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Synergistic-Unique-Redundant Decompositionof Causality (SURD)

Last update: 2025-10-30
Last run: 2025-10-30

Introduction

Understanding how causal influences combine and interact is essentialfor both temporal and spatial systems. TheSynergistic–Unique–Redundant Decomposition (SURD)framework provides a principled way to break down total information flowbetween variables into:

This vignette demonstrates how to perform SURD analysis using theinfocausality package. We will explore both temporal andspatial cases, showing howinfocausality unifiestime-series andspatialcross-sectional causal analysis via consistent interfaces todata.frame,sf, andSpatRasterobjects.

Utility Functions

To simplify post-processing and visualization, two helper functionsare provided below.

utils_process_surd_result() converts asurd_list object (as returned by an SURD computationroutine) into a tidytibble for plotting and furtheranalysis.

utils_process_surd_result= \(surd_list,threshold =0){  I_unique= surd_list$unique  I_synergistic= surd_list$synergistic  I_redundant= surd_list$redundant  info_leak= surd_list$info_leak  df= tibble::tibble(label =c(paste0("U[",names(I_unique),"]"),paste0("S[",names(I_synergistic),"]"),paste0("R[",names(I_redundant),"]")    ),value =c(as.numeric(I_unique),as.numeric(I_synergistic),as.numeric(I_redundant)),type =c(rep("unique",length(I_unique)),rep("synergistic",length(I_synergistic)),rep("redundant",length(I_redundant)))  )  df$value= df$value/sum(df$value)if(threshold>0){    df= dplyr::filter(df, value> threshold)  }return(dplyr::bind_rows(df,tibble::tribble(~label,~value,~type,"Info Leak", info_leak,"info leak")))}

utils_plot_surd() produces a SURD decomposition plot — agrouped bar chart ofunique (U),synergistic(S), andredundant (R) components, along witha side panel showing theinformation leak.

utils_plot_surd= \(surd_list,threshold =0,style ="shallow"){  I_unique= surd_list$unique  I_synergistic= surd_list$synergistic  I_redundant= surd_list$redundant  df= tibble::tibble(label =c(paste0("U[",names(I_unique),"]"),paste0("S[",names(I_synergistic),"]"),paste0("R[",names(I_redundant),"]")    ),value =c(as.numeric(I_unique),as.numeric(I_synergistic),as.numeric(I_redundant)),type =c(rep("unique",length(I_unique)),rep("synergistic",length(I_synergistic)),rep("redundant",length(I_redundant)))  )  df$value= df$value/sum(df$value)if(threshold>0){    df= dplyr::filter(df, value> threshold)  }  df$label=factor(df$label,levels = df$label)if (style=="shallow") {    colors=c(unique ="#ec9e9e",synergistic ="#fac58c",redundant ="#668392",infoleak ="#7f7f7f")  }else {    colors=c(unique ="#d62828",synergistic ="#f77f00",redundant ="#003049",infoleak ="gray")  }  p1= ggplot2::ggplot(df, ggplot2::aes(x = label,y = value,fill = type))+    ggplot2::geom_col(color ="black",linewidth =0.15,show.legend =FALSE)+    ggplot2::scale_fill_manual(name =NULL,values = colors)+    ggplot2::scale_x_discrete(name =NULL,labels =function(x)parse(text = x))+    ggplot2::scale_y_continuous(name =NULL,limits =c(0,1),expand =c(0,0),breaks =c(0,0.25,0.5,0.75,1),labels =c("0","0.25","0.5","0.75","1") )+    ggplot2::theme_bw()+    ggplot2::theme(axis.ticks.x = ggplot2::element_blank(),axis.text.x = ggplot2::element_text(angle =45,hjust =1),panel.grid = ggplot2::element_blank(),panel.border = ggplot2::element_rect(color ="black",linewidth =2)    )  df_leak= tibble::tibble(label ="Info Leak",value = surd_list$info_leak)  p2= ggplot2::ggplot(df_leak, ggplot2::aes(x = label,y = value))+    ggplot2::geom_col(fill = colors[4],color ="black",linewidth =0.15)+    ggplot2::scale_x_discrete(name =NULL,expand =c(0,0))+    ggplot2::scale_y_continuous(name =NULL,limits =c(0,1),expand =c(0,0),breaks =c(0,1),labels =c(0,1) )+    ggplot2::theme_bw()+    ggplot2::theme(axis.ticks.x = ggplot2::element_blank(),panel.grid = ggplot2::element_blank(),panel.border = ggplot2::element_rect(color ="black",linewidth =2)    )  patchwork::wrap_plots(p1, p2,ncol =2,widths =c(10,1))}

Example Cases

The following sections demonstrate SURD decomposition in differentcontexts, illustrating its flexibility across temporal and spatialcausal analyses.

Air Pollution and Cardiovascular Health in Hong Kong

cvd= readr::read_csv(system.file("case/cvd.csv",package ="tEDM"))head(cvd)## # A tibble: 6 × 5##     cvd   rsp   no2   so2    o3##   <dbl> <dbl> <dbl> <dbl> <dbl>## 1   214  73.7  74.5  19.1 17.4## 2   203  77.6  80.9  18.8 39.4## 3   202  64.8  67.1  13.8 56.4## 4   182  68.8  74.7  30.8  5.6## 5   181  49.4  62.3  23.1  3.6## 6   129  67.4  63.6  17.4  6.73
cvd_long= cvd|>  tibble::rowid_to_column("id")|>  tidyr::pivot_longer(cols =-id,names_to ="variable",values_to ="value")fig_cvds_ts= ggplot2::ggplot(cvd_long, ggplot2::aes(x = id,y = value,color = variable))+  ggplot2::geom_line(linewidth =0.5)+  ggplot2::labs(x ="Days (from 1995-3 to 1997-11)",y ="Concentrations or\nNO. of CVD admissions",color ="")+  ggplot2::theme_bw()+  ggplot2::theme(legend.direction ="horizontal",legend.position ="inside",legend.justification =c("center","top"),legend.background = ggplot2::element_rect(fill ="transparent",color =NA))fig_cvds_ts
Figure 1. Time series of air pollutants and confirmed CVD cases in Hong Kong from March 1995 to November 1997.
Figure 1. Time series of airpollutants and confirmed CVD cases in Hong Kong from March 1995 toNovember 1997.

To investigate the causal influences of air pollutants on theincidence of cardiovascular diseases, we performed SURD analysis using atime lag step3 and10 discretizationbins.

res_cvds= infocausality::surd(cvd,"cvd",c("rsp","no2","so2","o3"),lag =15,bin =10,cores =6)#### SURD Decomposition Results:##   Unique (U):##     X1             : 0.0022##     X2             : 0.0162##     X3             : 0.0024##     X4             : 0.0000##   Synergistic (S):##     X1-X2          : 0.0181##     X1-X3          : 0.0198##     X1-X4          : 0.0339##     X2-X3          : 0.0293##     X2-X4          : 0.0332##     X3-X4          : 0.0477##     X1-X2-X3       : 0.0343##     X1-X2-X4       : 0.1084##     X1-X3-X4       : 0.1540##     X2-X3-X4       : 0.0426##     X1-X2-X3-X4    : 0.4093##   Redundant (R):##     X1-X2          : 0.0034##     X1-X3          : 0.0000##     X1-X4          : 0.0007##     X2-X3          : 0.0006##     X2-X4          : 0.0034##     X3-X4          : 0.0000##     X1-X2-X3       : 0.0050##     X1-X2-X4       : 0.0100##     X1-X3-X4       : 0.0008##     X2-X3-X4       : 0.0014##     X1-X2-X3-X4    : 0.0230##   Information Leak:  52.82%

The SURD results are shown in the figure below:

utils_plot_surd(res_cvds)
Figure 2. SURD decomposition results between different air pollutants and cardiovascular diseases.
Figure 2. SURD decompositionresults between different air pollutants and cardiovasculardiseases.

Population Density and Its Drivers in Mainland China

popd_nb= spdep::read.gal(system.file("case/popd_nb.gal",package ="spEDM"))## Warning in spdep::read.gal(system.file("case/popd_nb.gal", package = "spEDM")):## neighbour object has 4 sub-graphspopd= readr::read_csv(system.file("case/popd.csv",package ="spEDM"))popd_sf= sf::st_as_sf(popd,coords =c("lon","lat"),crs =4326)popd_sf## Simple feature collection with 2806 features and 5 fields## Geometry type: POINT## Dimension:     XY## Bounding box:  xmin: 74.9055 ymin: 18.2698 xmax: 134.269 ymax: 52.9346## Geodetic CRS:  WGS 84## # A tibble: 2,806 × 6##     popd  elev   tem   pre slope          geometry##  * <dbl> <dbl> <dbl> <dbl> <dbl>       <POINT [°]>##  1  780.     8  17.4 1528. 0.452 (116.912 30.4879)##  2  395.    48  17.2 1487. 0.842 (116.755 30.5877)##  3  261.    49  16.0 1456. 3.56  (116.541 30.7548)##  4  258.    23  17.4 1555. 0.932  (116.241 30.104)##  5  211.   101  16.3 1494. 3.34   (116.173 30.495)##  6  386.    10  16.6 1382. 1.65  (116.935 30.9839)##  7  350.    23  17.5 1569. 0.346 (116.677 30.2412)##  8  470.    22  17.1 1493. 1.88  (117.066 30.6514)##  9 1226.    11  17.4 1526. 0.208 (117.171 30.5558)## 10  137.   598  13.9 1458. 5.92  (116.208 30.8983)## # ℹ 2,796 more rows
res_popd= infocausality::surd(popd_sf,"popd",c("elev","tem","pre","slope"),lag =3,bin =10,nb = popd_nb,cores =6)#### SURD Decomposition Results:##   Unique (U):##     X1             : 0.0046##     X2             : 0.0000##     X3             : 0.0398##     X4             : 0.0036##   Synergistic (S):##     X1-X2          : 0.0369##     X1-X3          : 0.0471##     X1-X4          : 0.0885##     X2-X3          : 0.0468##     X2-X4          : 0.0765##     X3-X4          : 0.0877##     X1-X2-X3       : 0.0384##     X1-X2-X4       : 0.0597##     X1-X3-X4       : 0.0463##     X2-X3-X4       : 0.1577##     X1-X2-X3-X4    : 0.1095##   Redundant (R):##     X1-X2          : 0.0000##     X1-X3          : 0.0089##     X1-X4          : 0.0000##     X2-X3          : 0.0130##     X2-X4          : 0.0027##     X3-X4          : 0.0138##     X1-X2-X3       : 0.0056##     X1-X2-X4       : 0.0000##     X1-X3-X4       : 0.0120##     X2-X3-X4       : 0.0110##     X1-X2-X3-X4    : 0.0897##   Information Leak:  64.64%
utils_plot_surd(res_popd)
Figure 3. SURD decomposition results between population density and its Drivers.
Figure 3. SURD decompositionresults between population density and its Drivers.

Influence of Climatic and Topographic Factors on Net PrimaryProductivity (NPP) in Mainland China

npp= terra::rast(system.file("case/npp.tif",package ="spEDM"))npp## class       : SpatRaster## size        : 404, 483, 5  (nrow, ncol, nlyr)## resolution  : 10000, 10000  (x, y)## extent      : -2625763, 2204237, 1877078, 5917078  (xmin, xmax, ymin, ymax)## coord. ref. : CGCS2000_Albers## source      : npp.tif## names       :      npp,        pre,      tem,      elev,         hfp## min values  :   164.00,   384.3409, -47.8194, -122.2004,  0.03390418## max values  : 16606.33, 23878.3555, 263.6938, 5350.4902, 44.90312195
res_npp= infocausality::surd(npp,"npp",c("pre","tem","elev","hfp"),lag =2,bin =10,cores =6)#### SURD Decomposition Results:##   Unique (U):##     X1             : 0.2060##     X2             : 0.0004##     X3             : 0.0001##     X4             : 0.0000##   Synergistic (S):##     X1-X2          : 0.0477##     X1-X3          : 0.0734##     X1-X4          : 0.0360##     X2-X3          : 0.0268##     X2-X4          : 0.0049##     X3-X4          : 0.0010##     X1-X2-X3       : 0.0363##     X1-X2-X4       : 0.0171##     X1-X3-X4       : 0.0380##     X2-X3-X4       : 0.0201##     X1-X2-X3-X4    : 0.0796##   Redundant (R):##     X1-X2          : 0.2658##     X1-X3          : 0.0020##     X1-X4          : 0.0000##     X2-X3          : 0.0000##     X2-X4          : 0.0000##     X3-X4          : 0.0000##     X1-X2-X3       : 0.0439##     X1-X2-X4       : 0.0000##     X1-X3-X4       : 0.0065##     X2-X3-X4       : 0.0000##     X1-X2-X3-X4    : 0.0941##   Information Leak:  40.78%
utils_plot_surd(res_npp)
Figure 4. SURD decomposition results between climatic and topographic factors and npp.
Figure 4. SURD decompositionresults between climatic and topographic factors and npp.

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