"""
Getis and Ord G statistic for spatial autocorrelation
"""
__author__ = "Sergio J. Rey <srey@asu.edu>, Myunghwa Hwang <mhwang4@gmail.com> "
__all__ = ['G', 'G_Local']
from pysal.lib.common import np, stats
from pysal.lib.weights.spatial_lag import lag_spatial as slag
from .tabular import _univariate_handler
PERMUTATIONS = 999
[文档]class G(object):
"""
Global G Autocorrelation Statistic
Parameters
----------
y : array (n,1)
Attribute values
w : W
DistanceBand W spatial weights based on distance band
permutations : int
the number of random permutations for calculating pseudo p_values
Attributes
----------
y : array
original variable
w : W
DistanceBand W spatial weights based on distance band
permutation : int
the number of permutations
G : float
the value of statistic
EG : float
the expected value of statistic
VG : float
the variance of G under normality assumption
z_norm : float
standard normal test statistic
p_norm : float
p-value under normality assumption (one-sided)
sim : array
(if permutations > 0)
vector of G values for permutated samples
p_sim : float
p-value based on permutations (one-sided)
null: spatial randomness
alternative: the observed G is extreme it is either extremely high or extremely low
EG_sim : float
average value of G from permutations
VG_sim : float
variance of G from permutations
seG_sim : float
standard deviation of G under permutations.
z_sim : float
standardized G based on permutations
p_z_sim : float
p-value based on standard normal approximation from
permutations (one-sided)
Notes
-----
Moments are based on normality assumption.
For technical details see :cite:`Getis_2010` and :cite:`Ord_2010`.
Examples
--------
>>> import pysal.lib
>>> import numpy
>>> numpy.random.seed(10)
Preparing a point data set
>>> points = [(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
Creating a weights object from points
>>> w = pysal.lib.weights.DistanceBand(points,threshold=15)
>>> w.transform = "B"
Preparing a variable
>>> y = numpy.array([2, 3, 3.2, 5, 8, 7])
Applying Getis and Ord G test
>>> from pysal.explore.esda.getisord import G
>>> g = G(y,w)
Examining the results
>>> round(g.G, 3)
0.557
>>> round(g.p_norm, 3)
0.173
"""
[文档] def __init__(self, y, w, permutations=PERMUTATIONS):
y = np.asarray(y).flatten()
self.n = len(y)
self.y = y
w.transform = "B"
self.w = w
self.permutations = permutations
self.__moments()
self.y2 = y * y
y = y.reshape(len(y), 1) # Ensure that y is an n by 1 vector, otherwise y*y.T == y*y
self.den_sum = (y * y.T).sum() - (y * y).sum()
self.G = self.__calc(self.y)
self.z_norm = (self.G - self.EG) / np.sqrt(self.VG)
self.p_norm = 1.0 - stats.norm.cdf(np.abs(self.z_norm))
if permutations:
sim = [self.__calc(np.random.permutation(self.y))
for i in range(permutations)]
self.sim = sim = np.array(sim)
above = sim >= self.G
larger = sum(above)
if (self.permutations - larger) < larger:
larger = self.permutations - larger
self.p_sim = (larger + 1.0) / (permutations + 1.)
self.EG_sim = sum(sim) / permutations
self.seG_sim = sim.std()
self.VG_sim = self.seG_sim ** 2
self.z_sim = (self.G - self.EG_sim) / self.seG_sim
self.p_z_sim = 1. - stats.norm.cdf(np.abs(self.z_sim))
def __moments(self):
y = self.y
n = self.n
w = self.w
n2 = n * n
s0 = w.s0
self.EG = s0 / (n * (n - 1))
s02 = s0 * s0
s1 = w.s1
s2 = w.s2
b0 = (n2 - 3 * n + 3) * s1 - n * s2 + 3 * s02
b1 = (-1.) * ((n2 - n) * s1 - 2 * n * s2 + 6 * s02)
b2 = (-1.) * (2 * n * s1 - (n + 3) * s2 + 6 * s02)
b3 = 4 * (n - 1) * s1 - 2 * (n + 1) * s2 + 8 * s02
b4 = s1 - s2 + s02
self.b0 = b0
self.b1 = b1
self.b2 = b2
self.b3 = b3
self.b4 = b4
y2 = y * y
y3 = y * y2
y4 = y2 * y2
EG2 = (b0 * (sum(
y2) ** 2) + b1 * sum(y4) + b2 * (sum(y) ** 2) * sum(y2))
EG2 += b3 * sum(y) * sum(y3) + b4 * (sum(y) ** 4)
EG2NUM = EG2
EG2DEN = (((sum(y) ** 2 - sum(y2)) ** 2) * n * (n - 1) * (
n - 2) * (n - 3))
self.EG2 = EG2NUM / EG2DEN
self.VG = self.EG2 - self.EG ** 2
def __calc(self, y):
yl = slag(self.w, y)
self.num = y * yl
return self.num.sum() / self.den_sum
@property
def _statistic(self):
""" Standardized accessor for pysal.explore.esda statistics"""
return self.G
@classmethod
def by_col(cls, df, cols, w=None, inplace=False, pvalue='sim', outvals=None, **stat_kws):
"""
Function to compute a G statistic on a dataframe
Arguments
---------
df : pandas.DataFrame
a pandas dataframe with a geometry column
cols : string or list of string
name or list of names of columns to use to compute the statistic
w : pysal weights object
a weights object aligned with the dataframe. If not provided, this
is searched for in the dataframe's metadata
inplace : bool
a boolean denoting whether to operate on the dataframe inplace or to
return a series contaning the results of the computation. If
operating inplace, the derived columns will be named 'column_g'
pvalue : string
a string denoting which pvalue should be returned. Refer to the
the G statistic's documentation for available p-values
outvals : list of strings
list of arbitrary attributes to return as columns from the
G statistic
**stat_kws : keyword arguments
options to pass to the underlying statistic. For this, see the
documentation for the G statistic.
Returns
-------
If inplace, None, and operation is conducted on dataframe in memory. Otherwise,
returns a copy of the dataframe with the relevant columns attached.
"""
return _univariate_handler(df, cols, w=w, inplace=inplace, pvalue=pvalue,
outvals=outvals, stat=cls,
swapname=cls.__name__.lower(), **stat_kws)
[文档]class G_Local(object):
"""
Generalized Local G Autocorrelation
Parameters
----------
y : array
variable
w : W
DistanceBand, weights instance that is based on threshold distance
and is assumed to be aligned with y
transform : {'R', 'B'}
the type of w, either 'B' (binary) or 'R' (row-standardized)
permutations : int
the number of random permutations for calculating
pseudo p values
star : boolean
whether or not to include focal observation in sums (default: False)
Attributes
----------
y : array
original variable
w : DistanceBand W
original weights object
permutations : int
the number of permutations
Gs : array
of floats, the value of the orginal G statistic in Getis & Ord (1992)
EGs : float
expected value of Gs under normality assumption
the values is scalar, since the expectation is identical
across all observations
VGs : array
of floats, variance values of Gs under normality assumption
Zs : array
of floats, standardized Gs
p_norm : array
of floats, p-value under normality assumption (one-sided)
for two-sided tests, this value should be multiplied by 2
sim : array
of arrays of floats (if permutations>0), vector of I values
for permutated samples
p_sim : array
of floats, p-value based on permutations (one-sided)
null - spatial randomness
alternative - the observed G is extreme it is either extremely high or extremely low
EG_sim : array
of floats, average value of G from permutations
VG_sim : array
of floats, variance of G from permutations
seG_sim : array
of floats, standard deviation of G under permutations.
z_sim : array
of floats, standardized G based on permutations
p_z_sim : array
of floats, p-value based on standard normal approximation from
permutations (one-sided)
Notes
-----
To compute moments of Gs under normality assumption,
PySAL considers w is either binary or row-standardized.
For binary weights object, the weight value for self is 1
For row-standardized weights object, the weight value for self is
1/(the number of its neighbors + 1).
For technical details see :cite:`Getis_2010` and :cite:`Ord_2010`.
Examples
--------
>>> import pysal.lib
>>> import numpy
>>> numpy.random.seed(10)
Preparing a point data set
>>> points = [(10, 10), (20, 10), (40, 10), (15, 20), (30, 20), (30, 30)]
Creating a weights object from points
>>> w = pysal.lib.weights.DistanceBand(points,threshold=15)
Prepareing a variable
>>> y = numpy.array([2, 3, 3.2, 5, 8, 7])
Applying Getis and Ord local G test using a binary weights object
>>> from pysal.explore.esda.getisord import G_Local
>>> lg = G_Local(y,w,transform='B')
Examining the results
>>> lg.Zs
array([-1.0136729 , -0.04361589, 1.31558703, -0.31412676, 1.15373986,
1.77833941])
>>> round(lg.p_sim[0], 3)
0.101
>>> numpy.random.seed(10)
Applying Getis and Ord local G* test using a binary weights object
>>> lg_star = G_Local(y,w,transform='B',star=True)
Examining the results
>>> lg_star.Zs
array([-1.39727626, -0.28917762, 0.65064964, -0.28917762, 1.23452088,
2.02424331])
>>> round(lg_star.p_sim[0], 3)
0.101
>>> numpy.random.seed(12345)
Applying Getis and Ord local G test using a row-standardized weights object
>>> lg = G_Local(y,w,transform='R')
Examining the results
>>> lg.Zs
array([-0.62074534, -0.01780611, 1.31558703, -0.12824171, 0.28843496,
1.77833941])
>>> round(lg.p_sim[0], 3)
0.103
>>> numpy.random.seed(10)
Applying Getis and Ord local G* test using a row-standardized weights object
>>> lg_star = G_Local(y,w,transform='R',star=True)
Examining the results
>>> lg_star.Zs
array([-0.62488094, -0.09144599, 0.41150696, -0.09144599, 0.24690418,
1.28024388])
>>> round(lg_star.p_sim[0], 3)
0.101
"""
[文档] def __init__(self, y, w, transform='R', permutations=PERMUTATIONS, star=False):
y = np.asarray(y).flatten()
self.n = len(y)
self.y = y
self.w = w
self.w_original = w.transform
self.w.transform = self.w_transform = transform.lower()
self.permutations = permutations
self.star = star
self.calc()
self.p_norm = np.array(
[1 - stats.norm.cdf(np.abs(i)) for i in self.Zs])
if permutations:
self.__crand()
sim = np.transpose(self.rGs)
above = sim >= self.Gs
larger = sum(above)
low_extreme = (self.permutations - larger) < larger
larger[low_extreme] = self.permutations - larger[low_extreme]
self.p_sim = (larger + 1.0) / (permutations + 1)
self.sim = sim
self.EG_sim = sim.mean()
self.seG_sim = sim.std()
self.VG_sim = self.seG_sim * self.seG_sim
self.z_sim = (self.Gs - self.EG_sim) / self.seG_sim
self.p_z_sim = 1 - stats.norm.cdf(np.abs(self.z_sim))
def __crand(self):
y = self.y
rGs = np.zeros((self.n, self.permutations))
n_1 = self.n - 1
rid = list(range(n_1))
prange = list(range(self.permutations))
k = self.w.max_neighbors + 1
rids = np.array([np.random.permutation(rid)[0:k] for i in prange])
ids = np.arange(self.w.n)
ido = self.w.id_order
wc = self.__getCardinalities()
if self.w_transform == 'r':
den = np.array(wc) + self.star
else:
den = np.ones(self.w.n)
for i in range(self.w.n):
idsi = ids[ids != i]
np.random.shuffle(idsi)
yi_star = y[i] * self.star
wci = wc[i]
rGs[i] = (y[idsi[rids[:, 0:wci]]]).sum(1) + yi_star
rGs[i] = (np.array(rGs[i]) / den[i]) / (
self.y_sum - (1 - self.star) * y[i])
self.rGs = rGs
def __getCardinalities(self):
ido = self.w.id_order
self.wc = np.array(
[self.w.cardinalities[ido[i]] for i in range(self.n)])
return self.wc
def calc(self):
y = self.y
y2 = y * y
self.y_sum = y_sum = sum(y)
y2_sum = sum(y2)
if not self.star:
yl = 1.0 * slag(self.w, y)
ydi = y_sum - y
self.Gs = yl / ydi
N = self.n - 1
yl_mean = ydi / N
s2 = (y2_sum - y2) / N - (yl_mean) ** 2
else:
self.w.transform = 'B'
yl = 1.0 * slag(self.w, y)
yl += y
if self.w_transform == 'r':
yl = yl / (self.__getCardinalities() + 1.0)
self.Gs = yl / y_sum
N = self.n
yl_mean = y.mean()
s2 = y.var()
EGs_num, VGs_num = 1.0, 1.0
if self.w_transform == 'b':
W = self.__getCardinalities()
W += self.star
EGs_num = W * 1.0
VGs_num = (W * (1.0 * N - W)) / (1.0 * N - 1)
self.EGs = (EGs_num * 1.0) / N
self.VGs = (VGs_num) * (1.0 / (N ** 2)) * ((s2 * 1.0) / (yl_mean ** 2))
self.Zs = (self.Gs - self.EGs) / np.sqrt(self.VGs)
self.w.transform = self.w_original
@property
def _statistic(self):
"""Standardized accessor for pysal.explore.esda statistics"""
return self.Gs
@classmethod
def by_col(cls, df, cols, w=None, inplace=False, pvalue='sim', outvals=None, **stat_kws):
"""
Function to compute a G_Local statistic on a dataframe
Arguments
---------
df : pandas.DataFrame
a pandas dataframe with a geometry column
cols : string or list of string
name or list of names of columns to use to compute the statistic
w : pysal weights object
a weights object aligned with the dataframe. If not provided, this
is searched for in the dataframe's metadata
inplace : bool
a boolean denoting whether to operate on the dataframe inplace or to
return a series contaning the results of the computation. If
operating inplace, the derived columns will be named 'column_g_local'
pvalue : string
a string denoting which pvalue should be returned. Refer to the
the G_Local statistic's documentation for available p-values
outvals : list of strings
list of arbitrary attributes to return as columns from the
G_Local statistic
**stat_kws : keyword arguments
options to pass to the underlying statistic. For this, see the
documentation for the G_Local statistic.
Returns
--------
If inplace, None, and operation is conducted on dataframe in memory. Otherwise,
returns a copy of the dataframe with the relevant columns attached.
See also
--------
G_Local
"""
return _univariate_handler(df, cols, w=w, inplace=inplace, pvalue=pvalue,
outvals=outvals, stat=cls,
swapname=cls.__name__.lower(), **stat_kws)