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"""
Directional Analysis of Dynamic LISAs
"""
__author__ = "Sergio J. Rey <sjsrey@gmail.com>"
__all__ = ['Rose']
import warnings
import numpy as np
from libpysal import weights
from libpysal.common import requires as _requires
_POS8 = np.array([1, 1, 0, 0, 1, 1, 0, 0])
_POS4 = np.array([1, 0, 1, 0])
_NEG8 = 1 - _POS8
_NEG4 = 1 - _POS4
class Rose(object):
"""
Rose diagram based inference for directional LISAs.
For n units with LISA values at two points in time, the Rose class provides
the LISA vectors, their visualization, and computationally based inference.
Parameters
----------
Y : array (n,2)
Columns correspond to end-point time periods to calculate LISA vectors for n object.
w : PySAL W
Spatial weights object.
k : int
Number of circular sectors in rose diagram.
Attributes
----------
cuts : (k, 1) ndarray
Radian cuts for rose diagram (circular histogram).
counts: (k, 1) ndarray
Number of vectors contained in each sector.
r : (n, 1) ndarray
Vector lengths.
theta : (n,1) ndarray
Signed radians for observed LISA vectors.
If self.permute is called the following attributes are available:
alternative : string
Form of the specified alternative hypothesis ['two-sided'(default) |
'positive' | 'negative']
counts_perm : (permutations, k) ndarray
Counts obtained for each sector for every permutation
expected_perm : (k, 1) ndarray
Average number of counts for each sector taken over all permutations.
p : (k, 1) ndarray
Psuedo p-values for the observed sector counts under the specified alternative.
larger_perm : (k, 1) ndarray
Number of times realized counts are as large as observed sector count.
smaller_perm : (k, 1) ndarray
Number of times realized counts are as small as observed sector count.
"""
def __init__(self, Y, w, k=8):
"""
Calculation of rose diagram for local indicators of spatial
association.
Parameters
----------
Y : (n, 2) ndarray
Variable observed on n spatial units over 2 time periods
w : W
Spatial weights object.
k : int
number of circular sectors in rose diagram (the default is 8).
Notes
-----
Based on :cite:`Rey2011`.
Examples
--------
Constructing data for illustration of directional LISA analytics.
Data is for the 48 lower US states over the period 1969-2009 and
includes per capita income normalized to the national average.
Load comma delimited data file in and convert to a numpy array
>>> import libpysal
>>> from giddy.directional import Rose
>>> import matplotlib.pyplot as plt
>>> file_path = libpysal.examples.get_path("spi_download.csv")
>>> f=open(file_path,'r')
>>> lines=f.readlines()
>>> f.close()
>>> lines=[line.strip().split(",") for line in lines]
>>> names=[line[2] for line in lines[1:-5]]
>>> data=np.array([list(map(int,line[3:])) for line in lines[1:-5]])
Bottom of the file has regional data which we don't need for this
example so we will subset only those records that match a state name
>>> sids=list(range(60))
>>> out=['"United States 3/"',
... '"Alaska 3/"',
... '"District of Columbia"',
... '"Hawaii 3/"',
... '"New England"',
... '"Mideast"',
... '"Great Lakes"',
... '"Plains"',
... '"Southeast"',
... '"Southwest"',
... '"Rocky Mountain"',
... '"Far West 3/"']
>>> snames=[name for name in names if name not in out]
>>> sids=[names.index(name) for name in snames]
>>> states=data[sids,:]
>>> us=data[0]
>>> years=np.arange(1969,2009)
Now we convert state incomes to express them relative to the national
average
>>> rel=states/(us*1.)
Create our contiguity matrix from an external GAL file and row
standardize the resulting weights
>>> gal=libpysal.io.open(libpysal.examples.get_path('states48.gal'))
>>> w=gal.read()
>>> w.transform='r'
Take the first and last year of our income data as the interval to do
the directional directional analysis
>>> Y=rel[:,[0,-1]]
Set the random seed generator which is used in the permutation based
inference for the rose diagram so that we can replicate our example
results
>>> np.random.seed(100)
Call the rose function to construct the directional histogram for the
dynamic LISA statistics. We will use four circular sectors for our
histogram
>>> r4=Rose(Y,w,k=4)
What are the cut-offs for our histogram - in radians
>>> r4.cuts
array([0. , 1.57079633, 3.14159265, 4.71238898, 6.28318531])
How many vectors fell in each sector
>>> r4.counts
array([32, 5, 9, 2])
We can test whether these counts are different than what would be
expected if there was no association between the movement of the
focal unit and its spatial lag.
To do so we call the `permute` method of the object
>>> r4.permute()
and then inspect the `p` attibute:
>>> r4.p
array([0.04, 0. , 0.02, 0. ])
Repeat the exercise but now for 8 rather than 4 sectors
>>> r8 = Rose(Y, w, k=8)
>>> r8.counts
array([19, 13, 3, 2, 7, 2, 1, 1])
>>> r8.permute()
>>> r8.p
array([0.86, 0.08, 0.16, 0. , 0.02, 0.2 , 0.56, 0. ])
The default is a two-sided alternative. There is an option for a
directional alternative reflecting positive co-movement of the focal
series with its spatial lag. In this case the number of vectors in
quadrants I and III should be much larger than expected, while the
counts of vectors falling in quadrants II and IV should be much lower
than expected.
>>> r8.permute(alternative='positive')
>>> r8.p
array([0.51, 0.04, 0.28, 0.02, 0.01, 0.14, 0.57, 0.03])
Finally, there is a second directional alternative for examining the
hypothesis that the focal unit and its lag move in opposite directions.
>>> r8.permute(alternative='negative')
>>> r8.p
array([0.69, 0.99, 0.92, 1. , 1. , 0.97, 0.74, 1. ])
We can call the plot method to visualize directional LISAs as a
rose diagram conditional on the starting relative income:
>>> fig1, _ = r8.plot(attribute=Y[:,0])
>>> plt.show(fig1)
"""
self.Y = Y
self.w = w
self.k = k
self.permtuations = 0
self.sw = 2 * np.pi / self.k
self.cuts = np.arange(0.0, 2 * np.pi + self.sw, self.sw)
observed = self._calc(Y, w, k)
self.theta = observed['theta']
self.bins = observed['bins']
self.counts = observed['counts']
self.r = observed['r']
self.lag = observed['lag']
self._dx = observed['dx']
self._dy = observed['dy']
def permute(self, permutations=99, alternative='two.sided'):
"""
Generate ransom spatial permutations for inference on LISA vectors.
Parameters
----------
permutations : int, optional
Number of random permutations of observations.
alternative : string, optional
Type of alternative to form in generating p-values.
Options are: `two-sided` which tests for difference between observed
counts and those obtained from the permutation distribution;
`positive` which tests the alternative that the focal unit and its
lag move in the same direction over time; `negative` which tests
that the focal unit and its lag move in opposite directions over
the interval.
"""
rY = self.Y.copy()
idxs = np.arange(len(rY))
counts = np.zeros((permutations, len(self.counts)))
for m in range(permutations):
np.random.shuffle(idxs)
res = self._calc(rY[idxs, :], self.w, self.k)
counts[m] = res['counts']
self.counts_perm = counts
self.larger_perm = np.array(
[(counts[:, i] >= self.counts[i]).sum() for i in range(self.k)])
self.smaller_perm = np.array(
[(counts[:, i] <= self.counts[i]).sum() for i in range(self.k)])
self.expected_perm = counts.mean(axis=0)
self.alternative = alternative
# pvalue logic
# if P is the proportion that are as large for a one sided test (larger
# than), then
# p=P.
#
# For a two-tailed test, if P < .5, p = 2 * P, else, p = 2(1-P)
# Source: Rayner, J. C. W., O. Thas, and D. J. Best. 2009. "Appendix B:
# Parametric Bootstrap P-Values." In Smooth Tests of Goodness of Fit,
# 247. John Wiley and Sons.
# Note that the larger and smaller counts would be complements (except
# for the shared equality, for
# a given bin in the circular histogram. So we only need one of them.
# We report two-sided p-values for each bin as the default
# since a priori there could # be different alternatives for each bin
# depending on the problem at hand.
alt = alternative.upper()
if alt == 'TWO.SIDED':
P = (self.larger_perm + 1) / (permutations + 1.)
mask = P < 0.5
self.p = mask * 2 * P + (1 - mask) * 2 * (1 - P)
elif alt == 'POSITIVE':
# NE, SW sectors are higher, NW, SE are lower
POS = _POS8
if self.k == 4:
POS = _POS4
L = (self.larger_perm + 1) / (permutations + 1.)
S = (self.smaller_perm + 1) / (permutations + 1.)
P = POS * L + (1 - POS) * S
self.p = P
elif alt == 'NEGATIVE':
# NE, SW sectors are lower, NW, SE are higher
NEG = _NEG8
if self.k == 4:
NEG = _NEG4
L = (self.larger_perm + 1) / (permutations + 1.)
S = (self.smaller_perm + 1) / (permutations + 1.)
P = NEG * L + (1 - NEG) * S
self.p = P
else:
print(('Bad option for alternative: %s.' % alternative))
def _calc(self, Y, w, k):
wY = weights.lag_spatial(w, Y)
dx = Y[:, -1] - Y[:, 0]
dy = wY[:, -1] - wY[:, 0]
self.wY = wY
self.Y = Y
r = np.sqrt(dx * dx + dy * dy)
theta = np.arctan2(dy, dx)
neg = theta < 0.0
utheta = theta * (1 - neg) + neg * (2 * np.pi + theta)
counts, bins = np.histogram(utheta, self.cuts)
results = {}
results['counts'] = counts
results['theta'] = theta
results['bins'] = bins
results['r'] = r
results['lag'] = wY
results['dx'] = dx
results['dy'] = dy
return results
@_requires('splot')
def plot(self, attribute=None, ax=None, **kwargs):
"""
Plot the rose diagram.
Parameters
----------
attribute : (n,) ndarray, optional
Variable to specify colors of the colorbars.
ax : Matplotlib Axes instance, optional
If given, the figure will be created inside this axis.
Default =None. Note, this axis should have a polar projection.
**kwargs : keyword arguments, optional
Keywords used for creating and designing the plot.
Note: 'c' and 'color' cannot be passed when attribute is not None
Returns
-------
fig : Matplotlib Figure instance
Moran scatterplot figure
ax : matplotlib Axes instance
Axes in which the figure is plotted
"""
from splot.giddy import dynamic_lisa_rose
fig, ax = dynamic_lisa_rose(self, attribute=attribute,
ax=ax, **kwargs)
return fig, ax
def plot_origin(self): # TODO add attribute option to color vectors
"""
Plot vectors of positional transition of LISA values starting
from the same origin.
"""
import matplotlib.cm as cm
import matplotlib.pyplot as plt
ax = plt.subplot(111)
xlim = [self._dx.min(), self._dx.max()]
ylim = [self._dy.min(), self._dy.max()]
for x, y in zip(self._dx, self._dy):
xs = [0, x]
ys = [0, y]
plt.plot(xs, ys, '-b') # TODO change this to scale with attribute
plt.axis('equal')
plt.xlim(xlim)
plt.ylim(ylim)
@_requires('splot')
def plot_vectors(self, arrows=True):
"""
Plot vectors of positional transition of LISA values
within quadrant in scatterplot in a polar plot.
Parameters
----------
ax : Matplotlib Axes instance, optional
If given, the figure will be created inside this axis.
Default =None.
arrows : boolean, optional
If True show arrowheads of vectors. Default =True
**kwargs : keyword arguments, optional
Keywords used for creating and designing the plot.
Note: 'c' and 'color' cannot be passed when attribute is not None
Returns
-------
fig : Matplotlib Figure instance
Moran scatterplot figure
ax : matplotlib Axes instance
Axes in which the figure is plotted
"""
from splot.giddy import dynamic_lisa_vectors
fig, ax = dynamic_lisa_vectors(self, arrows=arrows)
return fig, ax
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