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- Ripley’s K Function Estimators

# 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).

### Example

To use Ripley’s K Function Estimators from `astropy`’s stats sub-package:

    import numpy as np
    from matplotlib import pyplot as plt
    from astropy.stats import RipleysKEstimator

    rng = np.random.default_rng()
    z = rng.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'$K_{ripley}$')
    plt.legend()

(png, svg, pdf)

## References

[1]

Peebles, P.J.E. The large scale structure of the universe. <https://ui.adsabs.harvard.edu/abs/1980lssu.book…..P>

[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)>