class Kernel(W):
"""
Spatial weights based on kernel functions.
Parameters
----------
data : array
(n,k) or KDTree where KDtree.data is array (n,k)
n observations on k characteristics used to measure
distances between the n objects
bandwidth : float
or array-like (optional)
the bandwidth :math:`h_i` for the kernel.
fixed : binary
If true then :math:`h_i=h \\forall i`. If false then
bandwidth is adaptive across observations.
k : int
the number of nearest neighbors to use for determining
bandwidth. For fixed bandwidth, :math:`h_i=max(dknn) \\forall i`
where :math:`dknn` is a vector of k-nearest neighbor
distances (the distance to the kth nearest neighbor for each
observation). For adaptive bandwidths, :math:`h_i=dknn_i`
diagonal : boolean
If true, set diagonal weights = 1.0, if false (default),
diagonals weights are set to value according to kernel
function.
function : {'triangular','uniform','quadratic','quartic','gaussian'}
kernel function defined as follows with
.. math::
z_{i,j} = d_{i,j}/h_i
triangular
.. math::
K(z) = (1 - |z|) \ if |z| \le 1
uniform
.. math::
K(z) = 1/2 \ if |z| \le 1
quadratic
.. math::
K(z) = (3/4)(1-z^2) \ if |z| \le 1
quartic
.. math::
K(z) = (15/16)(1-z^2)^2 \ if |z| \le 1
gaussian
.. math::
K(z) = (2\pi)^{(-1/2)} exp(-z^2 / 2)
eps : float
adjustment to ensure knn distance range is closed on the
knnth observations
Attributes
----------
weights : dict
Dictionary keyed by id with a list of weights for each neighbor
neighbors : dict
of lists of neighbors keyed by observation id
bandwidth : array
array of bandwidths
Examples
--------
>>> points=[(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> kw=Kernel(points)
>>> kw.weights[0]
[1.0, 0.500000049999995, 0.4409830615267465]
>>> kw.neighbors[0]
[0, 1, 3]
>>> kw.bandwidth
array([[ 20.000002],
[ 20.000002],
[ 20.000002],
[ 20.000002],
[ 20.000002],
[ 20.000002]])
>>> kw15=Kernel(points,bandwidth=15.0)
>>> kw15[0]
{0: 1.0, 1: 0.33333333333333337, 3: 0.2546440075000701}
>>> kw15.neighbors[0]
[0, 1, 3]
>>> kw15.bandwidth
array([[ 15.],
[ 15.],
[ 15.],
[ 15.],
[ 15.],
[ 15.]])
Adaptive bandwidths user specified
>>> bw=[25.0,15.0,25.0,16.0,14.5,25.0]
>>> kwa=Kernel(points,bandwidth=bw)
>>> kwa.weights[0]
[1.0, 0.6, 0.552786404500042, 0.10557280900008403]
>>> kwa.neighbors[0]
[0, 1, 3, 4]
>>> kwa.bandwidth
array([[ 25. ],
[ 15. ],
[ 25. ],
[ 16. ],
[ 14.5],
[ 25. ]])
Endogenous adaptive bandwidths
>>> kwea=Kernel(points,fixed=False)
>>> kwea.weights[0]
[1.0, 0.10557289844279438, 9.99999900663795e-08]
>>> kwea.neighbors[0]
[0, 1, 3]
>>> kwea.bandwidth
array([[ 11.18034101],
[ 11.18034101],
[ 20.000002 ],
[ 11.18034101],
[ 14.14213704],
[ 18.02775818]])
Endogenous adaptive bandwidths with Gaussian kernel
>>> kweag=Kernel(points,fixed=False,function='gaussian')
>>> kweag.weights[0]
[0.3989422804014327, 0.2674190291577696, 0.2419707487162134]
>>> kweag.bandwidth
array([[ 11.18034101],
[ 11.18034101],
[ 20.000002 ],
[ 11.18034101],
[ 14.14213704],
[ 18.02775818]])
Diagonals to 1.0
>>> kq = Kernel(points,function='gaussian')
>>> kq.weights
{0: [0.3989422804014327, 0.35206533556593145, 0.3412334260702758], 1: [0.35206533556593145, 0.3989422804014327, 0.2419707487162134, 0.3412334260702758, 0.31069657591175387], 2: [0.2419707487162134, 0.3989422804014327, 0.31069657591175387], 3: [0.3412334260702758, 0.3412334260702758, 0.3989422804014327, 0.3011374490937829, 0.26575287272131043], 4: [0.31069657591175387, 0.31069657591175387, 0.3011374490937829, 0.3989422804014327, 0.35206533556593145], 5: [0.26575287272131043, 0.35206533556593145, 0.3989422804014327]}
>>> kqd = Kernel(points, function='gaussian', diagonal=True)
>>> kqd.weights
{0: [1.0, 0.35206533556593145, 0.3412334260702758], 1: [0.35206533556593145, 1.0, 0.2419707487162134, 0.3412334260702758, 0.31069657591175387], 2: [0.2419707487162134, 1.0, 0.31069657591175387], 3: [0.3412334260702758, 0.3412334260702758, 1.0, 0.3011374490937829, 0.26575287272131043], 4: [0.31069657591175387, 0.31069657591175387, 0.3011374490937829, 1.0, 0.35206533556593145], 5: [0.26575287272131043, 0.35206533556593145, 1.0]}
"""
def __init__(self,
data,
bandwidth=None,
fixed=True,
k=2,
function='triangular',
eps=1.0000001,
ids=None,
diagonal=False):
if isKDTree(data):
self.kdt = data
self.data = self.kdt.data
data = self.data
else:
self.data = data
self.kdt = KDTree(self.data)
self.k = k + 1
self.function = function.lower()
self.fixed = fixed
self.eps = eps
if bandwidth:
try:
bandwidth = np.array(bandwidth)
bandwidth.shape = (len(bandwidth), 1)
except:
bandwidth = np.ones((len(data), 1), 'float') * bandwidth
self.bandwidth = bandwidth
else:
self._set_bw()
self._eval_kernel()
neighbors, weights = self._k_to_W(ids)
if diagonal:
for i in neighbors:
weights[i][neighbors[i].index(i)] = 1.0
W.__init__(self, neighbors, weights, ids)
def _k_to_W(self, ids=None):
allneighbors = {}
weights = {}
if ids:
ids = np.array(ids)
else:
ids = np.arange(len(self.data))
for i, neighbors in enumerate(self.kernel):
if len(self.neigh[i]) == 0:
allneighbors[ids[i]] = []
weights[ids[i]] = []
else:
allneighbors[ids[i]] = list(ids[self.neigh[i]])
weights[ids[i]] = self.kernel[i].tolist()
return allneighbors, weights
def _set_bw(self):
dmat, neigh = self.kdt.query(self.data, k=self.k)
if self.fixed:
# use max knn distance as bandwidth
bandwidth = dmat.max() * self.eps
n = len(dmat)
self.bandwidth = np.ones((n, 1), 'float') * bandwidth
else:
# use local max knn distance
self.bandwidth = dmat.max(axis=1) * self.eps
self.bandwidth.shape = (self.bandwidth.size, 1)
# identify knn neighbors for each point
nnq = self.kdt.query(self.data, k=self.k)
self.neigh = nnq[1]
def _eval_kernel(self):
# get points within bandwidth distance of each point
if not hasattr(self, 'neigh'):
kdtq = self.kdt.query_ball_point
neighbors = [
kdtq(self.data[i], r=bwi[0])
for i, bwi in enumerate(self.bandwidth)
]
self.neigh = neighbors
# get distances for neighbors
bw = self.bandwidth
kdtq = self.kdt.query
z = []
for i, nids in enumerate(self.neigh):
di, ni = kdtq(self.data[i], k=len(nids))
if not isinstance(di, np.ndarray):
di = np.asarray([di] * len(nids))
ni = np.asarray([ni] * len(nids))
zi = np.array([dict(zip(ni, di))[nid] for nid in nids]) / bw[i]
z.append(zi)
zs = z
# functions follow Anselin and Rey (2010) table 5.4
if self.function == 'triangular':
self.kernel = [1 - zi for zi in zs]
elif self.function == 'uniform':
self.kernel = [np.ones(zi.shape) * 0.5 for zi in zs]
elif self.function == 'quadratic':
self.kernel = [(3. / 4) * (1 - zi**2) for zi in zs]
elif self.function == 'quartic':
self.kernel = [(15. / 16) * (1 - zi**2)**2 for zi in zs]
elif self.function == 'gaussian':
c = np.pi * 2
c = c**(-0.5)
self.kernel = [c * np.exp(-(zi**2) / 2.) for zi in zs]
else:
print('Unsupported kernel function', self.function)
def knnW(data, k=2, p=2, ids=None):
"""
Creates nearest neighbor weights matrix based on k nearest
neighbors.
Parameters
----------
kdtree : object
PySAL KDTree or ArcKDTree where KDtree.data is array (n,k)
n observations on k characteristics used to measure
distances between the n objects
k : int
number of nearest neighbors
p : float
Minkowski p-norm distance metric parameter:
1<=p<=infinity
2: Euclidean distance
1: Manhattan distance
Ignored if the KDTree is an ArcKDTree
ids : list
identifiers to attach to each observation
Returns
-------
w : W
instance
Weights object with binary weights
Examples
--------
>>> points = [(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> kd = pysal.cg.kdtree.KDTree(np.array(points))
>>> wnn2 = pysal.knnW(kd, 2)
>>> [1,3] == wnn2.neighbors[0]
True
ids
>>> wnn2 = knnW(kd,2)
>>> wnn2[0]
{1: 1.0, 3: 1.0}
>>> wnn2[1]
{0: 1.0, 3: 1.0}
now with 1 rather than 0 offset
>>> wnn2 = knnW(kd, 2, ids=range(1,7))
>>> wnn2[1]
{2: 1.0, 4: 1.0}
>>> wnn2[2]
{1: 1.0, 4: 1.0}
>>> 0 in wnn2.neighbors
False
Notes
-----
Ties between neighbors of equal distance are arbitrarily broken.
See Also
--------
pysal.weights.W
"""
if isKDTree(data):
kdt = data
data = kdt.data
else:
kdt = KDTree(data)
nnq = kdt.query(data, k=k + 1, p=p)
info = nnq[1]
neighbors = {}
for i, row in enumerate(info):
row = row.tolist()
if i in row:
row.remove(i)
focal = i
if ids:
row = [ids[j] for j in row]
focal = ids[i]
neighbors[focal] = row
return pysal.weights.W(neighbors, id_order=ids)
class Kernel(W):
"""
Spatial weights based on kernel functions.
Parameters
----------
data : array
(n,k) or KDTree where KDtree.data is array (n,k)
n observations on k characteristics used to measure
distances between the n objects
bandwidth : float
or array-like (optional)
the bandwidth :math:`h_i` for the kernel.
fixed : binary
If true then :math:`h_i=h \\forall i`. If false then
bandwidth is adaptive across observations.
k : int
the number of nearest neighbors to use for determining
bandwidth. For fixed bandwidth, :math:`h_i=max(dknn) \\forall i`
where :math:`dknn` is a vector of k-nearest neighbor
distances (the distance to the kth nearest neighbor for each
observation). For adaptive bandwidths, :math:`h_i=dknn_i`
diagonal : boolean
If true, set diagonal weights = 1.0, if false (default),
diagonals weights are set to value according to kernel
function.
function : {'triangular','uniform','quadratic','quartic','gaussian'}
kernel function defined as follows with
.. math::
z_{i,j} = d_{i,j}/h_i
triangular
.. math::
K(z) = (1 - |z|) \ if |z| \le 1
uniform
.. math::
K(z) = 1/2 \ if |z| \le 1
quadratic
.. math::
K(z) = (3/4)(1-z^2) \ if |z| \le 1
quartic
.. math::
K(z) = (15/16)(1-z^2)^2 \ if |z| \le 1
gaussian
.. math::
K(z) = (2\pi)^{(-1/2)} exp(-z^2 / 2)
eps : float
adjustment to ensure knn distance range is closed on the
knnth observations
Attributes
----------
weights : dict
Dictionary keyed by id with a list of weights for each neighbor
neighbors : dict
of lists of neighbors keyed by observation id
bandwidth : array
array of bandwidths
Examples
--------
>>> points=[(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> kw=Kernel(points)
>>> kw.weights[0]
[1.0, 0.500000049999995, 0.4409830615267465]
>>> kw.neighbors[0]
[0, 1, 3]
>>> kw.bandwidth
array([[ 20.000002],
[ 20.000002],
[ 20.000002],
[ 20.000002],
[ 20.000002],
[ 20.000002]])
>>> kw15=Kernel(points,bandwidth=15.0)
>>> kw15[0]
{0: 1.0, 1: 0.33333333333333337, 3: 0.2546440075000701}
>>> kw15.neighbors[0]
[0, 1, 3]
>>> kw15.bandwidth
array([[ 15.],
[ 15.],
[ 15.],
[ 15.],
[ 15.],
[ 15.]])
Adaptive bandwidths user specified
>>> bw=[25.0,15.0,25.0,16.0,14.5,25.0]
>>> kwa=Kernel(points,bandwidth=bw)
>>> kwa.weights[0]
[1.0, 0.6, 0.552786404500042, 0.10557280900008403]
>>> kwa.neighbors[0]
[0, 1, 3, 4]
>>> kwa.bandwidth
array([[ 25. ],
[ 15. ],
[ 25. ],
[ 16. ],
[ 14.5],
[ 25. ]])
Endogenous adaptive bandwidths
>>> kwea=Kernel(points,fixed=False)
>>> kwea.weights[0]
[1.0, 0.10557289844279438, 9.99999900663795e-08]
>>> kwea.neighbors[0]
[0, 1, 3]
>>> kwea.bandwidth
array([[ 11.18034101],
[ 11.18034101],
[ 20.000002 ],
[ 11.18034101],
[ 14.14213704],
[ 18.02775818]])
Endogenous adaptive bandwidths with Gaussian kernel
>>> kweag=Kernel(points,fixed=False,function='gaussian')
>>> kweag.weights[0]
[0.3989422804014327, 0.2674190291577696, 0.2419707487162134]
>>> kweag.bandwidth
array([[ 11.18034101],
[ 11.18034101],
[ 20.000002 ],
[ 11.18034101],
[ 14.14213704],
[ 18.02775818]])
Diagonals to 1.0
>>> kq = Kernel(points,function='gaussian')
>>> kq.weights
{0: [0.3989422804014327, 0.35206533556593145, 0.3412334260702758], 1: [0.35206533556593145, 0.3989422804014327, 0.2419707487162134, 0.3412334260702758, 0.31069657591175387], 2: [0.2419707487162134, 0.3989422804014327, 0.31069657591175387], 3: [0.3412334260702758, 0.3412334260702758, 0.3989422804014327, 0.3011374490937829, 0.26575287272131043], 4: [0.31069657591175387, 0.31069657591175387, 0.3011374490937829, 0.3989422804014327, 0.35206533556593145], 5: [0.26575287272131043, 0.35206533556593145, 0.3989422804014327]}
>>> kqd = Kernel(points, function='gaussian', diagonal=True)
>>> kqd.weights
{0: [1.0, 0.35206533556593145, 0.3412334260702758], 1: [0.35206533556593145, 1.0, 0.2419707487162134, 0.3412334260702758, 0.31069657591175387], 2: [0.2419707487162134, 1.0, 0.31069657591175387], 3: [0.3412334260702758, 0.3412334260702758, 1.0, 0.3011374490937829, 0.26575287272131043], 4: [0.31069657591175387, 0.31069657591175387, 0.3011374490937829, 1.0, 0.35206533556593145], 5: [0.26575287272131043, 0.35206533556593145, 1.0]}
"""
def __init__(self, data, bandwidth=None, fixed=True, k=2,
function='triangular', eps=1.0000001, ids=None,
diagonal=False):
if isKDTree(data):
self.kdt = data
self.data = self.kdt.data
data = self.data
else:
self.data = data
self.kdt = KDTree(self.data)
self.k = k + 1
self.function = function.lower()
self.fixed = fixed
self.eps = eps
if bandwidth:
try:
bandwidth = np.array(bandwidth)
bandwidth.shape = (len(bandwidth), 1)
except:
bandwidth = np.ones((len(data), 1), 'float') * bandwidth
self.bandwidth = bandwidth
else:
self._set_bw()
self._eval_kernel()
neighbors, weights = self._k_to_W(ids)
if diagonal:
for i in neighbors:
weights[i][neighbors[i].index(i)] = 1.0
W.__init__(self, neighbors, weights, ids)
@classmethod
def from_shapefile(cls, filepath, idVariable=None, **kwargs):
"""
Kernel based weights from shapefile
Arguments
---------
shapefile : string
shapefile name with shp suffix
idVariable : string
name of column in shapefile's DBF to use for ids
Returns
--------
Kernel Weights Object
See Also
---------
:class:`pysal.weights.Kernel`
:class:`pysal.weights.W`
"""
points = get_points_array_from_shapefile(filepath)
if idVariable is not None:
ids = get_ids(filepath, idVariable)
else:
ids = None
return cls.from_array(points, ids=ids, **kwargs)
@classmethod
def from_array(cls, array, **kwargs):
"""
Construct a Kernel weights from an array. Supports all the same options
as :class:`pysal.weights.Kernel`
See Also
--------
:class:`pysal.weights.Kernel`
:class:`pysal.weights.W`
"""
return cls(array, **kwargs)
@classmethod
def from_dataframe(cls, df, geom_col='geometry', ids=None, **kwargs):
"""
Make Kernel weights from a dataframe.
Parameters
----------
df : pandas.dataframe
a dataframe with a geometry column that can be used to
construct a W object
geom_col : string
column name of the geometry stored in df
ids : string or iterable
if string, the column name of the indices from the dataframe
if iterable, a list of ids to use for the W
if None, df.index is used.
See Also
--------
:class:`pysal.weights.Kernel`
:class:`pysal.weights.W`
"""
pts = get_points_array(df[geom_col])
if ids is None:
ids = df.index.tolist()
elif isinstance(ids, str):
ids = df[ids].tolist()
return cls(pts, ids=ids, **kwargs)
def _k_to_W(self, ids=None):
allneighbors = {}
weights = {}
if ids:
ids = np.array(ids)
else:
ids = np.arange(len(self.data))
for i, neighbors in enumerate(self.kernel):
if len(self.neigh[i]) == 0:
allneighbors[ids[i]] = []
weights[ids[i]] = []
else:
allneighbors[ids[i]] = list(ids[self.neigh[i]])
weights[ids[i]] = self.kernel[i].tolist()
return allneighbors, weights
def _set_bw(self):
dmat, neigh = self.kdt.query(self.data, k=self.k)
if self.fixed:
# use max knn distance as bandwidth
bandwidth = dmat.max() * self.eps
n = len(dmat)
self.bandwidth = np.ones((n, 1), 'float') * bandwidth
else:
# use local max knn distance
self.bandwidth = dmat.max(axis=1) * self.eps
self.bandwidth.shape = (self.bandwidth.size, 1)
# identify knn neighbors for each point
nnq = self.kdt.query(self.data, k=self.k)
self.neigh = nnq[1]
def _eval_kernel(self):
# get points within bandwidth distance of each point
if not hasattr(self, 'neigh'):
kdtq = self.kdt.query_ball_point
neighbors = [kdtq(self.data[i], r=bwi[0]) for i,
bwi in enumerate(self.bandwidth)]
self.neigh = neighbors
# get distances for neighbors
bw = self.bandwidth
kdtq = self.kdt.query
z = []
for i, nids in enumerate(self.neigh):
di, ni = kdtq(self.data[i], k=len(nids))
if not isinstance(di, np.ndarray):
di = np.asarray([di] * len(nids))
ni = np.asarray([ni] * len(nids))
zi = np.array([dict(zip(ni, di))[nid] for nid in nids]) / bw[i]
z.append(zi)
zs = z
# functions follow Anselin and Rey (2010) table 5.4
if self.function == 'triangular':
self.kernel = [1 - zi for zi in zs]
elif self.function == 'uniform':
self.kernel = [np.ones(zi.shape) * 0.5 for zi in zs]
elif self.function == 'quadratic':
self.kernel = [(3. / 4) * (1 - zi ** 2) for zi in zs]
elif self.function == 'quartic':
self.kernel = [(15. / 16) * (1 - zi ** 2) ** 2 for zi in zs]
elif self.function == 'gaussian':
c = np.pi * 2
c = c ** (-0.5)
self.kernel = [c * np.exp(-(zi ** 2) / 2.) for zi in zs]
else:
print('Unsupported kernel function', self.function)
class Kernel(W):
def __init__(self,
data,
bandwidth=None,
fixed=True,
k=2,
function='triangular',
eps=1.0000001,
ids=None,
diagonal=False,
ncores=1):
if issubclass(type(data), scipy.spatial.KDTree):
self.kdt = data
self.data = self.kdt.data
data = self.data
else:
self.data = data
self.kdt = KDTree(self.data)
self.k = k + 1
self.function = function.lower()
self.fixed = fixed
self.eps = eps
self.ncores = ncores
if bandwidth:
try:
bandwidth = np.array(bandwidth)
bandwidth.shape = (len(bandwidth), 1)
except:
bandwidth = np.ones((len(data), 1), 'float') * bandwidth
self.bandwidth = bandwidth
else:
self._set_bw()
self._eval_kernel()
neighbors, weights = self._k_to_W(ids)
if diagonal:
for i in neighbors:
weights[i][neighbors[i].index(i)] = 1.0
W.__init__(self, neighbors, weights, ids)
def _k_to_W(self, ids=None):
allneighbors = {}
weights = {}
if ids:
ids = np.array(ids)
else:
ids = np.arange(len(self.data))
for i, neighbors in enumerate(self.kernel):
if len(self.neigh[i]) == 0:
allneighbors[ids[i]] = []
weights[ids[i]] = []
else:
allneighbors[ids[i]] = list(ids[self.neigh[i]])
weights[ids[i]] = self.kernel[i].tolist()
return allneighbors, weights
def _set_bw(self):
dmat, neigh = self.kdt.query(self.data, k=self.k)
if self.fixed:
# use max knn distance as bandwidth
bandwidth = dmat.max() * self.eps
n = len(dmat)
self.bandwidth = np.ones((n, 1), 'float') * bandwidth
else:
# use local max knn distance
self.bandwidth = dmat.max(axis=1) * self.eps
self.bandwidth.shape = (self.bandwidth.size, 1)
# identify knn neighbors for each point
nnq = self.kdt.query(self.data, k=self.k)
self.neigh = nnq[1]
def _eval_kernel(self):
t1 = time.time()
# get points within bandwidth distance of each point
kdtbq = self.kdt.query_ball_point
kdtq = self.kdt.query
bw = self.bandwidth
if self.ncores > 1:
pool = mp.Pool(processes=self.ncores,
initializer=loadkd,
initargs=(kdtbq, kdtq, bw))
if not hasattr(self, 'neigh'):
if self.ncores > 1:
neighbors = pool.map(bqwrapper,
self.data,
chunksize=len(self.bandwidth) /
self.ncores)
else:
neighbors = [
kdtbq(self.data[i], r=bwi[0])
for i, bwi in enumerate(self.bandwidth)
]
self.neigh = neighbors
t2 = time.time()
print "Ball Point Query took {} seconds.".format(t2 - t1)
# get distances for neighbors
bw = self.bandwidth
#kdtq = self.kdt.query
z = []
t1 = time.time()
if self.ncores > 1:
iterable = [(i, nids, self.data[i])
for i, nids in enumerate(self.neigh)]
z = pool.map(qwrapper, iterable)
else:
for i, nids in enumerate(self.neigh):
di, ni = kdtq(self.data[i], k=len(nids))
zi = np.array([dict(zip(ni, di))[nid] for nid in nids]) / bw[i]
z.append(zi)
t2 = time.time()
print "Local query took: {} seconds".format(t2 - t1)
zs = z
# functions follow Anselin and Rey (2010) table 5.4
if self.function == 'triangular':
self.kernel = [1 - zi for zi in zs]
elif self.function == 'uniform':
self.kernel = [np.ones(zi.shape) * 0.5 for zi in zs]
elif self.function == 'quadratic':
self.kernel = [(3. / 4) * (1 - zi**2) for zi in zs]
elif self.function == 'quartic':
self.kernel = [(15. / 16) * (1 - zi**2)**2 for zi in zs]
elif self.function == 'gaussian':
c = np.pi * 2
c = c**(-0.5)
self.kernel = [c * np.exp(-(zi**2) / 2.) for zi in zs]
else:
print 'Unsupported kernel function', self.function
def knnW(data, k=2, p=2, ids=None, pct_unique=0.25):
"""
Creates nearest neighbor weights matrix based on k nearest
neighbors.
Parameters
----------
data : array (n,k) or KDTree where KDtree.data is array (n,k)
n observations on k characteristics used to measure
distances between the n objects
k : int
number of nearest neighbors
p : float
Minkowski p-norm distance metric parameter:
1<=p<=infinity
2: Euclidean distance
1: Manhattan distance
ids : list
identifiers to attach to each observation
pct_unique : float
threshold percentage of unique points in data. Below this
threshold tree is built on unique values only
Returns
-------
w : W instance
Weights object with binary weights
Examples
--------
>>> x,y=np.indices((5,5))
>>> x.shape=(25,1)
>>> y.shape=(25,1)
>>> data=np.hstack([x,y])
>>> wnn2=knnW(data,k=2)
>>> wnn4=knnW(data,k=4)
>>> set([1,5,6,2]) == set(wnn4.neighbors[0])
True
>>> set([0,6,10,1]) == set(wnn4.neighbors[5])
True
>>> set([1,5]) == set(wnn2.neighbors[0])
True
>>> set([0,6]) == set(wnn2.neighbors[5])
True
>>> "%.2f"%wnn2.pct_nonzero
'0.08'
>>> wnn4.pct_nonzero
0.16
>>> wnn3e=knnW(data,p=2,k=3)
>>> set([1,5,6]) == set(wnn3e.neighbors[0])
True
>>> wnn3m=knnW(data,p=1,k=3)
>>> a = set([1,5,2])
>>> b = set([1,5,6])
>>> c = set([1,5,10])
>>> w0n = set(wnn3m.neighbors[0])
>>> a==w0n or b==w0n or c==w0n
True
ids
>>> wnn2 = knnW(data,2)
>>> wnn2[0]
{1: 1.0, 5: 1.0}
>>> wnn2[1]
{0: 1.0, 2: 1.0}
now with 1 rather than 0 offset
>>> wnn2 = knnW(data,2, ids = range(1,26))
>>> wnn2[1]
{2: 1.0, 6: 1.0}
>>> wnn2[2]
{1: 1.0, 3: 1.0}
>>> 0 in wnn2.neighbors
False
Notes
-----
Ties between neighbors of equal distance are arbitrarily broken.
See Also
--------
pysal.weights.W
"""
if issubclass(type(data), scipy.spatial.KDTree):
kd = data
data = kd.data
nnq = kd.query(data, k=k+1, p=p)
info = nnq[1]
elif type(data).__name__ == 'ndarray':
# check if unique points are a small fraction of all points
ind = np.lexsort(data.T)
u = data[np.concatenate(([True],np.any(data[ind[1:]]!=data[ind[:-1]],axis=1)))]
pct_u = len(u)*1. / len(data)
if pct_u < pct_unique:
tree = KDTree(u)
nnq = tree.query(data, k=k+1, p=p)
info = nnq[1]
uid = [np.where((data == ui).all(axis=1))[0][0] for ui in u]
new_info = np.zeros((len(data), k + 1), 'int')
for i, row in enumerate(info):
new_info[i] = [uid[j] for j in row]
info = new_info
else:
kd = KDTree(data)
# calculate
nnq = kd.query(data, k=k + 1, p=p)
info = nnq[1]
else:
print 'Unsupported type'
return None
neighbors = {}
for i, row in enumerate(info):
row = row.tolist()
if i in row:
row.remove(i)
focal = i
if ids:
row = [ ids[j] for j in row]
focal = ids[i]
neighbors[focal] = row
return pysal.weights.W(neighbors, id_order=ids)
class KNN(W):
"""
Creates nearest neighbor weights matrix based on k nearest
neighbors.
Parameters
----------
kdtree : object
PySAL KDTree or ArcKDTree where KDtree.data is array (n,k)
n observations on k characteristics used to measure
distances between the n objects
k : int
number of nearest neighbors
p : float
Minkowski p-norm distance metric parameter:
1<=p<=infinity
2: Euclidean distance
1: Manhattan distance
Ignored if the KDTree is an ArcKDTree
ids : list
identifiers to attach to each observation
Returns
-------
w : W
instance
Weights object with binary weights
Examples
--------
>>> points = [(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> kd = pysal.cg.kdtree.KDTree(np.array(points))
>>> wnn2 = pysal.KNN(kd, 2)
>>> [1,3] == wnn2.neighbors[0]
True
ids
>>> wnn2 = KNN(kd,2)
>>> wnn2[0]
{1: 1.0, 3: 1.0}
>>> wnn2[1]
{0: 1.0, 3: 1.0}
now with 1 rather than 0 offset
>>> wnn2 = KNN(kd, 2, ids=range(1,7))
>>> wnn2[1]
{2: 1.0, 4: 1.0}
>>> wnn2[2]
{1: 1.0, 4: 1.0}
>>> 0 in wnn2.neighbors
False
Notes
-----
Ties between neighbors of equal distance are arbitrarily broken.
See Also
--------
:class:`pysal.weights.W`
"""
def __init__(self, data, k=2, p=2, ids=None, radius=None, distance_metric='euclidean'):
if isKDTree(data):
self.kdtree = data
self.data = data.data
else:
self.data = data
self.kdtree = KDTree(data, radius=radius, distance_metric=distance_metric)
self.k = k
self.p = p
this_nnq = self.kdtree.query(self.data, k=k+1, p=p)
to_weight = this_nnq[1]
if ids is None:
ids = list(range(to_weight.shape[0]))
neighbors = {}
for i,row in enumerate(to_weight):
row = row.tolist()
row.remove(i)
row = [ids[j] for j in row]
focal = ids[i]
neighbors[focal] = row
W.__init__(self, neighbors, id_order=ids)
@classmethod
def from_shapefile(cls, filepath, **kwargs):
"""
Nearest neighbor weights from a shapefile.
Parameters
----------
data : string
shapefile containing attribute data.
k : int
number of nearest neighbors
p : float
Minkowski p-norm distance metric parameter:
1<=p<=infinity
2: Euclidean distance
1: Manhattan distance
ids : list
identifiers to attach to each observation
radius : float
If supplied arc_distances will be calculated
based on the given radius. p will be ignored.
Returns
-------
w : KNN
instance; Weights object with binary weights.
Examples
--------
Polygon shapefile
>>> wc=knnW_from_shapefile(pysal.examples.get_path("columbus.shp"))
>>> "%.4f"%wc.pct_nonzero
'4.0816'
>>> set([2,1]) == set(wc.neighbors[0])
True
>>> wc3=pysal.knnW_from_shapefile(pysal.examples.get_path("columbus.shp"),k=3)
>>> set(wc3.neighbors[0]) == set([2,1,3])
True
>>> set(wc3.neighbors[2]) == set([4,3,0])
True
1 offset rather than 0 offset
>>> wc3_1=knnW_from_shapefile(pysal.examples.get_path("columbus.shp"),k=3,idVariable="POLYID")
>>> set([4,3,2]) == set(wc3_1.neighbors[1])
True
>>> wc3_1.weights[2]
[1.0, 1.0, 1.0]
>>> set([4,1,8]) == set(wc3_1.neighbors[2])
True
Point shapefile
>>> w=knnW_from_shapefile(pysal.examples.get_path("juvenile.shp"))
>>> w.pct_nonzero
1.1904761904761905
>>> w1=knnW_from_shapefile(pysal.examples.get_path("juvenile.shp"),k=1)
>>> "%.3f"%w1.pct_nonzero
Notes
-----
Ties between neighbors of equal distance are arbitrarily broken.
See Also
--------
:class:`pysal.weights.KNN`
:class:`pysal.weights.W`
"""
return cls(get_points_array_from_shapefile(filepath), **kwargs)
@classmethod
def from_array(cls, array, **kwargs):
"""
Creates nearest neighbor weights matrix based on k nearest
neighbors.
Parameters
----------
array : np.ndarray
(n, k) array representing n observations on
k characteristics used to measure distances
between the n objects
**kwargs : keyword arguments, see Rook
Returns
-------
w : W
instance
Weights object with binary weights
Examples
--------
>>> points = [(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> wnn2 = pysal.KNN.from_array(points, 2)
>>> [1,3] == wnn2.neighbors[0]
True
ids
>>> wnn2 = KNN.from_array(points,2)
>>> wnn2[0]
{1: 1.0, 3: 1.0}
>>> wnn2[1]
{0: 1.0, 3: 1.0}
now with 1 rather than 0 offset
>>> wnn2 = KNN.from_array(points, 2, ids=range(1,7))
>>> wnn2[1]
{2: 1.0, 4: 1.0}
>>> wnn2[2]
{1: 1.0, 4: 1.0}
>>> 0 in wnn2.neighbors
False
Notes
-----
Ties between neighbors of equal distance are arbitrarily broken.
See Also
--------
:class: `pysal.weights.KNN`
:class:`pysal.weights.W`
"""
return cls(array, **kwargs)
@classmethod
def from_dataframe(cls, df, geom_col='geometry', ids=None, **kwargs):
"""
Make KNN weights from a dataframe.
Parameters
----------
df : pandas.dataframe
a dataframe with a geometry column that can be used to
construct a W object
geom_col : string
column name of the geometry stored in df
ids : string or iterable
if string, the column name of the indices from the dataframe
if iterable, a list of ids to use for the W
if None, df.index is used.
See Also
--------
:class: `pysal.weights.KNN`
:class:`pysal.weights.W`
"""
pts = get_points_array(df[geom_col])
if ids is None:
ids = df.index.tolist()
elif isinstance(ids, str):
ids = df[ids].tolist()
return cls(pts, ids=ids, **kwargs)
def reweight(self, k=None, p=None, new_data=None, new_ids=None, inplace=True):
"""
Redo K-Nearest Neighbor weights construction using given parameters
Parameters
----------
new_data : np.ndarray
an array containing additional data to use in the KNN
weight
new_ids : list
a list aligned with new_data that provides the ids for
each new observation
inplace : bool
a flag denoting whether to modify the KNN object
in place or to return a new KNN object
k : int
number of nearest neighbors
p : float
Minkowski p-norm distance metric parameter:
1<=p<=infinity
2: Euclidean distance
1: Manhattan distance
Ignored if the KDTree is an ArcKDTree
Returns
-------
A copy of the object using the new parameterization, or None if the
object is reweighted in place.
"""
if (new_data is not None):
new_data = np.asarray(new_data).reshape(-1,2)
data = np.vstack((self.data, new_data)).reshape(-1,2)
if new_ids is not None:
ids = copy.deepcopy(self.id_order)
ids.extend(list(new_ids))
else:
ids = list(range(data.shape[0]))
elif (new_data is None) and (new_ids is None):
# If not, we can use the same kdtree we have
data = self.kdtree
ids = self.id_order
elif (new_data is None) and (new_ids is not None):
Warn('Remapping ids must be done using w.remap_ids')
if k is None:
k = self.k
if p is None:
p = self.p
if inplace:
self._reset()
self.__init__(data, ids=ids, k=k, p=p)
else:
return KNN(data, ids=ids, k=k, p=p)
class KNN(W):
"""
Creates nearest neighbor weights matrix based on k nearest
neighbors.
Parameters
----------
kdtree : object
PySAL KDTree or ArcKDTree where KDtree.data is array (n,k)
n observations on k characteristics used to measure
distances between the n objects
k : int
number of nearest neighbors
p : float
Minkowski p-norm distance metric parameter:
1<=p<=infinity
2: Euclidean distance
1: Manhattan distance
Ignored if the KDTree is an ArcKDTree
ids : list
identifiers to attach to each observation
Returns
-------
w : W
instance
Weights object with binary weights
Examples
--------
>>> points = [(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> kd = pysal.cg.kdtree.KDTree(np.array(points))
>>> wnn2 = pysal.KNN(kd, 2)
>>> [1,3] == wnn2.neighbors[0]
True
ids
>>> wnn2 = KNN(kd,2)
>>> wnn2[0]
{1: 1.0, 3: 1.0}
>>> wnn2[1]
{0: 1.0, 3: 1.0}
now with 1 rather than 0 offset
>>> wnn2 = KNN(kd, 2, ids=range(1,7))
>>> wnn2[1]
{2: 1.0, 4: 1.0}
>>> wnn2[2]
{1: 1.0, 4: 1.0}
>>> 0 in wnn2.neighbors
False
Notes
-----
Ties between neighbors of equal distance are arbitrarily broken.
See Also
--------
:class:`pysal.weights.W`
"""
def __init__(self,
data,
k=2,
p=2,
ids=None,
radius=None,
distance_metric='euclidean'):
if isKDTree(data):
self.kdtree = data
self.data = data.data
else:
self.data = data
self.kdtree = KDTree(data,
radius=radius,
distance_metric=distance_metric)
self.k = k
self.p = p
this_nnq = self.kdtree.query(self.data, k=k + 1, p=p)
to_weight = this_nnq[1]
if ids is None:
ids = list(range(to_weight.shape[0]))
neighbors = {}
for i, row in enumerate(to_weight):
row = row.tolist()
row.remove(i)
row = [ids[j] for j in row]
focal = ids[i]
neighbors[focal] = row
W.__init__(self, neighbors, id_order=ids)
@classmethod
def from_shapefile(cls, filepath, **kwargs):
"""
Nearest neighbor weights from a shapefile.
Parameters
----------
data : string
shapefile containing attribute data.
k : int
number of nearest neighbors
p : float
Minkowski p-norm distance metric parameter:
1<=p<=infinity
2: Euclidean distance
1: Manhattan distance
ids : list
identifiers to attach to each observation
radius : float
If supplied arc_distances will be calculated
based on the given radius. p will be ignored.
Returns
-------
w : KNN
instance; Weights object with binary weights.
Examples
--------
Polygon shapefile
>>> wc=knnW_from_shapefile(pysal.examples.get_path("columbus.shp"))
>>> "%.4f"%wc.pct_nonzero
'4.0816'
>>> set([2,1]) == set(wc.neighbors[0])
True
>>> wc3=pysal.knnW_from_shapefile(pysal.examples.get_path("columbus.shp"),k=3)
>>> set(wc3.neighbors[0]) == set([2,1,3])
True
>>> set(wc3.neighbors[2]) == set([4,3,0])
True
1 offset rather than 0 offset
>>> wc3_1=knnW_from_shapefile(pysal.examples.get_path("columbus.shp"),k=3,idVariable="POLYID")
>>> set([4,3,2]) == set(wc3_1.neighbors[1])
True
>>> wc3_1.weights[2]
[1.0, 1.0, 1.0]
>>> set([4,1,8]) == set(wc3_1.neighbors[2])
True
Point shapefile
>>> w=knnW_from_shapefile(pysal.examples.get_path("juvenile.shp"))
>>> w.pct_nonzero
1.1904761904761905
>>> w1=knnW_from_shapefile(pysal.examples.get_path("juvenile.shp"),k=1)
>>> "%.3f"%w1.pct_nonzero
Notes
-----
Ties between neighbors of equal distance are arbitrarily broken.
See Also
--------
:class:`pysal.weights.KNN`
:class:`pysal.weights.W`
"""
return cls(get_points_array_from_shapefile(filepath), **kwargs)
@classmethod
def from_array(cls, array, **kwargs):
"""
Creates nearest neighbor weights matrix based on k nearest
neighbors.
Parameters
----------
array : np.ndarray
(n, k) array representing n observations on
k characteristics used to measure distances
between the n objects
**kwargs : keyword arguments, see Rook
Returns
-------
w : W
instance
Weights object with binary weights
Examples
--------
>>> points = [(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> wnn2 = pysal.KNN.from_array(points, 2)
>>> [1,3] == wnn2.neighbors[0]
True
ids
>>> wnn2 = KNN.from_array(points,2)
>>> wnn2[0]
{1: 1.0, 3: 1.0}
>>> wnn2[1]
{0: 1.0, 3: 1.0}
now with 1 rather than 0 offset
>>> wnn2 = KNN.from_array(points, 2, ids=range(1,7))
>>> wnn2[1]
{2: 1.0, 4: 1.0}
>>> wnn2[2]
{1: 1.0, 4: 1.0}
>>> 0 in wnn2.neighbors
False
Notes
-----
Ties between neighbors of equal distance are arbitrarily broken.
See Also
--------
:class: `pysal.weights.KNN`
:class:`pysal.weights.W`
"""
return cls(array, **kwargs)
@classmethod
def from_dataframe(cls, df, geom_col='geometry', ids=None, **kwargs):
"""
Make KNN weights from a dataframe.
Parameters
----------
df : pandas.dataframe
a dataframe with a geometry column that can be used to
construct a W object
geom_col : string
column name of the geometry stored in df
ids : string or iterable
if string, the column name of the indices from the dataframe
if iterable, a list of ids to use for the W
if None, df.index is used.
See Also
--------
:class: `pysal.weights.KNN`
:class:`pysal.weights.W`
"""
pts = get_points_array(df[geom_col])
if ids is None:
ids = df.index.tolist()
elif isinstance(ids, str):
ids = df[ids].tolist()
return cls(pts, ids=ids, **kwargs)
def reweight(self,
k=None,
p=None,
new_data=None,
new_ids=None,
inplace=True):
"""
Redo K-Nearest Neighbor weights construction using given parameters
Parameters
----------
new_data : np.ndarray
an array containing additional data to use in the KNN
weight
new_ids : list
a list aligned with new_data that provides the ids for
each new observation
inplace : bool
a flag denoting whether to modify the KNN object
in place or to return a new KNN object
k : int
number of nearest neighbors
p : float
Minkowski p-norm distance metric parameter:
1<=p<=infinity
2: Euclidean distance
1: Manhattan distance
Ignored if the KDTree is an ArcKDTree
Returns
-------
A copy of the object using the new parameterization, or None if the
object is reweighted in place.
"""
if (new_data is not None):
new_data = np.asarray(new_data).reshape(-1, 2)
data = np.vstack((self.data, new_data)).reshape(-1, 2)
if new_ids is not None:
ids = copy.deepcopy(self.id_order)
ids.extend(list(new_ids))
else:
ids = list(range(data.shape[0]))
elif (new_data is None) and (new_ids is None):
# If not, we can use the same kdtree we have
data = self.kdtree
ids = self.id_order
elif (new_data is None) and (new_ids is not None):
Warn('Remapping ids must be done using w.remap_ids')
if k is None:
k = self.k
if p is None:
p = self.p
if inplace:
self._reset()
self.__init__(data, ids=ids, k=k, p=p)
else:
return KNN(data, ids=ids, k=k, p=p)
def knnW(data, k=2, p=2, ids=None):
"""
Creates nearest neighbor weights matrix based on k nearest
neighbors.
Parameters
----------
kdtree : object
PySAL KDTree or ArcKDTree where KDtree.data is array (n,k)
n observations on k characteristics used to measure
distances between the n objects
k : int
number of nearest neighbors
p : float
Minkowski p-norm distance metric parameter:
1<=p<=infinity
2: Euclidean distance
1: Manhattan distance
Ignored if the KDTree is an ArcKDTree
ids : list
identifiers to attach to each observation
Returns
-------
w : W
instance
Weights object with binary weights
Examples
--------
>>> points = [(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> kd = pysal.cg.kdtree.KDTree(np.array(points))
>>> wnn2 = pysal.knnW(kd, 2)
>>> [1,3] == wnn2.neighbors[0]
True
ids
>>> wnn2 = knnW(kd,2)
>>> wnn2[0]
{1: 1.0, 3: 1.0}
>>> wnn2[1]
{0: 1.0, 3: 1.0}
now with 1 rather than 0 offset
>>> wnn2 = knnW(kd, 2, ids=range(1,7))
>>> wnn2[1]
{2: 1.0, 4: 1.0}
>>> wnn2[2]
{1: 1.0, 4: 1.0}
>>> 0 in wnn2.neighbors
False
Notes
-----
Ties between neighbors of equal distance are arbitrarily broken.
See Also
--------
pysal.weights.W
"""
if isKDTree(data):
kdt = data
data = kdt.data
else:
kdt = KDTree(data)
nnq = kdt.query(data, k=k+1, p=p)
info = nnq[1]
neighbors = {}
for i, row in enumerate(info):
row = row.tolist()
if i in row:
row.remove(i)
focal = i
if ids:
row = [ ids[j] for j in row]
focal = ids[i]
neighbors[focal] = row
return pysal.weights.W(neighbors, id_order=ids)
class Kernel(W):
def __init__(self, data, bandwidth=None, fixed=True, k=2,
function='triangular', eps=1.0000001, ids=None,
diagonal=False, ncores=1):
if issubclass(type(data), scipy.spatial.KDTree):
self.kdt = data
self.data = self.kdt.data
data = self.data
else:
self.data = data
self.kdt = KDTree(self.data)
self.k = k + 1
self.function = function.lower()
self.fixed = fixed
self.eps = eps
self.ncores = ncores
if bandwidth:
try:
bandwidth = np.array(bandwidth)
bandwidth.shape = (len(bandwidth), 1)
except:
bandwidth = np.ones((len(data), 1), 'float') * bandwidth
self.bandwidth = bandwidth
else:
self._set_bw()
self._eval_kernel()
neighbors, weights = self._k_to_W(ids)
if diagonal:
for i in neighbors:
weights[i][neighbors[i].index(i)] = 1.0
W.__init__(self, neighbors, weights, ids)
def _k_to_W(self, ids=None):
allneighbors = {}
weights = {}
if ids:
ids = np.array(ids)
else:
ids = np.arange(len(self.data))
for i, neighbors in enumerate(self.kernel):
if len(self.neigh[i]) == 0:
allneighbors[ids[i]] = []
weights[ids[i]] = []
else:
allneighbors[ids[i]] = list(ids[self.neigh[i]])
weights[ids[i]] = self.kernel[i].tolist()
return allneighbors, weights
def _set_bw(self):
dmat, neigh = self.kdt.query(self.data, k=self.k)
if self.fixed:
# use max knn distance as bandwidth
bandwidth = dmat.max() * self.eps
n = len(dmat)
self.bandwidth = np.ones((n, 1), 'float') * bandwidth
else:
# use local max knn distance
self.bandwidth = dmat.max(axis=1) * self.eps
self.bandwidth.shape = (self.bandwidth.size, 1)
# identify knn neighbors for each point
nnq = self.kdt.query(self.data, k=self.k)
self.neigh = nnq[1]
def _eval_kernel(self):
t1 = time.time()
# get points within bandwidth distance of each point
kdtbq = self.kdt.query_ball_point
kdtq = self.kdt.query
bw = self.bandwidth
if self.ncores > 1:
pool = mp.Pool(processes=self.ncores, initializer=loadkd, initargs=(kdtbq,kdtq,bw))
if not hasattr(self, 'neigh'):
if self.ncores > 1:
neighbors = pool.map(bqwrapper,self.data, chunksize = len(self.bandwidth) / self.ncores)
else:
neighbors = [kdtbq(self.data[i], r=bwi[0]) for i,
bwi in enumerate(self.bandwidth)]
self.neigh = neighbors
t2 = time.time()
print "Ball Point Query took {} seconds.".format(t2 - t1)
# get distances for neighbors
bw = self.bandwidth
#kdtq = self.kdt.query
z = []
t1 = time.time()
if self.ncores > 1:
iterable = [(i,nids, self.data[i]) for i, nids in enumerate(self.neigh)]
z = pool.map(qwrapper, iterable)
else:
for i, nids in enumerate(self.neigh):
di, ni = kdtq(self.data[i], k=len(nids))
zi = np.array([dict(zip(ni, di))[nid] for nid in nids]) / bw[i]
z.append(zi)
t2 = time.time()
print "Local query took: {} seconds".format(t2 - t1)
zs = z
# functions follow Anselin and Rey (2010) table 5.4
if self.function == 'triangular':
self.kernel = [1 - zi for zi in zs]
elif self.function == 'uniform':
self.kernel = [np.ones(zi.shape) * 0.5 for zi in zs]
elif self.function == 'quadratic':
self.kernel = [(3. / 4) * (1 - zi ** 2) for zi in zs]
elif self.function == 'quartic':
self.kernel = [(15. / 16) * (1 - zi ** 2) ** 2 for zi in zs]
elif self.function == 'gaussian':
c = np.pi * 2
c = c ** (-0.5)
self.kernel = [c * np.exp(-(zi ** 2) / 2.) for zi in zs]
else:
print 'Unsupported kernel function', self.function
def knnW(data, k=2, p=2, ids=None, pct_unique=0.25):
"""
Creates nearest neighbor weights matrix based on k nearest
neighbors.
Parameters
----------
data : array (n,k) or KDTree where KDtree.data is array (n,k)
n observations on k characteristics used to measure
distances between the n objects
k : int
number of nearest neighbors
p : float
Minkowski p-norm distance metric parameter:
1<=p<=infinity
2: Euclidean distance
1: Manhattan distance
ids : list
identifiers to attach to each observation
pct_unique : float
threshold percentage of unique points in data. Below this
threshold tree is built on unique values only
Returns
-------
w : W instance
Weights object with binary weights
Examples
--------
>>> x,y=np.indices((5,5))
>>> x.shape=(25,1)
>>> y.shape=(25,1)
>>> data=np.hstack([x,y])
>>> wnn2=knnW(data,k=2)
>>> wnn4=knnW(data,k=4)
>>> set([1,5,6,2]) == set(wnn4.neighbors[0])
True
>>> set([0,6,10,1]) == set(wnn4.neighbors[5])
True
>>> set([1,5]) == set(wnn2.neighbors[0])
True
>>> set([0,6]) == set(wnn2.neighbors[5])
True
>>> "%.2f"%wnn2.pct_nonzero
'0.08'
>>> wnn4.pct_nonzero
0.16
>>> wnn3e=knnW(data,p=2,k=3)
>>> set([1,5,6]) == set(wnn3e.neighbors[0])
True
>>> wnn3m=knnW(data,p=1,k=3)
>>> a = set([1,5,2])
>>> b = set([1,5,6])
>>> c = set([1,5,10])
>>> w0n = set(wnn3m.neighbors[0])
>>> a==w0n or b==w0n or c==w0n
True
ids
>>> wnn2 = knnW(data,2)
>>> wnn2[0]
{1: 1.0, 5: 1.0}
>>> wnn2[1]
{0: 1.0, 2: 1.0}
now with 1 rather than 0 offset
>>> wnn2 = knnW(data,2, ids = range(1,26))
>>> wnn2[1]
{2: 1.0, 6: 1.0}
>>> wnn2[2]
{1: 1.0, 3: 1.0}
>>> 0 in wnn2.neighbors
False
Notes
-----
Ties between neighbors of equal distance are arbitrarily broken.
See Also
--------
pysal.weights.W
"""
if issubclass(type(data), scipy.spatial.KDTree):
kd = data
data = kd.data
nnq = kd.query(data, k=k+1, p=p)
info = nnq[1]
elif type(data).__name__ == 'ndarray':
# check if unique points are a small fraction of all points
ind = np.lexsort(data.T)
u = data[np.concatenate(([True],np.any(data[ind[1:]]!=data[ind[:-1]],axis=1)))]
pct_u = len(u)*1. / len(data)
if pct_u < pct_unique:
tree = KDTree(u)
nnq = tree.query(data, k=k+1, p=p)
info = nnq[1]
uid = [np.where((data == ui).all(axis=1))[0][0] for ui in u]
new_info = np.zeros((len(data), k + 1), 'int')
for i, row in enumerate(info):
new_info[i] = [uid[j] for j in row]
info = new_info
else:
kd = KDTree(data)
# calculate
nnq = kd.query(data, k=k + 1, p=p)
info = nnq[1]
else:
print 'Unsupported type'
return None
neighbors = {}
for i, row in enumerate(info):
row = row.tolist()
if i in row:
row.remove(i)
focal = i
if ids:
row = [ ids[j] for j in row]
focal = ids[i]
neighbors[focal] = row
return pysal.weights.W(neighbors, id_order=ids)
class Kernel(W):
"""Spatial weights based on kernel functions
Parameters
----------
data : array (n,k) or KDTree where KDtree.data is array (n,k)
n observations on k characteristics used to measure
distances between the n objects
bandwidth : float or array-like (optional)
the bandwidth :math:`h_i` for the kernel.
fixed : binary
If true then :math:`h_i=h \\forall i`. If false then
bandwidth is adaptive across observations.
k : int
the number of nearest neighbors to use for determining
bandwidth. For fixed bandwidth, :math:`h_i=max(dknn) \\forall i`
where :math:`dknn` is a vector of k-nearest neighbor
distances (the distance to the kth nearest neighbor for each
observation). For adaptive bandwidths, :math:`h_i=dknn_i`
diagonal : boolean
If true, set diagonal weights = 1.0, if false (default),
diagonals weights are set to value according to kernel
function.
function : string {'triangular','uniform','quadratic','quartic','gaussian'}
kernel function defined as follows with
.. math::
z_{i,j} = d_{i,j}/h_i
triangular
.. math::
K(z) = (1 - |z|) \ if |z| \le 1
uniform
.. math::
K(z) = 1/2 \ if |z| \le 1
quadratic
.. math::
K(z) = (3/4)(1-z^2) \ if |z| \le 1
quartic
.. math::
K(z) = (15/16)(1-z^2)^2 \ if |z| \le 1
gaussian
.. math::
K(z) = (2\pi)^{(-1/2)} exp(-z^2 / 2)
eps : float
adjustment to ensure knn distance range is closed on the
knnth observations
Examples
--------
>>> points=[(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> kw=Kernel(points)
>>> kw.weights[0]
[1.0, 0.500000049999995, 0.4409830615267465]
>>> kw.neighbors[0]
[0, 1, 3]
>>> kw.bandwidth
array([[ 20.000002],
[ 20.000002],
[ 20.000002],
[ 20.000002],
[ 20.000002],
[ 20.000002]])
>>> kw15=Kernel(points,bandwidth=15.0)
>>> kw15[0]
{0: 1.0, 1: 0.33333333333333337, 3: 0.2546440075000701}
>>> kw15.neighbors[0]
[0, 1, 3]
>>> kw15.bandwidth
array([[ 15.],
[ 15.],
[ 15.],
[ 15.],
[ 15.],
[ 15.]])
Adaptive bandwidths user specified
>>> bw=[25.0,15.0,25.0,16.0,14.5,25.0]
>>> kwa=Kernel(points,bandwidth=bw)
>>> kwa.weights[0]
[1.0, 0.6, 0.552786404500042, 0.10557280900008403]
>>> kwa.neighbors[0]
[0, 1, 3, 4]
>>> kwa.bandwidth
array([[ 25. ],
[ 15. ],
[ 25. ],
[ 16. ],
[ 14.5],
[ 25. ]])
Endogenous adaptive bandwidths
>>> kwea=Kernel(points,fixed=False)
>>> kwea.weights[0]
[1.0, 0.10557289844279438, 9.99999900663795e-08]
>>> kwea.neighbors[0]
[0, 1, 3]
>>> kwea.bandwidth
array([[ 11.18034101],
[ 11.18034101],
[ 20.000002 ],
[ 11.18034101],
[ 14.14213704],
[ 18.02775818]])
Endogenous adaptive bandwidths with Gaussian kernel
>>> kweag=Kernel(points,fixed=False,function='gaussian')
>>> kweag.weights[0]
[0.3989422804014327, 0.2674190291577696, 0.2419707487162134]
>>> kweag.bandwidth
array([[ 11.18034101],
[ 11.18034101],
[ 20.000002 ],
[ 11.18034101],
[ 14.14213704],
[ 18.02775818]])
Diagonals to 1.0
>>> kq = Kernel(points,function='gaussian')
>>> kq.weights
{0: [0.3989422804014327, 0.35206533556593145, 0.3412334260702758], 1: [0.35206533556593145, 0.3989422804014327, 0.2419707487162134, 0.3412334260702758, 0.31069657591175387], 2: [0.2419707487162134, 0.3989422804014327, 0.31069657591175387], 3: [0.3412334260702758, 0.3412334260702758, 0.3989422804014327, 0.3011374490937829, 0.26575287272131043], 4: [0.31069657591175387, 0.31069657591175387, 0.3011374490937829, 0.3989422804014327, 0.35206533556593145], 5: [0.26575287272131043, 0.35206533556593145, 0.3989422804014327]}
>>> kqd = Kernel(points, function='gaussian', diagonal=True)
>>> kqd.weights
{0: [1.0, 0.35206533556593145, 0.3412334260702758], 1: [0.35206533556593145, 1.0, 0.2419707487162134, 0.3412334260702758, 0.31069657591175387], 2: [0.2419707487162134, 1.0, 0.31069657591175387], 3: [0.3412334260702758, 0.3412334260702758, 1.0, 0.3011374490937829, 0.26575287272131043], 4: [0.31069657591175387, 0.31069657591175387, 0.3011374490937829, 1.0, 0.35206533556593145], 5: [0.26575287272131043, 0.35206533556593145, 1.0]}
"""
def __init__(
self, data, bandwidth=None, fixed=True, k=2, function="triangular", eps=1.0000001, ids=None, diagonal=False
):
if issubclass(type(data), scipy.spatial.KDTree):
self.kdt = data
self.data = self.kdt.data
data = self.data
else:
self.data = data
self.kdt = KDTree(self.data)
self.k = k + 1
self.function = function.lower()
self.fixed = fixed
self.eps = eps
if bandwidth:
try:
bandwidth = np.array(bandwidth)
bandwidth.shape = (len(bandwidth), 1)
except:
bandwidth = np.ones((len(data), 1), "float") * bandwidth
self.bandwidth = bandwidth
else:
self._set_bw()
self._eval_kernel()
neighbors, weights = self._k_to_W(ids)
if diagonal:
for i in neighbors:
weights[i][neighbors[i].index(i)] = 1.0
W.__init__(self, neighbors, weights, ids)
def _k_to_W(self, ids=None):
allneighbors = {}
weights = {}
if ids:
ids = np.array(ids)
else:
ids = np.arange(len(self.data))
for i, neighbors in enumerate(self.kernel):
if len(self.neigh[i]) == 0:
allneighbors[ids[i]] = []
weights[ids[i]] = []
else:
allneighbors[ids[i]] = list(ids[self.neigh[i]])
weights[ids[i]] = self.kernel[i].tolist()
return allneighbors, weights
def _set_bw(self):
dmat, neigh = self.kdt.query(self.data, k=self.k)
if self.fixed:
# use max knn distance as bandwidth
bandwidth = dmat.max() * self.eps
n = len(dmat)
self.bandwidth = np.ones((n, 1), "float") * bandwidth
else:
# use local max knn distance
self.bandwidth = dmat.max(axis=1) * self.eps
self.bandwidth.shape = (self.bandwidth.size, 1)
# identify knn neighbors for each point
nnq = self.kdt.query(self.data, k=self.k)
self.neigh = nnq[1]
def _eval_kernel(self):
# get points within bandwidth distance of each point
if not hasattr(self, "neigh"):
kdtq = self.kdt.query_ball_point
neighbors = [kdtq(self.data[i], r=bwi[0]) for i, bwi in enumerate(self.bandwidth)]
self.neigh = neighbors
# get distances for neighbors
bw = self.bandwidth
kdtq = self.kdt.query
z = []
for i, nids in enumerate(self.neigh):
di, ni = kdtq(self.data[i], k=len(nids))
zi = np.array([dict(zip(ni, di))[nid] for nid in nids]) / bw[i]
z.append(zi)
zs = z
# functions follow Anselin and Rey (2010) table 5.4
if self.function == "triangular":
self.kernel = [1 - zi for zi in zs]
elif self.function == "uniform":
self.kernel = [np.ones(zi.shape) * 0.5 for zi in zs]
elif self.function == "quadratic":
self.kernel = [(3.0 / 4) * (1 - zi ** 2) for zi in zs]
elif self.function == "quartic":
self.kernel = [(15.0 / 16) * (1 - zi ** 2) ** 2 for zi in zs]
elif self.function == "gaussian":
c = np.pi * 2
c = c ** (-0.5)
self.kernel = [c * np.exp(-(zi ** 2) / 2.0) for zi in zs]
else:
print "Unsupported kernel function", self.function