WASP:An R package for Wavelet System Prediction

library(WASP)
library(ggplot2)

library(FNN)
#> 
#> Attaching package: 'FNN'
#> The following object is masked from 'package:WASP':
#> 
#>     knn
library(synthesis)
#> 
#> Attaching package: 'synthesis'
#> The following objects are masked from 'package:WASP':
#> 
#>     data.gen.HL, data.gen.Rossler, data.gen.SW, data.gen.ar1,
#>     data.gen.ar4, data.gen.ar9, data.gen.tar1, data.gen.tar2
library(waveslim)
#> 
#> waveslim: Wavelet Method for 1/2/3D Signals (version = 1.8.2)
library(cowplot)
library(gridGraphics)
#> Loading required package: grid

1 DWT, MODWT and AT basic propertites

# data generation
x <- arima.sim(list(order = c(1, 0, 0), ar = 0.6), n = 512)
# x <- as.numeric(scale(data.gen.Rossler(time = seq(0, 50, length.out = 512))$x, scale=F))

# Daubechies wavelets
for (wf in c("haar", "d4", "d8", "d16")) {
  print(paste0("Wavelet filter: ", wf))
  #----------------------------------------------------------------------------
  # wavelet family, extension mode and package
  # wf <- "haar" # wavelet family D8 or db4
  boundary <- "periodic"
  if (wf != "haar") v <- as.integer(readr::parse_number(wf) / 2) else v <- 1

  # Maximum decomposition level J
  n <- length(x)
  J <- ceiling(log(n / (2 * v - 1)) / log(2)) - 1 # (Kaiser, 1994)

  cov <- rnorm(J + 1, sd = 2)
  Vr <- as.numeric(cov / norm(cov, type = "2") * sd(x))
  #----------------------------------------------------------------------------
  # DWT-MRA
  print("-----------DWT-MRA-----------")
  x.mra <- waveslim::mra(x, wf = wf, J = J, method = "dwt", boundary = boundary)
  x.mra.m <- matrix(unlist(x.mra), ncol = J + 1)

  x.n <- scale(x.mra.m) %*% Vr
  var(x.n) - var(x)

  message(paste0("Additive decompostion: ", isTRUE(all.equal(as.numeric(x), rowSums(x.mra.m)))))
  message(paste0("Variance decompostion: ", isTRUE(all.equal(var(x), sum(apply(x.mra.m, 2, var))))))

  #----------------------------------------------------------------------------
  # MODWT
  print("-----------MODWT-----------")
  x.modwt <- waveslim::modwt(x, wf = wf, n.levels = J, boundary = boundary)
  x.modwt.m <- matrix(unlist(x.modwt), ncol = J + 1)

  x.n <- scale(x.modwt.m) %*% Vr
  var(x.n) - var(x)

  message(paste0("Additive decompostion: ", isTRUE(all.equal(as.numeric(x), rowSums(x.modwt.m)))))
  message(paste0("Variance decompostion: ", isTRUE(all.equal(var(x), sum(apply(x.modwt.m, 2, var))))))

  #----------------------------------------------------------------------------
  # a trous
  print("-----------AT-----------")
  x.at <- at.wd(x, wf = wf, J = J, boundary = boundary)
  x.at.m <- matrix(unlist(x.at), ncol = J + 1)

  # x.mra.modwt <- waveslim::mra(x,wf=wf, J=J, method="modwt", boundary=boundary)
  # x.mra.modwt <- matrix(unlist(x.mra.modwt), ncol=J+1)
  #
  # print(sum(abs(x.at.m-x.mra.modwt)))

  message(paste0("Additive decompostion: ", isTRUE(all.equal(as.numeric(x), rowSums(x.at.m)))))
  message(paste0("Variance decompostion: ", isTRUE(all.equal(var(x), sum(apply(x.at.m, 2, var))))))

  if (isTRUE(all.equal(x.at.m, x.modwt.m))) {
    message(paste0("AT and MODWT is equivalent using the", wf, "!"))
  }
}
#> [1] "Wavelet filter: haar"
#> [1] "-----------DWT-MRA-----------"
#> Additive decompostion: TRUE
#> Variance decompostion: TRUE
#> [1] "-----------MODWT-----------"
#> Additive decompostion: TRUE
#> Variance decompostion: TRUE
#> [1] "-----------AT-----------"
#> Additive decompostion: TRUE
#> Variance decompostion: TRUE
#> AT and MODWT is equivalent using thehaar!
#> [1] "Wavelet filter: d4"
#> [1] "-----------DWT-MRA-----------"
#> Additive decompostion: TRUE
#> Variance decompostion: TRUE
#> [1] "-----------MODWT-----------"
#> Additive decompostion: FALSE
#> Variance decompostion: TRUE
#> [1] "-----------AT-----------"
#> Additive decompostion: TRUE
#> Variance decompostion: FALSE
#> [1] "Wavelet filter: d8"
#> [1] "-----------DWT-MRA-----------"
#> Additive decompostion: TRUE
#> Variance decompostion: TRUE
#> [1] "-----------MODWT-----------"
#> Additive decompostion: FALSE
#> Variance decompostion: TRUE
#> [1] "-----------AT-----------"
#> Additive decompostion: TRUE
#> Variance decompostion: FALSE
#> [1] "Wavelet filter: d16"
#> [1] "-----------DWT-MRA-----------"
#> Additive decompostion: TRUE
#> Variance decompostion: TRUE
#> [1] "-----------MODWT-----------"
#> Additive decompostion: FALSE
#> Variance decompostion: TRUE
#> [1] "-----------AT-----------"
#> Additive decompostion: TRUE
#> Variance decompostion: FALSE

1.1 Summary of various properties for the three DWT methods

tab1 <- data.frame(
  col1 = c("DWT-MRA", "MODWT", "AT"),
  col2 = c("$\\checkmark$", "", "$\\checkmark$"),
  col3 = c("$\\checkmark$", "$\\checkmark$", ""),
  col4 = c("", "$\\checkmark$", "$\\checkmark$"),
  col5 = c("$\\checkmark$", "", "")
)

colnames(tab1) <- c("Wavelet Method", "Additive decomposition", "Variance decomposition", "No dependence on future data", "Dyadic sample size")

kable(tab1, caption = "Summary of various properties for the three DWT methods", booktabs = T, escape = F) %>%
  kable_styling(latex_options = c("HOLD_position"), position = "center") %>%
  column_spec(1, width = "6em") %>%
  column_spec(2:5, width = "7em") %>%
  footnote(general = "When Haar wavelet filter is used, MODWT and AT are equivalent and both of them preserves additive and variance decomposition.", footnote_as_chunk = T)
Table 1.1: Summary of various properties for the three DWT methods
Wavelet Method Additive decomposition Variance decomposition No dependence on future data Dyadic sample size
DWT-MRA \(\checkmark\) \(\checkmark\) \(\checkmark\)
MODWT \(\checkmark\) \(\checkmark\)
AT \(\checkmark\) \(\checkmark\)
Note: When Haar wavelet filter is used, MODWT and AT are equivalent and both of them preserves additive and variance decomposition.

1.2 Illustration of three types of DWT methods

p.list <- NULL
wf.opts <- c("d16", "haar")
for (k in seq_along(wf.opts)) {
  # data generation
  x <- arima.sim(list(order = c(1, 0, 0), ar = 0.6), n = 128)

  #----------------------------------------------------------------------------
  # wavelet family, extension mode and package
  wf <- wf.opts[k] # wavelet family D8 or db4
  boundary <- "periodic"
  if (wf != "haar") v <- as.integer(readr::parse_number(wf) / 2) else v <- 1

  # Maximum decomposition level J
  n <- length(x)
  J <- ceiling(log(n / (2 * v - 1)) / log(2)) - 1 # (Kaiser, 1994)

  limits.x <- c(0, n)
  limits.y <- c(-3, 3)
  #----------------------------------------------------------------------------
  # DWT-MRA
  x.mra <- waveslim::mra(x, wf = wf, J = J, method = "dwt", boundary = boundary)
  x.mra.m <- matrix(unlist(x.mra), ncol = J + 1)

  p1 <- mra.plot(x, x.mra.m, limits.x, limits.y,
    ylab = "X", col = "red", type = "details",
    main = paste0("DWT-MRA", "(", wf, ")"), ps = 12
  )
  # p1 <- recordPlot()

  #----------------------------------------------------------------------------
  # MODWT
  x.modwt <- waveslim::modwt(x, wf = wf, n.levels = J, boundary = boundary)
  x.modwt.m <- matrix(unlist(x.modwt), ncol = J + 1)

  p2 <- mra.plot(x, x.modwt.m, limits.x, limits.y,
    ylab = "X", col = "red", type = "coefs",
    main = paste0("MODWT", "(", wf, ")"), ps = 12
  )

  #----------------------------------------------------------------------------
  # a trous
  x.at <- at.wd(x, wf = wf, J = J, boundary = boundary)
  x.at.m <- matrix(unlist(x.at), ncol = J + 1)

  p3 <- mra.plot(x, x.at.m, limits.x, limits.y,
    ylab = "X", col = "red", type = "coefs",
    main = paste0("AT", "(", wf, ")"), ps = 12
  )

  p.list[[k]] <- list(p1, p2, p3)
}

1.2.1 Daubechies 16 wavelet

#----------------------------------------------------------------------------
# plot and save
cowplot::plot_grid(
  plotlist = p.list[[1]], ncol = 3, labels = c("(a)", "(b)", "(c)"),
  label_size = 12
)
Illustration of three types of DWT methods

Figure 1.1: Illustration of three types of DWT methods

1.2.2 Haar wavelet filter

#----------------------------------------------------------------------------
# plot and save
cowplot::plot_grid(
  plotlist = p.list[[2]], ncol = 3, labels = c("(a)", "(b)", "(c)"),
  label_size = 12
)
Illustration of three types of DWT methods

Figure 1.2: Illustration of three types of DWT methods

1.3 Wavelet transform: decompostion level

sample <- seq(100, by = 200, length.out = 5)
v <- 2 # vanishing moment
tmp <- NULL
for (n in sample) {
  J1 <- floor(log(n / (2 * v - 1)) / log(2))
  J # (Kaiser, 1994)
  J2 <- floor(log2(n / (2 * v - 1) - 1))
  J # Cornish, C. R., Bretherton, C. S., & Percival, D. B. (2006)
  J3 <- floor(log10(n))
  J # (Nourani et al., 2008)

  tmp <- cbind(tmp, c(J1, J2, J3))
}

tab <- tmp
colnames(tab) <- sample
rownames(tab) <- paste0("Method", 1:3)

kable(tab,
  caption = "Decompostion level with varying sample size",
  booktabs = T, align = "c", digits = 3
) %>%
  kable_styling("striped", position = "center", full_width = FALSE) # %>%
Table 1.2: Decompostion level with varying sample size
100 300 500 700 900
Method1 5 6 7 7 8
Method2 5 6 7 7 8
Method3 2 2 2 2 2
# collapse_rows(columns = 1:2, valign = "middle")

2 Variance transformation

2.1 Optimal preditive accuracy (RMSE)

if (TRUE) {
  ### Synthetic example
  # data generation
  set.seed(2020)
  sample <- 512
  # frequency, sampled from a given range
  fd <- c(3, 5, 10, 15, 25, 30, 55, 70, 95)
  # data <- WASP::data.gen.SW(nobs=sample,fp=25,fd=fd)
  data <- WASP::data.gen.SW(nobs = sample, fp = c(15, 25, 30), fd = fd)

  # ts = data.gen.Rossler(time = seq(0, 50, length.out = sample))
  # data <- list(x=ts$z, dp=cbind(ts$x, ts$y))
} else {
  ### Real-world example
  data("obs.mon")
  data("rain.mon")

  if (1) { # SPI12 as response
    SPI.12 <- SPEI::spi(rain.mon[, 5], scale = 12)$fitted
    x <- window(SPI.12, start = c(1950, 1), end = c(2009, 12))
    dp <- window(obs.mon, start = c(1950, 1), end = c(2009, 12))
  } else { # rainfall as response
    x <- window(rain.mon[, 5], start = c(1950, 1), end = c(2009, 12))
    dp <- window(obs.mon, start = c(1950, 1), end = c(2009, 12))
  }
  data <- list(x = x, dp = dp)
  sample <- length(x)
}

# plot.ts(cbind(data$x,data$dp))

tab.list <- list()
mode.opts <- c("MRA", "MODWT", "AT")
for (mode in mode.opts) {
  print(mode)

  # cov.opt <- switch(2,"auto","pos","neg")
  if (mode == "MRA") {
    method <- switch(1,
      "dwt",
      "modwt"
    )
  }

  # wavelet family, extension mode and package
  # wf <- switch(mode, "MRA"="haar", "MODWT"="haar", "AT"="haar")
  wf <- "haar"
  pad <- "zero"
  boundary <- "periodic"
  if (wf != "haar") v <- as.integer(readr::parse_number(wf) / 2) else v <- 1

  # Maximum decomposition level J
  n <- sample
  J <- ceiling(log(n / (2 * v - 1)) / log(2)) - 1 # (Kaiser, 1994)

  tab <- NULL
  for (cov.opt in c("auto", "pos", "neg")) {
    # variance transform - calibration
    if (mode == "MRA") {
      dwt <- dwt.vt(data, wf, J, method, pad, boundary, cov.opt)
    } else if (mode == "MODWT") {
      dwt <- modwt.vt(data, wf, J, boundary, cov.opt)
    } else {
      dwt <- at.vt(data, wf, J, boundary, cov.opt)
    }

    # optimal prediction accuracy
    opti.rmse <- NULL
    dp.RMSE <- NULL
    dp.n.RMSE <- NULL
    S <- dwt$S
    ndim <- ncol(S)
    for (i in 1:ndim) {
      x <- dwt$x
      dp <- dwt$dp[, i]
      dp.n <- dwt$dp.n[, i]

      # ts.plot(cbind(dp,dp.n), col=1:2)

      dp.RMSE <- c(dp.RMSE, sqrt(mean(lm(x ~ dp)$residuals^2)))
      dp.n.RMSE <- c(dp.n.RMSE, sqrt(mean(lm(x ~ dp.n)$residuals^2)))

      # small difference due to the reconstruction
      opti.rmse <- c(opti.rmse, sqrt((n - 1) / n * (var(x) - sum(S[, i]^2) * var(dp) / var(dp.n))))
      # opti.rmse <- c(opti.rmse, sqrt((n-1)/n*(var(x)-sum(S[,i]^2))))
    }

    tab <- rbind(tab, data.frame(cov.opt, var=1:ndim, dp.RMSE, dp.n.RMSE, opti.rmse))
  }

  colnames(tab) <- c("Sign of covariance", "Variable", "Std", "VT", "Optimal")
  tab.list[[length(tab.list) + 1]] <- tab
}
#> [1] "MRA"
#> [1] "MODWT"
#> [1] "AT"

# print(tab.list)

kable(tab.list[[1]], caption = "Optimal RMSE using DWT-based VT",
      booktabs = T, align = "c", digits = 3) %>%
kable_styling("striped", position = "center", full_width = FALSE)  %>%
collapse_rows(columns = 1, valign = "middle")
Table 2.1: Optimal RMSE using DWT-based VT
Sign of covariance Variable Std VT Optimal
auto 1 1.235 1.231 1.231
auto 2 1.235 1.232 1.232
auto 3 1.235 1.232 1.232
auto 4 1.021 1.010 1.010
auto 5 1.016 1.013 1.013
auto 6 1.013 1.008 1.008
auto 7 1.235 1.222 1.222
auto 8 1.235 1.231 1.231
auto 9 1.235 1.220 1.220
pos 1 1.235 1.231 1.231
pos 2 1.235 1.232 1.232
pos 3 1.235 1.232 1.232
pos 4 1.021 1.010 1.010
pos 5 1.016 1.013 1.013
pos 6 1.013 1.008 1.008
pos 7 1.235 1.222 1.222
pos 8 1.235 1.231 1.231
pos 9 1.235 1.220 1.220
neg 1 1.235 1.231 1.231
neg 2 1.235 1.232 1.232
neg 3 1.235 1.232 1.232
neg 4 1.021 1.010 1.010
neg 5 1.016 1.013 1.013
neg 6 1.013 1.008 1.008
neg 7 1.235 1.222 1.222
neg 8 1.235 1.231 1.231
neg 9 1.235 1.220 1.220
kable(tab.list[[2]], caption = "Optimal RMSE using MODWT/AT-based VT",
      booktabs = T, align = "c", digits = 3) %>%
kable_styling("striped", position = "center", full_width = FALSE)  %>%
collapse_rows(columns = 1, valign = "middle")
Table 2.2: Optimal RMSE using MODWT/AT-based VT
Sign of covariance Variable Std VT Optimal
auto 1 1.235 1.235 1.235
auto 2 1.235 1.234 1.234
auto 3 1.235 1.235 1.235
auto 4 1.021 1.021 1.021
auto 5 1.016 1.022 1.022
auto 6 1.013 1.016 1.016
auto 7 1.235 1.221 1.221
auto 8 1.235 1.228 1.228
auto 9 1.235 1.234 1.234
pos 1 1.235 1.235 1.235
pos 2 1.235 1.234 1.234
pos 3 1.235 1.235 1.235
pos 4 1.021 1.021 1.021
pos 5 1.016 1.022 1.022
pos 6 1.013 1.016 1.016
pos 7 1.235 1.221 1.221
pos 8 1.235 1.228 1.228
pos 9 1.235 1.234 1.234
neg 1 1.235 1.235 1.235
neg 2 1.235 1.234 1.234
neg 3 1.235 1.235 1.235
neg 4 1.021 1.021 1.021
neg 5 1.016 1.022 1.022
neg 6 1.013 1.016 1.016
neg 7 1.235 1.221 1.221
neg 8 1.235 1.228 1.228
neg 9 1.235 1.234 1.234

2.2 Transformed predictor variables

#-------------------------------------------------------------------
if (TRUE) {
  set.seed(2020)
  ### synthetic example - Rossler
  sample <- 10000
  s <- 0.1
  ts.list <- list()
  for (i in seq_along(s)) {
    ts.r <- data.gen.Rossler(a = 0.2, b = 0.2, w = 5.7, start = c(-2, -10, 0.2), time = seq(0, 50, length.out = sample))

    # add noise
    ts.r$x <- ts(ts.r$x + rnorm(n = sample, mean = 0, sd = s[i]))
    ts.r$y <- ts(ts.r$y + rnorm(n = sample, mean = 0, sd = s[i]))
    ts.r$z <- ts(ts.r$z + rnorm(n = sample, mean = 0, sd = s[i]))

    ts.list[[i]] <- ts.r
  }

  data.list <- lapply(ts.list, function(ts) list(x = ts$z, dp = cbind(ts$x, ts$y)))

  lab.names <- c("x", "y")
  ylim <- c(-55, 55)
} else {

  ### Real-world example
  data("obs.mon")
  data("rain.mon")

  SPI.12 <- SPEI::spi(rain.mon[, 5], scale = 12)$fitted
  x <- window(SPI.12, start = c(1950, 1), end = c(2009, 12))
  dp <- window(obs.mon, start = c(1950, 1), end = c(2009, 12))

  data.list <- list(list(x = x, dp = dp))
  sample <- length(x)

  lab.names <- colnames(obs.mon)
  ylim <- NULL
}

#-------------------------------------------------------------------
p.list <- list()
dp.list <- list()
if (wf != "haar") mode.opts <- c("MRA", "MODWT", "AT")[1:3] else mode.opts <- c("MRA", "MODWT","AT")[1:2]
for (mode in mode.opts) {
  cov.opt <- switch(1,
    "auto",
    "pos",
    "neg"
  )
  flag <- switch(1,
    "biased",
    "unbiased"
  )
  if (mode == "MRA") {
    method <- switch(1,
      "dwt",
      "modwt"
    )
  }

  # wavelet family, extension mode and package
  # wf <- switch(mode, "MRA"="haar", "MODWT"="haar", "AT"="haar")
  wf <- "d16"
  pad <- "zero"
  boundary <- "periodic"
  if (wf != "haar") v <- as.integer(readr::parse_number(wf) / 2) else v <- 1

  # Maximum decomposition level J
  n <- sample
  J <- ceiling(log(n / (2 * v - 1)) / log(2)) - 1 # (Kaiser, 1994)
  # J <- floor(log(n/(2*v-1))/log(2))

  # variance transfrom - calibration
  if (mode == "MRA") {
    dwt.list <- lapply(data.list, function(x) dwt.vt(x, wf, J, method, pad, boundary, cov.opt, flag))
  } else if (mode == "MODWT") {
    dwt.list <- lapply(data.list, function(x) modwt.vt(x, wf, J, boundary, cov.opt, flag))
  } else {
    dwt.list <- lapply(data.list, function(x) at.vt(x, wf, J, boundary, cov.opt, flag))
  }

  for (j in 1:length(dwt.list)) {
    dwt <- dwt.list[[j]]

    par(
      mfrow = c(ncol(dwt$dp), 1), mar = c(0, 2.5, 2, 1),
      oma = c(2, 1, 0, 0), # move plot to the right and up
      mgp = c(1.5, 0.5, 0), # move axis labels closer to axis
      pty = "m", bg = "transparent",
      ps = 12
    )

    # plot(dwt$x, type="l", xlab=NA, ylab="SPI12", col="red")
    # plot(dwt$x, type="l", xlab=NA, ylab="Rain", col="red")
    for (i in 1:ncol(dwt$dp)) {
      ts.plot(cbind(dwt$dp[, i], dwt$dp.n[, i]),
        xlab = NA, ylab = paste0(lab.names[i]),
        xlim = c(0, n), ylim = ylim,
        col = c("black", "blue"), lwd = c(1, 2)
      )
    }

    p.list[[length(p.list) + 1]] <- recordPlot()

    dp.list[[length(dp.list) + 1]] <- dwt$dp.n
  }
}

#----------------------------------------------------------------------------
# plot and save
fig <- cowplot::plot_grid(plotlist = p.list, nrow = 1, labels = c("(a)", "(b)", "(c)"))
fig
Orignal and VT predictors. (a): DWT-MRA (b): MODWT/AT

Figure 2.1: Orignal and VT predictors. (a): DWT-MRA (b): MODWT/AT

3 Stepwise variance transfromation

#-------------------------------------------------------------------
### Real-world example
data("obs.mon")
data("rain.mon")
op <- par()
station.id <- 5
lab.names <- colnames(obs.mon)[c(1, 3, 4, 5, 7)]

if (TRUE) { # SPI12 as response
  SPI.12 <- SPEI::spi(rain.mon, scale = 12)$fitted
  x <- window(SPI.12, start = c(1950, 1), end = c(2009, 12))
  dp <- window(obs.mon[, lab.names], start = c(1950, 1), end = c(2009, 12))
} else { # rainfall as response
  x <- window(rain.mon, start = c(1950, 1), end = c(2009, 12))
  dp <- window(obs.mon[, lab.names], start = c(1950, 1), end = c(2009, 12))
}

data.list <- lapply(station.id, function(id) list(x = x[, id], dp = dp))


ylim <- data.frame(
  GPH = c(700, 900), TDP700 = c(5, 25), TDP500 = c(5, 25), EPT = c(300, 330),
  UWND = c(-5, 25), VWND = c(-5, 10), MSLP = c(-1, 1)
)[c(1, 3, 4, 5, 7)]

#-------------------------------------------------------------------
p.list <- list()
RMSE <- NULL
mode.opts <- c("MRA", "MODWT", "AT")[1:2]
for (mode in mode.opts) {
  cov.opt <- switch(1,
    "auto",
    "pos",
    "neg"
  )
  if (mode == "MRA") {
    method <- switch(1,
      "dwt",
      "modwt"
    )
  }

  # wavelet family, extension mode and package
  wf <- switch(mode,
    "MRA" = "d4",
    "MODWT" = "haar",
    "AT" = "haar"
  )
  pad <- "zero"
  boundary <- "periodic"
  if (wf != "haar") v <- as.integer(readr::parse_number(wf) / 2) else v <- 1

  # Maximum decomposition level J
  n <- nrow(x)
  J <- ceiling(log(n / (2 * v - 1)) / log(2)) - 1 # (Kaiser, 1994)

  # high order variance transformation
  dwt.list <- lapply(data.list, function(data) stepwise.VT(data, mode = mode, wf = wf, J=J))

  for (j in seq_len(length(dwt.list))) {
    dwt <- dwt.list[[j]]
    cpy <- dwt$cpy

    MSE <- NULL
    for (i in seq_len(length(cpy))) {
      m1 <- sqrt(FNN::knn.reg(train = dwt$dp[, 1:i], y = dwt$x)$PRESS / n)
      m2 <- sqrt(FNN::knn.reg(train = dwt$dp.n[, 1:i], y = dwt$x)$PRESS / n)

      MSE <- rbind(MSE, c(m1, m2))
    }

    RMSE <- cbind(RMSE, MSE)

    par(
      mfrow = c(length(cpy), 1), mar = c(0, 4, 2, 1),
      oma = c(2, 1, 0, 0), # move plot to the right and up
      mgp = c(1.5, 0.5, 0), # move axis labels closer to axis
      pty = "m", bg = "transparent",
      ps = 8
    )

    # plot(dwt$x, type="l", xlab=NA, ylab="SPI12", ylim=c(-3,3),col="red")
    # plot(dwt$x, type="l", xlab=NA, ylab="Rain", col="red")
    for (i in seq_len(length(cpy))) {
      ts.plot(dwt$dp[, i], dwt$dp.n[, i],
        xlab = NA, ylab = paste0(lab.names[cpy[i]]), # ylim=ylim[,i],
        col = c("black", "blue"), lwd = c(1, 2)
      )
    }

    p.list[[length(p.list) + 1]] <- recordPlot()
  }
}
par(op)
#-------------------------------------------------------------------
# plot and save
cowplot::plot_grid(plotlist = p.list, nrow = 1, labels = c("(a)", "(b)", "(c)"))
Orignal and SVT predictors. (a): DWT-MRA (b): MODWT/AT

Figure 3.1: Orignal and SVT predictors. (a): DWT-MRA (b): MODWT/AT




#-------------------------------------------------------------------
# RMSE when more predictors are included
tab1 <- round(RMSE, 3)
tab1 <- cbind(1:nrow(tab1), tab1)
colnames(tab1) <- c("No. of Predictors", rep(c("Original", "Transformed"), length(mode.opts)))

kable(tab1, caption = "Comparison of prediction accuracy using Std and SVT", booktabs = T) %>%
  kable_styling(latex_options = c("HOLD_position"), position = "center", full_width = FALSE)  %>%
  #  add_header_above(c(" " = 1, "DWT-MRA" = 2, "MODWT" = 2, "AT" = 2))
  add_header_above(c(" " = 1, "DWT-MRA" = 2, "MODWT/AT" = 2))
Table 3.1: Comparison of prediction accuracy using Std and SVT
DWT-MRA
MODWT/AT
No. of Predictors Original Transformed Original Transformed
1 1.114 1.007 1.112 0.995
2 1.093 0.876 1.100 0.934
3 1.089 0.743 1.050 0.821
4 1.086 0.744 1.053 0.824