def adaptive_kernelW(points, bandwidths=None, k=2, function='triangular',
radius=None, diagonal=False):
"""
Kernel weights with adaptive bandwidths.
Parameters
----------
points : array
(n,k)
n observations on k characteristics used to measure
distances between the n objects
bandwidths : float
or array-like (optional)
the bandwidth :math:`h_i` for the kernel.
if no bandwidth is specified k is used to determine the
adaptive bandwidth
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`
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) = |z| \ 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)
radius : float
If supplied arc_distances will be calculated
based on the given radius. p will be ignored.
diagonal : boolean
If true, set diagonal weights = 1.0, if false (default)
diagonal weights are set to value according to kernel
function
Returns
-------
w : W
instance of spatial weights
Examples
--------
User specified bandwidths
>>> points=[(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> bw=[25.0,15.0,25.0,16.0,14.5,25.0]
>>> kwa=adaptive_kernelW(points,bandwidths=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=adaptive_kernelW(points)
>>> 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=adaptive_kernelW(points,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]])
with diagonal
>>> kweag = pysal.adaptive_kernelW(points, function='gaussian')
>>> kweagd = pysal.adaptive_kernelW(points, function='gaussian', diagonal=True)
>>> kweag.neighbors[0]
[0, 1, 3]
>>> kweagd.neighbors[0]
[0, 1, 3]
>>> kweag.weights[0]
[0.3989422804014327, 0.2674190291577696, 0.2419707487162134]
>>> kweagd.weights[0]
[1.0, 0.2674190291577696, 0.2419707487162134]
"""
if radius is not None:
points = pysal.cg.KDTree(points, distance_metric='Arc', radius=radius)
return Kernel(points, bandwidth=bandwidths, fixed=False, k=k,
function=function, diagonal=diagonal)
def adaptive_kernelW_from_shapefile(shapefile, bandwidths=None, k=2, function='triangular',
idVariable=None, radius=None,
diagonal = False):
"""
Kernel weights with adaptive bandwidths.
Parameters
----------
shapefile : string
shapefile name with shp suffix
bandwidths : float
or array-like (optional)
the bandwidth :math:`h_i` for the kernel.
if no bandwidth is specified k is used to determine the
adaptive bandwidth
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`
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) = |z| \ 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)
idVariable : string
name of a column in the shapefile's DBF to use for ids
radius : float
If supplied arc_distances will be calculated
based on the given radius. p will be ignored.
diagonal : boolean
If true, set diagonal weights = 1.0, if false (default)
diagonal weights are set to value according to kernel
function
Returns
-------
w : W
instance of spatial weights
Examples
--------
>>> kwa = pysal.adaptive_kernelW_from_shapefile(pysal.examples.get_path("columbus.shp"), function='gaussian')
>>> kwad = pysal.adaptive_kernelW_from_shapefile(pysal.examples.get_path("columbus.shp"), function='gaussian', diagonal=True)
>>> kwa.neighbors[0]
[0, 2, 1]
>>> kwad.neighbors[0]
[0, 2, 1]
>>> kwa.weights[0]
[0.3989422804014327, 0.24966013701844503, 0.2419707487162134]
>>> kwad.weights[0]
[1.0, 0.24966013701844503, 0.2419707487162134]
>>>
Notes
-----
Supports polygon or point shapefiles. For polygon shapefiles, distance is
based on polygon centroids. Distances are defined using coordinates in
shapefile which are assumed to be projected and not geographical
coordinates.
"""
points = get_points_array_from_shapefile(shapefile)
if radius is not None:
points = pysal.cg.KDTree(points, distance_metric='Arc', radius=radius)
if idVariable:
ids = get_ids(shapefile, idVariable)
return Kernel(points, bandwidth=bandwidths, fixed=False, k=k,
function=function, ids=ids, diagonal=diagonal)
return adaptive_kernelW(points, bandwidths=bandwidths, k=k,
function=function, diagonal=diagonal)
def kernelW_from_shapefile(shapefile, k=2, function='triangular',
idVariable=None, fixed=True, radius=None, diagonal=False):
"""
Kernel based weights.
Parameters
----------
shapefile : string
shapefile name with shp suffix
k : int
the number of nearest neighbors to use for determining
bandwidth. Bandwidth taken as :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).
function : {'triangular','uniform','quadratic','epanechnikov', 'quartic','bisquare','gaussian'}
.. math::
z_{i,j} = d_{i,j}/h_i
triangular
.. math::
K(z) = (1 - |z|) \ if |z| \le 1
uniform
.. math::
K(z) = |z| \ if |z| \le 1
quadratic
.. math::
K(z) = (3/4)(1-z^2) \ if |z| \le 1
epanechnikov
.. math::
K(z) = (1-z^2) \ if |z| \le 1
quartic
.. math::
K(z) = (15/16)(1-z^2)^2 \ if |z| \le 1
bisquare
.. math::
K(z) = (1-z^2)^2 \ if |z| \le 1
gaussian
.. math::
K(z) = (2\pi)^{(-1/2)} exp(-z^2 / 2)
idVariable : string
name of a column in the shapefile's DBF to use for ids
fixed : binary
If true then :math:`h_i=h \\forall i`. If false then
bandwidth is adaptive across observations.
radius : float
If supplied arc_distances will be calculated
based on the given radius. p will be ignored.
diagonal : boolean
If true, set diagonal weights = 1.0, if false (default)
diagonal weights are set to value according to kernel
function
Returns
-------
w : W
instance of spatial weights
Examples
--------
>>> kw = pysal.kernelW_from_shapefile(pysal.examples.get_path("columbus.shp"),idVariable='POLYID', function = 'gaussian')
>>> kwd = pysal.kernelW_from_shapefile(pysal.examples.get_path("columbus.shp"),idVariable='POLYID', function = 'gaussian', diagonal = True)
>>> set(kw.neighbors[1]) == set([4, 2, 3, 1])
True
>>> set(kwd.neighbors[1]) == set([4, 2, 3, 1])
True
>>>
>>> set(kw.weights[1]) == set( [0.2436835517263174, 0.29090631630909874, 0.29671172124745776, 0.3989422804014327])
True
>>> set(kwd.weights[1]) == set( [0.2436835517263174, 0.29090631630909874, 0.29671172124745776, 1.0])
True
Notes
-----
Supports polygon or point shapefiles. For polygon shapefiles, distance is
based on polygon centroids. Distances are defined using coordinates in
shapefile which are assumed to be projected and not geographical
coordinates.
"""
points = get_points_array_from_shapefile(shapefile)
if radius is not None:
points = pysal.cg.KDTree(points, distance_metric='Arc', radius=radius)
if idVariable:
ids = get_ids(shapefile, idVariable)
return Kernel(points, function=function, k=k, ids=ids, fixed=fixed,
diagonal = diagonal)
return kernelW(points, k=k, function=function, fixed=fixed,
diagonal=diagonal)
def kernelW(points, k=2, function='triangular', fixed=True,
radius=None, diagonal=False):
"""
Kernel based weights.
Parameters
----------
points : array
(n,k)
n observations on k characteristics used to measure
distances between the n objects
k : int
the number of nearest neighbors to use for determining
bandwidth. Bandwidth taken as :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).
function : {'triangular','uniform','quadratic','epanechnikov','quartic','bisquare','gaussian'}
.. math::
z_{i,j} = d_{i,j}/h_i
triangular
.. math::
K(z) = (1 - |z|) \ if |z| \le 1
uniform
.. math::
K(z) = |z| \ if |z| \le 1
quadratic
.. math::
K(z) = (3/4)(1-z^2) \ if |z| \le 1
epanechnikov
.. math::
K(z) = (1-z^2) \ if |z| \le 1
quartic
.. math::
K(z) = (15/16)(1-z^2)^2 \ if |z| \le 1
bisquare
.. math::
K(z) = (1-z^2)^2 \ if |z| \le 1
gaussian
.. math::
K(z) = (2\pi)^{(-1/2)} exp(-z^2 / 2)
fixed : boolean
If true then :math:`h_i=h \\forall i`. If false then
bandwidth is adaptive across observations.
radius : float
If supplied arc_distances will be calculated
based on the given radius. p will be ignored.
diagonal : boolean
If true, set diagonal weights = 1.0, if false (default)
diagonal weights are set to value according to kernel
function
Returns
-------
w : W
instance of spatial weights
Examples
--------
>>> points=[(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> kw=kernelW(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]])
use different k
>>> kw=kernelW(points,k=3)
>>> kw.neighbors[0]
[0, 1, 3, 4]
>>> kw.bandwidth
array([[ 22.36068201],
[ 22.36068201],
[ 22.36068201],
[ 22.36068201],
[ 22.36068201],
[ 22.36068201]])
Diagonals to 1.0
>>> kq = kernelW(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 = kernelW(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]}
"""
if radius is not None:
points = pysal.cg.KDTree(points, distance_metric='Arc', radius=radius)
return Kernel(points, function=function, k=k, fixed=fixed,
diagonal=diagonal)
def adaptive_kernelW_from_shapefile(shapefile,
bandwidths=None,
k=2,
function='triangular',
idVariable=None,
radius=None):
"""
Kernel weights with adaptive bandwidths
Parameters
----------
shapefile : string
shapefile name with shp suffix
bandwidths : float or array-like (optional)
the bandwidth :math:`h_i` for the kernel.
if no bandwidth is specified k is used to determine the
adaptive bandwidth
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`
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) = |z| \ 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)
idVariable : string
name of a column in the shapefile's DBF to use for ids
radius : If supplied arc_distances will be calculated
based on the given radius. p will be ignored.
Returns
-------
w : W
instance of spatial weights
Examples
--------
>>> kwa = adaptive_kernelW_from_shapefile(pysal.examples.get_path("columbus.shp"))
>>> kwa.weights[0]
[1.0, 0.03178906767736345, 9.99999900663795e-08]
>>> kwa.bandwidth[:3]
array([[ 0.59871832],
[ 0.59871832],
[ 0.56095647]])
Notes
-----
Supports polygon or point shapefiles. For polygon shapefiles, distance is
based on polygon centroids. Distances are defined using coordinates in
shapefile which are assumed to be projected and not geographical
coordinates.
"""
points = get_points_array_from_shapefile(shapefile)
if radius is not None:
points = pysal.cg.KDTree(points, distance_metric='Arc', radius=radius)
if idVariable:
ids = get_ids(shapefile, idVariable)
return Kernel(points,
bandwidth=bandwidths,
fixed=False,
k=k,
function=function,
ids=ids)
return adaptive_kernelW(points,
bandwidths=bandwidths,
k=k,
function=function)
def kernelW_from_shapefile(shapefile,
k=2,
function='triangular',
idVariable=None,
fixed=True,
radius=None):
"""
Kernel based weights
Parameters
----------
shapefile : string
shapefile name with shp suffix
k : int
the number of nearest neighbors to use for determining
bandwidth. Bandwidth taken as :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).
function : string {'triangular','uniform','quadratic','epanechnikov',
'quartic','bisquare','gaussian'}
.. math::
z_{i,j} = d_{i,j}/h_i
triangular
.. math::
K(z) = (1 - |z|) \ if |z| \le 1
uniform
.. math::
K(z) = |z| \ if |z| \le 1
quadratic
.. math::
K(z) = (3/4)(1-z^2) \ if |z| \le 1
epanechnikov
.. math::
K(z) = (1-z^2) \ if |z| \le 1
quartic
.. math::
K(z) = (15/16)(1-z^2)^2 \ if |z| \le 1
bisquare
.. math::
K(z) = (1-z^2)^2 \ if |z| \le 1
gaussian
.. math::
K(z) = (2\pi)^{(-1/2)} exp(-z^2 / 2)
idVariable : string
name of a column in the shapefile's DBF to use for ids
fixed : binary
If true then :math:`h_i=h \\forall i`. If false then
bandwidth is adaptive across observations.
radius : If supplied arc_distances will be calculated
based on the given radius. p will be ignored.
Returns
-------
w : W
instance of spatial weights
Examples
--------
>>> kw = kernelW_from_shapefile(pysal.examples.get_path("columbus.shp"),idVariable='POLYID')
>>> kw.weights[1]
[0.2052478782400463, 0.007078773148450623, 1.0, 0.23051223027663237]
>>> kw.bandwidth[:3]
array([[ 0.75333961],
[ 0.75333961],
[ 0.75333961]])
Notes
-----
Supports polygon or point shapefiles. For polygon shapefiles, distance is
based on polygon centroids. Distances are defined using coordinates in
shapefile which are assumed to be projected and not geographical
coordinates.
"""
points = get_points_array_from_shapefile(shapefile)
if radius is not None:
points = pysal.cg.KDTree(points, distance_metric='Arc', radius=radius)
if idVariable:
ids = get_ids(shapefile, idVariable)
return Kernel(points, function=function, k=k, ids=ids, fixed=fixed)
return kernelW(points, k=k, function=function, fixed=fixed)
def kernelW(points, k=2, function='triangular', fixed=True, radius=None):
"""
Kernel based weights
Parameters
----------
points : array (n,k)
n observations on k characteristics used to measure
distances between the n objects
k : int
the number of nearest neighbors to use for determining
bandwidth. Bandwidth taken as :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).
function : string {'triangular','uniform','quadratic','epanechnikov',
'quartic','bisquare','gaussian'}
.. math::
z_{i,j} = d_{i,j}/h_i
triangular
.. math::
K(z) = (1 - |z|) \ if |z| \le 1
uniform
.. math::
K(z) = |z| \ if |z| \le 1
quadratic
.. math::
K(z) = (3/4)(1-z^2) \ if |z| \le 1
epanechnikov
.. math::
K(z) = (1-z^2) \ if |z| \le 1
quartic
.. math::
K(z) = (15/16)(1-z^2)^2 \ if |z| \le 1
bisquare
.. math::
K(z) = (1-z^2)^2 \ if |z| \le 1
gaussian
.. math::
K(z) = (2\pi)^{(-1/2)} exp(-z^2 / 2)
fixed : binary
If true then :math:`h_i=h \\forall i`. If false then
bandwidth is adaptive across observations.
radius : If supplied arc_distances will be calculated
based on the given radius. p will be ignored.
Returns
-------
w : W
instance of spatial weights
Examples
--------
>>> points=[(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
>>> kw=kernelW(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]])
use different k
>>> kw=kernelW(points,k=3)
>>> kw.neighbors[0]
[0, 1, 3, 4]
>>> kw.bandwidth
array([[ 22.36068201],
[ 22.36068201],
[ 22.36068201],
[ 22.36068201],
[ 22.36068201],
[ 22.36068201]])
"""
if radius is not None:
points = pysal.cg.KDTree(points, distance_metric='Arc', radius=radius)
return Kernel(points, function=function, k=k, fixed=fixed)
