# Local indicator of Colocation Quotient

Source:`R/local-colocation-quotient-impl.R`

`local_colocation.Rd`

The local indicator of the colocation quotient (LCLQ) is a Local Indicator of Spatial Association (LISA) that evaluates if a given observation's subcategory in A is colocated with subcategories in B. Like the CLQ, the LCLQ provides insight into the asymmetric relationships between subcategories of A and B (where B can also equal A) but at the local level.

The LCLQ is defined using Gaussian kernel weights and an adaptive bandwidth (see `st_kernel_weights()`

). However, any type of weights list can be used. Kernel weights are used to introduce a decay into the calculation of the CLQ. This ensures that points nearer to the focal point have more influence than those that are more distant.

## Arguments

- A
a character or factor vector.

- B
a character or factor vector.

- nb
a neighbors list e.g. created by

`st_knn()`

or`st_contiguity()`

- wt
a weights list. Recommended that it is a Gaussian kernel weights list using an adaptive bandwidth e.g. created by

`st_kernel_weights(nb, geometry, "gaussian", addaptive = TRUE)`

that does not include the self.- nsim
default

`99`

. An integer representing how many simulations to run for calculating the simulated p-values.

## Value

a data frame with as many rows as observations in A and two times as many columns as unique values in B. Columns contain each unique value of B as well as the simulated p-value for each value of B.

## Details

The LCLQ is defined as \(LCLQ_{A_i \to B} = \frac{N_{A_i \to B}}{N_B / (N - 1)}\) where \(N_{A_i \to B} = \sum_{j = 1(j \ne i)}^{N}(\frac{w_{ij}f_{ij}}{\sum_{j = 1(j \ne i)}^{N}w_{ij}})\). And the weights matrix, wij, uses adaptive bandwidth Gaussian kernel weights.

LCLQ is only calculated for those subcategories which are present in the neighbor list. If a subcategory is not present, then the resultant LCLQ and simulated p-value will be `NA`

.

## References

Fahui Wang, Yujie Hu, Shuai Wang & Xiaojuan Li (2017) Local Indicator of Colocation Quotient with a Statistical Significance Test: Examining Spatial Association of Crime and Facilities, The Professional Geographer, 69:1, 22-31, doi: 10.1080/00330124.2016.1157498

## Examples

```
A <- guerry$main_city
B <- guerry$region
geo <- sf::st_centroid(sf::st_geometry(guerry))
nb <- include_self(st_knn(geo, 5))
wt <- st_kernel_weights(nb, geo, "gaussian", adaptive = TRUE)
res <- local_colocation(A, B, nb, wt, 9)
tail(res)
#> C E N S W p_sim_C p_sim_E p_sim_N
#> 80 NA 1.770957 NA 3.170219 NA NA 0.2 NA
#> 81 NA NA NA NA 4.941176 NA NA NA
#> 82 2.564959 NA NA NA 2.376218 0.1 NA NA
#> 83 2.478538 NA NA NA 2.462639 0.2 NA NA
#> 84 NA 4.186106 0.7550705 NA NA NA 0.1 0.3
#> 85 2.321401 1.816223 0.8035523 NA NA 0.1 0.2 0.2
#> p_sim_S p_sim_W
#> 80 0.1 NA
#> 81 NA 0.1
#> 82 NA 0.1
#> 83 NA 0.1
#> 84 NA NA
#> 85 NA NA
```