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Cross-wavelet analysis allows obtaining high temporal and frequency resolution in heart rate synchrony analysis.

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  1. [1] § Principles of cross-wavelet power analysis ↔ CrossWaveletPower_RScript.Rmd, lines 161–230 · score 0.80 · imaginary part, mother wavelet, Morlet wavelet, daughter wavelet, sine waves, Wavelet transform
  2. [2] § Principles of cross-wavelet power analysis ↔ CrossWaveletPower_RScript.Rmd, lines 76–159 · score 0.64 · widely known Fourier, Fourier transform, frequency components, Wavelet transform, amplitudes, power
  3. [3] § Principles of cross-wavelet power analysis ↔ CrossWaveletPower_RScript.Rmd, lines 161–230 · score 0.57 · shifted version, Morlet wavelet, daughter wavelet, compressed

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The authors' code

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  1. ---
  2. title: "R Script for the manuscript ''Cross-wavelet power analysis allows obtaining high temporal and spatial resolution in physiological synchrony analysis"
  3. output: html_document
  4. date: "2026-04-20"
  5. ---
  7. ```{r setup, include=FALSE}
  8. knitr::opts_chunk$set(echo = TRUE)
  9. ```
  11. ## Synchrony analysis using cross-wavelet power
  13. Simulated data to visualize different types of synchrony using cross-wavelet power plots, compared to cross-correlation analysis.
  15. ### Load libraries and set working directory:
  17. ```{r lib}
  18. setwd("/home/bernadette/Documents/Papers/Wavelet_paper")
  19. library(WaveletComp)
  20. library(ggplot2)
  21. library(SUSY)
  22. library(grid)
  24. set.seed(100)
  25. ```
  27. ### Different synchrony types
  29. Interpersonal synchrony is defined as the correspondence between individuals' trajectories,which can take on different forms: in-phase, anti-phase, and shifted synchrony are basic synchrony types.
  31. ```{r synctype_examples}
  32. # base time series
  33. ts1 <- sin(seq(1,20, length.out = 1000))
  34. # in-phase synchrony
  35. ts2 <- ts1-0.6
  36. # anti-phase synchrony
  37. ts3 <- -ts1-0.2
  38. # shifted synchrony (person 1 leading)
  39. ts4 <- cos(seq(1, 20, length.out = 1000))-0.2
  40. # shifted synchrony (person 1 following)
  41. ts5 <- -cos(seq(1, 20, length.out = 1000))-0.2
  42. # in-phase synchrony with higher-frequency signal
  43. ts6 <- sin(seq(1,50, length.out = 1000))
  44. ts7 <- ts6-0.6
  45. # change from synchronized to de-synchronized
  46. ts8 <- periodic.series(start.period = 330,
  47. length = 1000, phase = 54,
  48. end.period = 115)-0.6
  50. # data frame to store time series combinations
  51. combs_df <- data.frame(ts_a = c(rep("ts1", 4), "ts6", "ts1"),
  52. ts_b = c(paste("ts", 2:5, sep = ""),
  53. "ts7", "ts8"))
  54. df <- data.frame(index = 1:1000,
  55. ts1 = ts1, ts2 = ts2, ts3 = ts3,
  56. ts4 = ts4, ts5 = ts5,
  57. ts6 = ts6, ts7 = ts7,
  58. ts8 = ts8)
  60. # Plot all relevant combinations of time series
  61. par(mfrow = c(4,2))
  62. plt_list1 <- list()
  63. for (comb in 1:nrow(combs_df)){
  64. ts_a_comb <- combs_df$ts_a[comb]
  65. ts_b_comb <- combs_df$ts_b[comb]
  66. plot(df$index, df[, ts_a_comb],
  67. type = "l", xlab = "Time",
  68. ylab = "Marker", ylim = c(-2, 2),
  69. cex.lab=1.5, cex.axis=1.5, cex.main = 2, col = "darkgrey",
  70. lwd = 1.5, main = LETTERS[comb])
  71. lines(df$index, df[, ts_b_comb], col = "#00A9E0",
  72. lwd = 1.5)
  73. }
  74. ```
  76. ### Principles of cross-wavelet power
  78. To understand cross-wavelet power, we start with the principle behind wavelet transform. Wavelet transform is similar to the more widely-known Fourier transform. In both analyses, a signal is decomposed into its frequency components.
  80. ```{r fourier_example}
  81. # Define periodic time series with different frequencies
  82. x1 <- periodic.series(start.period = 80, length = 1000)
  83. x2 <- periodic.series(start.period = 30, length = 1000)
  84. x3 <- periodic.series(start.period = 100, length = 1000)
  85. x4 <- periodic.series(start.period = 10, length = 1000)
  86. # Add x1-x4
  87. x_add <- 3*x1 + 1.4*x2 + 0.7*x3 + 1*x4
  89. # Plot the underlying frequency components and the
  90. # added resulting time series
  91. layout(matrix(c(0,0,0,0,0,0,
  92. 0,1,0,2,0,5,
  93. 0,0,0,0,0,5,
  94. 0,3,0,4,0,5,
  95. 0,0,0,0,0,0),
  96. 5, 6, byrow=TRUE),
  97. heights = c(0.4,1.1,0.1,1.1,0.1),
  98. widths = c(0.05,0.5,0.05,0.5,0.05, 0.7))
  99. par(mar = c(2.5, 2.5, 1, 1), oma = c(0, 0, 0, 0))
  100. plot(1:1000, 3*x1, type = "l", col = "black", lwd = 1,
  101. main = "",
  102. ylab = "Amplitude", xlab = "Time")
  103. par(xpd = TRUE)
  104. mtext("A", side = 3, line = 1, adj = -0.1, col = "grey40",
  105. padj = 0.1, cex = 1.5)
  106. par(xpd = FALSE)
  107. par(mar = c(2.5, 2.5, 1, 1))
  108. plot(1:1000, 1.4*x2, type = "l", col = "black", lwd = 1,
  109. main = "", ylim = c(-3,3),
  110. ylab = "Amplitude", xlab = "Time")
  111. mtext("B", side = 3, line = 1, adj = -0.1, col = "grey40",
  112. padj = 0.1, cex = 1.5)
  113. par(xpd = FALSE)
  114. par(mar = c(2.5, 2.5, 1, 1))
  115. plot(1:1000, 0.7*x3, type = "l", col = "black", lwd = 1,
  116. main = "", ylim = c(-3,3),
  117. ylab = "Amplitude", xlab = "Time")
  118. par(mar = c(2.5, 2.5, 1, 1))
  119. par(xpd = TRUE)
  120. mtext("C", side = 3, line = 1, adj = -0.05, col = "grey40",
  121. padj = -0.1, cex = 1.5)
  122. par(xpd = FALSE)
  123. par(mar = c(2.5, 2.5, 1, 1))
  124. plot(1:1000, 1*x4, type = "l", col = "black", lwd = 1,
  125. main = "",
  126. ylab = "Amplitude", xlab = "Time",
  127. ylim = c(-3,3))
  128. par(mar = c(2.5, 2.5, 1, 1))
  129. par(xpd = TRUE)
  130. mtext("D", side = 3, line = 1, adj = -0.05, col = "grey40",
  131. padj = -0.1, cex = 1.5)
  132. par(xpd = FALSE)
  133. par(mar = c(2.5, 2.5, 1, 1))
  134. plot(1:1000, x_add, type = "l", col = "black", lwd = 1,
  135. main = "",
  136. ylab = "Amplitude", xlab = "Time")
  137. par(mar = c(2.5, 2.5, 1, 1))
  138. par(xpd = TRUE)
  139. mtext("E", side = 3, line = 1, adj = -0.05, col = "grey40",
  140. padj = -0.1, cex = 1.5)
  141. par(xpd = FALSE)
  143. # Demonstrate the decomposition into frequency bands
  144. # 1. using Fourier transform
  145. spec.pgram(x_add,
  146. taper = 0.1,
  147. log = "no",
  148. xlab = "Frequency (Hz)",
  149. ylab = "Power",
  150. main = "Periodogram", xlim = c(0,0.2))
  151. # -> the peaks correspond to the frequencies used in the data
  152. # 2. using wavelet transform
  153. x_add_df <- data.frame(x_add = x_add)
  154. wp_x_add <- analyze.wavelet(x_add_df, loess.span = 0,
  155. verbose = F)
  156. wt.image(wp_x_add)
  157. # -> the red lines correspond to the frequencies used in the data
  159. ```
  161. Instead of decomposing the signal into sine waves, wavelet transform decomposes the signal into daughter wavelets. Below are the Morlet wavelet and some exemplary daughter wavelets.
  163. ```{r wavelets}
  164. # Define a function to create the wavelet
  165. morlet_func <- function(t, omega0 = 6) {
  166. pi^(-1/4) * exp(1i * omega0 * t) * exp(-0.5 * t^2)
  167. }
  168. # Cycling frequency set to 6
  169. omega0 = 6
  170. # Time vector
  171. t <- seq(-5, 5, length.out = 500)
  172. # Morlet mother wavelet
  173. psi <- morlet_func(t)
  174. # Compressed daugther wavelet
  175. # Scaling factor 0.5
  176. s = 0.5
  177. psi_compr <- pi^(-1/4) *
  178. (1 / sqrt(s)) *
  179. exp(1i * 6 * (t / s)) *
  180. exp(-0.5 * (t / s)^2)
  182. # Shifted version
  183. t2 <- seq(-5, 10, length.out = 550)
  184. psi2 <- morlet_func(t2)
  185. psi_shftd <-
  186. pi^(-1/4) * exp(1i * omega0 * (t2-4)) * exp(-0.5 * (t2-4)^2)
  188. # Plot the different wavelets
  189. layout(matrix(c(0,0,0,0,0,
  190. 0,1,0,2,0,
  191. 0,1,0,0,0,
  192. 0,1,0,3,0,
  193. 0,0,0,0,0),
  194. 5, 5, byrow=TRUE),
  195. heights = c(0.4,1.1,0.1,1.1,0.1),
  196. widths = c(0.05,0.5,0.05,0.5,0.05))
  197. par(mar = c(2.5, 2.5, 1, 1), oma = c(0, 0, 0, 0))
  198. plot(t, Im(psi), type = "l", col = "grey", lwd = 2,
  199. main = "",
  200. ylab = "Amplitude", xlab = "Time",
  201. ylim = c(-1.5, 1.5))
  202. lines(t, Re(psi), col = "black", lwd = 2)
  203. legend("topright", legend = c("Real part", "Imaginary part"),
  204. col = c("black", "grey"), lty = 1, lwd = 2)
  205. par(xpd = TRUE)
  206. mtext("A", side = 3, line = 1, adj = -0.1, col = "grey40",
  207. padj = 0.1, cex = 2)
  208. par(xpd = FALSE)
  209. par(mar = c(2.5, 2.5, 1, 1))
  210. plot(t, Re(psi), type = "l", col = "black", lwd = 2,
  211. main = "",
  212. ylab = "Amplitude", xlab = "Time",
  213. ylim = c(-1.5, 1.5))
  214. lines(t, Re(psi_compr), col = "#00A9E0", lwd = 2)
  215. mtext("B", side = 3, line = 1, adj = -0.1, col = "grey40",
  216. padj = 0.1, cex = 2)
  217. par(xpd = FALSE)
  218. par(mar = c(2.5, 2.5, 1, 1))
  219. plot(t2, Re(psi2), type = "l", col = "black", lwd = 2,
  220. main = "",
  221. ylab = "Amplitude", xlab = "Time",
  222. xlim = c(-5, 9), ylim = c(-1.5, 1.5))
  223. lines(t2, Re(psi_shftd), col = "#00A9E0", lwd = 2)
  224. par(mar = c(2.5, 2.5, 1, 1))
  225. par(xpd = TRUE)
  226. mtext("C", side = 3, line = 1, adj = -0.05, col = "grey40",
  227. padj = -0.1, cex = 2)
  228. par(xpd = FALSE)
  230. ```
  232. Cross-wavelet power is the overlap between two wavelet-transformed signals.
  234. ```{r create_wt_to_cwp}
  235. # Add noise to each time series (newly generated every time)
  236. noise_func <- function(sd = 1){
  237. 0.2*rnorm(1000, sd = sd)
  238. }
  239. # Simulate time series with increasing and decreasing period, respectively
  240. ts1 <- periodic.series(start.period = 2, end.period = 64,
  241. length = 1000)+noise_func()
  242. ts2 <- periodic.series(start.period = 64, end.period = 2,
  243. length = 1000)+noise_func()
  245. # Combine into a data frame
  246. df1 <- data.frame(time = 1:1000, ts1 = ts1,
  247. ts2 = ts2)
  249. # Calculate wavelet transform of each time series and cross-wavelet power
  250. w1 <- analyze.coherency(df1, c("ts1", "ts2"), verbose = F)
  252. # Plot the raw and wavelet-transformed time series as well as the cross-wavelet power
  253. layout(matrix(c(0,0,0,0,0,
  254. 0,1,0,0,0,
  255. 0,2,0,5,0,
  256. 0,0,0,5,0,
  257. 0,3,0,5,0,
  258. 0,4,0,0,0,
  259. 0,0,0,0,0),
  260. 7, 5, byrow=TRUE),
  261. heights = c(0.1,1.1, 1.1, 0.1,1.1, 1.1,0.1),
  262. widths = c(0.05,0.5,0.05, 0.5,0.05,0.05))
  263. par(mar = c(2.5, 2.5, 1, 1), oma = c(0, 0, 0, 0))
  264. plot(ts1, type = "l", cex.axis = 1.5)
  265. par(xpd = TRUE)
  266. mtext("A", side = 3, line = 1, adj = -0.1, col = "grey40",
  267. padj = 1.6, cex = 2)
  268. par(xpd = FALSE)
  269. par(mar = c(2.5, 2.5, 1, 1))
  270. wt.image(w1, my.series = 1, color.key = "i", graphics.reset = FALSE)
  271. par(mar = c(2.5, 2.5, 1, 1))
  272. plot(ts2, type = "l", cex.axis = 1.5)
  273. par(xpd = TRUE)
  274. mtext("B", side = 3, line = 1, adj = -0.1, col = "grey40",
  275. padj = 1.6, cex = 2)
  276. par(xpd = FALSE)
  277. par(mar = c(2.5, 2.5, 1, 1))
  278. wt.image(w1, my.series = 2, color.key = "i", graphics.reset = FALSE)
  279. par(mar = c(2.5, 2.5, 1, 1))
  280. wc.image(w1, color.key = "i", graphics.reset = FALSE)
  281. grid.rect(x = 0.25, y = 0.74, width = 0.48, height = 0.48,
  282. gp = gpar(col = "grey40", lwd = 1.3, fill = NA))
  283. grid.rect(x = 0.25, y = 0.25, width = 0.48, height = 0.48,
  284. gp = gpar(col = "grey40", lwd = 1.3, fill = NA))
  285. grid.rect(x = 0.73, y = 0.53, width = 0.47, height = 0.55,
  286. gp = gpar(col = "grey40", lwd = 1.3, fill = NA))
  287. par(xpd = TRUE)
  288. mtext("C", side = 3, line = 1, adj = -0.05, col = "grey40",
  289. padj = -0.1, cex = 2)
  290. par(xpd = FALSE)
  291. ```
  293. ### Synchrony analysis of simulated data
  295. Next, we apply cross-wavelet power and cross-correlational analysis to simulated data.
  297. ```{r data_sim}
  298. # Simulate time series
  299. ts1 = periodic.series(start.period = 64,
  300. length = 1000)+noise_func()
  301. # TS2: in-phase synchrony
  302. ts2 = ts1-0.9+noise_func()
  303. # TS3: antiphase synchrony
  304. ts3 = -ts1-0.2+ noise_func()
  305. # TS4: shifted synchrony
  306. ts4 = periodic.series(start.period = 64, phase = -20,
  307. length = 1000)+noise_func()
  308. # TS5: non-synchronous time series
  309. ts5 = periodic.series(start.period = 15, phase = -20,
  310. length = 1000)+noise_func()
  312. # Store time series in data frame
  313. df <- data.frame(time = 1:1000, ts1 = ts1,
  314. ts2 = ts2, ts3 = ts3,
  315. ts4 = ts4, ts5 = ts5)
  316. ```
  318. Calculate cross-wavelet power for the simulated time series
  320. ```{r types_cwp}
  321. wc1 <- analyze.coherency(df, c("ts1", "ts2"), verbose = F,
  322. lowerPeriod = 8, upperPeriod = 128)
  323. wc.image(wc1, color.key = "i")
  324. wc2 <- analyze.coherency(df, c("ts1", "ts3"), verbose = F,
  325. lowerPeriod = 8, upperPeriod = 128)
  326. wc.image(wc2, color.key = "i")
  327. wc3 <- analyze.coherency(df, c("ts1", "ts4"), verbose = F,
  328. lowerPeriod = 8, upperPeriod = 128)
  329. wc.image(wc3, color.key = "i")
  330. wc4 <- analyze.coherency(df, c("ts1", "ts5"), verbose = F,
  331. lowerPeriod = 8, upperPeriod = 128)
  332. wc.image(wc4, color.key = "i")
  333. ```
  335. Comparison to SUSY:
  337. Customized SUSY function (only plot title is changed).
  339. ```{r plot_susy_function}
  340. plot_susy_custom <- function (x, type = c(4, 5), ...)
  341. {
  342. if (!inherits(x, "susy"))
  343. stop("'x' must be an object of class 'susy'")
  344. if (!is.numeric(type))
  345. stop("'type' argument must be numeric")
  346. type = as.integer(type)
  347. if (anyNA(type) || any(type > 5L) || any(type < 1L))
  348. stop("'type' values must be in range of 1 to 5")
  349. if (anyDuplicated(type))
  350. stop("'type' values must be unique")
  351. first.plot = TRUE
  352. plot1 = function(x) {
  353. a = x$data[[1L]]
  354. b = x$data[[2L]]
  355. variablenname1 = names(x$data)[1L]
  356. variablenname2 = names(x$data)[2L]
  357. size = x$params$size
  358. segment = x$params$segment
  359. anzahlPseudosProEpoche = x$params$anzahlPseudosProEpoche
  360. numberEpochen = x$params$numberEpochen
  361. maxlagHz = x$params$maxlagHz
  362. meanccorrReal = x$lagtimes2.data$meanccorrReal
  363. meanccorrPseudo = x$lagtimes2.data$meanccorrPseudo
  364. meanccorrRealZ = x$lagtimes2.data$meanccorrRealZ
  365. meanccorrPseudoZ = x$lagtimes2.data$meanccorrPseudoZ
  366. meanccorrRealZNotAbs = x$lagtimes2.data$meanccorrRealZNotAbs
  367. meanccorrPseudoZNotAbs = x$lagtimes2.data$meanccorrPseudoZNotAbs
  368. nReal = x$segment.data$nReal
  369. nPseudo = x$segment.data$nPseudo
  370. for (t in type) {
  371. if (!first.plot)
  372. dev.new()
  373. else first.plot <<- FALSE
  374. if (t == 1L) {
  375. min0 = min(meanccorrReal, meanccorrPseudo)
  376. max0 = max(meanccorrReal, meanccorrPseudo)
  377. title0 = ""
  378. plot(seq(from = -maxlagHz, to = maxlagHz), meanccorrReal,
  379. ylim = c(min0, max0 + 0.19 * (max0 - min0)),
  380. main = title0, xlab = "lag", ylab = "correlation",
  381. type = "l", col = "green", lwd = 4)
  382. lines(seq(from = -maxlagHz, to = maxlagHz), meanccorrPseudo,
  383. col = "red", lwd = 4)
  384. legend("topright", pch = c(3), col = c("green",
  385. "red"), legend = c("meanccorr", "meanccorr pseudo"))
  386. }
  387. else if (t == 2L) {
  388. title0 = ""
  389. plot(seq(from = 1, to = numberEpochen), nReal,
  390. ylim = c(min(nReal, nPseudo), max(nReal, nPseudo)),
  391. main = title0, xlab = "segment", ylab = "correlation",
  392. type = "l", col = "green")
  393. lines(seq(from = 1, to = numberEpochen), nPseudo,
  394. col = "red", lwd = 2)
  395. legend("topright", pch = c(3), col = c("green",
  396. "red"), legend = c("real", "pseudo"))
  397. }
  398. else if (t == 3L) {
  399. min0 = min(meanccorrRealZ, meanccorrPseudoZ)
  400. max0 = max(meanccorrRealZ, meanccorrPseudoZ)
  401. title0 = ""
  402. plot(seq(from = -maxlagHz, to = maxlagHz), meanccorrRealZ,
  403. ylim = c(min0, max0 + 0.19 * (max0 - min0)),
  404. main = title0, xlab = "lag", ylab = "correlation",
  405. type = "l", col = "green", lwd = 4)
  406. lines(seq(from = -maxlagHz, to = maxlagHz), meanccorrPseudoZ,
  407. col = "red", lwd = 4)
  408. legend("topright", pch = c(3), col = c("green",
  409. "red"), legend = c("Z(meanccorr)", "Z(meanccorr pseudo)"))
  410. }
  411. else if (t == 4L) {
  412. title0 = ""
  413. plot(seq(from = 1, to = size), a, ylim = c(min(a,
  414. b), max(a, b)), main = "", xlab = "Zeit",
  415. ylab = "Wert", type = "l", pch = 20, col = "green")
  416. points(seq(from = 1, to = size), b, type = "l",
  417. pch = 20, col = "red", lwd = 2)
  418. legend("topright", pch = c(20, 20), col = c("green",
  419. "red"), legend = c(variablenname1, variablenname2))
  420. }
  421. else if (t == 5L) {
  422. min0 = min(meanccorrRealZNotAbs, meanccorrPseudoZNotAbs)
  423. max0 = max(meanccorrRealZNotAbs, meanccorrPseudoZNotAbs)
  424. title0 = ""
  425. plot(seq(from = -maxlagHz, to = maxlagHz), meanccorrRealZNotAbs,
  426. ylim = c(min0, max0 + 0.19 * (max0 - min0)),
  427. main = title0, xlab = "lag", ylab = "correlation",
  428. type = "l", col = "green", lwd = 4)
  429. lines(seq(from = -maxlagHz, to = maxlagHz), meanccorrPseudoZNotAbs,
  430. col = "red", lwd = 4)
  431. legend("topright", pch = c(3), col = c("green",
  432. "red"), legend = c("real", #"Z(meanccorr not ABS)",
  433. "pseudo"))#"Z(meanccorr pseudo not ABS)"))
  434. }
  435. }
  436. }
  437. lapply(x, plot1)
  438. invisible()
  439. }
  441. ```
  443. Make data frame compatible with SUSY package:
  445. ```{r susy_example_1}
  446. susy_df1 <- data.frame(ts1 = ts1,
  447. ts2 = ts2,
  448. ts3 = ts3,
  449. ts4 = ts4,
  450. ts5 = ts5)
  451. ```
  453. Plot of simulated data with cross-wavelet power and cross-correlation outcomes:
  455. ```{r plot_cwp_susy_results}
  456. layout(matrix(c(0,0,0,0,0,0,0,
  457. 0,1,0,5,0,9,0,
  458. 0,2,0,6,0,10,0,
  459. 0,3,0,7,0,11,0,
  460. 0,4,0,8,0,12,0,
  461. 0,0,0,0,0,0,0),
  462. 6, 7, byrow=TRUE),
  463. heights = c(0.1,1.1, 1.1, 1.1,1.1,0.1),
  464. widths = c(0.05,0.5,0.05, 0.5,0.05,0.5,0.05))
  465. par(mar = c(2.5, 2.5, 1, 1), oma = c(0, 0, 0, 0))
  466. plot(ts1, type = "l",
  467. ylim = c(min(c(ts1, ts2)),
  468. max(c(ts1, ts2))))
  469. lines(ts2, col = "#00A9E0")
  470. par(xpd = TRUE)
  471. mtext("A", side = 3, line = 1, adj = -0.2, col = "grey40",
  472. padj = 1.6, cex = 1.6)
  473. par(xpd = FALSE)
  474. par(mar = c(2.5, 2.5, 1, 1))
  475. plot(ts1, type = "l",
  476. ylim = c(min(c(ts1, ts3)),
  477. max(c(ts1, ts3))))
  478. lines(ts3, col = "#00A9E0")
  479. par(xpd = TRUE)
  480. mtext("B", side = 3, line = 1, adj = -0.2, col = "grey40",
  481. padj = 1.6, cex = 1.6)
  482. par(xpd = FALSE)
  483. par(mar = c(2.5, 2.5, 1, 1))
  484. plot(ts1, type = "l",
  485. ylim = c(min(c(ts1, ts4)),
  486. max(c(ts1, ts4))))
  487. lines(ts4, col = "#00A9E0")
  488. par(xpd = TRUE)
  489. mtext("C", side = 3, line = 1, adj = -0.2, col = "grey40",
  490. padj = 1.6, cex = 1.6)
  491. par(xpd = FALSE)
  492. par(mar = c(2.5, 2.5, 1, 1))
  493. plot(ts1, type = "l",
  494. ylim = c(min(c(ts1, ts5)),
  495. max(c(ts1, ts5))))
  496. lines(ts5, col = "#00A9E0")
  497. par(xpd = TRUE)
  498. mtext("D", side = 3, line = 1, adj = -0.2, col = "grey40",
  499. padj = 1.6, cex = 1.6)
  500. par(xpd = FALSE)
  501. par(mar = c(2.5, 2.5, 1, 1))
  502. wc.image(wc1, color.key = "i", graphics.reset = FALSE)
  503. par(mar = c(2.5, 2.5, 1, 1))
  504. wc.image(wc2, color.key = "i", graphics.reset = FALSE)
  505. par(mar = c(2.5, 2.5, 1, 1))
  506. wc.image(wc3, color.key = "i", graphics.reset = FALSE)
  507. par(mar = c(2.5, 2.5, 1, 1))
  508. wc.image(wc4, color.key = "i", graphics.reset = FALSE)
  509. par(xpd = FALSE)
  510. par(mar = c(2.5, 2.5, 1, 1))
  511. plot_susy_custom(susy(susy_df1[, c(1,2)], segment = 60, Hz = 1, maxlag = 25), type = 5)
  512. par(mar = c(2.5, 2.5, 1, 1))
  513. plot_susy_custom(susy(susy_df1[, c(1,3)], segment = 60, Hz = 1, maxlag = 25), type = 5)
  514. par(mar = c(2.5, 2.5, 1, 1))
  515. plot_susy_custom(susy(susy_df1[, c(1,4)], segment = 60, Hz = 1, maxlag = 25), type = 5)
  516. par(mar = c(2.5, 2.5, 1, 1))
  517. plot_susy_custom(susy(susy_df1[, c(1,5)], segment = 60, Hz = 1, maxlag = 25), type = 5)
  518. ```
  520. Next, we consider simulated data containing changes in synchrony over time, specifically:
  522. - changes from non-synchronized to synchronized
  524. - changes from synchronized on 1 frequency band to synchronized on 2 frequency bands
  526. - changes from synchronized on some frequency bands to synchronized on other frequency bands
  528. ```{r cwp_temp}
  529. # define base TS
  530. tsb <- periodic.series(start.period = 64,
  531. length = 1000)+
  532. periodic.series(start.period = 16,
  533. length = 1000)+
  534. periodic.series(start.period = 4,
  535. length = 1000)+
  536. noise_func()
  538. # changes from non-synchronized to synchronized
  539. ts6 <- periodic.series(start.period = 30,
  540. end.period = 64,
  541. length = 1000)+
  542. periodic.series(start.period = 8,
  543. end.period = 16,
  544. length = 1000)+
  545. periodic.series(start.period = 2,
  546. end.period = 4,
  547. length = 1000)+
  548. noise_func()
  550. # changes from synchronized on 1 frequency band to synchronized on 2 frequency bands
  551. ts7 <- periodic.series(start.period = 64,
  552. length = 1000)+
  553. periodic.series(start.period = 8,
  554. length = 1000)+
  555. periodic.series(start.period = 2,
  556. end.period = 4,
  557. length = 1000)+
  558. noise_func()
  560. # changes from synchronized on some frequency bands to synchronized on other frequency bands
  561. # define other base time series
  562. tsb_2 <- periodic.series(start.period = 64,
  563. end.period = 16,
  564. length = 1000)+
  565. periodic.series(start.period = 2,
  566. end.period = 2,
  567. length = 1000)+
  568. noise_func()
  569. # define synchronized time series
  570. ts8 <- periodic.series(start.period = 64,
  571. end.period = 16,
  572. length = 1000)+
  573. periodic.series(start.period = 8,
  574. length = 1000)+
  575. periodic.series(start.period = 4,
  576. end.period = 4,
  577. length = 1000)+
  578. noise_func()
  580. # Store simulated data in data frame
  581. df_time <- data.frame(time = 1:1000,
  582. tsb = tsb,
  583. tsb_2 = tsb_2,
  584. ts6 = ts6,
  585. ts7 = ts7,
  586. ts8 = ts8)
  588. # Calculate cross-wavelet power
  589. wc5 <- analyze.coherency(df_time, c("tsb", "ts6"),
  590. verbose = F)
  591. wc6 <- analyze.coherency(df_time, c("tsb", "ts7"),
  592. verbose = F)
  593. wc7 <- analyze.coherency(df_time, c("tsb_2", "ts8"),
  594. verbose = F)
  595. ```
  597. ```{r ccf_examples}
  598. # Store the simulated data in a data frame suitable for analysis with SUSY
  599. susy_df <- data.frame(f1 = tsb,
  600. f2 = ts6,
  601. f1 = tsb,
  602. f3 = ts7,
  603. f4 = tsb_2,
  604. f5 = ts8)
  605. ```
  607. ```{r cwp_ccf_time_freq_plot}
  608. # Plot synchrony changes over time using cross-wavelet power and cross-corelation analyses
  609. layout(matrix(c(0,0,0,0,0,0,0,
  610. 0,1,0,4,0,7,0,
  611. 0,2,0,5,0,8,0,
  612. 0,3,0,6,0,9,0,
  613. 0,0,0,0,0,0,0),
  614. 5, 7, byrow=TRUE),
  615. heights = c(0.1,1.1, 1.1, 1.1,1.1,0.1),
  616. widths = c(0.05,0.5,0.05, 0.5,0.05,0.5, 0.05))
  617. par(mar = c(2.5, 2.5, 1, 1), oma = c(0, 0, 0, 0))
  618. plot(tsb, type = "l",
  619. ylim = c(min(c(tsb, ts6)),
  620. max(c(tsb, ts6))))
  621. lines(ts6, col = "#00A9E0")
  622. par(xpd = TRUE)
  623. mtext("A", side = 3, line = 1, adj = -0.2, col = "grey40",
  624. padj = 1.6, cex = 1.6)
  625. par(xpd = FALSE)
  626. par(mar = c(2.5, 2.5, 1, 1))
  627. plot(tsb, type = "l",
  628. ylim = c(min(c(tsb, ts7)),
  629. max(c(tsb, ts7))))
  630. lines(ts7, col = "#00A9E0")
  631. par(xpd = TRUE)
  632. mtext("B", side = 3, line = 1, adj = -0.2, col = "grey40",
  633. padj = 1.6, cex = 1.6)
  634. par(xpd = FALSE)
  635. par(mar = c(2.5, 2.5, 1, 1))
  636. plot(tsb_2, type = "l",
  637. ylim = c(min(c(tsb_2, ts8)),
  638. max(c(tsb_2, ts8))))
  639. lines(ts8, col = "#00A9E0")
  640. par(xpd = TRUE)
  641. mtext("C", side = 3, line = 1, adj = -0.2, col = "grey40",
  642. padj = 1.6, cex = 1.6)
  643. par(xpd = FALSE)
  644. par(mar = c(2.5, 2.5, 1, 1))
  645. wc.image(wc5, color.key = "i", graphics.reset = FALSE)
  646. par(mar = c(2.5, 2.5, 1, 1))
  647. wc.image(wc6, color.key = "i", graphics.reset = FALSE)
  648. par(mar = c(2.5, 2.5, 1, 1))
  649. wc.image(wc7, color.key = "i", graphics.reset = FALSE)
  650. par(xpd = FALSE)
  651. par(mar = c(2.5, 2.5, 1, 1))
  652. plot_susy_custom(susy(susy_df[, c(1,2)], segment = 60, Hz = 1,
  653. maxlag = 25),
  654. type = 2)
  655. par(mar = c(2.5, 2.5, 1, 1))
  656. plot_susy_custom(susy(susy_df[, c(3,4)], segment = 60, Hz = 1,
  657. maxlag = 25),
  658. type = 2)
  659. par(mar = c(2.5, 2.5, 1, 1))
  660. plot_susy_custom(susy(susy_df[, c(5,6)], segment = 60, Hz = 1,
  661. maxlag = 25),
  662. type = 2)
  663. ```

CrossWaveletPower_RScript.Rmd at commit 3b960c7, no license · at the source

Overview

Authors: Bernadette F. Denk1,2, Stella Wienhold1,2, Nina Volkmer1,2, Nikolaus F. Troje3, Maria Meier1,4, Jens C. Pruessner1,2
ORCID iDs: Jens C. Pruessner
  1. Department of Psychology, University of Konstanz, Konstanz, Germany
  2. Centre for the Advanced Study of Collective Behaviour, University of Konstanz, Konstanz, Germany
  3. Department of Biology, Centre for Vision Research, York University, Toronto, ON, Canada
  4. Department of Child and Adolescent Psychiatry/Psychotherapy, University of Ulm, Ulm, Germany
Institutions: University of Konstanz (Germany); York University (Canada); University of York (United Kingdom); Universität Ulm (Germany)
Journal: Frontiers in network physiology, volume 6, article 1869004
Dates: received 29 April 2026; accepted 8 June 2026; published online 8 July 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.3389/fnetp.2026.1869004 · PMID 42488130 · PMCID PMC13389767 · OpenAlex W7167712128
Open access: gold, a free copy (OpenAlex)
Status: code verified
Categories: systems (subfield)
Methods: Connectivity, Machine learning, Preprocessing, Spectral & time-frequency
Keywords: autonomic nervous system, cross-correlation, cross-wavelet power, interpersonal synchrony, network physiology, social interaction, synchrony methods, time-frequency analysis
Journal subjects: Technology and Code
Topic: Action Observation and Synchronization (Social Psychology, Psychology), according to OpenAlex
Funding: Deutsche Forschungsgemeinschaft (DFG)
Citations: not cited yet (Europe PMC); 55 references in the paper

Abstract

Interpersonal synchrony, the temporal correspondence of repeated behavioral, physiological, or neural measures between individuals, has been analyzed using a variety of different methods. However, the analysis method needs to be carefully chosen to ensure an adequate interpretation of the results. Here, we suggest using cross-wavelet power for interpersonal synchrony analysis due to several advantages. We demonstrate the proposed method using the example of heart rate synchrony. In cross-wavelet power analysis, synchrony is determined per frequency band and time point, allowing for both fine-grained frequency and temporal resolution. We argue that applying this approach to analyze synchrony in heart rate data provides additional information about underlying processes that may influence overall heart rate alignment between individuals. We describe the principles of cross-wavelet power analysis and compare the method to the frequently used cross-correlational approach using simulated and real data. Cross-correlation is a linear measure of the similarity between time series, which includes the quantification of leader-follower relationships. We illustrate different implications for which data series are considered to be synchronous and describe the advantages and drawbacks of using cross-wavelet analysis across various possible synchronization scenarios. The main advantage of cross-wavelet power is its high time- and frequency resolution, whereas cross-correlation is more suitable when researchers aim to differentiate between synchrony types. Finally, we provide recommendations for implementing cross-wavelet power analysis, including R code to facilitate the application to one’s own data.

Reproduced under the paper's license (CC BY), from the paper cited above.

Repository

Its files are read in the Code ↔ Paper reader above, with 3 matches between paragraphs and lines of code.

gitlab.inf.uni-konstanz.de/bernadette.denk/cwp_for_sync_analysis

License: none: the authors keep all their rights
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Commit: 3b960c7d8262fd4498ff2ffe6bdb1a134ca750a6, 20 April 2026
Languages: R (1)
Size: 2 files, 1 script
Software Heritage: not archived
Found in: “Data availability statement”
Holds: README, 1 notebook
Not found: license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: ggplot2 (1 file)
Availability: 1 check, the latest on 27 September 2026: the link answers
  • 27 September 2026: the link answers
2 files
At the source: gitlab.inf.uni-konstanz.de/bernadette.denk/cwp_for_sync_analysis

The paper's code and data availability statement is in the Data section.

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Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material. The code for data simulation and analysis is available under https://gitlab.inf.uni-konstanz.de/bernadette.denk/cwp_for_sync_analysis.

Reproduced under the paper's license (CC BY), from the paper cited above.

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Version 1, 27 September 2026: the first record

Recorded: type, language, journal, volume, pages, dates, 6 authors, 8 keywords, 1 funder, 49 references.

Cite

This paper

Denk, B. F., Wienhold, S., Volkmer, N., Troje, N. F., Meier, M., & Pruessner, J. C. (2026). Cross-wavelet analysis allows obtaining high temporal and frequency resolution in heart rate synchrony analysis. Frontiers in network physiology, 6, 1869004. https://doi.org/10.3389/fnetp.2026.1869004

BibTeX

@article{denk2026cross,
author = {Denk, Bernadette F. and Wienhold, Stella and Volkmer, Nina and Troje, Nikolaus F. and Meier, Maria and Pruessner, Jens C.},
title = {{Cross-wavelet analysis allows obtaining high temporal and frequency resolution in heart rate synchrony analysis}},
journal = {Frontiers in network physiology},
year = {2026},
month = jul,
volume = {6},
pages = {1869004},
publisher = {Frontiers Media SA},
issn = {2674-0109},
doi = {10.3389/fnetp.2026.1869004},
url = {https://doi.org/10.3389/fnetp.2026.1869004},
pmid = {42488130},
pmcid = {PMC13389767}
}

RIS

TY - JOUR
AU - Denk, Bernadette F.
AU - Wienhold, Stella
AU - Volkmer, Nina
AU - Troje, Nikolaus F.
AU - Meier, Maria
AU - Pruessner, Jens C.
TI - Cross-wavelet analysis allows obtaining high temporal and frequency resolution in heart rate synchrony analysis
T2 - Frontiers in network physiology
J2 - Front Netw Physiol
PY - 2026
DA - 2026/07/08
VL - 6
SP - 1869004
SN - 2674-0109
PB - Frontiers Media SA
DO - 10.3389/fnetp.2026.1869004
UR - https://doi.org/10.3389/fnetp.2026.1869004
LA - en
ER -

CSL-JSON

{
"id": "10.3389/fnetp.2026.1869004",
"type": "article-journal",
"title": "Cross-wavelet analysis allows obtaining high temporal and frequency resolution in heart rate synchrony analysis",
"container-title": "Frontiers in network physiology",
"author": [
{
"family": "Denk",
"given": "Bernadette F."
},
{
"family": "Wienhold",
"given": "Stella"
},
{
"family": "Volkmer",
"given": "Nina"
},
{
"family": "Troje",
"given": "Nikolaus F."
},
{
"family": "Meier",
"given": "Maria"
},
{
"family": "Pruessner",
"given": "Jens C."
}
],
"container-title-short": "Front Netw Physiol",
"volume": "6",
"page": "1869004",
"DOI": "10.3389/fnetp.2026.1869004",
"PMID": "42488130",
"PMCID": "PMC13389767",
"ISSN": "2674-0109",
"publisher": "Frontiers Media SA",
"URL": "https://doi.org/10.3389/fnetp.2026.1869004",
"language": "en",
"issued": {
"date-parts": [
[
2026,
7,
8
]
]
}
}

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