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A method for evaluating the degree of spatial and temporal avoidance in spatial point patterns

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 dA, i.e., the average distance to the nearest point when points appear. Let τ(t1, t2) be a binary variable that evaluates the order of two time points t1 and t2:

$$\tau \left( {t_{1} ,t_{2} } \right) = \left\{ {\begin{array}{*{20}c} 1 & {{\text{if}}\;t_{1} \le t_{2} } \\ 0 & {{\text{otherwise}}} \\ \end{array} } \right.$$
(4)

Using the function, we can evaluate whether Pi exists at time t:

$$p_{i} \left( t \right) = \tau \left( {T_{i}^{A} ,t} \right)\tau \left( {t,T_{i}^{D} } \right)$$
(5)

It is one if Pi exists at time t while it is zero if Pi does not exist.

The distance to the nearest point when Pi appears is mathematically given by

$$d_{i}^{A} = \left( {1 + \varepsilon } \right)\mathop {\min }\limits_{j \ne i} \frac{{\left\| {{\mathbf{z}}_{j} - {\mathbf{z}}_{i} } \right\|}}{{p_{j} \left( {T_{i}^{A} } \right) + \varepsilon }}$$
(6)

where

$$0 < \varepsilon \ll 1$$
(7)

The denominator of Eq. (6) is 1 +  ε if Pj exists when Pi appears at time TiA, while it is ε if Pj does not exist. We added ε to avoid dividing ||zj-zi|| by pj(TiA) = 0 when if Pj does not exist. Since ε is very small, ||zj-zi||/ε is very large for the points that do not exist when Pi appears. The minimum value of ||zj-zi||/pj(TiA) + ε is chosen only from the points that exist when Pi 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

$$n^{A} = \sum\limits_{i} {\sigma \left( {P_{i} ,C} \right)\tau \left( {T_{1} ,T_{i}^{A} } \right)\tau \left( {T_{i}^{A} ,T_{2} } \right)}$$
(8)

where σ(Pi, C) is a binary variable that indicates whether Pi is contained in C:

$$\sigma \left( {P_{i} ,C} \right) = \left\{ {\begin{array}{*{20}c} 1 & {{\text{if}}\;P_{i} \subset C} \\ 0 & {{\text{otherwise}}} \\ \end{array} } \right.$$
(9)

The term τ(T1, TiA) τ(TiA, T2) is one if Pi appears between T1 and T2, and thus nA indicates the number of appearances in C⊗Ω. Using diA, we can calculate the average distance to the nearest point when points appear:

$$d^{A} = \frac{{\sum\limits_{i} {\sigma \left( {P_{i} ,C} \right)\tau \left( {T_{1} ,T_{i}^{A} } \right)\tau \left( {T_{i}^{A} ,T_{2} } \right)d_{i}^{A} } }}{{n^{A} }}$$
(10)

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

The statistic dD, i.e., the average distance to the nearest point when points disappear, is calculated as follows. The distance to the nearest point when Pi disappears is given by

$$d_{i}^{D} = \left( {1 + \varepsilon } \right)\mathop {\min }\limits_{j \ne i} \frac{{\left\| {{\mathbf{z}}_{j} - {\mathbf{z}}_{i} } \right\|}}{{p_{j} \left( {T_{i}^{D} } \right) + \varepsilon }}$$
(11)

The statistic dD is

$$d^{D} = \frac{{\sum\limits_{i} {\sigma \left( {P_{i} ,C} \right)\tau \left( {T_{1} ,T_{i}^{D} } \right)\tau \left( {T_{i}^{D} ,T_{2} } \right)d_{i}^{D} } }}{{n^{D} }}$$
(12)

where nD is given by

$$n^{D} = \sum\limits_{i} {\sigma \left( {P_{i} ,C} \right)\tau \left( {T_{1} ,T_{i}^{D} } \right)\tau \left( {T_{i}^{D} ,T_{2} } \right)}$$
(13)

The calculation of ν is as follows. The expected number of points at time t in C under the random process R2 is given by

$${\text{E}}\left[ {n\left( t \right)} \right] = n\left( {T_{1} } \right) + \frac{{t - T_{1} }}{{T_{2} - T_{1} }}\left( {n^{A} - n^{D} } \right)$$
(14)

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

$$n\left( t \right) = \sum\limits_{i} {p_{i} \left( t \right)}$$
(15)

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

$$\nu = \frac{{\int_{{t = T_{1} }}^{{T_{2} }} {\left( {n\left( t \right) - {\text{E}}\left[ {n\left( t \right)} \right]} \right)^{2} {\text{d}}t} }}{{T_{2} - T_{1} }}$$
(16)

We finally consider the calculation of η. The number of appearances between the appearance of Pi and the disappearance of Pj is given by

$$\kappa_{ij}^{A} = \sum\limits_{k} {\sigma \left( {P_{k} ,C} \right)\left\{ {\tau \left( {T_{i}^{A} ,T_{k}^{A} } \right)\tau \left( {T_{k}^{A} ,T_{j}^{D} } \right) + \tau \left( {T_{j}^{D} ,T_{k}^{A} } \right)\tau \left( {T_{k}^{A} ,T_{i}^{A} } \right)} \right\}}$$
(17)

The term τ(TiA, TkA) τ(TkA, TjD) +  τ(TjD, TkA) τ(TkA, TiA) is one if Pi appears between TiA and TjD. Consequently, KijA indicates the number of appearances between the appearance of Pi and the disappearance of Pj. The number of disappearances is

$$\kappa_{ij}^{D} = \sum\limits_{k} {\sigma \left( {P_{k} ,C} \right)\left\{ {\tau \left( {T_{i}^{A} ,T_{k}^{D} } \right)\tau \left( {T_{k}^{D} ,T_{j}^{D} } \right) + \tau \left( {T_{j}^{D} ,T_{k}^{D} } \right)\tau \left( {T_{k}^{D} ,T_{i}^{A} } \right)} \right\}}$$
(18)

The number of the sequences of appearances and disappearances is

$$\eta = 1 + \sum\limits_{i < j} {\left\{ {\rho \left( {\kappa_{ij}^{A} + \kappa_{ij}^{D} } \right) + \rho \left( {\kappa_{ji}^{A} + \kappa_{ji}^{D} } \right)} \right\}}$$
(19)

where

$$\rho \left( x \right) = \left\{ {\begin{array}{*{20}c} 1 & {{\text{if}}\;x = 0} \\ 0 & {{\text{otherwise}}} \\ \end{array} } \right.$$
(20)

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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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Keywords

  • Spatial point pattern
  • Spatial avoidance
  • Temporal avoidance
  • Statistical analysis

JEL Classification

  • C600
  • C650
  • C690