Dataset for "Covering one point process with another"
We simulate a situation where we place a large number of transmitters and receivers in a given area. The question we are interested in is: what transmission radius is required so that each receiver can receive signals from at least k transmitters? This dataset contains the minimum radius needed for randomly-chosen locations of the transmitters and receivers.
Fix a set A as a subset of Euclidean space (for example, a disc or a polygon), and a subset B contained in A (we often consider the case B=A).
Place n points X_1, ..., X_n in A, with the locations chosen independently and uniformly at random. We think of these as "transmitters".
Place another m points Y_1, ..., Y_m in B contained in A, which we think of as "receivers".
At each X_i, place a Euclidean ball of radius r.
We define R_{n,m,k} to be the smallest r such that every receiver Y_j has at least k transmitters within distance r.
In the paper "Covering one point process with another" we proved that if m/n tends to tau as n tends to infinity, then the quantity n R_{n,m,k}^d - c_1 log(n) - c_2 loglog(n) (for constants c_1,c_2 which we give in the paper) converges to a random variable (whose distribution we also give). These datasets include large numbers of independent samples of n R_{n,m,k}^d - c_1 log(n) - c_2 loglog(n).
The dataset is separated into files, and each file into rows. All the data in a given file are generated using fixed sets A and B, and parameters n, m, d, k. Each row in this given file is a single number: the outcome of an experiment, conducted independently of the other rows. In each experiment we place n points at random locations in A, place m points at random locations in B, calculate R_{n,m,k} as described above (and as detailed formally in the paper) and record the value of n R_{n,m,k}^d - c_1 log(n) - c_2 loglog(n) on a row. For the next row, we remove the existing points, and place n points in A, m points in B, etc. for the same n,m, A, B, but with the random points chosen independently of previous experiments.
In probabilisitic terms, the rows of a given file are independent and identically distributed random variables with a common distribution, which is the distribution of n R_{n,m,k}^d - c_1 log(n) - c_2 loglog(n).
The distribution depends on A, B, n, m, d and k. Different files were generated using different choices of A, B, n, m, d and k.
The paper was written by Frankie Higgs, Mathew D. Penrose and Xiaochuan Yang.
We thank Keith Briggs for suggesting the problem and advice on the simulations.
Cite this dataset as:
Higgs, F.,
Penrose, M.,
Yang, X.,
2025.
Dataset for "Covering one point process with another".
Bath: University of Bath Research Data Archive.
Available from: https://doi.org/10.15125/BATH-01359.
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Data
n10000-tau1.csv
text/plain (1MB)
Creative Commons: Attribution 4.0
Simulation for the paper "Covering one point process with another" in the unit square. Every row is the outcome of an independent experiment, and each experiment had the same parameters (using the notation of the paper): d=2, k=1, B=A=[0,1]^2, n = 10^4, m = 10^4
n1000000-dim2-tau1-k1-s0.9.csv
text/plain (811kB)
Creative Commons: Attribution 4.0
Simulation for the paper "Covering one point process with another". Every row is the outcome of an independent experiment, and each experiment had the same parameters (using the notation of the paper): d=2, k=1, B = B(o,0.9), A = B(o,1), n = 10^6, m = 10^6.
d2-k3-tau1.0-n10000.csv
text/plain (538kB)
Creative Commons: Attribution 4.0
Simulation for the paper "Covering one point process with another" in the unit square. Every row is the outcome of an independent experiment, and each experiment had the same parameters (using the notation of the paper): d = 2, k = 3, n = 10^4, m = 10^4. A is a torus, and B = A.
2d-ball.zip
application/zip (1MB)
Creative Commons: Attribution 4.0
Simulation for the paper "Covering one point process with another" in the unit square. There are two files. In each file, every row is the outcome of an independent experiment, and each experiment in a given file had the same parameters as the other experiments recorded in the same file. One file had d=2, k=1, A=B(o,1), n=10^4, m = 10^4. The other had d=2, k=2, A=B(o,1), n=10^4, m = 10^4.
3d-ball.zip
application/zip (1MB)
Creative Commons: Attribution 4.0
Simulation for the paper "Covering one point process with another" in the unit square. There are three files. In each file, every row is the outcome of an independent experiment, and each experiment in a given file used the same parameters as the other experiments in that file. One file (with "tau1") had d=3,k=1, A=B(o,1), n=10^4, m = 10^4. The next (with "tau10") had d=3,k=1, A=B(o,1), n=10^4, m = 10^5. The final (with "tau100") had d=3,k=1, A=B(o,1), n=10^4, m = 10^6.
Code
CovXY-v1.0.0.zip
application/zip (14kB)
Software: MIT License
The code used to generate all the data samples in this database, and plot the diagrams based on these data in the paper. This is commit 4c4c086 (and release v1.0.0) of the code.
The code used to generate the data and corresponding diagrams (which used commit 4c4c086)
Creators
Frankie Higgs
University of Bath
Mathew Penrose
University of Bath
Xiaochuan Yang
Brunel University London
Contributors
University of Bath
Rights Holder
Documentation
Data collection method:
The dataset were generated using computer random number generation to sample two sets of random points (X_1, ..., X_n) and (Y_1, ..., Y_m), and then the two-sample coverage threshold R_{n,m,k} was calculated for these random points by measuring the Euclidean distances between pairs |X_i - Y_j|. This was implemented by the code available at https://github.com/frankiehiggs/CovXY (commit 4c4c086).
Technical details and requirements:
The data was generated on a laptop computer using Python 3.
Additional information:
Each csv file contains a number of rows. Every row in a given file was generated using the same sets A and B, and the same parameters n, m, d and k. The entry on one row is independent of the entries on every other row, as we performed the same experiment repeatedly with the same parameters, but sampling the random points independently each time.
Funders
Engineering and Physical Sciences Research Council (EPSRC)
https://doi.org/10.13039/501100000266
Coverage and connectivity in stochastic geometry
EP/T028653/1
Publication details
Publication date: 29 April 2025
by: University of Bath
Version: 1
DOI: https://doi.org/10.15125/BATH-01359
URL for this record: https://researchdata.bath.ac.uk/1359
Related datasets and code
Higgs, F., Penrose, M. D., and Yang, X., 2025. Covering One Point Process with Another. Methodology and Computing in Applied Probability, 27(2). Available from: https://doi.org/10.1007/s11009-025-10165-7.
Contact information
Please contact the Research Data Service in the first instance for all matters concerning this item.
Contact person: Mathew Penrose
Faculty of Science
Mathematical Sciences
Research Centres & Institutes
Probability Laboratory at Bath
