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Geographical Pattern Causality

Wenbo Lyu

Last update: 2026-04-03
Last run: 2026-04-05

Methodological Background

Geographical Pattern Causality (GPC) infers causal relations from spatial cross-sectional data by reconstructing a symbolic approximation of the underlying spatial dynamical system.

Let \(x(s)\) and \(y(s)\) denote two spatial cross-sections over spatial units \(s \in \mathcal{S}\).

(1) Spatial Embedding

For each spatial unit \(s_i\), GPC constructs an embedding vector

\[ \mathbf{E}_{x(s_i)} = \big( x(s_{i}^{(1)}), x(s_{i}^{(2)}), \dots, x(s_{i}^{(E\tau)}) \big), \]

where \(s_{i}^{(k)}\) denotes the \(k\)-th spatially lagged value of the spatial unit \(s_i\), determined by embedding dimension \(E\) and lag \(\tau\). This yields two reconstructed state spaces \(\mathcal{M}_x, \mathcal{M}_y \subset \mathbb{R}^E\).

(2) Symbolic Pattern Extraction

Local geometric transitions in each manifold are mapped to symbols

\[ \sigma_x(s_i),; \sigma_y(s_i) \in \mathcal{A}, \]

encoding increasing, decreasing, or non-changing modes. These symbolic trajectories summarize local pattern evolution.

(3) Cross-Pattern Mapping

Causality from \(x \to y\) is assessed by predicting:

\[ \hat{\sigma}_y(s_i) = F\big( \sigma_x(s_j): s_j \in \mathcal{N}_k(s_i) \big), \]

where \(\mathcal{N}_k\) denotes the set of \(k\) nearest neighbors in \(\mathcal{M}_x\). The agreement structure between \(\hat{\sigma}_y(s_i)\) and \(\sigma_y(s_i)\) determines the causal mode:

(4) Causal Strength

The global causal strength is the normalized consistency of symbol matches:

\[ C_{x \to y} = \frac{1}{|\mathcal{S}|} \sum_{s_i \in \mathcal{S}} \mathbb{I}\big[ \hat{\sigma}_y(s_i) \bowtie \sigma_y(s_i) \big], \]

where \(\bowtie\) encodes positive, negative, or dark matching rules.

Usage examples

Example of spatial vector data

Load the spEDM package and its columbus spatial analysis data:

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, …
##  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

The false nearest neighbours (FNN) method helps identify the appropriate minimal embedding dimension for reconstructing the state space of a time series or spatial cross-sectional data.

spEDM::fnn(columbus, "crime", E = 1:10, eps = stats::sd(columbus$crime))
## [spEDM] Output 'E:i' corresponds to the i-th valid embedding dimension.
## [spEDM] Input E values exceeding max embeddable dimension were truncated.
## [spEDM] Please map output indices to original E inputs before interpretation.
##        E:1        E:2        E:3        E:4        E:5        E:6        E:7 
## 0.59183673 0.04081633 0.04081633 0.10204082 0.00000000 0.00000000 0.00000000 
##        E:8 
## 0.00000000

The false nearest neighbours (FNN) ratio decreased to 0 when the embedding dimension E reached 5, and remained relatively stable thereafter. Therefore, we adopted \(E = 5\) as the minimal embedding dimension for subsequent parameter search.

Then, search optimal parameters:

# determine the type of causality using correlation
stats::cor.test(columbus$hoval,columbus$crime)
## 
##  Pearson's product-moment correlation
## 
## data:  columbus$hoval and columbus$crime
## t = -4.8117, df = 47, p-value = 1.585e-05
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.7366777 -0.3497978
## sample estimates:
##        cor 
## -0.5744867

# since the correlation is -0.574, negative causality is selected as the metric to maximize in the optimal parameter search
spEDM::pc(columbus, "crime", "hoval", E = 5:9, k = 7:10, tau = 1, maximize = "negative")
## The suggested E,k,tau for variable hoval is 7, 9 and 1

Run geographical pattern causality analysis

spEDM::gpc(columbus, "crime", "hoval", E = 7, k = 9)
##       type   strength      direction
## 1 positive 0.00000000 crime -> hoval
## 2 negative 0.34035656 crime -> hoval
## 3     dark 0.02786836 crime -> hoval
## 4 positive 0.00000000 hoval -> crime
## 5 negative 0.00000000 hoval -> crime
## 6     dark 0.05337361 hoval -> crime

Convergence diagnostics

crime_convergence = spEDM::gpc(columbus, "crime", "hoval", 
                               libsizes = seq(9, 50, by = 1),
                               E = 7, k = 9, progressbar = FALSE)
crime_convergence
##     libsizes     type    strength      direction
## 1         10 positive 0.000000000 crime -> hoval
## 2         11 positive 0.000000000 crime -> hoval
## 3         12 positive 0.000000000 crime -> hoval
## 4         13 positive 0.000000000 crime -> hoval
## 5         14 positive 0.000000000 crime -> hoval
## 6         15 positive 0.000000000 crime -> hoval
## 7         16 positive 0.000000000 crime -> hoval
## 8         17 positive 0.000000000 crime -> hoval
## 9         18 positive 0.000000000 crime -> hoval
## 10        19 positive 0.000000000 crime -> hoval
## 11        20 positive 0.000000000 crime -> hoval
## 12        21 positive 0.000000000 crime -> hoval
## 13        22 positive 0.000000000 crime -> hoval
## 14        23 positive 0.000000000 crime -> hoval
## 15        24 positive 0.000000000 crime -> hoval
## 16        25 positive 0.000000000 crime -> hoval
## 17        26 positive 0.000000000 crime -> hoval
## 18        27 positive 0.000000000 crime -> hoval
## 19        28 positive 0.000000000 crime -> hoval
## 20        29 positive 0.000000000 crime -> hoval
## 21        30 positive 0.000000000 crime -> hoval
## 22        31 positive 0.000000000 crime -> hoval
## 23        32 positive 0.000000000 crime -> hoval
## 24        33 positive 0.000000000 crime -> hoval
## 25        34 positive 0.000000000 crime -> hoval
## 26        35 positive 0.000000000 crime -> hoval
## 27        36 positive 0.000000000 crime -> hoval
## 28        37 positive 0.000000000 crime -> hoval
## 29        38 positive 0.000000000 crime -> hoval
## 30        39 positive 0.000000000 crime -> hoval
## 31        40 positive 0.000000000 crime -> hoval
## 32        41 positive 0.000000000 crime -> hoval
## 33        42 positive 0.000000000 crime -> hoval
## 34        43 positive 0.000000000 crime -> hoval
## 35        44 positive 0.000000000 crime -> hoval
## 36        45 positive 0.000000000 crime -> hoval
## 37        46 positive 0.000000000 crime -> hoval
## 38        47 positive 0.000000000 crime -> hoval
## 39        48 positive 0.000000000 crime -> hoval
## 40        49 positive 0.000000000 crime -> hoval
## 41        10 negative 0.000000000 crime -> hoval
## 42        11 negative 0.000000000 crime -> hoval
## 43        12 negative 0.000000000 crime -> hoval
## 44        13 negative 0.000000000 crime -> hoval
## 45        14 negative 0.000000000 crime -> hoval
## 46        15 negative 0.000000000 crime -> hoval
## 47        16 negative 0.000000000 crime -> hoval
## 48        17 negative 0.000000000 crime -> hoval
## 49        18 negative 0.000000000 crime -> hoval
## 50        19 negative 0.079758388 crime -> hoval
## 51        20 negative 0.095663420 crime -> hoval
## 52        21 negative 0.104704018 crime -> hoval
## 53        22 negative 0.083390936 crime -> hoval
## 54        23 negative 0.155933157 crime -> hoval
## 55        24 negative 0.160468454 crime -> hoval
## 56        25 negative 0.159841305 crime -> hoval
## 57        26 negative 0.153965990 crime -> hoval
## 58        27 negative 0.211917428 crime -> hoval
## 59        28 negative 0.186256958 crime -> hoval
## 60        29 negative 0.174366043 crime -> hoval
## 61        30 negative 0.172561491 crime -> hoval
## 62        31 negative 0.189445238 crime -> hoval
## 63        32 negative 0.212568790 crime -> hoval
## 64        33 negative 0.192410926 crime -> hoval
## 65        34 negative 0.193216971 crime -> hoval
## 66        35 negative 0.192956642 crime -> hoval
## 67        36 negative 0.209433064 crime -> hoval
## 68        37 negative 0.203325807 crime -> hoval
## 69        38 negative 0.203325807 crime -> hoval
## 70        39 negative 0.205654746 crime -> hoval
## 71        40 negative 0.205127658 crime -> hoval
## 72        41 negative 0.209042309 crime -> hoval
## 73        42 negative 0.209042309 crime -> hoval
## 74        43 negative 0.226222782 crime -> hoval
## 75        44 negative 0.226700257 crime -> hoval
## 76        45 negative 0.226904376 crime -> hoval
## 77        46 negative 0.238064699 crime -> hoval
## 78        47 negative 0.312430855 crime -> hoval
## 79        48 negative 0.340356565 crime -> hoval
## 80        49 negative 0.340356565 crime -> hoval
## 81        10     dark 0.000000000 crime -> hoval
## 82        11     dark 0.000000000 crime -> hoval
## 83        12     dark 0.000000000 crime -> hoval
## 84        13     dark 0.000000000 crime -> hoval
## 85        14     dark 0.000000000 crime -> hoval
## 86        15     dark 0.000000000 crime -> hoval
## 87        16     dark 0.000000000 crime -> hoval
## 88        17     dark 0.000000000 crime -> hoval
## 89        18     dark 0.000000000 crime -> hoval
## 90        19     dark 0.000000000 crime -> hoval
## 91        20     dark 0.000000000 crime -> hoval
## 92        21     dark 0.000000000 crime -> hoval
## 93        22     dark 0.000000000 crime -> hoval
## 94        23     dark 0.000000000 crime -> hoval
## 95        24     dark 0.000000000 crime -> hoval
## 96        25     dark 0.000000000 crime -> hoval
## 97        26     dark 0.000000000 crime -> hoval
## 98        27     dark 0.000000000 crime -> hoval
## 99        28     dark 0.000000000 crime -> hoval
## 100       29     dark 0.000000000 crime -> hoval
## 101       30     dark 0.000000000 crime -> hoval
## 102       31     dark 0.000000000 crime -> hoval
## 103       32     dark 0.000000000 crime -> hoval
## 104       33     dark 0.000000000 crime -> hoval
## 105       34     dark 0.000000000 crime -> hoval
## 106       35     dark 0.000000000 crime -> hoval
## 107       36     dark 0.000000000 crime -> hoval
## 108       37     dark 0.000000000 crime -> hoval
## 109       38     dark 0.000000000 crime -> hoval
## 110       39     dark 0.000000000 crime -> hoval
## 111       40     dark 0.000000000 crime -> hoval
## 112       41     dark 0.003273597 crime -> hoval
## 113       42     dark 0.009965908 crime -> hoval
## 114       43     dark 0.017923335 crime -> hoval
## 115       44     dark 0.019163588 crime -> hoval
## 116       45     dark 0.019163588 crime -> hoval
## 117       46     dark 0.026656692 crime -> hoval
## 118       47     dark 0.026656692 crime -> hoval
## 119       48     dark 0.027868360 crime -> hoval
## 120       49     dark 0.027868360 crime -> hoval
## 121       10 positive 0.000000000 hoval -> crime
## 122       11 positive 0.000000000 hoval -> crime
## 123       12 positive 0.000000000 hoval -> crime
## 124       13 positive 0.000000000 hoval -> crime
## 125       14 positive 0.000000000 hoval -> crime
## 126       15 positive 0.000000000 hoval -> crime
## 127       16 positive 0.000000000 hoval -> crime
## 128       17 positive 0.000000000 hoval -> crime
## 129       18 positive 0.000000000 hoval -> crime
## 130       19 positive 0.000000000 hoval -> crime
## 131       20 positive 0.000000000 hoval -> crime
## 132       21 positive 0.000000000 hoval -> crime
## 133       22 positive 0.000000000 hoval -> crime
## 134       23 positive 0.000000000 hoval -> crime
## 135       24 positive 0.000000000 hoval -> crime
## 136       25 positive 0.000000000 hoval -> crime
## 137       26 positive 0.000000000 hoval -> crime
## 138       27 positive 0.000000000 hoval -> crime
## 139       28 positive 0.000000000 hoval -> crime
## 140       29 positive 0.000000000 hoval -> crime
## 141       30 positive 0.000000000 hoval -> crime
## 142       31 positive 0.000000000 hoval -> crime
## 143       32 positive 0.000000000 hoval -> crime
## 144       33 positive 0.000000000 hoval -> crime
## 145       34 positive 0.000000000 hoval -> crime
## 146       35 positive 0.000000000 hoval -> crime
## 147       36 positive 0.000000000 hoval -> crime
## 148       37 positive 0.000000000 hoval -> crime
## 149       38 positive 0.000000000 hoval -> crime
## 150       39 positive 0.000000000 hoval -> crime
## 151       40 positive 0.000000000 hoval -> crime
## 152       41 positive 0.000000000 hoval -> crime
## 153       42 positive 0.000000000 hoval -> crime
## 154       43 positive 0.000000000 hoval -> crime
## 155       44 positive 0.000000000 hoval -> crime
## 156       45 positive 0.000000000 hoval -> crime
## 157       46 positive 0.000000000 hoval -> crime
## 158       47 positive 0.000000000 hoval -> crime
## 159       48 positive 0.000000000 hoval -> crime
## 160       49 positive 0.000000000 hoval -> crime
## 161       10 negative 0.000000000 hoval -> crime
## 162       11 negative 0.000000000 hoval -> crime
## 163       12 negative 0.000000000 hoval -> crime
## 164       13 negative 0.000000000 hoval -> crime
## 165       14 negative 0.000000000 hoval -> crime
## 166       15 negative 0.000000000 hoval -> crime
## 167       16 negative 0.000000000 hoval -> crime
## 168       17 negative 0.000000000 hoval -> crime
## 169       18 negative 0.000000000 hoval -> crime
## 170       19 negative 0.000000000 hoval -> crime
## 171       20 negative 0.000000000 hoval -> crime
## 172       21 negative 0.000000000 hoval -> crime
## 173       22 negative 0.000000000 hoval -> crime
## 174       23 negative 0.000000000 hoval -> crime
## 175       24 negative 0.000000000 hoval -> crime
## 176       25 negative 0.000000000 hoval -> crime
## 177       26 negative 0.000000000 hoval -> crime
## 178       27 negative 0.000000000 hoval -> crime
## 179       28 negative 0.000000000 hoval -> crime
## 180       29 negative 0.000000000 hoval -> crime
## 181       30 negative 0.000000000 hoval -> crime
## 182       31 negative 0.000000000 hoval -> crime
## 183       32 negative 0.000000000 hoval -> crime
## 184       33 negative 0.000000000 hoval -> crime
## 185       34 negative 0.000000000 hoval -> crime
## 186       35 negative 0.000000000 hoval -> crime
## 187       36 negative 0.000000000 hoval -> crime
## 188       37 negative 0.000000000 hoval -> crime
## 189       38 negative 0.000000000 hoval -> crime
## 190       39 negative 0.000000000 hoval -> crime
## 191       40 negative 0.000000000 hoval -> crime
## 192       41 negative 0.000000000 hoval -> crime
## 193       42 negative 0.000000000 hoval -> crime
## 194       43 negative 0.000000000 hoval -> crime
## 195       44 negative 0.000000000 hoval -> crime
## 196       45 negative 0.000000000 hoval -> crime
## 197       46 negative 0.000000000 hoval -> crime
## 198       47 negative 0.000000000 hoval -> crime
## 199       48 negative 0.000000000 hoval -> crime
## 200       49 negative 0.000000000 hoval -> crime
## 201       10     dark 0.050816175 hoval -> crime
## 202       11     dark 0.043383185 hoval -> crime
## 203       12     dark 0.041382131 hoval -> crime
## 204       13     dark 0.040519557 hoval -> crime
## 205       14     dark 0.048272151 hoval -> crime
## 206       15     dark 0.040230136 hoval -> crime
## 207       16     dark 0.044464149 hoval -> crime
## 208       17     dark 0.044665447 hoval -> crime
## 209       18     dark 0.042914041 hoval -> crime
## 210       19     dark 0.041468145 hoval -> crime
## 211       20     dark 0.038849769 hoval -> crime
## 212       21     dark 0.038819148 hoval -> crime
## 213       22     dark 0.038858502 hoval -> crime
## 214       23     dark 0.039451135 hoval -> crime
## 215       24     dark 0.040338936 hoval -> crime
## 216       25     dark 0.038705221 hoval -> crime
## 217       26     dark 0.036890764 hoval -> crime
## 218       27     dark 0.037987604 hoval -> crime
## 219       28     dark 0.039793512 hoval -> crime
## 220       29     dark 0.037765229 hoval -> crime
## 221       30     dark 0.038233561 hoval -> crime
## 222       31     dark 0.036447303 hoval -> crime
## 223       32     dark 0.039415083 hoval -> crime
## 224       33     dark 0.037622935 hoval -> crime
## 225       34     dark 0.038474108 hoval -> crime
## 226       35     dark 0.037545521 hoval -> crime
## 227       36     dark 0.041351817 hoval -> crime
## 228       37     dark 0.039936088 hoval -> crime
## 229       38     dark 0.044526449 hoval -> crime
## 230       39     dark 0.041276856 hoval -> crime
## 231       40     dark 0.044994931 hoval -> crime
## 232       41     dark 0.044496909 hoval -> crime
## 233       42     dark 0.049654135 hoval -> crime
## 234       43     dark 0.050765691 hoval -> crime
## 235       44     dark 0.052651619 hoval -> crime
## 236       45     dark 0.053726028 hoval -> crime
## 237       46     dark 0.053726628 hoval -> crime
## 238       47     dark 0.053373606 hoval -> crime
## 239       48     dark 0.053373606 hoval -> crime
## 240       49     dark 0.053373606 hoval -> crime
plot(crime_convergence, ylimits = c(-0.01,0.61),
     xlimits = c(9,50), xbreaks = seq(10, 50, 5))
Figure 1. Convergence curves of causal strengths among house value and crime.
Figure 1. Convergence curves of causal strengths among house value and crime.


Example of spatial raster 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

# Check the validated cell number
nnamat = terra::as.matrix(npp[[1]], wide = TRUE)
nnaindice = which(!is.na(nnamat), arr.ind = TRUE)
dim(nnaindice)
## [1] 6920    2

Determining minimal embedding dimension:

spEDM::fnn(npp, "npp", E = 1:15,
           eps = stats::sd(terra::values(npp[["npp"]]),na.rm = TRUE))
## [spEDM] Output 'E:i' corresponds to the i-th valid embedding dimension.
## [spEDM] Input E values exceeding max embeddable dimension were truncated.
## [spEDM] Please map output indices to original E inputs before interpretation.
##          E:1          E:2          E:3          E:4          E:5          E:6 
## 0.9813070569 0.5309427415 0.1322254335 0.0167630058 0.0017341040 0.0000000000 
##          E:7          E:8          E:9         E:10         E:11         E:12 
## 0.0001445087 0.0000000000 0.0000000000 0.0000000000 0.0002890173 0.0000000000 
##         E:13         E:14 
## 0.0001445087 0.0001445087

At \(E = 6\), the false nearest neighbor ratio stabilizes approximately at 0.0001 and remains constant thereafter. Therefore, \(E = 6\) is selected as minimal embedding dimension for the subsequent GPC analysis.

Then, search optimal parameters:

stats::cor.test(~ pre + npp,
                data = terra::values(npp[[c("pre","npp")]],
                                     dataframe = TRUE, na.rm = TRUE))
## 
##  Pearson's product-moment correlation
## 
## data:  pre and npp
## t = 108.74, df = 6912, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.7855441 0.8029426
## sample estimates:
##       cor 
## 0.7944062


g1 = spEDM::pc(npp, "pre", "npp", E = 6:10, k = 7:12, tau = 1:5, maximize = "positive")
g1
## The suggested E,k,tau for variable npp is 6, 9 and 5

Run geographical pattern causality analysis

spEDM::gpc(npp, "pre", "npp", E = 6, k = 9, tau = 5)
##       type  strength  direction
## 1 positive 0.5222514 pre -> npp
## 2 negative 0.3180282 pre -> npp
## 3     dark 0.3795653 pre -> npp
## 4 positive 0.4491962 npp -> pre
## 5 negative 0.6663956 npp -> pre
## 6     dark 0.4303680 npp -> pre

Convergence diagnostics

npp_convergence = spEDM::gpc(npp, "pre", "npp",
                             libsizes = matrix(rep(seq(10,80,10),2), ncol = 2),
                             E = 6, k = 9, tau = 5, progressbar = FALSE)
npp_convergence
##    libsizes     type   strength  direction
## 1       100 positive 0.13573984 pre -> npp
## 2       400 positive 0.18147283 pre -> npp
## 3       900 positive 0.24614666 pre -> npp
## 4      1600 positive 0.33710096 pre -> npp
## 5      2500 positive 0.40395843 pre -> npp
## 6      3600 positive 0.45524137 pre -> npp
## 7      4900 positive 0.48874459 pre -> npp
## 8      6400 positive 0.51020746 pre -> npp
## 9       100 negative 0.00000000 pre -> npp
## 10      400 negative 0.00000000 pre -> npp
## 11      900 negative 0.00000000 pre -> npp
## 12     1600 negative 0.00000000 pre -> npp
## 13     2500 negative 0.05496496 pre -> npp
## 14     3600 negative 0.12023142 pre -> npp
## 15     4900 negative 0.23867121 pre -> npp
## 16     6400 negative 0.30815296 pre -> npp
## 17      100     dark 0.06048052 pre -> npp
## 18      400     dark 0.11029673 pre -> npp
## 19      900     dark 0.16639601 pre -> npp
## 20     1600     dark 0.22696194 pre -> npp
## 21     2500     dark 0.27885166 pre -> npp
## 22     3600     dark 0.31852011 pre -> npp
## 23     4900     dark 0.35338378 pre -> npp
## 24     6400     dark 0.37870193 pre -> npp
## 25      100 positive 0.13945164 npp -> pre
## 26      400 positive 0.19473737 npp -> pre
## 27      900 positive 0.25254635 npp -> pre
## 28     1600 positive 0.31327780 npp -> pre
## 29     2500 positive 0.36524631 npp -> pre
## 30     3600 positive 0.41042253 npp -> pre
## 31     4900 positive 0.43152740 npp -> pre
## 32     6400 positive 0.45137261 npp -> pre
## 33      100 negative 0.16639620 npp -> pre
## 34      400 negative 0.24974175 npp -> pre
## 35      900 negative 0.49955110 npp -> pre
## 36     1600 negative 0.66653262 npp -> pre
## 37     2500 negative 0.83295414 npp -> pre
## 38     3600 negative 0.99940859 npp -> pre
## 39     4900 negative 0.78807485 npp -> pre
## 40     6400 negative 0.66641031 npp -> pre
## 41      100     dark 0.08290872 npp -> pre
## 42      400     dark 0.14873362 npp -> pre
## 43      900     dark 0.22827942 npp -> pre
## 44     1600     dark 0.28941655 npp -> pre
## 45     2500     dark 0.32906970 npp -> pre
## 46     3600     dark 0.36571275 npp -> pre
## 47     4900     dark 0.39113087 npp -> pre
## 48     6400     dark 0.41925841 npp -> pre
plot(npp_convergence, ylimits = c(-0.01,1.05),
     xlimits = c(0,6500), xbreaks = seq(100, 6400, 500))
Figure 2. Convergence curves of causal strengths among precipitation and NPP.
Figure 2. Convergence curves of causal strengths among precipitation and NPP.

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