In the previous sections we examined several distance measures to judge whether a point pattern data set is completely random. The general procedure was to compare the observed point pattern with a completely random pattern, the Poisson point process.

Based on the graphical representation it was fairly easy to distinguish between the clustered data set (berlin_locations) and the theoretical Poisson process, as well as between the regular data set (berlin_regular) and the theoretical Poisson process. However, in some cases even the random data point data set (berlin_random) shows large discrepancies from the theoretical Poisson process. Because of random variability the random point data set was not a completely random pattern.

We load the required data sets and packages.

In [2]:
# First, let's import the needed libraries.
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import geopandas as gpd

from pointpats import PointPattern, PoissonPointProcess, as_window
from pointpats.distance_statistics import Genv, Fenv, Kenv, Lenv
In [3]:
# load berlin shapefile
berlin_district = gpd.read_file(
    "http://userpage.fu-berlin.de/soga/soga-py/300/30800_spatial_point_patterns/berlin_district.zip"
)
# load data from the previous section

berlin_random = gpd.read_file(
    "http://userpage.fu-berlin.de/soga/soga-py/300/30800_spatial_point_patterns/berlin_random.geojson"
)

Monte Carlo tests¶

In order to assess if a observed point pattern is significantly different from a Poisson process, we may use the language of hypothesis testing. The null hypothesis $(H_0)$ is that the data point pattern is a realization of complete spatial randomness. The alternative hypothesis $(H_A)$ is that the data pattern is a realization of another, unspecified point process.

In order the test the hypothesis we apply a randomization test based on simulations, denoted as Monte Carlo test.

Consider one of the distance measure functions, e.g. the $K$ function, we discussed in the previous section. We generate $M$ independent simulations of CSR inside the study region $W$. We compute the estimated $K$ functions for each of these realizations, $\hat K^{(j)}(r)$ for $j = 1,2,3,...M$ and we obtain the point-wise upper and lower envelopes of these simulated curves, given by

$$L(r) = \min_j \hat K^{(j)}(r)$$$$L(r) = \max_j \hat K^{(j)}(r)$$

For any fixed value of $r$, if the data came from a uniform Poisson process, the probability of $\hat K(r)$ lying outside the interval $[L(r),U(r)]$ is $2/(M + 1)$. Instead of the point-wise maximum and minimum, one could use the point-wise order statistics (the point-wise kth largest and kth smallest values) giving a test of exact size $2k/(M + 1)$.

Simultaneous Monte Carlo test¶

In pointpats we can perform a Monte Carlo test by applying the Genv(), Kenv(),... functions. In order to compute the point-wise envelopes we provide the function an object containing point pattern data (pp), the number of simulated point patterns to be generated when computing the envelopes (interval), and the argument realizations, which takes point process instance as Input.

First we will have to create that point process instance using PoissonPointProcess(), which returns a Poisson point process including N-conditioned CSR process and λ-conditioned CSR process. We set the argument n (Size of each realization) and samples (Number of realizations).

In [4]:
## reshape coordinates to right input format ([x1,y1], [x2,y2],....[xn, yn])
## extract coordinates from GeoPandas as array
coord_list_ran = np.array(
    [(x, y) for x, y in zip(berlin_random.geometry.x, berlin_random.geometry.y)]
)

## extract coordinates from GeoPandas as PointPattern
pp_ran = PointPattern(coord_list_ran)

C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:1492: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:1208: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)

Let us apply the Kenv() function as our desired summary statistic, on the random point data set (from berlin_random). Consider that we want the envelopes to correspond to the critical values of 5%. We will have to use 0.05 as the pct argument, which is already the default setting.

In [5]:
# creating the point process instance for each set of coordinates
# asPP=True to return a PointPattern instead of an array

crs_ran = PoissonPointProcess(pp_ran.window, n=40, samples=40, asPP=True)
kenv_ran = Kenv(
    pp_ran,
    realizations=crs_ran,
    pct=0.05,
)
kenv_ran.plot()

C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:1923: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)

Note that the simulation envelopes are no confidence intervals, they are the critical values for a test of the hypothesis that the process is CSR (Baddeley 2010). The width of the envelopes reflect the variability of the process under the null hypothesis of CSR.

If so, we compute for each estimate $\hat K(r)$, its maximum deviation from the Poisson $K$ function, $D = \max_r \vert \hat K(r) - K_{pois}(r) \vert$, and we obtain for each of the $M$ simulated data sets the maximum value $D_{max}$. Then the upper and lower limits are

$$L(r) = \pi r^2 - D_{max}$$$$U(r) = \pi r^2 + D_{max}$$

Under $H_0$ this occurs with probability $k/(M + 1)$. Thus, critical bands of $5$% are constructed by for instance taking $M = 19$ and $k=1$.

In [6]:
crs_ran = PoissonPointProcess(pp_ran.window, n=20, samples=19, asPP=True)
kenv_ran = Kenv(pp_ran, realizations=crs_ran, pct=0.05)
kenv_ran.plot()

C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)

There is a set of point patterns functions that work on the same principle of simualtion envelopes. For example Genv() (for the $G$ function), Fenv, Lenv and many more. See here for a more detailed overview.

Let us apply the Genv() function on the random point data set ppp.random to compute the simultaneous critical bands of 10%, using the $G$ function as summary statistics.

In [7]:
crs_ran = PoissonPointProcess(pp_ran.window, n=90, samples=90, asPP=True)
genv_ran = Genv(pp_ran, realizations=crs_ran, pct=0.10)
genv_ran.plot()

C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)

Let us finally apply the Fenv() function on the random point data set from berlin_random to compute the simultaneous critical bands of 1%, using the $F$ function as summary statistics.

In [8]:
crs_ran = PoissonPointProcess(pp_ran.window, n=10, samples=99, asPP=True)
fenv_ran = Fenv(pp_ran, realizations=crs_ran, pct=0.10)
fenv_ran.plot()

C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)
C:\Users\mceck\miniconda3\envs\rasterdata\lib\site-packages\libpysal\cg\shapes.py:103: FutureWarning: Objects based on the `Geometry` class will deprecated and removed in a future version of libpysal.
  warnings.warn(dep_msg, FutureWarning)

Citation

The E-Learning project SOGA-Py was developed at the Department of Earth Sciences by Annette Rudolph, Joachim Krois and Kai Hartmann. You can reach us via mail by soga[at]zedat.fu-berlin.de.

Creative Commons License
You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Rudolph, A., Krois, J., Hartmann, K. (2023): Statistics and Geodata Analysis using Python (SOGA-Py). Department of Earth Sciences, Freie Universitaet Berlin.