Ripley’s K Function Estimators¶
Spatial correlation functions have been used in the astronomical context to estimate the probability of finding an object, e.g. a galaxy, within a given distance of another object 1.
Ripley’s K function is a type of estimator used to characterize the correlation
of such spatial point processes
2, 3, 4, 5, 6.
More precisely, it describes correlation among objects in a given field.
The RipleysKEstimator
class implements some
estimators for this function which provides several methods for
edge-effects correction.
Basic Usage¶
The actual implementation of Ripley’s K function estimators lie in the method
evaluate
which take the following arguments data
, radii
, and,
optionally, mode
.
The data
argument is a 2D array which represents the set of observed
points (events) in the area of study. The radii
argument corresponds to a
set of distances for which the estimator will be evaluated. The mode
argument takes a value on the following linguistic set
{none, translation, ohser, var-width, ripley}
; each keyword represents a
different method to perform correction due to edge-effects. See the API
documentation and references for details about these methods.
Instances of RipleysKEstimator
can also be used as
callables (which is equivalent to calling the evaluate
method).
A minimal usage example is shown as follows:
import numpy as np
from matplotlib import pyplot as plt
from astropy.stats import RipleysKEstimator
z = np.random.uniform(low=5, high=10, size=(100, 2))
Kest = RipleysKEstimator(area=25, x_max=10, y_max=10, x_min=5, y_min=5)
r = np.linspace(0, 2.5, 100)
plt.plot(r, Kest.poisson(r), color='green', ls=':', label=r'$K_{pois}$')
plt.plot(r, Kest(data=z, radii=r, mode='none'), color='red', ls='--',
label=r'$K_{un}$')
plt.plot(r, Kest(data=z, radii=r, mode='translation'), color='black',
label=r'$K_{trans}$')
plt.plot(r, Kest(data=z, radii=r, mode='ohser'), color='blue', ls='-.',
label=r'$K_{ohser}$')
plt.plot(r, Kest(data=z, radii=r, mode='var-width'), color='green',
label=r'$K_{var-width}$')
plt.plot(r, Kest(data=z, radii=r, mode='ripley'), color='yellow',
label=r'$K_{ripley}$')
References¶
- 1
Peebles, P.J.E. The large scale structure of the universe. <http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1980lssu.book…..P&db_key=AST>
- 2
Ripley, B.D. The second-order analysis of stationary point processes. Journal of Applied Probability. 13: 255–266, 1976.
- 3
Spatial descriptive statistics. <https://en.wikipedia.org/wiki/Spatial_descriptive_statistics>
- 4
Cressie, N.A.C. Statistics for Spatial Data, Wiley, New York.
- 5
Stoyan, D., Stoyan, H. Fractals, Random Shapes and Point Fields, Akademie Verlag GmbH, Chichester, 1992.
- 6
Correlation function. <https://en.wikipedia.org/wiki/Correlation_function_(astronomy)>