = np.array(col)
data = np.array(data)
s = scipy.sparse.csr_matrix((data, (row, col)), shape=(self.n, self.n))
return s
@property
def id2i(self):
"""
Dictionary where the key is an ID and the value is that ID's
index in W.id_order.
"""
if 'id2i' not in self._cache:
self._id2i = {}
for i, id in enumerate(self._id_order):
self._id2i[id] = i
self._id2i = ROD(self._id2i)
self._cache['id2i'] = self._id2i
return self._id2i
@property
def n(self):
"""
number of units
"""
if "n" not in self._cache:
self._n = len(self.neighbors)
self._cache['n'] = self._n
return self._n
@property
def s0(self):
"""
float
.. math::
s0=\sum_i \sum_j w_{i,j}
"""
if 's0' not in self._cache:
self._s0 = self.sparse.sum()
self._cache['s0'] = self._s0
return self._s0
@property
def s1(self):
"""
float
.. math::
s1=1/2 \sum_i \sum_j (w_{i,j} + w_{j,i})^2
"""
if 's1' not in self._cache:
t = self.sparse.transpose()
t = t + self.sparse
t2 = t.multiply(t) # element-wise square
self._s1 = t2.sum() / 2.
self._cache['s1'] = self._s1
return self._s1
@property
def s2array(self):
"""
individual elements comprising s2
See Also
--------
s2
"""
if 's2array' not in self._cache:
s = self.sparse
self._s2array = np.array(s.sum(1) + s.sum(0).transpose()) ** 2
self._cache['s2array'] = self._s2array
return self._s2array
@property
def s2(self):
"""
float
.. math::
s2=\sum_j (\sum_i w_{i,j} + \sum_i w_{j,i})^2
"""
if 's2' not in self._cache:
self._s2 = self.s2array.sum()
self._cache['s2'] = self._s2
return self._s2
@property
def trcW2(self):
"""
Trace of :math:`WW`
See Also
--------
diagW2
"""
if 'trcW2' not in self._cache:
self._trcW2 = self.diagW2.sum()
self._cache['trcw2'] = self._trcW2
return self._trcW2
@property
def diagW2(self):
"""
Diagonal of :math:`WW` : array
See Also
--------
trcW2
"""
if 'diagw2' not in self._cache:
self._diagW2 = (self.sparse * self.sparse).diagonal()
self._cache['diagW2'] = self._diagW2
return self._diagW2
@property
def diagWtW(self):
"""
Diagonal of :math:`W^{'}W` : array
See Also
--------
trcWtW
"""
if 'diagWtW' not in self._cache:
self._diagWtW = (self.sparse.transpose() * self.sparse).diagonal()
self._cache['diagWtW'] = self._diagWtW
return self._diagWtW
@property
def trcWtW(self):
"""
Trace of :math:`W^{'}W` : float
See Also
--------
diagWtW
"""
if 'trcWtW' not in self._cache:
self._trcWtW = self.diagWtW.sum()
self._cache['trcWtW'] = self._trcWtW
return self._trcWtW
@property
def diagWtW_WW(self):
"""
diagonal of :math:`W^{'}W + WW`
"""
if 'diagWtW_WW' not in self._cache:
wt = self.sparse.transpose()
w = self.sparse
self._diagWtW_WW = (wt * w + w * w).diagonal()
self._cache['diagWtW_WW'] = self._diagWtW_WW
return self._diagWtW_WW
@property
def trcWtW_WW(self):
"""
trace of :math:`W^{'}W + WW`
"""
if 'trcWtW_WW' not in self._cache:
self._trcWtW_WW = self.diagWtW_WW.sum()
self._cache['trcWtW_WW'] = self._trcWtW_WW
return self._trcWtW_WW
@property
def pct_nonzero(self):
"""
percentage of nonzero weights
"""
if 'pct_nonzero' not in self._cache:
self._pct_nonzero = self.sparse.nnz / (1. * self._n ** 2)
self._cache['pct_nonzero'] = self._pct_nonzero
return self._pct_nonzero
@property
def cardinalities(self):
"""
number of neighbors for each observation : dict
"""
if 'cardinalities' not in self._cache:
c = {}
for i in self._id_order:
c[i] = len(self.neighbors[i])
self._cardinalities = c
self._cache['cardinalities'] = self._cardinalities
return self._cardinalities
@property
def max_neighbors(self):
"""
largest number of neighbors
"""
if 'max_neighbors' not in self._cache:
self._max_neighbors = max(self.cardinalities.values())
self._cache['max_neighbors'] = self._max_neighbors
return self._max_neighbors
@property
def mean_neighbors(self):
"""
average number of neighbors
"""
if 'mean_neighbors' not in self._cache:
self._mean_neighbors = np.mean(self.cardinalities.values())
self._cache['mean_neighbors'] = self._mean_neighbors
return self._mean_neighbors
@property
def min_neighbors(self):
"""
minimum number of neighbors
"""
if 'min_neighbors' not in self._cache:
self._min_neighbors = min(self.cardinalities.values())
self._cache['min_neighbors'] = self._min_neighbors
return self._min_neighbors
@property
def nonzero(self):
"""
number of nonzero weights
"""
if 'nonzero' not in self._cache:
self._nonzero = self.sparse.nnz
self._cache['nonzero'] = self._nonzero
return self._nonzero
@property
def sd(self):
"""
standard deviation of number of neighbors : float
"""
if 'sd' not in self._cache:
self._sd = np.std(self.cardinalities.values())
self._cache['sd'] = self._sd
return self._sd
@property
def asymmetries(self):
"""
list of id pairs with asymmetric weights
"""
if 'asymmetries' not in self._cache:
self._asymmetries = self.asymmetry()
self._cache['asymmetries'] = self._asymmetries
return self._asymmetries
@property
def islands(self):
"""
list of ids without any neighbors
"""
if 'islands' not in self._cache:
self._islands = [i for i,
c in self.cardinalities.items() if c == 0]
self._cache['islands'] = self._islands
return self._islands
@property
def histogram(self):
"""
cardinality histogram as a dictionary, key is the id, value is the
number of neighbors for that unit
"""
if 'histogram' not in self._cache:
ct, bin = np.histogram(self.cardinalities.values(),
range(self.min_neighbors, self.max_neighbors + 2))
self._histogram = zip(bin, ct)
self._cache['histogram'] = self._histogram
return self._histogram
def __getitem__(self, key):
"""
Allow a dictionary like interaction with the weights class.
Examples
--------
>>> from pysal import rook_from_shapefile as rfs
>>> from pysal import lat2W
>>> w = rfs(pysal.examples.get_path('10740.shp'))
>>> w[0]
{1: 1.0, 4: 1.0, 101: 1.0, 85: 1.0, 5: 1.0}
>>> w = lat2W()
>>> w[1]
{0: 1.0, 2: 1.0, 6: 1.0}
>>> w[0]
{1: 1.0, 5: 1.0}
"""
return dict(zip(self.neighbors[key], self.weights[key]))
def __iter__(self):
"""
Support iteration over weights
Examples
--------
>>> import pysal
>>> w=pysal.lat2W(3,3)
>>> for i,wi in enumerate(w):
... print i,wi
...
0 {1: 1.0, 3: 1.0}
1 {0: 1.0, 2: 1.0, 4: 1.0}
2 {1: 1.0, 5: 1.0}
3 {0: 1.0, 4: 1.0, 6: 1.0}
4 {1: 1.0, 3: 1.0, 5: 1.0, 7: 1.0}
5 {8: 1.0, 2: 1.0, 4: 1.0}
6 {3: 1.0, 7: 1.0}
7 {8: 1.0, 4: 1.0, 6: 1.0}
8 {5: 1.0, 7: 1.0}
>>>
"""
class _W_iter:
def __init__(self, w):
self.w = w
self.n = len(w._id_order)
self._idx = 0
def next(self):
if self._idx >= self.n:
self._idx = 0
raise StopIteration
value = self.w.__getitem__(self.w._id_order[self._idx])
self._idx += 1
return value
return _W_iter(self)
def __set_id_order(self, ordered_ids):
"""
Set the iteration order in w.
W can be iterated over. On construction the iteration order is set to
the lexicographic order of the keys in the w.weights dictionary. If a specific order
is required it can be set with this method.
Parameters
----------
ordered_ids : sequence
identifiers for observations in specified order
Notes
-----
ordered_ids is checked against the ids implied by the keys in
w.weights. If they are not equivalent sets an exception is raised and
the iteration order is not changed.
Examples
--------
>>> import pysal
>>> w=pysal.lat2W(3,3)
>>> for i,wi in enumerate(w):
... print i,wi
...
0 {1: 1.0, 3: 1.0}
1 {0: 1.0, 2: 1.0, 4: 1.0}
2 {1: 1.0, 5: 1.0}
3 {0: 1.0, 4: 1.0, 6: 1.0}
4 {1: 1.0, 3: 1.0, 5: 1.0, 7: 1.0}
5 {8: 1.0, 2: 1.0, 4: 1.0}
6 {3: 1.0, 7: 1.0}
7 {8: 1.0, 4: 1.0, 6: 1.0}
8 {5: 1.0, 7: 1.0}
>>> w.id_order
[0, 1, 2, 3, 4, 5, 6, 7, 8]
>>> w.id_order=range(8,-1,-1)
>>> w.id_order
[8, 7, 6, 5, 4, 3, 2, 1, 0]
>>> for i,w_i in enumerate(w):
... print i,w_i
...
0 {5: 1.0, 7: 1.0}
1 {8: 1.0, 4: 1.0, 6: 1.0}
2 {3: 1.0, 7: 1.0}
3 {8: 1.0, 2: 1.0, 4: 1.0}
4 {1: 1.0, 3: 1.0, 5: 1.0, 7: 1.0}
5 {0: 1.0, 4: 1.0, 6: 1.0}
6 {1: 1.0, 5: 1.0}
7 {0: 1.0, 2: 1.0, 4: 1.0}
8 {1: 1.0, 3: 1.0}
>>>
"""
if set(self._id_order) == set(ordered_ids):
self._id_order = ordered_ids
self._idx = 0
self._id_order_set = True
self._reset()
else:
raise Exception('ordered_ids do not align with W ids')
def __get_id_order(self):
"""returns the ids for the observations in the order in which they
would be encountered if iterating over the weights."""
return self._id_order
id_order = property(__get_id_order, __set_id_order)
@property
def id_order_set(self):
"""returns True if user has set id_order, False if not.
Examples
--------
>>> from pysal import lat2W
>>> w=lat2W()
>>> w.id_order_set
True
"""
return self._id_order_set
@property
def neighbor_offsets(self):
"""
Given the current id_order, neighbor_offsets[id] is the offsets of the
id's neighbors in id_order
Examples
--------
>>> from pysal 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]
"""
if "neighbors_0" not in self._cache:
self.__neighbors_0 = {}
id2i = self.id2i
for id, neigh_list in self.neighbors.iteritems():
self.__neighbors_0[id] = [id2i[neigh] for neigh in neigh_list]
self._cache['neighbors_0'] = self.__neighbors_0
return self.__neighbors_0
def get_transform(self):
"""
Getter for transform property
Returns
-------
transformation : string (or none)
Examples
--------
>>> from pysal 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]
>>>
"""
return self._transform
def set_transform(self, value="B"):
"""
Transformations of weights.
Notes
-----
Transformations are applied only to the value of the weights at
instantiation. Chaining of transformations cannot be done on a W
instance.
Parameters
----------
transform : string (not case sensitive)
B: Binary
R: Row-standardization (global sum=n)
D: Double-standardization (global sum=1)
V: Variance stabilizing
O: Restore original transformation (from instantiation)
Examples
--------
>>> from pysal 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]
>>>
"""
value = value.upper()
self._transform = value
if value in self.transformations:
self.weights = self.transformations[value]
self._reset()
else:
if value == "R":
# row standardized weights
weights = {}
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
row_sum = sum(wijs) * 1.0
weights[i] = [wij / row_sum for wij in wijs]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "D":
# doubly-standardized weights
# update current chars before doing global sum
self._reset()
s0 = self.s0
ws = 1.0 / s0
weights = {}
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
weights[i] = [wij * ws for wij in wijs]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "B":
# binary transformation
weights = {}
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
weights[i] = [1.0 for wij in wijs]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "V":
# variance stabilizing
weights = {}
q = {}
k = self.cardinalities
s = {}
Q = 0.0
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
q[i] = math.sqrt(sum([wij * wij for wij in wijs]))
s[i] = [wij / q[i] for wij in wijs]
Q += sum([si for si in s[i]])
nQ = self.n / Q
for i in self.weights:
weights[i] = [w * nQ for w in s[i]]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "O":
# put weights back to original transformation
weights = {}
original = self.transformations[value]
self.weights = original
self._reset()
else:
print 'unsupported weights transformation'
transform = property(get_transform, set_transform)
def asymmetry(self, intrinsic=True):
"""
Asymmetry check
Arguments
---------
intrinsic: boolean (default=True)
intrinsic symmetry:
:math:`w_{i,j} == w_{j,i}`
if intrisic is False:
symmetry is defined as :math:`i \in N_j \ AND \ j \in N_i` where
:math:`N_j` is the set of neighbors for j.
Returns
-------
asymmetries : list
empty if no asymmetries are found
if asymmetries, first list is row indices, second
list is column indices of asymmetric cells
Examples
--------
>>> from pysal import lat2W
>>> w=lat2W(3,3)
>>> w.asymmetry()
[]
>>> w.transform='r'
>>> result = w.asymmetry()[0:2]
>>> print result[0]
[1 3 0 2 4 1 5 0 4 6 1 3 5 7 2 4 8 3 7 4 6 8 5 7]
>>> print result[1]
[0 0 1 1 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 7 7 8 8]
>>> 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)
>>> result = w.asymmetry()
>>> print result[0]
[1 0]
>>> print result[1]
[0 1]
"""
if intrinsic:
wd = self.sparse.transpose() - self.sparse
else:
transform = self.transform
self.transform = 'b'
wd = self.sparse.transpose() - self.sparse
self.transform = transform
ids = np.nonzero(wd)
if len(ids[0]) == 0:
return []
else:
return ids
def full(self):
"""
Generate a full numpy array
Returns
-------
implicit : tuple
first element being the full numpy array and second element
keys being the ids associated with each row in the array.
Examples
--------
>>> from pysal import W
>>> neighbors={'first':['second'],'second':['first','third'],'third':['second']}
>>> weights={'first':[1],'second':[1,1],'third':[1]}
>>> w=W(neighbors,weights)
>>> wf,ids=w.full()
>>> wf
array([[ 0., 1., 0.],
[ 1., 0., 1.],
[ 0., 1., 0.]])
>>> ids
['first', 'second', 'third']
See also
--------
full
"""
return util.full(self)
def set_transform(self, value="B"):
"""
Transformations of weights.
Notes
-----
Transformations are applied only to the value of the weights at
instantiation. Chaining of transformations cannot be done on a W
instance.
Parameters
----------
transform : string (not case sensitive)
B: Binary
R: Row-standardization (global sum=n)
D: Double-standardization (global sum=1)
V: Variance stabilizing
O: Restore original transformation (from instantiation)
Examples
--------
>>> from pysal 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]
>>>
"""
value = value.upper()
self._transform = value
if value in self.transformations:
self.weights = self.transformations[value]
self._reset()
else:
if value == "R":
# row standardized weights
weights = {}
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
row_sum = sum(wijs) * 1.0
weights[i] = [wij / row_sum for wij in wijs]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "D":
# doubly-standardized weights
# update current chars before doing global sum
self._reset()
s0 = self.s0
ws = 1.0 / s0
weights = {}
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
weights[i] = [wij * ws for wij in wijs]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "B":
# binary transformation
weights = {}
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
weights[i] = [1.0 for wij in wijs]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "V":
# variance stabilizing
weights = {}
q = {}
k = self.cardinalities
s = {}
Q = 0.0
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
q[i] = math.sqrt(sum([wij * wij for wij in wijs]))
s[i] = [wij / q[i] for wij in wijs]
Q += sum([si for si in s[i]])
nQ = self.n / Q
for i in self.weights:
weights[i] = [w * nQ for w in s[i]]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "O":
# put weights back to original transformation
weights = {}
original = self.transformations[value]
self.weights = original
self._reset()
else:
print 'unsupported weights transformation'
class W(object):
"""
Spatial weights
Parameters
----------
neighbors : dictionary
key is region ID, value is a list of neighbor IDS
Example: {'a':['b'],'b':['a','c'],'c':['b']}
weights = None : dictionary
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.
Example: {'a':[0.5],'b':[0.5,1.5],'c':[1.5]}
id_order = None : list
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.
Attributes
----------
asymmetries
cardinalities
diagW2
diagWtW
diagWtW_WW
histogram
id2i
id_order
id_order_set
islands
max_neighbors
mean_neighbors
min_neighbors
n
neighbor_offsets
nonzero
pct_nonzero
s0
s1
s2
s2array
sd
sparse
trcW2
trcWtW
trcWtW_WW
transform
Examples
--------
>>> from pysal import W, lat2W
>>> 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)
>>> w.pct_nonzero
0.29629629629629628
Read from external gal file
>>> import pysal
>>> w = pysal.open(pysal.examples.get_path("stl.gal")).read()
>>> w.n
78
>>> w.pct_nonzero
0.065417488494411577
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)
>>> w.pct_nonzero
0.29629629629629628
>>> w = lat2W(100, 100)
>>> w.trcW2
39600.0
>>> w.trcWtW
39600.0
>>> w.transform='r'
>>> w.trcW2
2530.7222222222586
>>> w.trcWtW
2533.6666666666774
Cardinality Histogram
>>> w=pysal.rook_from_shapefile(pysal.examples.get_path("sacramentot2.shp"))
>>> w.histogram
[(1, 1), (2, 6), (3, 33), (4, 103), (5, 114), (6, 73), (7, 35), (8, 17), (9, 9), (10, 4), (11, 4), (12, 3), (13, 0), (14, 1)]
"""
def __init__(self, neighbors, weights=None, id_order=None):
self.transformations = {}
self.neighbors = ROD(neighbors)
if not weights:
weights = {}
for key in neighbors:
weights[key] = [1.] * len(neighbors[key])
self.weights = ROD(weights)
self.transformations['O'] = self.weights # original weights
self.transform = 'O'
if id_order is None:
self._id_order = self.neighbors.keys()
self._id_order.sort()
self._id_order_set = False
else:
self._id_order = id_order
self._id_order_set = True
self._reset()
self._n = len(self.weights)
def _reset(self):
"""
Reset properties
"""
self._cache = {}
@property
def sparse(self):
"""
Sparse matrix object
For any matrix manipulations required for w, w.sparse should be
used. This is based on scipy.sparse.
"""
if 'sparse' not in self._cache:
self._sparse = self._build_sparse()
self._cache['sparse'] = self._sparse
return self._sparse
def _build_sparse(self):
"""
construct the sparse attribute
"""
row = []
col = []
data = []
gc.disable()
id2i = self.id2i
for id, neigh_list in self.neighbor_offsets.iteritems():
card = self.cardinalities[id]
row.extend([id2i[id]] * card)
col.extend(neigh_list)
data.extend(self.weights[id])
gc.enable()
row = np.array(row)
col = np.array(col)
data = np.array(data)
s = scipy.sparse.csr_matrix((data, (row, col)), shape=(self.n, self.n))
return s
@property
def id2i(self):
"""
Dictionary where the key is an ID and the value is that ID's
index in W.id_order.
"""
if 'id2i' not in self._cache:
self._id2i = {}
for i, id in enumerate(self._id_order):
self._id2i[id] = i
self._id2i = ROD(self._id2i)
self._cache['id2i'] = self._id2i
return self._id2i
@property
def n(self):
"""
number of units
"""
if "n" not in self._cache:
self._n = len(self.neighbors)
self._cache['n'] = self._n
return self._n
@property
def s0(self):
"""
float
.. math::
s0=\sum_i \sum_j w_{i,j}
"""
if 's0' not in self._cache:
self._s0 = self.sparse.sum()
self._cache['s0'] = self._s0
return self._s0
@property
def s1(self):
"""
float
.. math::
s1=1/2 \sum_i \sum_j (w_{i,j} + w_{j,i})^2
"""
if 's1' not in self._cache:
t = self.sparse.transpose()
t = t + self.sparse
t2 = t.multiply(t) # element-wise square
self._s1 = t2.sum() / 2.
self._cache['s1'] = self._s1
return self._s1
@property
def s2array(self):
"""
individual elements comprising s2
See Also
--------
s2
"""
if 's2array' not in self._cache:
s = self.sparse
self._s2array = np.array(s.sum(1) + s.sum(0).transpose())**2
self._cache['s2array'] = self._s2array
return self._s2array
@property
def s2(self):
"""
float
.. math::
s2=\sum_j (\sum_i w_{i,j} + \sum_i w_{j,i})^2
"""
if 's2' not in self._cache:
self._s2 = self.s2array.sum()
self._cache['s2'] = self._s2
return self._s2
@property
def trcW2(self):
"""
Trace of :math:`WW`
See Also
--------
diagW2
"""
if 'trcW2' not in self._cache:
self._trcW2 = self.diagW2.sum()
self._cache['trcw2'] = self._trcW2
return self._trcW2
@property
def diagW2(self):
"""
Diagonal of :math:`WW` : array
See Also
--------
trcW2
"""
if 'diagw2' not in self._cache:
self._diagW2 = (self.sparse * self.sparse).diagonal()
self._cache['diagW2'] = self._diagW2
return self._diagW2
@property
def diagWtW(self):
"""
Diagonal of :math:`W^{'}W` : array
See Also
--------
trcWtW
"""
if 'diagWtW' not in self._cache:
self._diagWtW = (self.sparse.transpose() * self.sparse).diagonal()
self._cache['diagWtW'] = self._diagWtW
return self._diagWtW
@property
def trcWtW(self):
"""
Trace of :math:`W^{'}W` : float
See Also
--------
diagWtW
"""
if 'trcWtW' not in self._cache:
self._trcWtW = self.diagWtW.sum()
self._cache['trcWtW'] = self._trcWtW
return self._trcWtW
@property
def diagWtW_WW(self):
"""
diagonal of :math:`W^{'}W + WW`
"""
if 'diagWtW_WW' not in self._cache:
wt = self.sparse.transpose()
w = self.sparse
self._diagWtW_WW = (wt * w + w * w).diagonal()
self._cache['diagWtW_WW'] = self._diagWtW_WW
return self._diagWtW_WW
@property
def trcWtW_WW(self):
"""
trace of :math:`W^{'}W + WW`
"""
if 'trcWtW_WW' not in self._cache:
self._trcWtW_WW = self.diagWtW_WW.sum()
self._cache['trcWtW_WW'] = self._trcWtW_WW
return self._trcWtW_WW
@property
def pct_nonzero(self):
"""
percentage of nonzero weights
"""
if 'pct_nonzero' not in self._cache:
self._pct_nonzero = self.sparse.nnz / (1. * self._n**2)
self._cache['pct_nonzero'] = self._pct_nonzero
return self._pct_nonzero
@property
def cardinalities(self):
"""
number of neighbors for each observation : dict
"""
if 'cardinalities' not in self._cache:
c = {}
for i in self._id_order:
c[i] = len(self.neighbors[i])
self._cardinalities = c
self._cache['cardinalities'] = self._cardinalities
return self._cardinalities
@property
def max_neighbors(self):
"""
largest number of neighbors
"""
if 'max_neighbors' not in self._cache:
self._max_neighbors = max(self.cardinalities.values())
self._cache['max_neighbors'] = self._max_neighbors
return self._max_neighbors
@property
def mean_neighbors(self):
"""
average number of neighbors
"""
if 'mean_neighbors' not in self._cache:
self._mean_neighbors = np.mean(self.cardinalities.values())
self._cache['mean_neighbors'] = self._mean_neighbors
return self._mean_neighbors
@property
def min_neighbors(self):
"""
minimum number of neighbors
"""
if 'min_neighbors' not in self._cache:
self._min_neighbors = min(self.cardinalities.values())
self._cache['min_neighbors'] = self._min_neighbors
return self._min_neighbors
@property
def nonzero(self):
"""
number of nonzero weights
"""
if 'nonzero' not in self._cache:
self._nonzero = self.sparse.nnz
self._cache['nonzero'] = self._nonzero
return self._nonzero
@property
def sd(self):
"""
standard deviation of number of neighbors : float
"""
if 'sd' not in self._cache:
self._sd = np.std(self.cardinalities.values())
self._cache['sd'] = self._sd
return self._sd
@property
def asymmetries(self):
"""
list of id pairs with asymmetric weights
"""
if 'asymmetries' not in self._cache:
self._asymmetries = self.asymmetry()
self._cache['asymmetries'] = self._asymmetries
return self._asymmetries
@property
def islands(self):
"""
list of ids without any neighbors
"""
if 'islands' not in self._cache:
self._islands = [
i for i, c in self.cardinalities.items() if c == 0
]
self._cache['islands'] = self._islands
return self._islands
@property
def histogram(self):
"""
cardinality histogram as a dictionary, key is the id, value is the
number of neighbors for that unit
"""
if 'histogram' not in self._cache:
ct, bin = np.histogram(
self.cardinalities.values(),
range(self.min_neighbors, self.max_neighbors + 2))
self._histogram = zip(bin, ct)
self._cache['histogram'] = self._histogram
return self._histogram
def __getitem__(self, key):
"""
Allow a dictionary like interaction with the weights class.
Examples
--------
>>> from pysal import rook_from_shapefile as rfs
>>> from pysal import lat2W
>>> w = rfs(pysal.examples.get_path('10740.shp'))
>>> w[0]
{1: 1.0, 4: 1.0, 101: 1.0, 85: 1.0, 5: 1.0}
>>> w = lat2W()
>>> w[1]
{0: 1.0, 2: 1.0, 6: 1.0}
>>> w[0]
{1: 1.0, 5: 1.0}
"""
return dict(zip(self.neighbors[key], self.weights[key]))
def __iter__(self):
"""
Support iteration over weights
Examples
--------
>>> import pysal
>>> w=pysal.lat2W(3,3)
>>> for i,wi in enumerate(w):
... print i,wi
...
0 {1: 1.0, 3: 1.0}
1 {0: 1.0, 2: 1.0, 4: 1.0}
2 {1: 1.0, 5: 1.0}
3 {0: 1.0, 4: 1.0, 6: 1.0}
4 {1: 1.0, 3: 1.0, 5: 1.0, 7: 1.0}
5 {8: 1.0, 2: 1.0, 4: 1.0}
6 {3: 1.0, 7: 1.0}
7 {8: 1.0, 4: 1.0, 6: 1.0}
8 {5: 1.0, 7: 1.0}
>>>
"""
class _W_iter:
def __init__(self, w):
self.w = w
self.n = len(w._id_order)
self._idx = 0
def next(self):
if self._idx >= self.n:
self._idx = 0
raise StopIteration
value = self.w.__getitem__(self.w._id_order[self._idx])
self._idx += 1
return value
return _W_iter(self)
def __set_id_order(self, ordered_ids):
"""
Set the iteration order in w.
W can be iterated over. On construction the iteration order is set to
the lexicographic order of the keys in the w.weights dictionary. If a specific order
is required it can be set with this method.
Parameters
----------
ordered_ids : sequence
identifiers for observations in specified order
Notes
-----
ordered_ids is checked against the ids implied by the keys in
w.weights. If they are not equivalent sets an exception is raised and
the iteration order is not changed.
Examples
--------
>>> import pysal
>>> w=pysal.lat2W(3,3)
>>> for i,wi in enumerate(w):
... print i,wi
...
0 {1: 1.0, 3: 1.0}
1 {0: 1.0, 2: 1.0, 4: 1.0}
2 {1: 1.0, 5: 1.0}
3 {0: 1.0, 4: 1.0, 6: 1.0}
4 {1: 1.0, 3: 1.0, 5: 1.0, 7: 1.0}
5 {8: 1.0, 2: 1.0, 4: 1.0}
6 {3: 1.0, 7: 1.0}
7 {8: 1.0, 4: 1.0, 6: 1.0}
8 {5: 1.0, 7: 1.0}
>>> w.id_order
[0, 1, 2, 3, 4, 5, 6, 7, 8]
>>> w.id_order=range(8,-1,-1)
>>> w.id_order
[8, 7, 6, 5, 4, 3, 2, 1, 0]
>>> for i,w_i in enumerate(w):
... print i,w_i
...
0 {5: 1.0, 7: 1.0}
1 {8: 1.0, 4: 1.0, 6: 1.0}
2 {3: 1.0, 7: 1.0}
3 {8: 1.0, 2: 1.0, 4: 1.0}
4 {1: 1.0, 3: 1.0, 5: 1.0, 7: 1.0}
5 {0: 1.0, 4: 1.0, 6: 1.0}
6 {1: 1.0, 5: 1.0}
7 {0: 1.0, 2: 1.0, 4: 1.0}
8 {1: 1.0, 3: 1.0}
>>>
"""
if set(self._id_order) == set(ordered_ids):
self._id_order = ordered_ids
self._idx = 0
self._id_order_set = True
self._reset()
else:
raise Exception('ordered_ids do not align with W ids')
def __get_id_order(self):
"""returns the ids for the observations in the order in which they
would be encountered if iterating over the weights."""
return self._id_order
id_order = property(__get_id_order, __set_id_order)
@property
def id_order_set(self):
"""returns True if user has set id_order, False if not.
Examples
--------
>>> from pysal import lat2W
>>> w=lat2W()
>>> w.id_order_set
True
"""
return self._id_order_set
@property
def neighbor_offsets(self):
"""
Given the current id_order, neighbor_offsets[id] is the offsets of the
id's neighbors in id_order
Examples
--------
>>> from pysal 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]
"""
if "neighbors_0" not in self._cache:
self.__neighbors_0 = {}
id2i = self.id2i
for id, neigh_list in self.neighbors.iteritems():
self.__neighbors_0[id] = [id2i[neigh] for neigh in neigh_list]
self._cache['neighbors_0'] = self.__neighbors_0
return self.__neighbors_0
def get_transform(self):
"""
Getter for transform property
Returns
-------
transformation : string (or none)
Examples
--------
>>> from pysal 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]
>>>
"""
return self._transform
def set_transform(self, value="B"):
"""
Transformations of weights.
Notes
-----
Transformations are applied only to the value of the weights at
instantiation. Chaining of transformations cannot be done on a W
instance.
Parameters
----------
transform : string (not case sensitive)
B: Binary
R: Row-standardization (global sum=n)
D: Double-standardization (global sum=1)
V: Variance stabilizing
O: Restore original transformation (from instantiation)
Examples
--------
>>> from pysal 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]
>>>
"""
value = value.upper()
self._transform = value
if value in self.transformations:
self.weights = self.transformations[value]
self._reset()
else:
if value == "R":
# row standardized weights
weights = {}
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
row_sum = sum(wijs) * 1.0
weights[i] = [wij / row_sum for wij in wijs]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "D":
# doubly-standardized weights
# update current chars before doing global sum
self._reset()
s0 = self.s0
ws = 1.0 / s0
weights = {}
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
weights[i] = [wij * ws for wij in wijs]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "B":
# binary transformation
weights = {}
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
weights[i] = [1.0 for wij in wijs]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "V":
# variance stabilizing
weights = {}
q = {}
k = self.cardinalities
s = {}
Q = 0.0
self.weights = self.transformations['O']
for i in self.weights:
wijs = self.weights[i]
q[i] = math.sqrt(sum([wij * wij for wij in wijs]))
s[i] = [wij / q[i] for wij in wijs]
Q += sum([si for si in s[i]])
nQ = self.n / Q
for i in self.weights:
weights[i] = [w * nQ for w in s[i]]
weights = ROD(weights)
self.transformations[value] = weights
self.weights = weights
self._reset()
elif value == "O":
# put weights back to original transformation
weights = {}
original = self.transformations[value]
self.weights = original
self._reset()
else:
print 'unsupported weights transformation'
transform = property(get_transform, set_transform)
def asymmetry(self, intrinsic=True):
"""
Asymmetry check
Arguments
---------
intrinsic: boolean (default=True)
intrinsic symmetry:
:math:`w_{i,j} == w_{j,i}`
if intrisic is False:
symmetry is defined as :math:`i \in N_j \ AND \ j \in N_i` where
:math:`N_j` is the set of neighbors for j.
Returns
-------
asymmetries : list
empty if no asymmetries are found
if asymmetries, first list is row indices, second
list is column indices of asymmetric cells
Examples
--------
>>> from pysal import lat2W
>>> w=lat2W(3,3)
>>> w.asymmetry()
[]
>>> w.transform='r'
>>> result = w.asymmetry()[0:2]
>>> print result[0]
[1 3 0 2 4 1 5 0 4 6 1 3 5 7 2 4 8 3 7 4 6 8 5 7]
>>> print result[1]
[0 0 1 1 1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7 7 7 8 8]
>>> 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)
>>> result = w.asymmetry()
>>> print result[0]
[1 0]
>>> print result[1]
[0 1]
"""
if intrinsic:
wd = self.sparse.transpose() - self.sparse
else:
transform = self.transform
self.transform = 'b'
wd = self.sparse.transpose() - self.sparse
self.transform = transform
ids = np.nonzero(wd)
if len(ids[0]) == 0:
return []
else:
return ids
def full(self):
"""
Generate a full numpy array
Returns
-------
implicit : tuple
first element being the full numpy array and second element
keys being the ids associated with each row in the array.
Examples
--------
>>> from pysal import W
>>> neighbors={'first':['second'],'second':['first','third'],'third':['second']}
>>> weights={'first':[1],'second':[1,1],'third':[1]}
>>> w=W(neighbors,weights)
>>> wf,ids=w.full()
>>> wf
array([[ 0., 1., 0.],
[ 1., 0., 1.],
[ 0., 1., 0.]])
>>> ids
['first', 'second', 'third']
See also
--------
full
"""
return util.full(self)
