release 2.0.0 pysal developers

pointpats DocumentationRelease 2.0.0

pysal developers

Jun 22, 2019

CONTENTS:

1 Installation 31.1 Installing released version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Installing development version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 API reference 52.1 Point Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 pointpats.PointPattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Point Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 pointpats.PointProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 pointpats.PoissonPointProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 pointpats.PoissonClusterPointProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Centrography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 pointpats.mbr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 pointpats.hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.3 pointpats.mean_center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.4 pointpats.weighted_mean_center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.5 pointpats.manhattan_median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.6 pointpats.std_distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.7 pointpats.euclidean_median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.8 pointpats.ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.9 pointpats.skyum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.10 pointpats.dtot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.11 pointpats._circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Quadrat Based Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.1 pointpats.RectangleM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.2 pointpats.HexagonM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.3 pointpats.QStatistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Distance Based Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.1 pointpats.DStatistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.2 pointpats.G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.3 pointpats.F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.4 pointpats.J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.5 pointpats.K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5.6 pointpats.L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5.7 pointpats.Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5.8 pointpats.Genv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.9 pointpats.Fenv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5.10 pointpats.Jenv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5.11 pointpats.Kenv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.12 pointpats.Lenv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

i

2.6 Window functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6.1 pointpats.Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6.2 pointpats.as_window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6.3 pointpats.poly_from_bbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6.4 pointpats.to_ccf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 References 37

Bibliography 39

Index 41

ii

pointpats Documentation, Release 2.0.0

pointpats is an open-source python library for the statistical analysis of planar point patterns. It is a subpackage ofPySAL (Python Spatial Analysis Library) and is under active development for the inclusion of many newly proposedanalytics for point patterns.

CONTENTS: 1

https://github.com/pysal/pysal


2 CONTENTS:

CHAPTER

ONE

INSTALLATION

pointpats supports python 3.5 and 3.6 only. Please make sure that you are operating in a python 3 environment.

1.1 Installing released version

pointpats is available on the Python Package Index. Therefore, you can either install directly with pip from thecommand line:

pip install -U pointpats

or download the source distribution (.tar.gz) and decompress it to your selected destination. Open a command shelland navigate to the decompressed folder. Type:

pip install .

1.2 Installing development version

Potentially, you might want to use the newest features in the development version of pointpats on github -pysal/pointpats while have not been incorporated in the Pypi released version. You can achieve that by installingpysal/pointpats by running the following from a command shell:

pip install git+https://github.com/pysal/pointpats.git

You can also fork the pysal/pointpats repo and create a local clone of your fork. By making changes to your localclone and submitting a pull request to pysal/pointpats, you can contribute to the pointpats development.

3

https://docs.python.org/3.5/

https://docs.python.org/3.6/

https://pypi.org/project/pointpats/

https://github.com/pysal/pointpats


https://help.github.com/articles/fork-a-repo/




4 Chapter 1. Installation

CHAPTER

TWO

API REFERENCE

2.1 Point Pattern

PointPattern(points[, window, names, . . . ]) Planar Point Pattern Class 2-D.

2.1.1 pointpats.PointPattern

class pointpats.PointPattern(points, window=None, names=None, coord_names=None)Planar Point Pattern Class 2-D.

Parameters

points: array (n,p), n points with p >= 2 attributes on each point. Two attributes must comprisethe spatial coordinate pair. Default is that the first two attributes are the x and y spatialcoordinates.

window: :class:‘.Window‘ Bounding geometric object for the point pattern. If not specified,window will be set to the minimum bounding rectangle of the point pattern.

names: list The names of the attributes.

coord_names: list The names of the attributes defining the two spatial coordinates.

Examples

>>> from pointpats import PointPattern>>> points = [[66.22, 32.54], [22.52, 22.39], [31.01, 81.21],... [9.47, 31.02], [30.78, 60.10], [75.21, 58.93],... [79.26, 7.68], [8.23, 39.93], [98.73, 77.17],... [89.78, 42.53], [65.19, 92.08], [54.46, 8.48]]>>> pp = PointPattern(points)>>> pp.n12>>> pp.mean_nnd21.612139802089246>>> pp.lambda_mbb0.0015710507711240867>>> pp.lambda_hull0.0022667153468973137>>> pp.hull_area5294.00395>>> pp.mbb_area7638.200000000001

5


Attributes

hull Points defining convex hull in counterclockwise order

hull_area Area of convex hull

lambda_hull Intensity based on convex hull

lambda_mbb Intensity based on minimum bounding box

lambda_window Intensity estimate based on area of window

max_nnd Max nearest neighbor distance

mbb Minimum bounding box

mbb_area Area of minimum bounding box

mean_nnd Mean nearest neighbor distance

min_nnd Min nearest neighbor distance

n Number of points

nnd Nearest neighbor distances

tree

window Bounding geometry for the point pattern

Methods

explode(self, mark) Explode a marked point pattern into a sequence ofindividual point patterns.

find_pairs(self, r) Find all pairs of points in the pattern that are withinr units of each other

flip_coordinates(self) Flips the coordinates of a point pattern.get_window(self) Bounding geometry for the point patternknn(self[, k]) Find k nearest neighbors for each point in the patternknn_other(self, other[, k]) Find k nearest neighbors in the pattern for each point

in otherplot(self[, window, title, hull, get_ax]) Plot function for a point pattern.summary(self) Description of the point pattern.superimpose(self, point_pattern) Returns a superimposed point pattern.unique(self) Remove duplicate points in the point pattern.

pointpats.PointPattern

PointPattern.explode(self, mark)Explode a marked point pattern into a sequence of individual point patterns. If the mark has k uniquevalues, then the sequence will be of length k.

Parameters

mark: string The label of the mark to use for the subsetting

Returns

pps: list sequence of PointPattern instances

6 Chapter 2. API reference



PointPattern.find_pairs(self, r)Find all pairs of points in the pattern that are within r units of each other

Parameters

r: float diameter of pair circle

Returns

s: set pairs of points within r units of each other


PointPattern.flip_coordinates(self)Flips the coordinates of a point pattern.

Doesn’t change the structure of data frame. This function swaps _x and _y variables, which are used torepresent coordinates.


PointPattern.get_window(self)Bounding geometry for the point pattern

window.Window


PointPattern.knn(self, k=1)Find k nearest neighbors for each point in the pattern

Parameters

k: int number of nearest neighbors to find

Returns

nn: array (n x k) row i column j contains the id for i’s jth nearest neighbor

nnd: array(n x k) row i column j contains the distance between i and its jth nearest neighbor


PointPattern.knn_other(self, other, k=1)Find k nearest neighbors in the pattern for each point in other

Parameters

other: :class:‘PointPattern‘

k: int number of nearest neighbors to find

Returns

nn: array (n x k) row i column j contains the id for i’s jth nearest neighbor

nnd: array(n x k) row i column j contains the distance between i and its jth nearest neighbor

2.1. Point Pattern 7



PointPattern.plot(self, window=False, title=’Point Pattern’, hull=False, get_ax=False)Plot function for a point pattern.

Parameters

window [boolean] If window is True, plot window of the point pattern. If not, don’t plotwindow.

title [string] Name of the figure.

hull [boolean] If hull is True, plot convex hull of the point pattern. If not, don’t plot convexhull.

get_ax [boolean] If get_ax is True, return the current plot ax.

Returns

ax [matplotlib.axes._subplots.AxesSubplot] Current plot ax. Only return it when get_ax isTrue.


PointPattern.summary(self)Description of the point pattern.


PointPattern.superimpose(self, point_pattern)Returns a superimposed point pattern.

Parameters

point_pattern: PointPattern instance

Returns

superimposed : PointPattern instance

Examples

>>> from pointpats import PointPattern>>> points1 = [[1, 3], [4, 5], [0, 0]]>>> points2 = [[5, 6], [1, 4], [0, 0]]>>> pp1 = PointPattern(points1)>>> pp2 = PointPattern(points2)>>> pp1.superimpose(pp2).points

x y0 1 31 4 52 0 00 5 61 1 4




PointPattern.unique(self)Remove duplicate points in the point pattern.

Two points in a point pattern are deemed to be identical if their coordinates are the same, and their marksare the same (if any)

Returns

pp: list A deduplicated PointPattern instance

Examples

>>> from pointpats import PointPattern>>> points = [[1.2, 2.1], [1.2, 2.1], [0, 1], [1, 2]]>>> pp = PointPattern(points)>>> pp.unique().df

x y0 1.2 2.12 0.0 1.03 1.0 2.0

add_marksset_window

__init__(self, points, window=None, names=None, coord_names=None)Initialize self. See help(type(self)) for accurate signature.

2.2 Point Processes

PointProcess(window, n, samples[, asPP]) Point Process base class.PoissonPointProcess(window, n, samples[, . . . ]) Poisson point process including 𝑁 -conditioned CSR

process and 𝜆-conditioned CSR process.PoissonClusterPointProcess(window, n,. . . [, . . . ])

Poisson cluster point process (Neyman Scott).

2.2.1 pointpats.PointProcess

class pointpats.PointProcess(window, n, samples, asPP=False, **args)Point Process base class.

Parameters

window [Window] Bounding geometric object to contain point process realizations.

n [int] Size of each realization.

samples [list] Number of realizations.

asPP [bool] Control the data type of value in the “realizations” dictionary. If True, the data typeis point pattern as defined in pointpattern.py; if False, the data type is an two-dimensionalarray.

2.2. Point Processes 9


Attributes

realizations [dictionary] The key is the index of each realization, and the value is simulatedevent points for each realization. The data type of the value is controlled by the parameter“asPP”.

parameters [dictionary] Dictionary of a dictionary. The key is the index of each realization,and the value is a dictionary with the key ‘n’ and the value size of each realization.

Methods

draw(self, parameter) Generate a series of point coordinates within thegiven window.

pointpats.PointProcess

PointProcess.draw(self, parameter)Generate a series of point coordinates within the given window.

Parameters

parameter [dictionary] Key: ‘n’. Value: size of the realization.

Returns

: array A series of point coordinates.

realizesetup

__init__(self, window, n, samples, asPP=False, **args)Initialize self. See help(type(self)) for accurate signature.

2.2.2 pointpats.PoissonPointProcess

class pointpats.PoissonPointProcess(window, n, samples, conditioning=False, asPP=False)Poisson point process including 𝑁 -conditioned CSR process and 𝜆-conditioned CSR process.

Parameters




conditioning [bool] If True, use the 𝜆-conditioned CSR process, number of events would varyacross realizations; if False, use the 𝑁 -conditioned CSR process.




Examples

>>> import libpysal as ps>>> import numpy as np>>> from pointpats import Window>>> from libpysal.cg import shapely_ext

Open the virginia polygon shapefile

>>> va = ps.io.open(ps.examples.get_path("virginia.shp"))

Create the exterior polygons for VA from the union of the county shapes

>>> polys = [shp for shp in va]>>> state = shapely_ext.cascaded_union(polys)

Create window from virginia state boundary

>>> window = Window(state.parts)

1. Simulate a 𝑁 -conditioned csr process in the same window (10 points, 2 realizations)

>>> np.random.seed(5)>>> samples1 = PoissonPointProcess(window, 10, 2, conditioning=False, asPP=False)>>> samples1.realizations[0] # the first realized event pointsarray([[-81.80326547, 36.77687577],

[-78.5166233 , 37.34055832],[-77.21660795, 37.7491503 ],[-79.30361037, 37.40467853],[-78.61625258, 36.61234487],[-81.43369537, 37.13784646],[-80.91302108, 36.60834063],[-76.90806444, 37.95525903],[-76.33475868, 36.62635347],[-79.71621808, 37.27396618]])

2. Simulate a 𝜆-conditioned csr process in the same window (10 points, 2 realizations)

>>> np.random.seed(5)>>> samples2 = PoissonPointProcess(window, 10, 2, conditioning=True, asPP=True)>>> samples2.realizations[0].n # the size of first realized point pattern10>>> samples2.realizations[1].n # the size of second realized point pattern13

Attributes


parameters [dictionary] Dictionary of a dictionary. The key is the index of each realization,and the value is a dictionary with the key ‘n’ and the value: 1. always equal to the parametern in the case of N-conditioned process. For example, {0:{‘n’:100},1:{‘n’:100},2:{‘n’:100}}2. randomly generated from a Possion process in the case of lambda-conditioned process.For example, {0:{‘n’:97},1:{‘n’:100},2:{‘n’:98}}



Methods


realize(self, n) Generate n points which are randomly and indepen-dently distributed in the minimum bounding box of“window”.

setup(self) Generate the number of events for each realization.

pointpats.PoissonPointProcess

PoissonPointProcess.draw(self, parameter)Generate a series of point coordinates within the given window.

Parameters


Returns



PoissonPointProcess.realize(self, n)Generate n points which are randomly and independently distributed in the minimum bounding box of“window”.

Parameters

n [int] Number of point events.

Returns

: array (n,2), n point coordinates.


PoissonPointProcess.setup(self)Generate the number of events for each realization. If “conditioning” is False, all the event numbers arethe same; if it is True, the event number is a random variable following a Poisson distribution.

__init__(self, window, n, samples, conditioning=False, asPP=False)Initialize self. See help(type(self)) for accurate signature.

2.2.3 pointpats.PoissonClusterPointProcess

class pointpats.PoissonClusterPointProcess(window, n, parents, radius, samples,keep=False, asPP=False, condition-ing=False)

Poisson cluster point process (Neyman Scott). Two stages: 1. parent CSR process: 𝑁 -conditioned or 𝜆-conditioned. If parent events follow a 𝜆-conditioned CSR process, the number of parent events varies acrossrealizations. 2. child process: fixed number of points in circle centered on each parent.

Parameters





parents [int] Number of parents.

radius [float] Radius of the circle centered on each parent.



conditioning [bool] If True, use the 𝑙𝑎𝑚𝑏𝑑𝑎-conditioned CSR process for parent events, lead-ing to varied number of parent events across realizations; if False, use the 𝑁 -conditionedCSR process.

Examples

>>> import libpysal as ps>>> import numpy as np>>> from pointpats import Window>>> from libpysal.cg import shapely_ext

Open the virginia polygon shapefile

>>> va = ps.io.open(ps.examples.get_path("virginia.shp"))

Create the exterior polygons for VA from the union of the county shapes

>>> polys = [shp for shp in va]>>> state = shapely_ext.cascaded_union(polys)

Create window from virginia state boundary

>>> window = Window(state.parts)

1. Simulate a Poisson cluster process of size 200 with 10 parents and 20 children within 0.5 units of each parent(parent events: 𝑁 -conditioned CSR)

>>> np.random.seed(10)>>> samples1 = PoissonClusterPointProcess(window, 200, 10, 0.5, 1, asPP=True,→˓conditioning=False)>>> samples1.parameters # number of events for the realization{0: {'n': 200}}>>> samples1.num_parents #number of parent events for each realization{0: 10}>>> samples1.children # number of children events centered on each parent event20

2. Simulate a Poisson cluster process of size 200 with 10 parents and 20 children within 0.5 units of each parent(parent events: 𝜆-conditioned CSR)

>>> np.random.seed(10)>>> samples2 = PoissonClusterPointProcess(window, 200, 10, 0.5, 1, asPP=True,→˓conditioning=True)>>> samples2.parameters # number of events for the realization might not be equal→˓to 200

(continues on next page)



(continued from previous page)

{0: {'n': 260}}>>> samples2.num_parents #number of parent events for each realization{0: 13}>>> samples2.children # number of children events centered on each parent event20

Attributes

children [int] Number of childrens centered on each parent. Can be considered as local inten-sity.

num_parents [dictionary] The key is the index of each realization. The value is the number ofparent events for each realization.


parameters [dictionary] Dictionary of a dictionary. The key is the index of each realization,and the value is a dictionary with the key ‘n’ and the value always equal to the parameter nin the case of N-conditioned process. For example, {0:{‘n’:100},1:{‘n’:100},2:{‘n’:100}}2. randomly generated from a Possion process in the case of lambda-conditioned process.For example, {0:{‘n’:97},1:{‘n’:100},2:{‘n’:98}}

Methods


realize(self, n) Generate n points which are distributed in a clusteredfashion in the minimum bounding box of “window”.

setup(self) Generate the number of events for each realization.

pointpats.PoissonClusterPointProcess

PoissonClusterPointProcess.draw(self, parameter)Generate a series of point coordinates within the given window.

Parameters


Returns



PoissonClusterPointProcess.realize(self, n)Generate n points which are distributed in a clustered fashion in the minimum bounding box of “window”.

Parameters

n [int] Number of point events.

Returns



res [array] (n,2), n point coordinates.


PoissonClusterPointProcess.setup(self)Generate the number of events for each realization. If “conditioning” is False, all the event numbers are thesame; if it is True, the number of parents is a random variable following a Poisson distribution, resultingin varied number of events.

__init__(self, window, n, parents, radius, samples, keep=False, asPP=False, conditioning=False)Initialize self. See help(type(self)) for accurate signature.

2.3 Centrography

mbr(points) Find minimum bounding rectangle of a point array.hull(points) Find convex hull of a point array.mean_center(points) Find mean center of a point array.weighted_mean_center(points, weights) Find weighted mean center of a marked point pattern.manhattan_median(points) Find manhattan median of a point array.std_distance(points) Calculate standard distance of a point array.euclidean_median(points) Calculate the Euclidean median for a point pattern.ellipse(points) Calculate parameters of standard deviational ellipse for

a point pattern.skyum(points[, not_hull]) Implements Skyum (1990)’s algorithm for the minimum

bounding circle in R^2.dtot(coord, points) Sum of Euclidean distances between event points and a

selected point._circle(p, q, r[, dmetric]) Returns (radius, (center_x, center_y)) of the circum-

scribed circle by the triangle pqr.

2.3.1 pointpats.mbr

pointpats.mbr(points)Find minimum bounding rectangle of a point array.

Parameters

points [arraylike] (n,2), (x,y) coordinates of a series of event points.

Returns

min_x [float] leftmost value of the vertices of minimum bounding rectangle.

min_y [float] downmost value of the vertices of minimum bounding rectangle.

max_x [float] rightmost value of the vertices of minimum bounding rectangle.

max_y [float] upmost value of the vertices of minimum bounding rectangle.

2.3.2 pointpats.hull

pointpats.hull(points)Find convex hull of a point array.

2.3. Centrography 15


Parameters

points: arraylike (n,2), (x,y) coordinates of a series of event points.

Returns

_ [array] (h,2), points defining the hull in counterclockwise order.

2.3.3 pointpats.mean_center

pointpats.mean_center(points)Find mean center of a point array.

Parameters


Returns

_ [array] (2,), (x,y) coordinates of the mean center.

2.3.4 pointpats.weighted_mean_center

pointpats.weighted_mean_center(points, weights)Find weighted mean center of a marked point pattern.

Parameters


weights [arraylike] a series of attribute values of length n.

Returns

_ [array] (2,), (x,y) coordinates of the weighted mean center.

2.3.5 pointpats.manhattan_median

pointpats.manhattan_median(points)Find manhattan median of a point array.

Parameters


Returns

_ [array] (2,), (x,y) coordinates of the manhattan median.

2.3.6 pointpats.std_distance

pointpats.std_distance(points)Calculate standard distance of a point array.

Parameters


Returns

_ [float] standard distance.



2.3.7 pointpats.euclidean_median

pointpats.euclidean_median(points)Calculate the Euclidean median for a point pattern.

Parameters


Returns

_ [array] (2,), (x,y) coordinates of the Euclidean median.

2.3.8 pointpats.ellipse

pointpats.ellipse(points)Calculate parameters of standard deviational ellipse for a point pattern.

Parameters


Returns

_ [float] semi-major axis.

_ [float] semi-minor axis.

theta [float] clockwise rotation angle of the ellipse.

Notes

Implements approach from:

https://www.icpsr.umich.edu/CrimeStat/files/CrimeStatChapter.4.pdf

2.3.9 pointpats.skyum

pointpats.skyum(points, not_hull=True)Implements Skyum (1990)’s algorithm for the minimum bounding circle in R^2.

0. Store points clockwise.

1. Find p in S that maximizes angle(prec(p), p, succ(p) THEN radius(prec(p), p, succ(p)). This is also calledthe lexicographic maximum, and is the last entry of a list of (radius, angle) in lexicographical order. 2a. Ifangle(prec(p), p, succ(p)) <= 90 degrees, then finish. 2b. If not, remove p from set.

2.3.10 pointpats.dtot

pointpats.dtot(coord, points)Sum of Euclidean distances between event points and a selected point.

Parameters

coord [arraylike] (x,y) coordinates of a point.


Returns

2.3. Centrography 17

https://www.icpsr.umich.edu/CrimeStat/files/CrimeStatChapter.4.pdf


d [float] sum of Euclidean distances.

2.3.11 pointpats._circle

pointpats._circle(p, q, r, dmetric=<function euclidean at 0x7f55a53f80d0>)Returns (radius, (center_x, center_y)) of the circumscribed circle by the triangle pqr.

note, this does not assume that p!=q!=r

2.4 Quadrat Based Statistics

RectangleM (pp[, count_column, count_row, . . . ]) Rectangle grid structure for quadrat-based method.HexagonM (pp, lh) Hexagon grid structure for quadrat-based method.QStatistic(pp[, shape, nx, ny, lh, realizations]) Quadrat analysis of point pattern.

2.4.1 pointpats.RectangleM

class pointpats.RectangleM(pp, count_column=3, count_row=3, rectangle_width=0, rectan-gle_height=0)

Rectangle grid structure for quadrat-based method.

Parameters

pp [PointPattern] Point Pattern instance.

count_column [integer] Number of rectangles in the horizontal direction. Use in pair withcount_row to fully specify a rectangle. Incompatible with rectangle_width and rectan-gle_height.

count_row [integer] Number of rectangles in the vertical direction. Use in pair withcount_column to fully specify a rectangle. Incompatible with rectangle_width and rect-angle_height.

rectangle_width [float] Rectangle width. Use in pair with rectangle_height to fully specify arectangle. Incompatible with count_column & count_row.

rectangle_height [float] Rectangle height. Use in pair with rectangle_width to fully specify arectangle. Incompatible with count_column & count_row.

Attributes


mbb [array] Minimum bounding box for the point pattern.

points [array] x,y coordinates of the point points.

count_column [integer] Number of columns.

count_row [integer] Number of rows.

num [integer] Number of rectangular quadrats.

Methods



plot(self[, title]) Plot rectangle tessellation as well as the number ofpoints falling in each rectangle.

point_location_sta(self) Count the point events in each cell.

pointpats.RectangleM

RectangleM.plot(self, title=’Quadrat Count’)Plot rectangle tessellation as well as the number of points falling in each rectangle.

Parameters

title: str, optional Title of the plot. Default is “Quadrat Count”.

pointpats.RectangleM

RectangleM.point_location_sta(self)Count the point events in each cell.

Returns

dict_id_count [dict] keys: rectangle id, values: number of point events in each cell.

__init__(self, pp, count_column=3, count_row=3, rectangle_width=0, rectangle_height=0)Initialize self. See help(type(self)) for accurate signature.

2.4.2 pointpats.HexagonM

class pointpats.HexagonM(pp, lh)Hexagon grid structure for quadrat-based method.

Parameters


lh [float] Hexagon length (hexagon).

Attributes


h_length [float] Hexagon length (hexagon).

mbb [array] Minimum bounding box for the point pattern.

points [array] x,y coordinates of the point points.

h_length [float] Hexagon length (hexagon).

count_row_even [integer] Number of even rows.

count_row_odd [integer] Number of odd rows.

count_column [integer] Number of columns.

num [integer] Number of hexagonal quadrats.

Methods

2.4. Quadrat Based Statistics 19


plot(self[, title]) Plot hexagon quadrats as well as the number ofpoints falling in each quadrat.

point_location_sta(self) Count the point events in each hexagon cell.

pointpats.HexagonM

HexagonM.plot(self, title=’Quadrat Count’)Plot hexagon quadrats as well as the number of points falling in each quadrat.

Parameters


pointpats.HexagonM

HexagonM.point_location_sta(self)Count the point events in each hexagon cell.

Returns

dict_id_count [dict] keys: rectangle id, values: number of point events in each hexagoncell.

__init__(self, pp, lh)Initialize self. See help(type(self)) for accurate signature.

2.4.3 pointpats.QStatistic

class pointpats.QStatistic(pp, shape=’rectangle’, nx=3, ny=3, lh=10, realizations=0)Quadrat analysis of point pattern.

Parameters


shape [string] Grid structure. Either “rectangle” or “hexagon”. Default is “rectangle”.

nx [integer] Number of rectangles in the horizontal direction. Only when shape is specified as“rectangle” will nx be considered.

ny [integer] Number of rectangles in the vertical direction. Only when shape is specified as“rectangle” will ny be considered.

lh [float] Hexagon length (hexagon). Only when shape is specified as “hexagon” will lh beconsidered. Incompatible with nx & ny.

realizations [PointProcess] Point process instance with more than 1 point pattern realiza-tions which would be used for simulation based inference. Default is 0 where no simulationbased inference is performed.

Attributes


mr [RectangleM or HexagonM ] RectangleM or HexagonM instance.

chi2 [float] Chi-squared test statistic for the observed point pattern pp.

df [integer] Degree of freedom.



chi2_pvalue [float] p-value based on analytical chi-squared distribution.

chi2_r_pvalue [float] p-value based on simulated sampling distribution. Only available whenrealizations is correctly specified.

chi2_realizations [array] Chi-squared test statistics calculated for all of the simulated csr pointpatterns.

Methods

plot(self[, title]) Plot quadrats as well as the number of points fallingin each quadrat.

pointpats.QStatistic

QStatistic.plot(self, title=’Quadrat Count’)Plot quadrats as well as the number of points falling in each quadrat.

Parameters


__init__(self, pp, shape=’rectangle’, nx=3, ny=3, lh=10, realizations=0)Initialize self. See help(type(self)) for accurate signature.

2.5 Distance Based Statistics

DStatistic(name) Abstract Base Class for distance statistics.G(pp[, intervals, dmin, dmax, d]) Estimates the nearest neighbor distance distribution

function G for a point pattern.F(pp[, n, intervals, dmin, dmax, d]) Estimates the empty space distribution function for a

point pattern: F(d).J(pp[, n, intervals, dmin, dmax, d]) Estimates the J function for a point pattern [LB96]K(pp[, intervals, dmin, dmax, d]) Estimates the K function for a point pattern.L(pp[, intervals, dmin, dmax, d]) Estimates the l function for a point pattern.Envelopes(*args, **kwargs) Abstract base class for simulation envelopes.Genv(pp[, intervals, dmin, dmax, d, pct, . . . ]) Simulation envelope for G function.Fenv(pp[, n, intervals, dmin, dmax, d, pct, . . . ]) Simulation envelope for F function.Jenv(pp[, n, intervals, dmin, dmax, d, pct, . . . ]) Simulation envelope for J function.Kenv(pp[, intervals, dmin, dmax, d, pct, . . . ]) Simulation envelope for K function.Lenv(pp[, intervals, dmin, dmax, d, pct, . . . ]) Simulation envelope for L function.

2.5.1 pointpats.DStatistic

class pointpats.DStatistic(name)Abstract Base Class for distance statistics.

Parameters

name [string] Name of the function. (“G”, “F”, “J”, “K” or “L”)

Attributes

2.5. Distance Based Statistics 21


d [array] The distance domain sequence.

Methods

plot(self[, qq]) Plot the distance function

pointpats.DStatistic

DStatistic.plot(self, qq=False)Plot the distance function

Parameters

qq: Boolean If False the statistic is plotted against distance. If Frue, the quantile-quantileplot is generated, observed vs. CSR.

__init__(self, name)Initialize self. See help(type(self)) for accurate signature.

2.5.2 pointpats.G

class pointpats.G(pp, intervals=10, dmin=0.0, dmax=None, d=None)Estimates the nearest neighbor distance distribution function G for a point pattern.

Parameters


intervals [int] The length of distance domain sequence.

dmin [float] The minimum of the distance domain.

dmax [float] The maximum of the distance domain.

d [sequence] The distance domain sequence. If d is specified, intervals, dmin and dmax areignored.

Notes

In the analysis of planar point processes, the estimate of 𝐺 is typically compared to the value expected from acompletely spatial random (CSR) process given as:

𝐺(𝑑) = 1− 𝑒−𝜆𝜋𝑑2

where 𝜆 is the intensity (points per unit area) of the point process and 𝑑 is distance.

For a clustered pattern, the empirical function will be above the expectation, while for a uniform pattern theempirical function falls below the expectation.

Attributes



G [array] The cumulative nearest neighbor distance distribution over d.



Methods


pointpats.G

G.plot(self, qq=False)Plot the distance function

Parameters


__init__(self, pp, intervals=10, dmin=0.0, dmax=None, d=None)Initialize self. See help(type(self)) for accurate signature.

2.5.3 pointpats.F

class pointpats.F(pp, n=100, intervals=10, dmin=0.0, dmax=None, d=None)Estimates the empty space distribution function for a point pattern: F(d).

Parameters


n [int] Number of empty space points (random points).





Notes

In the analysis of planar point processes, the estimate of 𝐹 is typically compared to the value expected from aprocess that displays complete spatial randomness (CSR):

𝐹 (𝑑) = 1− 𝑒−𝜆𝜋𝑑2

where 𝜆 is the intensity (points per unit area) of the point process and 𝑑 is distance.

The expectation is identical to the expectation for the G function for a CSR process. However, for a clustered pat-tern, the empirical G function will be below the expectation, while for a uniform pattern the empirical functionfalls above the expectation.

Attributes


G [array] The cumulative empty space nearest event distance distribution over d.

Methods




pointpats.F

F.plot(self, qq=False)Plot the distance function

Parameters


__init__(self, pp, n=100, intervals=10, dmin=0.0, dmax=None, d=None)Initialize self. See help(type(self)) for accurate signature.

2.5.4 pointpats.J

class pointpats.J(pp, n=100, intervals=10, dmin=0.0, dmax=None, d=None)Estimates the J function for a point pattern [LB96]

Parameters







Notes

The 𝐽 function is a ratio of the hazard functions defined for 𝐺 and 𝐹 :

𝐽(𝑑) =1−𝐺(𝑑)

1− 𝐹 (𝑑)

where 𝐺(𝑑) is the nearest neighbor distance distribution function (see G) and 𝐹 (𝑑) is the empty space function(see F).

For a CSR process the J function equals 1. Empirical values larger than 1 are indicative of uniformity, whilevalues below 1 suggest clustering.

Attributes


j [array] F function over d.

Methods




pointpats.J

J.plot(self, qq=False)Plot the distance function

Parameters


__init__(self, pp, n=100, intervals=10, dmin=0.0, dmax=None, d=None)Initialize self. See help(type(self)) for accurate signature.

2.5.5 pointpats.K

class pointpats.K(pp, intervals=10, dmin=0.0, dmax=None, d=None)Estimates the K function for a point pattern.

Parameters






Attributes


j [array] K function over d.

Methods


pointpats.K

K.plot(self, qq=False)Plot the distance function

Parameters





2.5.6 pointpats.L

class pointpats.L(pp, intervals=10, dmin=0.0, dmax=None, d=None)Estimates the l function for a point pattern.

Parameters






Attributes


l [array] L function over d.

Methods

plot(self) Plot the distance function

pointpats.L

L.plot(self)Plot the distance function

Parameters



2.5.7 pointpats.Envelopes

class pointpats.Envelopes(*args, **kwargs)Abstract base class for simulation envelopes.

Parameters


intervals [int] The length of distance domain sequence. Default is 10.




pct [float] 1-alpha, alpha is the significance level. Default is 0.05, 1-alpha is the confidencelevel for the envelope.



realizations: :class:‘.PointProcess‘ Point process instance with more than 1 realizations.

Attributes


observed [array] A 2-dimensional numpy array of 2 columns. The first column is the distancedomain sequence for the observed point pattern. The second column is the specific func-tion (“G”, “F”, “J”, “K” or “L”) over the distance domain sequence for the observed pointpattern.

low [array] A 1-dimensional numpy array. Lower bound of the simulation envelope.

high [array] A 1-dimensional numpy array. Higher bound of the simulation envelope.

mean [array] A 1-dimensional numpy array. Mean values of the simulation envelope.

Methods

calcmapperplot

__init__(self, *args, **kwargs)Initialize self. See help(type(self)) for accurate signature.

2.5.8 pointpats.Genv

class pointpats.Genv(pp, intervals=10, dmin=0.0, dmax=None, d=None, pct=0.05, realiza-tions=None)

Simulation envelope for G function.

Parameters




dmax [float] Upper limit of distance range. If dmax is None, dmax will be set to maximumnearest neighbor distance.


pct [float] 1-alpha, alpha is the significance level. Default is 0.05, which means 95% confidencelevel for the envelopes.


Examples

>>> import libpysal as ps>>> from pointpats import Genv, PoissonPointProcess, Window>>> from libpysal.cg import shapely_ext>>> va = ps.io.open(ps.examples.get_path("vautm17n.shp"))

(continues on next page)



(continued from previous page)

>>> polys = [shp for shp in va]>>> state = shapely_ext.cascaded_union(polys)>>> pp = PoissonPointProcess(Window(state.parts), 100, 1, asPP=True).→˓realizations[0]>>> csrs = PoissonPointProcess(pp.window, 100, 100, asPP=True)>>> genv_bb = Genv(pp, realizations=csrs)>>> genv_bb.plot()

Attributes


observed [array] A 2-dimensional numpy array of 2 columns. The first column is the distancedomain sequence for the observed point pattern. The second column is cumulative nearestneighbor distance distribution (G function) for the observed point pattern.




Methods

calcmapperplot

__init__(self, pp, intervals=10, dmin=0.0, dmax=None, d=None, pct=0.05, realizations=None)Initialize self. See help(type(self)) for accurate signature.

2.5.9 pointpats.Fenv

class pointpats.Fenv(pp, n=100, intervals=10, dmin=0.0, dmax=None, d=None, pct=0.05, realiza-tions=None)

Simulation envelope for F function.

Parameters











Examples

>>> import libpysal as ps>>> from libpysal.cg import shapely_ext>>> from pointpats import PoissonPointProcess,Window,Fenv>>> va = ps.io.open(ps.examples.get_path("vautm17n.shp"))>>> polys = [shp for shp in va]>>> state = shapely_ext.cascaded_union(polys)>>> pp = PoissonPointProcess(Window(state.parts), 100, 1, asPP=True).→˓realizations[0]>>> csrs = PoissonPointProcess(pp.window, 100, 100, asPP=True)>>> fenv = Fenv(pp, realizations=csrs)>>> fenv.plot()

Attributes


observed [array] A 2-dimensional numpy array of 2 columns. The first column is the distancedomain sequence for the observed point pattern. The second column is F function for theobserved point pattern.




Methods

calcmapperplot

__init__(self, pp, n=100, intervals=10, dmin=0.0, dmax=None, d=None, pct=0.05, realiza-tions=None)

Initialize self. See help(type(self)) for accurate signature.

Methods

__init__(self, pp[, n, intervals, dmin, . . . ]) Initialize self.calc(self, \*args, \*\*kwargs)mapper(self, realizations)plot(self)

2.5.10 pointpats.Jenv

class pointpats.Jenv(pp, n=100, intervals=10, dmin=0.0, dmax=None, d=None, pct=0.05, realiza-tions=None)

Simulation envelope for J function.

Parameters











Examples

>>> import libpysal as ps>>> from pointpats import Jenv, PoissonPointProcess, Window>>> from libpysal.cg import shapely_ext>>> va = ps.io.open(ps.examples.get_path("vautm17n.shp"))>>> polys = [shp for shp in va]>>> state = shapely_ext.cascaded_union(polys)>>> pp = PoissonPointProcess(Window(state.parts), 100, 1, asPP=True).→˓realizations[0]>>> csrs = PoissonPointProcess(pp.window, 100, 100, asPP=True)>>> jenv = Jenv(pp, realizations=csrs)>>> jenv.plot()

Attributes


observed [array] A 2-dimensional numpy array of 2 columns. The first column is the distancedomain sequence for the observed point pattern. The second column is J function for theobserved point pattern.




Methods

calcmapperplot

__init__(self, pp, n=100, intervals=10, dmin=0.0, dmax=None, d=None, pct=0.05, realiza-tions=None)

Initialize self. See help(type(self)) for accurate signature.



Methods

__init__(self, pp[, n, intervals, dmin, . . . ]) Initialize self.calc(self, \*args, \*\*kwargs)mapper(self, realizations)plot(self)

2.5.11 pointpats.Kenv

class pointpats.Kenv(pp, intervals=10, dmin=0.0, dmax=None, d=None, pct=0.05, realiza-tions=None)

Simulation envelope for K function.

Parameters






pct [float] 1-alpha, alpha is the significance level. Default is 0.05, which means 95% confidencelevel for the envelope.


Examples

>>> import libpysal as ps>>> from pointpats import Kenv, PoissonPointProcess, Window>>> from libpysal.cg import shapely_ext>>> va = ps.io.open(ps.examples.get_path("vautm17n.shp"))>>> polys = [shp for shp in va]>>> state = shapely_ext.cascaded_union(polys)>>> pp = PoissonPointProcess(Window(state.parts), 100, 1, asPP=True).→˓realizations[0]>>> csrs = PoissonPointProcess(pp.window, 100, 100, asPP=True)>>> kenv = Kenv(pp, realizations=csrs)>>> kenv.plot()

Attributes


observed [array] A 2-dimensional numpy array of 2 columns. The first column is the distancedomain sequence for the observed point pattern. The second column is K function for theobserved point pattern.






Methods

calcmapperplot


Methods

__init__(self, pp[, intervals, dmin, dmax, . . . ]) Initialize self.calc(self, \*args, \*\*kwargs)mapper(self, realizations)plot(self)

2.5.12 pointpats.Lenv

class pointpats.Lenv(pp, intervals=10, dmin=0.0, dmax=None, d=None, pct=0.05, realiza-tions=None)

Simulation envelope for L function.

Parameters








Examples

>>> import libpysal as ps>>> from pointpats import Lenv, PoissonPointProcess, Window>>> from libpysal.cg import shapely_ext>>> va = ps.io.open(ps.examples.get_path("vautm17n.shp"))>>> polys = [shp for shp in va]>>> state = shapely_ext.cascaded_union(polys)>>> pp = PoissonPointProcess(Window(state.parts), 100, 1, asPP=True).→˓realizations[0]>>> csrs = PoissonPointProcess(pp.window, 100, 100, asPP=True)>>> lenv = Lenv(pp, realizations=csrs)>>> lenv.plot()



Attributes


observed [array] A 2-dimensional numpy array of 2 columns. The first column is the distancedomain sequence for the observed point pattern. The second column is L function for theobserved point pattern.




Methods

calcmapperplot


Methods

__init__(self, pp[, intervals, dmin, dmax, . . . ]) Initialize self.calc(self, \*args, \*\*kwargs)mapper(self, realizations)plot(self)

2.6 Window functions

Window(parts[, holes]) Geometric container for point patterns.as_window(pysal_polygon) Convert a libpysal polygon to a Window.poly_from_bbox(bbox)to_ccf(poly)

2.6.1 pointpats.Window

class pointpats.Window(parts, holes=[])Geometric container for point patterns.

A window is used to define the area over which the pattern is observed. This area is used in estimating theintensity of the point pattern. See PointPattern.lambda_window.

Parameters

parts: sequence A sequence of rings which bound the positive space point pattern.

holes: sequence A sequence of rings which bound holes in the polygons that bound the pointpattern.

Attributes

2.6. Window functions 33


area Returns the area of the polygon.

bbox Returns the bounding box of the polygon as a list

bounding_box Returns the bounding box of the polygon.

centroid Returns the centroid of the polygon

holes Returns the holes of the polygon in clockwise order.

len Returns the number of vertices in the polygon.

parts Returns the parts of the polygon in clockwise order.

perimeter Returns the perimeter of the polygon.

vertices Returns the vertices of the polygon in clockwise order.

Methods

build_quad_tree_structure(self) Build the quad tree structure for this polygon.contains_point(self, point) Test if polygon contains point

pointpats.Window

Window.build_quad_tree_structure(self)Build the quad tree structure for this polygon. Once the structure is built, speed for testing if a point isinside the ring will be inscreased significantly. :return:

pointpats.Window

Window.contains_point(self, point)Test if polygon contains point

Notes

Points falling exactly on polygon edges may yield unpredictable results

Examples

>>> p = Polygon([Point((0,0)), Point((4,0)), Point((4,5)), Point((2,3)),→˓Point((0,5))])>>> p.contains_point((3,3))1>>> p.contains_point((0,6))0>>> p.contains_point((2,2.9))1>>> p.contains_point((4,5))0>>> p.contains_point((4,0))0>>>

Handles holes


https://docs.python.org/3.6/library/functions.html#len


>>> p = Polygon([Point((0, 0)), Point((0, 10)), Point((10, 10)), Point((10,→˓0))], [Point((2, 2)), Point((4, 2)), Point((4, 4)), Point((2, 4))])>>> p.contains_point((3.0,3.0))False>>> p.contains_point((1.0,1.0))True>>>

filter_contained

__init__(self, parts, holes=[])Returns a polygon created from the objects specified.

__init__(Point list or list of Point lists, holes list ) -> Polygon

Parameters

vertices [list – a list of vertices or a list of lists of vertices.]

holes [list – a list of sub-polygons to be considered as holes.]

is_quad_tree_structure_built : bool – record if the quad tree structure has been built forthis polygon. This quad tree structure could help speed up the contains_point test

Examples

>>> p1 = Polygon([Point((0, 0)), Point((1, 0)), Point((1, 1)), Point((0, 1))])

2.6.2 pointpats.as_window

pointpats.as_window(pysal_polygon)Convert a libpysal polygon to a Window.

Parameters

pysal_polygon: libpysal.cg.shapes.Polygon libpysal Polygon instance.

Returns

Window A Window instance.

2.6.3 pointpats.poly_from_bbox

pointpats.poly_from_bbox(bbox)

2.6.4 pointpats.to_ccf

pointpats.to_ccf(poly)

2.6. Window functions 35

CHAPTER

THREE

REFERENCES

37


38 Chapter 3. References

BIBLIOGRAPHY

[LB96] M. N. M. Lieshout and A. J. Baddeley. A nonparametric measure of spatial interaction in point patterns.Statistica Neerlandica, 50(3):344–361, 1996. doi:10.1111/j.1467-9574.1996.tb01501.x.

39

https://doi.org/10.1111/j.1467-9574.1996.tb01501.x


40 Bibliography

INDEX

Symbols__init__() (pointpats.DStatistic method), 22__init__() (pointpats.Envelopes method), 27__init__() (pointpats.F method), 24__init__() (pointpats.Fenv method), 29__init__() (pointpats.G method), 23__init__() (pointpats.Genv method), 28__init__() (pointpats.HexagonM method), 20__init__() (pointpats.J method), 25__init__() (pointpats.Jenv method), 30__init__() (pointpats.K method), 25__init__() (pointpats.Kenv method), 32__init__() (pointpats.L method), 26__init__() (pointpats.Lenv method), 33__init__() (pointpats.PointPattern method), 9__init__() (pointpats.PointProcess method), 10__init__() (pointpats.PoissonClusterPointProcess

method), 15__init__() (pointpats.PoissonPointProcess method),

12__init__() (pointpats.QStatistic method), 21__init__() (pointpats.RectangleM method), 19__init__() (pointpats.Window method), 35_circle() (in module pointpats), 18

Aas_window() (in module pointpats), 35

Bbuild_quad_tree_structure() (point-

pats.Window method), 34

Ccontains_point() (pointpats.Window method), 34

Ddraw() (pointpats.PointProcess method), 10draw() (pointpats.PoissonClusterPointProcess

method), 14draw() (pointpats.PoissonPointProcess method), 12DStatistic (class in pointpats), 21

dtot() (in module pointpats), 17

Eellipse() (in module pointpats), 17Envelopes (class in pointpats), 26euclidean_median() (in module pointpats), 17explode() (pointpats.PointPattern method), 6

FF (class in pointpats), 23Fenv (class in pointpats), 28find_pairs() (pointpats.PointPattern method), 7flip_coordinates() (pointpats.PointPattern

method), 7

GG (class in pointpats), 22Genv (class in pointpats), 27get_window() (pointpats.PointPattern method), 7

HHexagonM (class in pointpats), 19hull() (in module pointpats), 15

JJ (class in pointpats), 24Jenv (class in pointpats), 29

KK (class in pointpats), 25Kenv (class in pointpats), 31knn() (pointpats.PointPattern method), 7knn_other() (pointpats.PointPattern method), 7

LL (class in pointpats), 26Lenv (class in pointpats), 32

Mmanhattan_median() (in module pointpats), 16mbr() (in module pointpats), 15

41


mean_center() (in module pointpats), 16

Pplot() (pointpats.DStatistic method), 22plot() (pointpats.F method), 24plot() (pointpats.G method), 23plot() (pointpats.HexagonM method), 20plot() (pointpats.J method), 25plot() (pointpats.K method), 25plot() (pointpats.L method), 26plot() (pointpats.PointPattern method), 8plot() (pointpats.QStatistic method), 21plot() (pointpats.RectangleM method), 19point_location_sta() (pointpats.HexagonM

method), 20point_location_sta() (pointpats.RectangleM

method), 19PointPattern (class in pointpats), 5PointProcess (class in pointpats), 9PoissonClusterPointProcess (class in point-

pats), 12PoissonPointProcess (class in pointpats), 10poly_from_bbox() (in module pointpats), 35

QQStatistic (class in pointpats), 20

Rrealize() (pointpats.PoissonClusterPointProcess

method), 14realize() (pointpats.PoissonPointProcess method),

12RectangleM (class in pointpats), 18

Ssetup() (pointpats.PoissonClusterPointProcess

method), 15setup() (pointpats.PoissonPointProcess method), 12skyum() (in module pointpats), 17std_distance() (in module pointpats), 16summary() (pointpats.PointPattern method), 8superimpose() (pointpats.PointPattern method), 8

Tto_ccf() (in module pointpats), 35

Uunique() (pointpats.PointPattern method), 9

Wweighted_mean_center() (in module pointpats),

16Window (class in pointpats), 33

42 Index

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