# Source code for astropy.stats.spatial

```# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""
This module implements functions and classes for spatial statistics.
"""

import math

import numpy as np

__all__ = ["RipleysKEstimator"]

[docs]
class RipleysKEstimator:
"""
Estimators for Ripley's K function for two-dimensional spatial data.
See [1]_, [2]_, [3]_, [4]_, [5]_ for detailed mathematical and
practical aspects of those estimators.

Parameters
----------
area : float
Area of study from which the points where observed.
x_max, y_max : float, float, optional
Maximum rectangular coordinates of the area of study.
Required if ``mode == 'translation'`` or ``mode == ohser``.
x_min, y_min : float, float, optional
Minimum rectangular coordinates of the area of study.
Required if ``mode == 'variable-width'`` or ``mode == ohser``.

Examples
--------
>>> import numpy as np
>>> from matplotlib import pyplot as plt # doctest: +SKIP
>>> from astropy.stats import RipleysKEstimator
>>> z = np.random.uniform(low=5, high=10, size=(100, 2))
>>> Kest = RipleysKEstimator(area=25, x_max=10, y_max=10,
... x_min=5, y_min=5)
>>> r = np.linspace(0, 2.5, 100)
>>> plt.plot(r, Kest.poisson(r)) # doctest: +SKIP
>>> plt.plot(r, Kest(data=z, radii=r, mode='none')) # doctest: +SKIP
>>> plt.plot(r, Kest(data=z, radii=r, mode='translation')) # doctest: +SKIP
>>> plt.plot(r, Kest(data=z, radii=r, mode='ohser')) # doctest: +SKIP
>>> plt.plot(r, Kest(data=z, radii=r, mode='var-width')) # doctest: +SKIP
>>> plt.plot(r, Kest(data=z, radii=r, mode='ripley')) # doctest: +SKIP

References
----------
.. [1] Peebles, P.J.E. *The large scale structure of the universe*.
.. [2] Spatial descriptive statistics.
<https://en.wikipedia.org/wiki/Spatial_descriptive_statistics>
.. [3] Package spatstat.
<https://cran.r-project.org/web/packages/spatstat/spatstat.pdf>
.. [4] Cressie, N.A.C. (1991). Statistics for Spatial Data,
Wiley, New York.
.. [5] Stoyan, D., Stoyan, H. (1992). Fractals, Random Shapes and
Point Fields, Akademie Verlag GmbH, Chichester.
"""

def __init__(self, area, x_max=None, y_max=None, x_min=None, y_min=None):
self.area = area
self.x_max = x_max
self.y_max = y_max
self.x_min = x_min
self.y_min = y_min

@property
def area(self):
return self._area

@area.setter
def area(self, value):
if isinstance(value, (float, int)) and value > 0:
self._area = value
else:
raise ValueError(f"area is expected to be a positive number. Got {value}.")

@property
def y_max(self):
return self._y_max

@y_max.setter
def y_max(self, value):
if value is None or isinstance(value, (float, int)):
self._y_max = value
else:
raise ValueError(
f"y_max is expected to be a real number or None. Got {value}."
)

@property
def x_max(self):
return self._x_max

@x_max.setter
def x_max(self, value):
if value is None or isinstance(value, (float, int)):
self._x_max = value
else:
raise ValueError(
f"x_max is expected to be a real number or None. Got {value}."
)

@property
def y_min(self):
return self._y_min

@y_min.setter
def y_min(self, value):
if value is None or isinstance(value, (float, int)):
self._y_min = value
else:
raise ValueError(f"y_min is expected to be a real number. Got {value}.")

@property
def x_min(self):
return self._x_min

@x_min.setter
def x_min(self, value):
if value is None or isinstance(value, (float, int)):
self._x_min = value
else:
raise ValueError(f"x_min is expected to be a real number. Got {value}.")

[docs]
def __call__(self, data, radii, mode="none"):

def _pairwise_diffs(self, data):
npts = len(data)
diff = np.zeros(shape=(npts * (npts - 1) // 2, 2), dtype=np.double)
k = 0
for i in range(npts - 1):
size = npts - i - 1
diff[k : k + size] = abs(data[i] - data[i + 1 :])
k += size

return diff

[docs]
"""
Evaluates the Ripley K function for the homogeneous Poisson process,
also known as Complete State of Randomness (CSR).

Parameters
----------
radii : 1D array
Set of distances in which Ripley's K function will be evaluated.

Returns
-------
output : 1D array
Ripley's K function evaluated at ``radii``.
"""

[docs]
def Lfunction(self, data, radii, mode="none"):
"""
Evaluates the L function at ``radii``. For parameter description
see ``evaluate`` method.
"""
return np.sqrt(self.evaluate(data, radii, mode=mode) / np.pi)

[docs]
def Hfunction(self, data, radii, mode="none"):
"""
Evaluates the H function at ``radii``. For parameter description
see ``evaluate`` method.
"""

[docs]
def evaluate(self, data, radii, mode="none"):
"""
Evaluates the Ripley K estimator for a given set of values ``radii``.

Parameters
----------
data : 2D array
Set of observed points in as a n by 2 array which will be used to
estimate Ripley's K function.
radii : 1D array
Set of distances in which Ripley's K estimator will be evaluated.
Usually, it's common to consider max(radii) < (area/2)**0.5.
mode : str
Keyword which indicates the method for edge effects correction.
Available methods are 'none', 'translation', 'ohser', 'var-width',
and 'ripley'.

* 'none'
this method does not take into account any edge effects
whatsoever.
* 'translation'
computes the intersection of rectangular areas centered at
the given points provided the upper bounds of the
dimensions of the rectangular area of study. It assumes that
all the points lie in a bounded rectangular region satisfying
x_min < x_i < x_max; y_min < y_i < y_max. A detailed
description of this method can be found on ref [4].
* 'ohser'
this method uses the isotropized set covariance function of
the window of study as a weight to correct for
edge-effects. A detailed description of this method can be
found on ref [4].
* 'var-width'
this method considers the distance of each observed point to
the nearest boundary of the study window as a factor to
account for edge-effects. See [3] for a brief description of
this method.
* 'ripley'
this method is known as Ripley's edge-corrected estimator.
The weight for edge-correction is a function of the
proportions of circumferences centered at each data point
which crosses another data point of interest. See [3] for
a detailed description of this method.

Returns
-------
ripley : 1D array
Ripley's K function estimator evaluated at ``radii``.
"""
data = np.asarray(data)

if not data.shape[1] == 2:
raise ValueError(
"data must be an n by 2 array, where n is the "
"number of observed points."
)

npts = len(data)

if mode == "none":
diff = self._pairwise_diffs(data)
distances = np.hypot(diff[:, 0], diff[:, 1])
for r in range(len(radii)):
ripley[r] = (distances < radii[r]).sum()

ripley = self.area * 2.0 * ripley / (npts * (npts - 1))
# eq. 15.11 Stoyan book page 283
elif mode == "translation":
diff = self._pairwise_diffs(data)
distances = np.hypot(diff[:, 0], diff[:, 1])
intersec_area = ((self.x_max - self.x_min) - diff[:, 0]) * (
(self.y_max - self.y_min) - diff[:, 1]
)

for r in range(len(radii)):
dist_indicator = distances < radii[r]
ripley[r] = ((1 / intersec_area) * dist_indicator).sum()

ripley = (self.area**2 / (npts * (npts - 1))) * 2 * ripley
# Stoyan book page 123 and eq 15.13
elif mode == "ohser":
diff = self._pairwise_diffs(data)
distances = np.hypot(diff[:, 0], diff[:, 1])
a = self.area
b = max(
(self.y_max - self.y_min) / (self.x_max - self.x_min),
(self.x_max - self.x_min) / (self.y_max - self.y_min),
)
x = distances / math.sqrt(a / b)
u = np.sqrt((x * x - 1) * (x > 1))
v = np.sqrt((x * x - b**2) * (x < math.sqrt(b**2 + 1)) * (x > b))
c1 = np.pi - 2 * x * (1 + 1 / b) + x * x / b
c2 = 2 * np.arcsin((1 / x) * (x > 1)) - 1 / b - 2 * (x - u)
c3 = (
2
* np.arcsin(
((b - u * v) / (x * x)) * (x > b) * (x < math.sqrt(b**2 + 1))
)
+ 2 * u
+ 2 * v / b
- b
- (1 + x * x) / b
)

cov_func = (a / np.pi) * (
c1 * (x >= 0) * (x <= 1)
+ c2 * (x > 1) * (x <= b)
+ c3 * (b < x) * (x < math.sqrt(b**2 + 1))
)

for r in range(len(radii)):
dist_indicator = distances < radii[r]
ripley[r] = ((1 / cov_func) * dist_indicator).sum()

ripley = (self.area**2 / (npts * (npts - 1))) * 2 * ripley
# Cressie book eq 8.2.20 page 616
elif mode == "var-width":
lt_dist = np.minimum(
np.minimum(self.x_max - data[:, 0], self.y_max - data[:, 1]),
np.minimum(data[:, 0] - self.x_min, data[:, 1] - self.y_min),
)

for r in range(len(radii)):
for i in range(npts):
for j in range(npts):
if i != j:
diff = abs(data[i] - data[j])
dist = math.sqrt((diff * diff).sum())
if dist < radii[r] < lt_dist[i]:
ripley[r] = ripley[r] + 1
lt_dist_sum = (lt_dist > radii[r]).sum()
if not lt_dist_sum == 0:
ripley[r] = ripley[r] / lt_dist_sum

ripley = self.area * ripley / npts
# Cressie book eq 8.4.22 page 640
elif mode == "ripley":
hor_dist = np.zeros(shape=(npts * (npts - 1)) // 2, dtype=np.double)
ver_dist = np.zeros(shape=(npts * (npts - 1)) // 2, dtype=np.double)

for k in range(npts - 1):
min_hor_dist = min(self.x_max - data[k][0], data[k][0] - self.x_min)
min_ver_dist = min(self.y_max - data[k][1], data[k][1] - self.y_min)
start = (k * (2 * (npts - 1) - (k - 1))) // 2
end = ((k + 1) * (2 * (npts - 1) - k)) // 2
hor_dist[start:end] = min_hor_dist * np.ones(npts - 1 - k)
ver_dist[start:end] = min_ver_dist * np.ones(npts - 1 - k)

diff = self._pairwise_diffs(data)
dist = np.hypot(diff[:, 0], diff[:, 1])
dist_ind = dist <= np.hypot(hor_dist, ver_dist)

w1 = (
1
- (
np.arccos(np.minimum(ver_dist, dist) / dist)
+ np.arccos(np.minimum(hor_dist, dist) / dist)
)
/ np.pi
)
w2 = (
3 / 4
- 0.5
* (
np.arccos(ver_dist / dist * ~dist_ind)
+ np.arccos(hor_dist / dist * ~dist_ind)
)
/ np.pi
)

weight = dist_ind * w1 + ~dist_ind * w2

for r in range(len(radii)):
ripley[r] = ((dist < radii[r]) / weight).sum()

ripley = self.area * 2.0 * ripley / (npts * (npts - 1))
else:
raise ValueError(f"mode {mode} is not implemented.")

return ripley

```