Author: Wenbo Lv
Date: 2024-12-01

# Information Consistency-Based Measures for Spatial Stratified Heterogeneity

#### Wenbo Lv

#### 2024-12-01

## 1. Introduction to `sshicm` package

### 1.1 The `sshicm` package can be used to address following issues:

- Information consistency-based measures of spatial stratified heterogeneity intensity for continuous and nominal variables.

- Strength of spatial pattern associations based on information consistency measures.

### 1.2 Example data in the `sshicm` package

#### baltim data

“baltim” consists of Baltimore home sale prices and hedonics. In total, there are 221 instances in “baltim” data. The explanatory variables are whether it is a detached unit (DWELL), whether it has a patio (PATIO), whether it has a fireplace (FIREPL), whether it has air conditioning (AC), and whether the dwelling is in Baltimore County (CITCOU, while the target variable is the sale price of the home (PRICE).

#### cinc data

“cinc” is derived from the 2008 Cincinnati Crime + Socio-Demographics dataset. It includes spatial data on 457 objects located on an irregular lattice. The explanatory variables are male population (MALE), female population (FEMALE), median age (MEDIAN_AGE), average family size (AVG_FAMSIZ), and population density (DENSITY), while the target variable is the existence of theft (THEFT_D).

Figure 1. Maps of the baltim and cinc data sets. (Bai et al. 2023)

### 1.3 Functions in the `sshicm` package

#### Two functions for vector-type inputs of dependent and independent variables.

- `sshic()` for continuous dependent variable

- `sshin()` for continuous nominal variable

#### Regression-style data frame modeling function

A function `sshicm()` that yields all results in a single line, with the `type` parameter set to `IC` (Continuous) or `IN` (Nominal) to specify whether the dependent variable is a continuous or nominal variable.

## 2. The principle of measuring spatial stratified heterogeneity based on information consistency

Note: All explanatory variables must be discretized in advance or inherently be discrete nominal variables.

### 2.1 When the dependent variable is a continuous variable:

\[ I_{C}\left(d,s\right) = \sum_{s_{i} \in S}p\left(s_{i}\right)\frac{ \arctan \left(\textbf{RelE} \left( f_{d_{i}} \mid \mid f \right) \right)}{\pi / 2} \]

where \(d_i\) is the random variable corresponding to the target variable in stratum \(s_i\) , and \(f_{d_i}\) and \(f\) are the density functions of \(d_i\) and \(d\), respectively. Additionally, \(\textbf{RelE} \left( f_{d_{i}} \mid \mid f \right)\) is the relative entropy of \(f_{d_i}\) and \(f\).

\[ \textbf{RelE} \left( f_{d_{i}} \mid \mid f \right) = H \left(f_{d_{i}} , f\right) - H \left(f_{d_{i}}\right) = \sum_{i = 1}^{n} f_{d_{i}} \log \frac{1}{f} - \sum_{i = 1}^{n} f_{d_{i}} \log \frac{1}{f_{d_{i}}} = \sum_{i = 1}^{n} f_{d_{i}} \log \frac{f_{d_{i}}}{f} \]

### 2.2 When the dependent variable is a nominal variable:

\[ I_{N}\left(d,s\right) = \frac{I \left(d,s\right)}{I \left(d\right)} = \frac{I \left(d\right) - I \left(d \mid s\right)}{I \left(d\right)} = 1 - \frac{\sum_{s_i \in S}\sum_{x \in V_d} p\left(s_i,x\right) \log p\left(x \mid s_i\right)}{\sum_{x \in V_d} p\left(x\right) \log p\left(x\right)} \]

where \(p\left(x\right)\) is the probability of observing \(x\) in \(U\), \(p\left(s_i,x\right)\) is the probability of observing \(s_i\) and \(x\) in \(U\), and \(p\left(x \mid s_i\right)\) is the probability of observing \(x\) given that the stratum is \(s_i\).

## 3. Examples of the `sshicm` package

    install.packages("sshicm", dep = TRUE)

    library(sshicm)

    baltim = sf::read_sf(system.file("extdata/baltim.gpkg",package = "sshicm"))
    sshicm(PRICE ~ .,baltim,type = "IC")
    ## # A tibble: 5 × 3
    ##   Variable     Ic      Pv
    ##   <chr>     <dbl>   <dbl>
    ## 1 DWELL    0.648  0.00801
    ## 2 AC       0.223  0.0591
    ## 3 PATIO    0.168  0.556
    ## 4 FIREPL   0.135  0.667
    ## 5 CITCOU   0.0898 0.988

    cinc = sf::read_sf(system.file("extdata/cinc.gpkg",package = "sshicm"))
    sshicm(THEFT_D ~ .,cinc,type = "IN")
    ## # A tibble: 5 × 3
    ##   Variable        In      Pv
    ##   <chr>        <dbl>   <dbl>
    ## 1 DENSITY    0.776   0.0681
    ## 2 MEDIAN_AGE 0.228   0.0230
    ## 3 MALE       0.0367  0
    ## 4 AVG_FAMSIZ 0.0205  0.00300
    ## 5 FEMALE     0.00584 0.0200

    ntds = gdverse::NTDs
    sshicm(incidence ~ watershed + elevation + soiltype,data = ntds)
    ## # A tibble: 3 × 3
    ##   Variable     Ic     Pv
    ##   <chr>     <dbl>  <dbl>
    ## 1 elevation 0.293 0.0250
    ## 2 watershed 0.177 0.0521
    ## 3 soiltype  0.117 0.0671

## Reference

Wang, J., Haining, R., Zhang, T., Xu, C., Hu, M., Yin, Q., … Chen, H. (2024). Statistical Modeling of Spatially Stratified Heterogeneous Data. Annals of the American Association of Geographers, 114(3), 499–519. https://doi.org/10.1080/24694452.2023.2289982.

Bai, H., Wang, H., Li, D., & Ge, Y. (2023). Information Consistency-Based Measures for Spatial Stratified Heterogeneity. Annals of the American Association of Geographers, 113(10), 2512–2524. https://doi.org/10.1080/24694452.2023.2223700.

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