# Spatial Causality Test

#### Wenbo Lv

####
Last update: 2025-11-20
Last run: 2025-11-30

## Methodological Background

Let us begin by defining two spatial cross-sectional series \(\{x_s\}_{s \in S}\) and \(\{y_s\}_{s \in S}\), where \(S\) represents the study area.

We use:

\[ \mathcal{X}_W = \{ W_i x \mid W_i \in \mathcal{W}(x, y) \} \]

\[ \mathcal{Y}_W = \{ W_i y \mid W_i \in \mathcal{W}(x, y) \} \]

to denote the sets of spatial lags of \(x\) and \(y\) given by all the weighting matrices in \(\mathcal{W}(x, y)\)(that is the set of spatial dependence structures between \(x\) and \(y\)).

We say that \(\{x_s\}_{s \in S}\) does not cause \(\{y_s\}_{s \in S}\) under the spatial structures \(\mathcal{X}_W\) and \(\mathcal{Y}_W\) if

\[ h_{y |\mathcal{Y}_W}(m) = h_{y | \mathcal{Y}_W,\mathcal{X}_W}(m) \] A unilateral non-parametric test can be applied to assess the spatial causality via the following null hypothesis:

\[ H_0 : \{x_s\}_{s \in S} \text{ does not cause } \{y_s\}_{s \in S} \text{ under the spatial structures } \mathcal{X}_W \text{ and } \mathcal{Y}_W, \]

with the following statistic:

\[ \hat{\delta}(\mathcal{Y}_W, \mathcal{X}_W) = \hat{h}_{y |\mathcal{Y}_W}(m) - \hat{h}_{y | \mathcal{Y}_W,\mathcal{X}_W}(m) \]

where \(\hat{h}_*(m)\) is the estimated conditional symbolic entropy using Shannon’s entropy with \(m-1\) nearest neighbors. The alternative is that the null hypothesis of is not true.

If \(\mathcal{X}_W\) does not contain extra information about \(y\) then \(\hat{\delta}(\mathcal{Y}_W, \mathcal{X}_W) = 0\), otherwise, $ (_W, _W) > 0$.

\(h_{y |\mathcal{Y}_W}(m)\) measures the uncertainty of the distribution of symbols of \(y\), conditional to the symbols of its spatial lag, $ _W$. Moreover, \(h_{y | \mathcal{Y}_W,\mathcal{X}_W}(m)\) measures the uncertainty of the distribution of symbols of \(y\), conditional to the symbols of the spatial lags of \(y\), \(\mathcal{Y}_W\), and of \(x\), $ _W$. If the second variable, \(x\), indeed causes the first one then there should be a significant decrease in the entropy, and the statistic \(\hat{\delta}(\mathcal{Y}_W, \mathcal{X}_W)\) will take on high positive values. If there is only a spatial correlation, but not causation, the difference between both entropies will be small. The statistical significance of \(\hat{\delta}(\mathcal{Y}_W, \mathcal{X}_W)\) is assessed using spatial block bootstrap.

## Usage examples

### An example of spatial lattice data

Load the `spEDM` package and the columbus OH dataset:

    library(spEDM)

    columbus = sf::read_sf(system.file("case/columbus.gpkg", package="spEDM"))
    columbus
    ## Simple feature collection with 49 features and 6 fields
    ## Geometry type: POLYGON
    ## Dimension:     XY
    ## Bounding box:  xmin: 5.874907 ymin: 10.78863 xmax: 11.28742 ymax: 14.74245
    ## Projected CRS: Undefined Cartesian SRS with unknown unit
    ## # A tibble: 49 × 7
    ##    hoval   inc  crime  open plumb discbd                                     geom
    ##    <dbl> <dbl>  <dbl> <dbl> <dbl>  <dbl>                                <POLYGON>
    ##  1  80.5 19.5  15.7   2.85  0.217   5.03 ((8.624129 14.23698, 8.5597 14.74245, 8…
    ##  2  44.6 21.2  18.8   5.30  0.321   4.27 ((8.25279 14.23694, 8.282758 14.22994, …
    ##  3  26.4 16.0  30.6   4.53  0.374   3.89 ((8.653305 14.00809, 8.81814 14.00205, …
    ##  4  33.2  4.48 32.4   0.394 1.19    3.7  ((8.459499 13.82035, 8.473408 13.83227,…
    ##  5  23.2 11.3  50.7   0.406 0.625   2.83 ((8.685274 13.63952, 8.677577 13.72221,…
    ##  6  28.8 16.0  26.1   0.563 0.254   3.78 ((9.401384 13.5504, 9.434411 13.69427, …
    ##  7  75    8.44  0.178 0     2.40    2.74 ((8.037741 13.60752, 8.062716 13.60452,…
    ##  8  37.1 11.3  38.4   3.48  2.74    2.89 ((8.247527 13.58651, 8.2795 13.5965, 8.…
    ##  9  52.6 17.6  30.5   0.527 0.891   3.17 ((9.333297 13.27242, 9.671007 13.27361,…
    ## 10  96.4 13.6  34.0   1.55  0.558   4.33 ((10.08251 13.03377, 10.0925 13.05275, …
    ## # ℹ 39 more rows

Detect spatial causality among the variables inc, crime, and hoval :

    # house value and crime
    sc.test(columbus, "hoval", "crime", k = 15)
    ## spatial causality test
    ## hoval -> crime: statistic = 1.114, p value = 0.549
    ## crime -> hoval: statistic = 1.555, p value = 0.008

    # household income and crime
    sc.test(columbus, "inc", "crime", k = 15)
    ## spatial causality test
    ## inc -> crime: statistic = 0.739, p value = 0.99
    ## crime -> inc: statistic = 0.946, p value = 0.762

    # household income and house value
    sc.test(columbus, "inc", "hoval", k = 15)
    ## spatial causality test
    ## inc -> hoval: statistic = 1.211, p value = 0.311
    ## hoval -> inc: statistic = 0.992, p value = 0.702

### An example of spatial grid data

Load the `spEDM` package and its farmland NPP data:

    library(spEDM)

    npp = terra::rast(system.file("case/npp.tif", package = "spEDM"))
    # To save the computation time, we will aggregate the data by 3 times
    npp = terra::aggregate(npp, fact = 3, na.rm = TRUE)
    npp
    ## class       : SpatRaster
    ## size        : 135, 161, 5  (nrow, ncol, nlyr)
    ## resolution  : 30000, 30000  (x, y)
    ## extent      : -2625763, 2204237, 1867078, 5917078  (xmin, xmax, ymin, ymax)
    ## coord. ref. : CGCS2000_Albers
    ## source(s)   : memory
    ## names       :      npp,        pre,      tem,      elev,         hfp
    ## min values  :   187.50,   390.3351, -47.8194, -110.1494,  0.04434316
    ## max values  : 15381.89, 23734.5330, 262.8576, 5217.6431, 42.68803711

Detect spatial causality among the variables pre, tem, and npp :

    # precipitation and npp
    sc.test(npp,"pre","npp",k = 30)
    ## spatial causality test
    ## pre -> npp: statistic = 0.227, p value = 0
    ## npp -> pre: statistic = 0.16, p value = 0

    # temperature and npp
    sc.test(npp,"tem","npp",k = 30)
    ## spatial causality test
    ## tem -> npp: statistic = 0.168, p value = 0
    ## npp -> tem: statistic = 0.169, p value = 0

    # precipitation and temperature
    sc.test(npp,"pre","tem",k = 30)
    ## spatial causality test
    ## pre -> tem: statistic = 0.19, p value = 0
    ## tem -> pre: statistic = 0.147, p value = 0