Basis functions

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captioner <- function(caption) {
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library(tidyr)
library(dplyr)
library(plotly)
library(MetricGraph)

gets.graph.basis <- function(h, cont = TRUE){
  #F
  edge1 <- rbind(c(0,0), c(0,2))
  edge2 <- rbind(c(0,2), c(0,4))
  edge3 <- rbind(c(0,2), c(1,2))
  edge4 <- rbind(c(0,4), c(2,4))
  #U
  edge5 <- rbind(c(2,4), c(2,1))
  thetau <- seq(pi, 2*pi, length.out = 100)
  edge6 <- cbind(3+1*cos(thetau), 1+1*sin(thetau))
  edge7 <- rbind(c(4,1), c(4,4))
  #N
  edge8 <- rbind(c(4,1), c(4,0))
  edge9 <- rbind(c(4,4), c(6,0))
  edge10 <- rbind(c(6,0), c(6,4))
  #C
  thetac1 <- seq(pi, pi/2, length.out = 100)
  edge11 <- cbind(8+2*cos(thetac1), 2+2*sin(thetac1))
  thetac2 <- seq(pi, 3*pi/2, length.out = 100)
  edge12 <- cbind(8+2*cos(thetac2), 2+2*sin(thetac2))
  #T
  edge13 <- rbind(c(8,4), c(10,4))
  edge14 <- rbind(c(9,4), c(9,0))
  #I
  edge15 <- rbind(c(10,4), c(12,4))
  edge16 <- rbind(c(10,0), c(12,0))
  edge17 <- rbind(c(11,0), c(11,4))
  #O
  thetao1 <- seq(pi, 2*pi, length.out = 100)
  edge18 <- cbind(13+1*cos(thetao1), 1+1*sin(thetao1))
  thetao2 <- seq(0, pi, length.out = 100)
  edge19 <- cbind(13+1*cos(thetao2), 3+1*sin(thetao2))
  edge20 <- rbind(c(12,3), c(12,1))
  edge21 <- rbind(c(14,1), c(14,4))
  #N
  edge22 <- rbind(c(14,4), c(16,0))
  edge23 <- rbind(c(16,0), c(16,4))
  edge56 <- rbind(c(14,0), c(14,1))
  #S
  edge24 <- rbind(c(16,0), c(17,0))
  tethas1 <- seq(-pi/2, pi/2, length.out = 100)
  edge25 <- cbind(17+1*cos(tethas1), 1+1*sin(tethas1))
  thetas2 <- seq(3*pi/2, pi/2, length.out = 100)
  edge26 <- cbind(17+1*cos(thetas2), 3+1*sin(thetas2))
  edge27 <- rbind(c(17,4), c(18,4))
  #H
  edge28 <- rbind(c(0,4), c(0,6))
  edge29 <- rbind(c(0,6), c(0,8))
  edge30 <- rbind(c(0,6), c(2,6))
  edge31 <- rbind(c(2,4), c(2,8))
  #A
  edge32 <- rbind(c(2,4), c(3,8))
  edge33 <- rbind(c(3,8), c(4,4))
  edge34 <- rbind(c(2.5,6), c(3.5,6))
  #T
  edge35 <- rbind(c(3,8), c(6,8))
  edge36 <- rbind(c(5,8), c(5,4))
  #B
  edge37 <- rbind(c(8,8), c(8,4))
  thetab1 <- seq(-pi/2, pi/2, length.out = 100)
  edge38 <- cbind(9+1*cos(thetab1), 5+1*sin(thetab1))
  edge39 <- rbind(c(8,6), c(9,6))
  edge40 <- rbind(c(9,8), c(8,8))
  thetab2 <- seq(-pi/2, pi/2, length.out = 100)
  edge41 <- cbind(9+1*cos(thetab2), 7+1*sin(thetab2))
  #A
  edge42 <- rbind(c(10,4), c(11,8))
  edge43 <- rbind(c(11,8), c(12,4))
  edge44 <- rbind(c(10.5,6), c(11.5,6))
  #S
  
  #I
  edge45 <- rbind(c(14,4), c(16,4))
  edge46 <- rbind(c(15,4), c(15,8))
  edge47 <- rbind(c(14,8), c(16,8))
  #S
  edge48 <- rbind(c(16,4), c(17,4))
  edge49 <- cbind(17+1*cos(tethas1), 5+1*sin(tethas1))
  edge50 <- cbind(17+1*cos(thetas2), 7+1*sin(thetas2))
  edge51 <- rbind(c(17,8), c(18,8))
  #S
  edge52 <- rbind(c(13,8), c(14,8))
  edge53 <-cbind(13+1*cos(thetas2), 7+1*sin(thetas2))
  edge54 <- cbind(13+1*cos(tethas1), 5+1*sin(tethas1))
  edge55 <- rbind(c(12,4), c(13,4))
  
  edges <- list(edge1, edge2, edge3, edge4, edge5, edge6, edge7,
                edge8, edge9, edge10, edge11, edge12, edge13, edge14,
                edge15, edge16, edge17, edge18, edge19, edge20, edge21,
                edge22, edge23, edge24, edge25, edge26, edge27,
                edge28, edge29, edge30, edge31, edge32, edge33, edge34,
                edge35, edge36, edge37, edge38, edge39, edge40, edge41,
                edge42, edge43, edge44, edge45, edge46, edge47,
                edge48, edge49, edge50, edge51, edge52, edge53, edge54, edge55, edge56)
  graph <- metric_graph$new(edges = edges, perform_merges = TRUE)
  graph$prune_vertices()
  graph$build_mesh(h = h, continuous = cont)
  return(graph)
}

add_group_boundaries <- function(mat) {
  # Unique group identifiers in the first column
  groups <- unique(mat[, 1])
  
  # Initialize list to store results
  result_list <- vector("list", length(groups))
  
  for (i in seq_along(groups)) {
    grp <- groups[i]
    group_rows <- mat[mat[, 1] == grp, , drop = FALSE]
    
    # Add boundary rows
    augmented <- rbind(
      c(grp, 0),
      group_rows,
      c(grp, 1)
    )
    result_list[[i]] <- augmented
  }
  
  # Combine all groups
  result <- do.call(rbind, result_list)
  rownames(result) <- NULL
  return(result)
}

# Function to insert NA row between groups
insert_na_between_groups <- function(mat, group_vec) {
  # Split the matrix by group
  mat_split <- split(as.data.frame(mat), group_vec)
  
  # Add NA rows after each group
  with_na <- lapply(mat_split, function(x) rbind(as.matrix(x), rep(NA, ncol(mat))))
  
  # Combine everything into one matrix (removing the last NA if not needed)
  mat <- do.call(rbind, with_na)
  return(mat) #mat[-nrow(mat), ]
}

fill_between_NA <- function(vec) {
  
  # Find the indices of the NA values
  na_indices <- which(is.na(vec))
  
  for (i in seq_along(na_indices)[-length(na_indices)]) {
    start <- na_indices[i]
    end <- na_indices[i + 1]
    
    # Work on the elements between two NA values
    if (end - start > 1) {
      segment <- vec[(start + 1):(end - 1)]
      if (all(segment == 0, na.rm = TRUE)) {
        vec[(start + 1):(end - 1)] <- NA
      }
    }
  }
  return(vec)
}

keep_nonzeros_and_border_zeros <- function(vec) {
  n <- length(vec)
  keep <- rep(FALSE, n)
  
  # Identify nonzero values (ignoring NAs)
  is_nonzero <- !is.na(vec) & vec != 0
  
  for (i in which(is_nonzero)) {
    keep[i] <- TRUE
    if (i > 1 && !is.na(vec[i - 1]) && vec[i - 1] == 0) keep[i - 1] <- TRUE
    if (i < n && !is.na(vec[i + 1]) && vec[i + 1] == 0) keep[i + 1] <- TRUE
  }
  
  # Replace zeros that are not marked for keeping with NA
  vec[!is.na(vec) & vec == 0 & !keep] <- NA
  return(vec)
}

graph <- gets.graph.basis(h = 1/1, cont = TRUE)
graph_cont <- gets.graph.basis(h = 1/50, cont = TRUE)

# discontinuous mesh
V <- graph_cont$mesh$V
VtE <- graph_cont$mesh$VtE


new_VtE <- add_group_boundaries(VtE[(graph$nV+1):nrow(VtE),])
new_V <- graph_cont$coordinates(PtE = new_VtE, normalized = TRUE)
V_with_NA <- rbind(c(NA,NA), insert_na_between_groups(new_V, new_VtE[,1]))
x <- V_with_NA[,1]
y <- V_with_NA[,2]


A <- as.matrix(graph$fem_basis(new_VtE))
A_with_NA <- rbind(rep(NA, ncol(A)), insert_na_between_groups(A, new_VtE[,1]))
A_with_NA_cleaned <- apply(A_with_NA, 2, fill_between_NA)
A_with_NA_zeroed <- apply(A_with_NA, 2, keep_nonzeros_and_border_zeros)

x_range <- range(x, na.rm = TRUE)
y_range <- range(y, na.rm = TRUE)
z_range <- c(0,1)

# Get all z values for vertical lines
z_vals <- apply(A_with_NA_zeroed, 1, max, na.rm = TRUE)
# Subsample every 5th index
idx <- seq(1, nrow(A_with_NA_zeroed), by = 20)

# Subsample x, y, and z for gray lines
Z_red <- unlist(lapply(z_vals[idx], function(zj) c(0, zj, NA)))
X_red <- rep(x[idx], each = 3)
Y_red <- rep(y[idx], each = 3)

df3 <- data.frame(x = graph$mesh$V[, 1], 
                  y = graph$mesh$V[, 2], 
                  z = rep(0, length(graph$mesh$V[, 1])))

# Start plot
p_basis <- plot_ly() %>% 
  add_trace(x = rep(x, times = graph$nV), 
            y = rep(y, times = graph$nV), 
            z = as.vector(A_with_NA_zeroed[, 1:graph$nV]), 
            type = "scatter3d",
            mode = "lines", 
            showlegend = FALSE, 
            line = list(color = "red", width = 2)) %>%
  add_trace(x = X_red, y = Y_red, z = Z_red,
            type = "scatter3d", mode = "lines",
            line = list(color = "gray", width = 0.5),
            showlegend = FALSE) %>%
  add_trace(x = rep(x, times = ncol(A_with_NA_zeroed) - graph$nV), 
            y = rep(y, times = ncol(A_with_NA_zeroed) - graph$nV), 
            z = as.vector(A_with_NA_zeroed[, (graph$nV+1):ncol(A_with_NA_zeroed)]), 
            type = "scatter3d",
            mode = "lines", 
            showlegend = FALSE, 
            line = list(color = "blue", width = 2)) %>%
  add_trace(x = x, 
            y = y, 
            z = x*0, 
            type = "scatter3d",
            mode = "lines", 
            showlegend = FALSE, 
            line = list(color = "black", width = 4)) %>% 
  add_trace(x = rep(x, times = graph$nV), 
            y = rep(y, times = graph$nV), 
            z = as.vector(A_with_NA_zeroed[, 1:graph$nV])*0, 
            type = "scatter3d",
            mode = "lines", 
            showlegend = FALSE, 
            line = list(color = "green", width = 4)) %>%
  add_trace(data = df3, x = ~x, y = ~y, z = ~z, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "gray", symbol = 104)) %>%
  layout(scene = list(
    xaxis = list(title = "x", range = x_range),
    yaxis = list(title = "y", range = y_range),
    zaxis = list(title = "z", range = z_range),
    aspectratio = list(x = 2.4, y = 1.2, z = 0.06),
    camera = list(eye = list(x = -2, y = -3, z = 1.5), 
                  center = list(x = 0, y = 0, z = 0))))

1 Illustration

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p_basis

Figure 1: Illustration of the basis function system \(\{\psi^i_h\}_{i=1}^{N_h}\) on the hat-basis-functions graph (in black). Standard hat functions associated with internal edge nodes are shown in blue, while special vertex-centered functions are highlighted in red.

save(p_basis, file = here::here("data_files/p_basis.RData"))

1.1 References

grateful::cite_packages(output = "paragraph", out.dir = ".")

We used R version 4.5.2 (R Core Team 2025a) and the following R packages: akima v. 0.6.3.6 (Akima and Gebhardt 2025), expm v. 1.0.0 (Maechler, Dutang, and Goulet 2024), fmesher v. 0.5.0 (Lindgren 2025), gsignal v. 0.3.7 (Van Boxtel, G.J.M., et al. 2021), here v. 1.0.1 (Müller 2020), htmltools v. 0.5.8.1 (Cheng et al. 2024), INLA v. 25.11.22 (Rue, Martino, and Chopin 2009; Lindgren, Rue, and Lindström 2011; Martins et al. 2013; Lindgren and Rue 2015; De Coninck et al. 2016; Rue et al. 2017; Verbosio et al. 2017; Bakka et al. 2018; Kourounis, Fuchs, and Schenk 2018), inlabru v. 2.13.0 (Yuan et al. 2017; Bachl et al. 2019), knitr v. 1.50 (Xie 2014, 2015, 2025a), Matrix v. 1.7.3 (Bates, Maechler, and Jagan 2025), MetricGraph v. 1.5.0.9000 (Bolin, Simas, and Wallin 2023a, 2023b, 2024, 2025; Bolin et al. 2024), neuralnet v. 1.44.2 (Fritsch, Guenther, and Wright 2019), orthopolynom v. 1.0.6.1 (Novomestky 2022), parallel v. 4.5.2 (R Core Team 2025b), patchwork v. 1.3.1 (Pedersen 2025), pbmcapply v. 1.5.1 (Kuang, Kong, and Napolitano 2022), plotly v. 4.11.0 (Sievert 2020), posterdown v. 1.0 (Thorne 2019), pracma v. 2.4.4 (Borchers 2023), qrcode v. 0.3.0 (Onkelinx and Teh 2024), RColorBrewer v. 1.1.3 (Neuwirth 2022), RefManageR v. 1.4.0 (McLean 2014, 2017), renv v. 1.1.5 (Ushey and Wickham 2025), reshape2 v. 1.4.4 (Wickham 2007), reticulate v. 1.44.1 (Ushey, Allaire, and Tang 2025), rmarkdown v. 2.30 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2025), rSPDE v. 2.5.1.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2024), RSpectra v. 0.16.2 (Qiu and Mei 2024), scales v. 1.4.0 (Wickham, Pedersen, and Seidel 2025), slackr v. 3.4.0 (Kaye et al. 2025), tidyverse v. 2.0.0 (Wickham et al. 2019), viridisLite v. 0.4.2 (Garnier et al. 2023), xaringan v. 0.31 (Xie 2025b), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024), xaringanthemer v. 0.4.4 (Aden-Buie 2025).

Aden-Buie, Garrick. 2025. xaringanthemer: Custom “xaringan” CSS Themes. https://doi.org/10.32614/CRAN.package.xaringanthemer.

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Allaire, JJ, Yihui Xie, Christophe Dervieux, Jonathan McPherson, Javier Luraschi, Kevin Ushey, Aron Atkins, et al. 2025. rmarkdown: Dynamic Documents for r. https://github.com/rstudio/rmarkdown.

Bachl, Fabian E., Finn Lindgren, David L. Borchers, and Janine B. Illian. 2019. “inlabru: An R Package for Bayesian Spatial Modelling from Ecological Survey Data.” Methods in Ecology and Evolution 10: 760–66. https://doi.org/10.1111/2041-210X.13168.

Bakka, Haakon, Håvard Rue, Geir-Arne Fuglstad, Andrea I. Riebler, David Bolin, Janine Illian, Elias Krainski, Daniel P. Simpson, and Finn K. Lindgren. 2018. “Spatial Modelling with INLA: A Review.” WIRES (Invited Extended Review) xx (Feb): xx–. http://arxiv.org/abs/1802.06350.

Bates, Douglas, Martin Maechler, and Mikael Jagan. 2025. Matrix: Sparse and Dense Matrix Classes and Methods. https://doi.org/10.32614/CRAN.package.Matrix.

Bolin, David, and Kristin Kirchner. 2020. “The Rational SPDE Approach for Gaussian Random Fields with General Smoothness.” Journal of Computational and Graphical Statistics 29 (2): 274–85. https://doi.org/10.1080/10618600.2019.1665537.

Bolin, David, Mihály Kovács, Vivek Kumar, and Alexandre B. Simas. 2024. “Regularity and Numerical Approximation of Fractional Elliptic Differential Equations on Compact Metric Graphs.” Mathematics of Computation 93 (349): 2439–72. https://doi.org/10.1090/mcom/3929.

Bolin, David, and Alexandre B. Simas. 2023. rSPDE: Rational Approximations of Fractional Stochastic Partial Differential Equations. https://CRAN.R-project.org/package=rSPDE.

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———. 2023b. “Statistical Inference for Gaussian Whittle-Matérn Fields on Metric Graphs.” arXiv Preprint arXiv:2304.10372. https://doi.org/10.48550/arXiv.2304.10372.

———. 2024. “Gaussian Whittle-Matérn Fields on Metric Graphs.” Bernoulli 30 (2): 1611–39. https://doi.org/10.3150/23-BEJ1647.

———. 2025. “Markov Properties of Gaussian Random Fields on Compact Metric Graphs.” Bernoulli. https://doi.org/10.48550/arXiv.2304.03190.

Bolin, David, Alexandre B. Simas, and Zhen Xiong. 2024. “Covariance-Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference.” Journal of Computational and Graphical Statistics 33 (1): 64–74. https://doi.org/10.1080/10618600.2023.2231051.

Borchers, Hans W. 2023. pracma: Practical Numerical Math Functions. https://doi.org/10.32614/CRAN.package.pracma.

Cheng, Joe, Carson Sievert, Barret Schloerke, Winston Chang, Yihui Xie, and Jeff Allen. 2024. htmltools: Tools for HTML. https://github.com/rstudio/htmltools.

De Coninck, Arne, Bernard De Baets, Drosos Kourounis, Fabio Verbosio, Olaf Schenk, Steven Maenhout, and Jan Fostier. 2016. “Needles: Toward Large-Scale Genomic Prediction with Marker-by-Environment Interaction.” Genetics 203 (1): 543–55. https://doi.org/10.1534/genetics.115.179887.

Fritsch, Stefan, Frauke Guenther, and Marvin N. Wright. 2019. neuralnet: Training of Neural Networks. https://doi.org/10.32614/CRAN.package.neuralnet.

Garnier, Simon, Ross, Noam, Rudis, Robert, Camargo, et al. 2023. viridis(Lite) - Colorblind-Friendly Color Maps for r. https://doi.org/10.5281/zenodo.4678327.

Kaye, Matt, Bob Rudis, Andrie de Vries, and Jonathan Sidi. 2025. slackr: Send Messages, Images, r Objects and Files to “Slack” Channels/Users. https://github.com/mrkaye97/slackr.

Kourounis, D., A. Fuchs, and O. Schenk. 2018. “Towards the Next Generation of Multiperiod Optimal Power Flow Solvers.” IEEE Transactions on Power Systems PP (99): 1–10. https://doi.org/10.1109/TPWRS.2017.2789187.

Kuang, Kevin, Quyu Kong, and Francesco Napolitano. 2022. pbmcapply: Tracking the Progress of Mc*pply with Progress Bar. https://doi.org/10.32614/CRAN.package.pbmcapply.

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Lindgren, Finn, and Håvard Rue. 2015. “Bayesian Spatial Modelling with R-INLA.” Journal of Statistical Software 63 (19): 1–25. http://www.jstatsoft.org/v63/i19/.

Lindgren, Finn, Håvard Rue, and Johan Lindström. 2011. “An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach (with Discussion).” Journal of the Royal Statistical Society B 73 (4): 423–98.

Maechler, Martin, Christophe Dutang, and Vincent Goulet. 2024. expm: Matrix Exponential, Log, “etc”. https://doi.org/10.32614/CRAN.package.expm.

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Novomestky, Frederick. 2022. orthopolynom: Collection of Functions for Orthogonal and Orthonormal Polynomials. https://doi.org/10.32614/CRAN.package.orthopolynom.

Onkelinx, Thierry, and Victor Teh. 2024. qrcode: Generate QRcodes with r. Version 0.3.0. https://doi.org/10.5281/zenodo.5040088.

Pedersen, Thomas Lin. 2025. patchwork: The Composer of Plots. https://doi.org/10.32614/CRAN.package.patchwork.

Qiu, Yixuan, and Jiali Mei. 2024. RSpectra: Solvers for Large-Scale Eigenvalue and SVD Problems. https://doi.org/10.32614/CRAN.package.RSpectra.

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———. 2025b. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

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Rue, Håvard, Andrea I. Riebler, Sigrunn H. Sørbye, Janine B. Illian, Daniel P. Simpson, and Finn K. Lindgren. 2017. “Bayesian Computing with INLA: A Review.” Annual Reviews of Statistics and Its Applications 4 (March): 395–421. http://arxiv.org/abs/1604.00860.

Sievert, Carson. 2020. Interactive Web-Based Data Visualization with r, Plotly, and Shiny. Chapman; Hall/CRC. https://plotly-r.com.

Thorne, W. Brent. 2019. posterdown: An r Package Built to Generate Reproducible Conference Posters for the Academic and Professional World Where Powerpoint and Pages Just Won’t Cut It. https://github.com/brentthorne/posterdown.

Ushey, Kevin, JJ Allaire, and Yuan Tang. 2025. reticulate: Interface to “Python”. https://doi.org/10.32614/CRAN.package.reticulate.

Ushey, Kevin, and Hadley Wickham. 2025. renv: Project Environments. https://rstudio.github.io/renv/.

Van Boxtel, G.J.M., et al. 2021. gsignal: Signal Processing. https://github.com/gjmvanboxtel/gsignal.

Verbosio, Fabio, Arne De Coninck, Drosos Kourounis, and Olaf Schenk. 2017. “Enhancing the Scalability of Selected Inversion Factorization Algorithms in Genomic Prediction.” Journal of Computational Science 22 (Supplement C): 99–108. https://doi.org/10.1016/j.jocs.2017.08.013.

Wickham, Hadley. 2007. “Reshaping Data with the reshape Package.” Journal of Statistical Software 21 (12): 1–20. http://www.jstatsoft.org/v21/i12/.

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