## Abstract

This paper develops a new method for evaluating the degree of spatial and temporal avoidance in spatial point patterns. We consider point patterns that change over time, where points represent spatial objects that appear at certain locations, stay there for certain periods, and may finally disappear, such as buildings in cities, plants in fields, and birds' nests in forests. Spatial avoidance in this paper refers to the phenomenon that points appear in sparse spaces while points disappear in dense spaces. Spatial avoidance often leads to dispersed point patterns, which are observed in the distributions of drug stores, gas stations, and animal burrows. Temporal avoidance refers to the phenomenon that close points avoid the overlap of their lifetime. Temporal avoidance is found in the relationships between preys and predators, animal species that share the same water resources, and restaurants in shopping malls. The paper develops four measures to evaluate the spatial and temporal patterns of avoidance. Two measures consider the avoidance from a spatial perspective, while the other two focus on the temporal aspect of avoidance. To test the validity of the proposed method, this paper applies it to the analysis of the convenience stores in Shibuya-ku, Tokyo. The results indicated the proposed method's effectiveness and revealed the spatial and temporal patterns of avoidance of convenience stores that existing methods cannot detect.

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## Acknowledgements

This research was supported by JSPS KAKENHI.

## Funding

Japan Society for the Promotion of Science, 16H01830, Yukio Sadahiro, 18K18535, Yukio Sadahiro, 19H02375, Yukio Sadahiro

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## Appendix

### Appendix

This appendix describes the mathematical definition of the four statistics used in the paper. We hope this will help the computational implementation of the proposed method.

We first consider *d*^{A}, i.e., the average distance to the nearest point when points appear. Let * τ*(*t*_{1}, *t*_{2}) be a binary variable that evaluates the order of two time points *t*_{1} and *t*_{2}:

Using the function, we can evaluate whether *P*_{i} exists at time *t*:

It is one if *P*_{i} exists at time *t* while it is zero if *P*_{i} does not exist.

The distance to the nearest point when *P*_{i} appears is mathematically given by

where

The denominator of Eq. (6) is 1 + *ε* if *P*_{j} exists when *P*_{i} appears at time *T*_{i}^{A}, while it is* ε* if *P*_{j} does not exist. We added *ε* to avoid dividing ||*z*_{j}-*z*_{i}|| by *p*_{j}(*T*_{i}^{A}) = 0 when if *P*_{j} does not exist. Since *ε* is very small, ||*z*_{j}-*z*_{i}||/*ε* is very large for the points that do not exist when *P*_{i} appears. The minimum value of ||*z*_{j}-*z*_{i}||/*p*_{j}(*T*_{i}^{A}) + *ε* is chosen only from the points that exist when *P*_{i} appears. The minimum value divided by 1 + *ε* is finally multiplied by 1 + *ε*, and consequently, Eq. (6) gives the distance to the nearest point.

The number of appearances in *C*⊗Ω is mathematically given by

where *σ*(*P*_{i}, *C*) is a binary variable that indicates whether *P*_{i} is contained in *C*:

The term * τ*(*T*_{1}, *T*_{i}^{A})* τ*(*T*_{i}^{A}, *T*_{2}) is one if *P*_{i} appears between *T*_{1} and *T*_{2}, and thus *n*^{A} indicates the number of appearances in *C*⊗Ω. Using *d*_{i}^{A}, we can calculate the average distance to the nearest point when points appear:

where the numerator is the summation of the distance to the nearest point when points appear.

The statistic *d*^{D}, i.e., the average distance to the nearest point when points disappear, is calculated as follows. The distance to the nearest point when *P*_{i} disappears is given by

The statistic *d*^{D} is

where *n*^{D} is given by

The calculation of ν is as follows. The expected number of points at time *t* in *C* under the random process *R*_{2} is given by

where *n*(*t*) is the number of points at time *t*:

We evaluate the fluctuation in the number of points by the average variance over the time period Ω:

We finally consider the calculation of* η*. The number of appearances between the appearance of *P*_{i} and the disappearance of *P*_{j} is given by

The term * τ*(*T*_{i}^{A}, *T*_{k}^{A})* τ*(*T*_{k}^{A}, *T*_{j}^{D}) + * τ*(*T*_{j}^{D}, *T*_{k}^{A})* τ*(*T*_{k}^{A}, *T*_{i}^{A}) is one if *P*_{i} appears between *T*_{i}^{A} and *T*_{j}^{D}. Consequently, *K*_{ij}^{A} indicates the number of appearances between the appearance of *P*_{i} and the disappearance of *P*_{j}. The number of disappearances is

The number of the sequences of appearances and disappearances is

where

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Sadahiro, Y. A method for evaluating the degree of spatial and temporal avoidance in spatial point patterns.
*J Geogr Syst* **24**, 241–260 (2022). https://doi.org/10.1007/s10109-022-00373-x

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DOI: https://doi.org/10.1007/s10109-022-00373-x

### Keywords

- Spatial point pattern
- Spatial avoidance
- Temporal avoidance
- Statistical analysis

### JEL Classification

- C600
- C650
- C690