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.

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(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 P_i 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 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

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

(6)

where

$$0 < \varepsilon \ll 1$$

(7)

The denominator of Eq. (6) is 1 +  ε if 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

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

(8)

where σ(P_i, C) is a binary variable that indicates whether P_i 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, T_i^A) τ(T_i^A, T₂) is one if P_i appears between T₁ and T₂, 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:

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

(10)

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

The statistic 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

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

(11)

The statistic d^D 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 n^D 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 P_i and the disappearance of P_j 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(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

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

(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)