Point Processes

Author: Serge Rey sjsrey@gmail.com and Wei Kang weikang9009@gmail.com

Introduction

One philosophy of applying inferential statistics to spatial data is to think in terms of spatial processes and their possible realizations. In this view, an observed map pattern is one of the possible patterns that might have been generated by a hypothesized process. In this notebook, we are going to regard point patterns as the outcome of point processes. There are three major types of point process, which will result in three types of point patterns:

We will investigate how to generate these point patterns via simulation (Data Generating Processes (DGP) is the correponding point process), and inspect how these resulting point patterns differ from each other visually. In Quadrat statistics notebook and distance statistics notebook, we will adpot some statistics to infer whether it is a Complete Spaital Randomness (CSR) process.

A python file named “process.py” contains several point process classes with which we can generate point patterns of different types.

from pointpats import PoissonPointProcess, PoissonClusterPointProcess, Window, poly_from_bbox, PointPattern
import libpysal as ps
from libpysal.cg import shapely_ext
%matplotlib inline
import numpy as np
#import matplotlib.pyplot as plt

Random Patterns

Random point patterns are the outcome of CSR. CSR has two major characteristics:

  1. Uniform: each location has equal probability of getting a point (where an event happens)
  2. Independent: location of event points are independent

It usually serves as the null hypothesis in testing whether a point pattern is the outcome of a random process.

There are two types of CSR:

  • $N$-conditioned CSR: $N$ is fixed
    • Given the total number of events $N$ occurring within an area $A$, the locations of the $N$ events represent an independent random sample of $N$ locations where each location is equally likely to be chosen as an event.
  • $\lambda$-conditioned CSR: $N$ is randomly generated from a Poisson process.
    • The number of events occurring within a finite region $A$ is a random variable $\dot{N}$ following a Poisson distribution with mean $\lambda A $, with $ A $ denoting area of $A$ and $\lambda$ denoting the intensity of the point pattern.
    • Given the total number of events $\dot{N}$ occurring within an area $A$, the locations of the $\dot{N}$ events represent an independent random sample of $\dot{N}$ locations where each location is equally likely to be chosen as an event.

Simulating CSR

We are going to generate several point patterns (200 events) from CSR within Virginia state boundary.

# open the virginia polygon shapefile
va = ps.io.open(ps.examples.get_path("virginia.shp"))
polys = [shp for shp in va]
# Create the exterior polygons for VA from the union of the county shapes
state = shapely_ext.cascaded_union(polys)
# create window from virginia state boundary
window = Window(state.parts)

1. Generate a point series from N-conditioned CSR

# simulate a csr process in the same window (200 points, 1 realization)
# by specifying "asPP" false, we can generate a point series
# by specifying "conditioning" false, we can simulate a N-conditioned CSR
np.random.seed(5)
samples = PoissonPointProcess(window, 200, 1, conditioning=False, asPP=False)
samples
<pointpats.process.PoissonPointProcess at 0x1b122d7b70>
samples.realizations[0] # simulated event points
array([[-76.3326571 ,  36.57893856],
       [-81.93206633,  37.0243966 ],
       [-79.55664806,  37.35242254],
       [-78.5166233 ,  37.55701527],
       [-77.21660795,  38.26514268],
       [-82.09226973,  37.00701809],
       [-77.44823305,  38.6714618 ],
       [-79.95384378,  37.99268412],
       [-81.36397951,  37.03187854],
       [-78.37289635,  38.78731506],
       [-78.78567678,  37.07057506],
       [-78.61625258,  36.89065808],
       [-82.45957129,  36.95802405],
       [-77.77370645,  37.01124988],
       [-78.80401745,  38.78711285],
       [-78.2800756 ,  37.64869473],
       [-76.33475868,  37.41755285],
       [-79.71621808,  37.22853963],
       [-80.31108929,  36.90895769],
       [-76.81331831,  37.13340462],
       [-77.17489798,  37.62727824],
       [-79.58599474,  37.17903022],
       [-78.34778262,  37.31439593],
       [-76.78862975,  36.56978482],
       [-78.90167513,  38.14339807],
       [-78.31750801,  38.6042175 ],
       [-80.63065732,  37.39418456],
       [-79.0679983 ,  38.20670684],
       [-76.97054563,  37.43228348],
       [-78.28781253,  38.80715429],
       [-79.51445209,  37.05899454],
       [-78.75479687,  36.94362059],
       [-76.56183486,  37.14352317],
       [-82.8250185 ,  36.77946552],
       [-75.57058942,  37.84115351],
       [-83.57095518,  36.62173807],
       [-82.73830125,  36.76063464],
       [-79.64158321,  36.68065846],
       [-76.92510237,  36.92895731],
       [-80.20619679,  37.80498897],
       [-77.74757811,  37.55846008],
       [-80.89034305,  36.65326387],
       [-79.23362271,  38.12835879],
       [-78.10257435,  36.59838991],
       [-76.14501324,  36.74210979],
       [-77.18266793,  36.84631799],
       [-77.27402385,  37.404438  ],
       [-77.68764731,  37.34273888],
       [-77.20460382,  37.91075633],
       [-81.8075696 ,  36.94054096],
       [-76.84214327,  37.82794034],
       [-76.86353526,  36.82540288],
       [-76.8056485 ,  37.18366661],
       [-78.70526218,  38.89314571],
       [-78.11460871,  39.00196074],
       [-78.77656343,  37.61506248],
       [-78.1748728 ,  37.29187339],
       [-78.33817368,  38.36125302],
       [-79.22973299,  37.54091061],
       [-75.5696782 ,  37.90756983],
       [-77.02179404,  36.72052872],
       [-79.48678905,  38.05681891],
       [-78.58205854,  37.41314179],
       [-77.36276352,  37.04549049],
       [-77.30130598,  37.53465902],
       [-79.10398846,  38.25305909],
       [-82.54221637,  36.90375945],
       [-77.94793345,  38.75199833],
       [-78.9452502 ,  37.54473244],
       [-79.048031  ,  37.65131473],
       [-78.25853547,  38.2769875 ],
       [-77.54648155,  36.66647077],
       [-78.48230503,  38.91668951],
       [-78.71263077,  38.2499848 ],
       [-78.57345575,  37.83448379],
       [-78.57683725,  38.88609472],
       [-81.60393528,  37.14655422],
       [-80.41085679,  37.23246613],
       [-79.45003004,  37.75664579],
       [-78.00505977,  39.15971417],
       [-79.5153296 ,  38.36726525],
       [-79.69680058,  37.87120598],
       [-77.87487209,  37.85508792],
       [-76.78504902,  36.99733473],
       [-81.76182614,  36.88421234],
       [-81.96888594,  36.99263064],
       [-77.86127482,  37.16191786],
       [-79.91539534,  36.57794908],
       [-82.27493376,  36.93038256],
       [-75.89989311,  37.49087981],
       [-80.83633012,  37.38476674],
       [-77.93737278,  37.73587757],
       [-78.80405416,  38.2423914 ],
       [-80.09426594,  36.77163754],
       [-78.55997549,  36.9372054 ],
       [-80.74982401,  36.69837703],
       [-79.89144123,  37.27287164],
       [-77.53568375,  38.42234669],
       [-79.36034573,  37.9199658 ],
       [-78.39624506,  39.22046697],
       [-77.32624847,  37.32763411],
       [-77.14780326,  38.1270279 ],
       [-80.24638938,  37.5142178 ],
       [-77.41027396,  36.97299833],
       [-78.73229552,  37.60233533],
       [-79.50446982,  38.04796476],
       [-77.34484259,  37.05615369],
       [-80.66964982,  37.07084403],
       [-77.15297781,  37.29870784],
       [-78.28959166,  39.29418715],
       [-77.10310375,  37.3812618 ],
       [-78.11943302,  37.92836454],
       [-80.31267194,  36.665347  ],
       [-76.6777552 ,  36.75870423],
       [-79.31751436,  38.06910198],
       [-77.02234401,  38.29308642],
       [-77.44257801,  38.34724139],
       [-77.54221373,  37.50425399],
       [-75.67749041,  37.81841772],
       [-80.21661887,  37.67742691],
       [-78.75115924,  38.71767437],
       [-78.95485683,  36.59501015],
       [-76.86872936,  38.02181925],
       [-79.21340288,  37.13898883],
       [-80.00081862,  36.81108808],
       [-77.77941742,  36.66281858],
       [-77.3124049 ,  38.04905423],
       [-77.9213301 ,  36.92944526],
       [-77.66093307,  38.33654176],
       [-77.21170491,  38.93214783],
       [-76.68169985,  36.70007358],
       [-77.30664489,  37.89347582],
       [-82.34535364,  36.75272866],
       [-76.86645645,  37.8687134 ],
       [-77.3709068 ,  38.3866506 ],
       [-78.8063798 ,  37.3391586 ],
       [-80.03257936,  37.4129918 ],
       [-78.68101007,  38.44562521],
       [-77.63204774,  36.87786176],
       [-78.88306754,  38.49431963],
       [-77.45255789,  37.83099746],
       [-79.5298396 ,  37.78361184],
       [-81.55542816,  36.61994736],
       [-78.47299022,  36.8579181 ],
       [-79.02176971,  36.65214546],
       [-78.28147935,  37.80847913],
       [-79.58518375,  38.20539312],
       [-77.77610921,  37.82863786],
       [-80.58914692,  37.04572831],
       [-81.84584342,  36.68964681],
       [-79.5701122 ,  37.36705848],
       [-76.7064535 ,  36.5658754 ],
       [-79.68195266,  37.01713442],
       [-76.22771852,  36.73171684],
       [-77.16980606,  38.0809812 ],
       [-77.10609198,  37.20993371],
       [-77.83263118,  37.30642911],
       [-77.3096478 ,  38.04267336],
       [-80.09196435,  37.69627213],
       [-77.06346097,  37.66044069],
       [-78.39635026,  38.35692905],
       [-80.70881825,  37.33395262],
       [-77.77980079,  36.81863702],
       [-77.48032587,  37.53013036],
       [-80.64284755,  37.29151092],
       [-78.31970329,  39.03988516],
       [-77.99991705,  38.62963975],
       [-81.39136576,  36.6361113 ],
       [-76.2500645 ,  36.58381878],
       [-77.75281574,  38.09955844],
       [-79.18848841,  36.86516089],
       [-78.10679754,  37.23406281],
       [-77.72774175,  37.75365148],
       [-80.79353455,  36.66466322],
       [-79.09248227,  38.11065381],
       [-79.43627162,  36.9317042 ],
       [-80.67513179,  36.98716053],
       [-79.23362918,  37.89815733],
       [-78.88007206,  38.63625233],
       [-77.102715  ,  36.9571268 ],
       [-79.16601272,  37.50364778],
       [-78.17995667,  37.56372944],
       [-78.55397235,  38.94719771],
       [-82.21842212,  37.31977937],
       [-75.70804637,  37.73079071],
       [-76.86774363,  37.59858498],
       [-79.2410832 ,  36.73533614],
       [-75.63397197,  37.85672189],
       [-78.43974651,  36.73714428],
       [-79.63776485,  38.06933981],
       [-78.32258504,  38.01500577],
       [-77.85944265,  36.88932439],
       [-77.86902482,  39.14909625],
       [-81.97464747,  36.8508439 ],
       [-78.99980174,  37.44186754],
       [-77.36680988,  38.99916544],
       [-79.9150312 ,  37.36377025],
       [-80.36600514,  36.67015317],
       [-77.42381708,  37.2241776 ],
       [-77.93652737,  38.17731926]])
# build a point pattern from the simulated point series
pp_csr = PointPattern(samples.realizations[0])
pp_csr
<pointpats.pointpattern.PointPattern at 0x1b1231dcc0>
pp_csr.plot(window=True, hull=True, title='Random Point Pattern')

png

pp_csr.n
200

2. Generate a point series from $\lambda$-conditioned CSR

# simulate a csr process in the same window (200 points, 1 realization)
# by specifying "asPP" false, we can generate a point series
# by specifying "conditioning" True, we can simulate a lamda-conditioned CSR
np.random.seed(5)
samples = PoissonPointProcess(window, 200, 1, conditioning=True, asPP=False)
samples
<pointpats.process.PoissonPointProcess at 0x1b1aaf25c0>
samples.realizations[0] # simulated points
array([[-79.55664806,  38.25720619],
       [-78.5166233 ,  38.30532302],
       [-77.21660795,  37.57582983],
       [-77.44823305,  36.74925133],
       [-79.95384378,  37.55525777],
       [-81.36397951,  36.71747455],
       [-80.18215183,  37.35242254],
       [-78.37289635,  38.26514268],
       [-78.78567678,  38.80097021],
       [-78.61625258,  38.36290302],
       [-81.43369537,  37.00701809],
       [-81.95031135,  37.07057506],
       [-77.77370645,  36.89065808],
       [-78.80401745,  38.72566936],
       [-79.32844778,  36.95802405],
       [-78.2800756 ,  38.90863979],
       [-81.49037579,  36.99262809],
       [-76.90806444,  37.72828737],
       [-77.17489798,  37.64869473],
       [-79.58599474,  37.9904752 ],
       [-78.34778262,  37.22853963],
       [-76.78862975,  37.8189276 ],
       [-78.90167513,  36.90895769],
       [-78.31750801,  37.13340462],
       [-75.86084614,  37.62727824],
       [-75.97937462,  37.17903022],
       [-79.0679983 ,  38.62297192],
       [-76.96164707,  37.31439593],
       [-76.97054563,  37.09732441],
       [-78.28781253,  39.27775751],
       [-79.51445209,  37.2942029 ],
       [-78.75479687,  37.39418456],
       [-75.75445326,  37.78241231],
       [-79.37160467,  37.43228348],
       [-79.64158321,  38.491645  ],
       [-79.33464281,  37.14352317],
       [-79.90916235,  37.50562343],
       [-76.92510237,  36.77946552],
       [-76.91730122,  36.62173807],
       [-82.06902468,  36.76063464],
       [-79.8601599 ,  36.68065846],
       [-77.74757811,  39.05919615],
       [-79.23362271,  36.56074643],
       [-78.10257435,  37.80365574],
       [-77.18266793,  37.63777248],
       [-77.27402385,  37.57885055],
       [-77.68764731,  37.21503901],
       [-77.75929316,  36.61739054],
       [-77.20460382,  38.67318986],
       [-80.52384818,  36.65326387],
       [-80.60914236,  36.74210979],
       [-76.84214327,  38.07338722],
       [-79.88271974,  37.404438  ],
       [-76.81805705,  37.34273888],
       [-82.02798408,  36.94054096],
       [-76.84238532,  37.82794034],
       [-78.70526218,  38.83337292],
       [-76.71235091,  37.95435389],
       [-78.11460871,  36.82540288],
       [-78.77656343,  37.18366661],
       [-78.1748728 ,  38.89314571],
       [-81.63574287,  37.11455891],
       [-79.08592614,  38.45792181],
       [-78.33817368,  37.64663718],
       [-79.22973299,  37.19362583],
       [-79.48678905,  37.29187339],
       [-78.58205854,  37.40208832],
       [-77.36276352,  38.76896337],
       [-77.30130598,  37.47745852],
       [-79.64493717,  36.72052872],
       [-79.10398846,  38.05681891],
       [-81.12955413,  37.04549049],
       [-77.94793345,  37.59843086],
       [-78.60434598,  38.89933921],
       [-79.048031  ,  37.77049293],
       [-78.25853547,  39.16083334],
       [-78.72462602,  37.17396404],
       [-77.29314545,  37.9220836 ],
       [-77.28147432,  38.79060528],
       [-76.66696623,  36.58417386],
       [-80.46513532,  36.64153932],
       [-78.65541452,  38.74225794],
       [-76.805115  ,  36.78532921],
       [-79.93532984,  37.08143783],
       [-78.29407639,  39.34140688],
       [-78.14216323,  38.3665535 ],
       [-78.00505977,  36.7069097 ],
       [-81.18975884,  36.62150498],
       [-80.79290321,  37.05030562],
       [-79.5153296 ,  37.75664579],
       [-77.87487209,  37.40994143],
       [-76.78504902,  38.11807362],
       [-80.09560083,  37.35129105],
       [-75.85714598,  37.57460507],
       [-77.86127482,  38.34868698],
       [-82.27493376,  36.99263064],
       [-80.83633012,  37.16191786],
       [-77.93737278,  36.57794908],
       [-78.55997549,  37.60246875],
       [-80.74982401,  36.93038256],
       [-81.89152421,  37.32986349],
       [-76.60039544,  37.49087981],
       [-79.89144123,  38.026734  ],
       [-77.53568375,  37.64300141],
       [-79.36034573,  36.77163754],
       [-77.32624847,  36.9372054 ],
       [-82.34166665,  36.69837703],
       [-77.14780326,  37.60598513],
       [-80.24638938,  37.27287164],
       [-77.41027396,  37.9199658 ],
       [-78.73229552,  37.32763411],
       [-79.50446982,  37.98941026],
       [-77.34484259,  37.24072148],
       [-80.34358018,  36.97299833],
       [-78.28959166,  38.70289952],
       [-79.23938464,  37.60233533],
       [-77.10310375,  38.04796476],
       [-78.11943302,  37.5521639 ],
       [-82.63246378,  37.05615369],
       [-76.6777552 ,  36.99804193],
       [-79.31751436,  37.29870784],
       [-77.02234401,  37.83495451],
       [-77.54221373,  37.3812618 ],
       [-75.71630753,  37.92836454],
       [-80.21661887,  36.75870423],
       [-78.75115924,  38.06910198],
       [-76.86872936,  37.50425399],
       [-79.21340288,  37.25364651],
       [-77.77941742,  39.02429976],
       [-77.3124049 ,  38.7084587 ],
       [-77.9591418 ,  38.05055288],
       [-77.9213301 ,  38.71767437],
       [-77.66093307,  36.59501015],
       [-77.21170491,  37.13898883],
       [-81.91956901,  36.81108808],
       [-76.68169985,  36.66281858],
       [-77.30664489,  36.92944526],
       [-81.85918569,  36.80543902],
       [-76.86645645,  37.57131542],
       [-80.68521276,  37.18782022],
       [-81.41932788,  36.70007358],
       [-78.8063798 ,  38.30071805],
       [-77.14952496,  37.89347582],
       [-79.07890941,  37.5601488 ],
       [-83.00313472,  36.75272866],
       [-78.68101007,  38.87234553],
       [-77.63204774,  37.61550741],
       [-78.88306754,  38.05667036],
       [-77.45255789,  37.3391586 ],
       [-78.47299022,  37.4129918 ],
       [-79.02176971,  37.37256883],
       [-78.28147935,  37.91147075],
       [-79.58518375,  37.78361184],
       [-77.77610921,  37.97641435],
       [-80.99224637,  36.61994736],
       [-80.58914692,  36.8579181 ],
       [-77.80923356,  37.58006348],
       [-78.86249696,  36.73878708],
       [-82.21675753,  36.74153288],
       [-83.24670184,  36.69004968],
       [-81.28732939,  36.87113389],
       [-75.84594392,  37.54487864],
       [-79.22235094,  36.87297199],
       [-78.48547431,  38.52479965],
       [-78.20476584,  38.54799414],
       [-77.89852106,  38.1201838 ],
       [-81.37405706,  37.04456699],
       [-78.02587914,  37.64685616],
       [-78.42418842,  36.88536942],
       [-76.68842544,  36.74893391],
       [-80.81565379,  36.84864385],
       [-77.83641513,  37.17473795],
       [-77.50393746,  38.0087299 ],
       [-82.66226354,  37.12507361],
       [-81.50584616,  36.91097209],
       [-78.50980015,  38.35200834],
       [-77.99991705,  39.2235099 ],
       [-77.75281574,  38.65985656],
       [-79.18848841,  37.81637356],
       [-75.99527387,  36.67900083],
       [-78.10679754,  38.48234391],
       [-77.72774175,  37.44640927],
       [-79.16298721,  38.41845743],
       [-80.79353455,  36.6361113 ],
       [-79.09248227,  38.09955844],
       [-79.43627162,  37.75365148],
       [-82.44550608,  36.66466322],
       [-78.88007206,  38.4383976 ]])
# build a point pattern from the simulated point series
pp_csr = PointPattern(samples.realizations[0])
pp_csr
<pointpats.pointpattern.PointPattern at 0x1b1ab022e8>
pp_csr.plot(window=True, hull=True, title='Random Point Pattern')

png

pp_csr.n
188

The simulated point pattern has $194$ events rather than the Possion mean $200$.

3. Generate a point pattern from N-conditioned CSR

# simulate a csr process in the same window (200 points, 1 realization)
# by specifying "asPP" True, we can generate a point pattern
# by specifying "conditioning" false, we can simulate a N-conditioned CSR
np.random.seed(5)
samples = PoissonPointProcess(window, 200, 1, conditioning=False, asPP=True)
samples
<pointpats.process.PoissonPointProcess at 0x1b1abceeb8>
pp_csr = samples.realizations[0] # simulated point pattern
pp_csr
<pointpats.pointpattern.PointPattern at 0x1b1abc6550>
pp_csr.plot(window=True, hull=True, title='Random Point Pattern')

png

pp_csr.n
200

4. Generate a point pattern of size 200 from a $\lambda$-conditioned CSR

# simulate a csr process in the same window (200 points, 1 realization)
# by specifying "asPP" True, we can generate a point pattern
# by specifying "conditioning" True, we can simulate a lamda-conditioned CSR
np.random.seed(5)
samples = PoissonPointProcess(window, 200, 1, conditioning=True, asPP=True)
samples
<pointpats.process.PoissonPointProcess at 0x1b1acb4dd8>
pp_csr = samples.realizations[0] # simulated point pattern
pp_csr
<pointpats.pointpattern.PointPattern at 0x1b11e26ac8>
pp_csr.plot(window=True, hull=True, title='Random Point Pattern')

png

pp_csr.n
188

Clustered Patterns

Clustered Patterns are more grouped than random patterns. Visually, we can observe more points at short distances. There are two sources of clustering:

  • Contagion: presence of events at one location affects probability of events at another location (correlated point process)
  • Heterogeneity: intensity $\lambda$ varies with location (heterogeneous Poisson point process)

We are going to focus on simulating correlated point process in this notebook. One example of correlated point process is Poisson cluster process. Two stages are involved in simulating a Poisson cluster process. First, parent events are simulted from a $\lambda$-conditioned or $N$-conditioned CSR. Second, $n$ offspring events for each parent event are simulated within a circle of radius $r$ centered on the parent. Offspring events are independently and identically distributed.

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(5)
csamples = PoissonClusterPointProcess(window, 200, 10, 0.5, 1, asPP=True, conditioning=False)
csamples
<pointpats.process.PoissonClusterPointProcess at 0x1b1ad8af28>
csamples.parameters #number of total events for each realization 
{0: {'n': 200}}
csamples.num_parents #number of parent events for each realization 
{0: 10}
csamples.children # number of children events centered on each parent event
20
pp_pcp = csamples.realizations[0]
pp_pcp
<pointpats.pointpattern.PointPattern at 0x1b1ad5e080>
pp_pcp.plot(window=True, hull=True, title='Clustered Point Pattern') #plot the first realization

png

It is obvious that there are several clusters in the above point pattern.

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

import numpy as np
np.random.seed(10)
csamples = PoissonClusterPointProcess(window, 200, 10, 0.5, 1, asPP=True, conditioning=True)
csamples
<pointpats.process.PoissonClusterPointProcess at 0x1b1ae4c6a0>
csamples.parameters #number of events for the realization might not be equal to 200
{0: {'n': 260}}
csamples.num_parents #number of parent events for the realization, not equal to 10
{0: 13}
csamples.children # number of children events centered on each parent event
20
pp_pcp = csamples.realizations[0]
pp_pcp.plot(window=True, hull=True, title='Clustered Point Pattern')

png

3. Simulate a Poisson cluster process of size 200 with 5 parents and 40 children within 0.5 units of each parent (parent events: $N$-conditioned CSR)

np.random.seed(10)
csamples = PoissonClusterPointProcess(window, 200, 5, 0.5, 1, asPP=True)
pp_pcp = csamples.realizations[0]
pp_pcp.plot(window=True, hull=True, title='Clustered Point Pattern')

png