Compute the Global Bivariate Moran's I

Given two continuous numeric variables, calculate the bivariate Moran's I. See details for more.

Usage

moran_bv(x, y, listw, nsim = 499, scale = TRUE)

Arguments

x: a numeric vector of same length as y.
y: a numeric vector of same length as x.
listw: a listw object for example as created by nb2listw().
nsim: the number of simulations to run.
scale: default TRUE.

Value

An object of class "boot", with the observed statistic in component t0.

Details

The Global Bivariate Moran is defined as

\( I_B = \frac{\Sigma_i(\Sigma_j{w_{ij}y_j\times x_i})}{\Sigma_i{x_i^2}} \)

It is important to note that this is a measure of autocorrelation of X with the spatial lag of Y. As such, the resultant measure may overestimate the amount of spatial autocorrelation which may be a product of the inherent correlation of X and Y. The output object is of class "boot", so that plots and confidence intervals are available using appropriate methods.

References

Wartenberg, D. (1985), Multivariate Spatial Correlation: A Method for Exploratory Geographical Analysis. Geographical Analysis, 17: 263-283. doi:10.1111/j.1538-4632.1985.tb00849.x

Author

Josiah Parry josiah.parry@gmail.com

Examples

data(boston, package = "spData")
x <- boston.c$CRIM
y <- boston.c$NOX
listw <- nb2listw(boston.soi)
set.seed(1)
res_xy <- moran_bv(x, y, listw, nsim=499)
res_xy$t0
#> [1] 0.4082073
boot::boot.ci(res_xy, conf=c(0.99, 0.95, 0.9), type="basic")
#> BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
#> Based on 499 bootstrap replicates
#> 
#> CALL : 
#> boot::boot.ci(boot.out = res_xy, conf = c(0.99, 0.95, 0.9), type = "basic")
#> 
#> Intervals : 
#> Level      Basic         
#> 99%   ( 0.7443,  0.8789 )   
#> 95%   ( 0.7639,  0.8673 )   
#> 90%   ( 0.7739,  0.8615 )  
#> Calculations and Intervals on Original Scale
#> Some basic intervals may be unstable
plot(res_xy)

set.seed(1)
lee_xy <- lee.mc(x, y, listw, nsim=499, return_boot=TRUE)
lee_xy$t0
#> [1] 0.3993476
boot::boot.ci(lee_xy, conf=c(0.99, 0.95, 0.9), type="basic")
#> BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
#> Based on 499 bootstrap replicates
#> 
#> CALL : 
#> boot::boot.ci(boot.out = lee_xy, conf = c(0.99, 0.95, 0.9), type = "basic")
#> 
#> Intervals : 
#> Level      Basic         
#> 99%   ( 0.6363,  0.7236 )   
#> 95%   ( 0.6538,  0.7131 )   
#> 90%   ( 0.6584,  0.7098 )  
#> Calculations and Intervals on Original Scale
#> Some basic intervals may be unstable
plot(lee_xy)

set.seed(1)
res_yx <- moran_bv(y, x, listw, nsim=499)
res_yx$t0
#> [1] 0.4029821
boot::boot.ci(res_yx, conf=c(0.99, 0.95, 0.9), type="basic")
#> BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
#> Based on 499 bootstrap replicates
#> 
#> CALL : 
#> boot::boot.ci(boot.out = res_yx, conf = c(0.99, 0.95, 0.9), type = "basic")
#> 
#> Intervals : 
#> Level      Basic         
#> 99%   ( 0.7301,  0.8729 )   
#> 95%   ( 0.7574,  0.8584 )   
#> 90%   ( 0.7664,  0.8541 )  
#> Calculations and Intervals on Original Scale
#> Some basic intervals may be unstable
plot(res_yx)

set.seed(1)
lee_yx <- lee.mc(y, x, listw, nsim=499, return_boot=TRUE)
lee_yx$t0
#> [1] 0.3993476
boot::boot.ci(lee_yx, conf=c(0.99, 0.95, 0.9), type="basic")
#> BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
#> Based on 499 bootstrap replicates
#> 
#> CALL : 
#> boot::boot.ci(boot.out = lee_yx, conf = c(0.99, 0.95, 0.9), type = "basic")
#> 
#> Intervals : 
#> Level      Basic         
#> 99%   ( 0.6363,  0.7236 )   
#> 95%   ( 0.6538,  0.7131 )   
#> 90%   ( 0.6584,  0.7098 )  
#> Calculations and Intervals on Original Scale
#> Some basic intervals may be unstable
plot(lee_yx)