R/Phi-plot.R

In rebeccakuiper/CTmeta: Continuous-time meta-analysis (CTmeta) on standarized lagged effects taking into account the various time-intervals used in the primary studies

#' Phi-plot of Phi based on its underlying drift matrix
#' This function makes a Phi-plot of Phi(DeltaT) for a range of time intervals based on its underlying drift matrix. There is also an interactive web application on my website to create a Phi-plot: Phi-and-Psi-Plots and Find DeltaT (\url{https://www.uu.nl/staff/RMKuiper/Websites\%20\%2F\%20Shiny\%20apps}).
#'
#' @param DeltaT Optional. The time interval used. By default, DeltaT = 1.
#' @param Phi Matrix of size q times q of (un)standardized lagged effects. Note that the Phi (or Drift) matrix should be standardized to make a fair comparison between cross-lagged effects.
#' It also takes a fitted object from the classes "varest" (from the VAR() function in vars package) and "ctsemFit" (from the ctFit() function in the ctsem package); see example below. From such an object, the (standardized) Drift matrix is calculated/extracted.
#' @param Drift Optional (either Phi or Drift). Underlying continuous-time lagged effects matrix (i.e., Drift matrix) of the discrete-time lagged effects matrix Phi(DeltaT). By default, input for Phi is used; only when Phi = NULL, Drift will be used.
#' @param Stand Optional. Indicator for whether Phi (or Drift) should be standardized (1) or not (0). In case Stand = 1, one of the following matrices should be input as well: SigmaVAR, Sigma, or Gamma  (or it is subtracted from a varest or ctsemFit object). By default, Stand = 0.
#' @param SigmaVAR Optional (if Stand = 1, then either SigmaVAR, Sigma, or Gamma needed). Residual covariance matrix of the first-order discrete-time vector autoregressive (DT-VAR(1)) model.
#' @param Sigma Optional (if Stand = 1, then either SigmaVAR, Sigma, or Gamma needed). Residual covariance matrix of the first-order continuous-time (CT-VAR(1)) model, that is, the diffusion matrix.
#' @param Gamma Optional (either SigmaVAR, Sigma, or Gamma). Stationary covariance matrix, that is, the contemporaneous covariance matrix of the data.
#' @param Min Optional. Minimum time interval used in the Phi-plot. By default, Min = 0.
#' @param Max Optional. Maximum time interval used in the Phi-plot. By default, Max = 10.
#' @param Step Optional. The step-size taken in the time intervals. By default, Step = 0.05. Hence, using the defaults, the Phi-plots is based on the values of Phi(DeltaT) for DeltaT = 0, 0.05, 0.10, ..., 10. Note: Especially in case of complex eigenvalues, this step size should be very small (then, the oscillating behavior can be seen best).
#' @param WhichElements Optional. Matrix of same size as Drift denoting which element/line should be plotted (1) or not (0). By default, WhichElements = NULL. Note that even though not all lines have to be plotted, the full Drift matrix is needed to determine the selected lines.
#' @param Labels Optional. Vector with (character) labels of the lines to be plotted. The length of this vector equals the number of 1s in WhichElements (or equals q*q). By default, Labels = NULL, which renders labels with Greek letter of Phi (as a function of the time interval) together with the indices (of outcome and predictor variables).
#' @param Col Optional. Vector with color values (integers) of the lines to be plotted. The length of this vector equals the number of 1s in WhichElements (or equals q*q). By default, Col = NULL, which renders the same color for effects that belong to the same outcome variable (i.e. a row in the Drift matrix). See \url{https://www.statmethods.net/advgraphs/parameters.html} for more information about the values.
#' @param Lty Optional. Vector with line type values (integers) of the lines to be plotted. The length of this vector equals the number of 1s in WhichElements (or equals q*q). By default, Lty = NULL, which renders solid lines for the autoregressive effects and the same type of dashed line for reciprocal effects (i.e., same type for Phi_ij as for Phi_ji). See \url{https://www.statmethods.net/advgraphs/parameters.html} for more information about the values.
#' @param Title Optional. A character or a list consisting of maximum 3 character-strings or 'expression' class objects that together represent the title of the Phi-plot. By default, Title = NULL, then the following code will be used for the title: as.list(expression(Phi(Delta[t])~plot), "How do the lagged parameters vary", "as a function of the time interval?").
#' @param MaxMinPhi Optional. An indicator (TRUE/FALSE) whether vertical lines for the optimum (maximum or minimum) should be added to the plot (TRUE) or not (FALSE). These values are obtained by the function MaxDeltaT(). By default, MaxMinPhi = FALSE; hence, by default, no vertical are added.
#'
#' @return This function returns a Phi-plot for a range of time intervals.
#' @importFrom expm expm
#' @export
#' @examples
#'
#' # library(CTmeta)
#'
#' ### Make Phi-plot ###
#'
#' ## Example 1 ##
#'
#' # Phi(DeltaT)
#' DeltaT <- 1
#' Phi <- myPhi[1:2,1:2]
#' # or: Drift
#' Drift <- myDrift
#'
#' # Example 1.1: unstandardized Phi #
#' #
#' # Make plot of Phi
#' PhiPlot(DeltaT, Phi)
#' PhiPlot(DeltaT, Phi, Min = 0, Max = 10, Step = 0.01)           # Specifying range x-axis and precision
#' PhiPlot(DeltaT, Drift = Drift, Min = 0, Max = 10, Step = 0.01) # Using Drift instead of Phi
#'
#'
#' # Example 1.2: standardized Phi #
#' q <- dim(Phi)[1]
#' SigmaVAR <- diag(q) # for ease
#' PhiPlot(DeltaT, Phi, Stand = 1, SigmaVAR = SigmaVAR)
#'
#'
#' ## Example 2: input from fitted object of class "varest" ##
#'
#' DeltaT <- 1
#' data <- myData
#' if (!require("vars")) install.packages("vars")
#' library(vars)
#' out_VAR <- VAR(data, p = 1)
#'
#' # Example 2.1: unstandardized Phi #
#' PhiPlot(DeltaT, out_VAR)
#'
#' # Example 2.2: standardized Phi #
#' PhiPlot(DeltaT, out_VAR, Stand = 1)
#'
#'
#' ## Example 3: Change plot options ##
#' DeltaT <- 1
#' Phi <- myPhi[1:2,1:2]
#' q <- dim(Phi)[1]
#' SigmaVAR <- diag(q) # for ease
#' #
#' WhichElements <- matrix(1, ncol = q, nrow = q) # Now, all elements are 1
#' diag(WhichElements) <- 0 # Now, the autoregressive parameters are excluded by setting the diagonals to 0.
#' Lab <- c("12", "21")
#' Labels <- NULL
#' for(i in 1:length(Lab)){
#'  e <- bquote(expression(Phi(Delta[t])[.(Lab[i])]))
#'  Labels <- c(Labels, eval(e))
#' }
#' Col <- c(1,2)
#' Lty <- c(1,2)
#' # Standardized Phi
#' PhiPlot(DeltaT = 1, Phi, Stand = 1, SigmaVAR = SigmaVAR, Min = 0, Max = 10, Step = 0.05, WhichElements = WhichElements, Labels = Labels, Col = Col, Lty = Lty)
#'


PhiPlot <- function(DeltaT = 1, Phi = NULL, Drift = NULL, Stand = 0, SigmaVAR = NULL, Sigma = NULL, Gamma = NULL, Min = 0, Max = 10, Step = 0.05, WhichElements = NULL, Labels = NULL, Col = NULL, Lty = NULL, Title = NULL, MaxMinPhi = FALSE) {
  #Min = 0; Max = 10; Step = 0.05; WhichElements = NULL; Labels = NULL; Col = NULL; Lty = NULL; Title = NULL; MaxMinPhi = FALSE
  #DeltaT = 1; Drift = NULL; Stand = 0; SigmaVAR = NULL; Sigma = NULL; Gamma = NULL; Min = 0; Max = 10; Step = 0.05; WhichElements = NULL; Labels = NULL; Col = NULL; Lty = NULL; Title = NULL; MaxMinPhi = FALSE

  #  #######################################################################################################################
  #
  #  #if (!require("expm")) install.packages("expm")
  #  library(expm)
  #
  #  #######################################################################################################################

  # Checks:
  if(length(DeltaT) != 1){
    #ErrorMessage <- (paste0("The argument DeltaT should be a scalar (i.e., one number or a vector with one element). In the given input, DeltaT = ", DeltaT))
    if(is.matrix(DeltaT)){
      ErrorMessage1 <- (paste0("The argument DeltaT should be a scalar (i.e., one number or a vector/matrix with one element). In the given input, DeltaT is a matrix with dimensions "))
      Value1 <- paste(dim(DeltaT), collapse = " x ")
      ErrorMessage2 <- (paste0(" and contains the values "))
      Value2 <- paste(DeltaT, collapse = "|")
      stop(ErrorMessage1, Value1, ErrorMessage2, Value2)
    }else{
      ErrorMessage <- (paste0("The argument DeltaT should be a scalar (i.e., one number or a vector/matrix with one element). In the given input, DeltaT contains the values "))
      Value <- paste(DeltaT, collapse = "|")
      stop(ErrorMessage, Value)
    }
  }
  if(Stand != 0 & Stand != 1){
    ErrorMessage <- (paste0("The argument Stand should be a 0 or a 1, not ", Stand))
    stop(ErrorMessage)
  }
  if(length(Min) != 1){
    ErrorMessage <- (paste0("The argument Min should be a scalar (i.e., one number or a vector with one element). In the given input, Min = ", Min))
    stop(ErrorMessage)
  }
  if(length(Max) != 1){
    ErrorMessage <- (paste0("The argument Max should be a scalar (i.e., one number or a vector with one element). In the given input, Max = ", Max))
    stop(ErrorMessage)
  }
  if(length(Step) != 1){
    ErrorMessage <- (paste0("The argument Step should be a scalar (i.e., one number or a vector with one element). In the given input, Step = ", Step))
    stop(ErrorMessage)
  }
  if(!is.logical(MaxMinPhi) & MaxMinPhi != FALSE & MaxMinPhi != TRUE){
    ErrorMessage <- (paste0("The argument 'MaxMinPhi' should be T(RUE) or F(ALSE) (or 1 or 0), not ", MaxMinPhi))
    stop(ErrorMessage)
  }
  #
  # Check on Phi
  if(any(class(Phi) == "varest")){
    Phi_VARest <- Acoef(Phi)[[1]]
    CTMp <- CTMparam(DeltaT, Phi_VARest)
    if(is.null(CTMp$ErrorMessage)){
      Drift <- CTMp$Drift  # Drift <- logm(Phi)/DeltaT  # Phi <- expm(Drift * DeltaT)
    }else{
      ErrorMessage <- CTMp$ErrorMessage
      stop(ErrorMessage)
    }
  } else if(any(class(Phi) == "ctsemFit")){
    Drift <- summary(Phi)$DRIFT
  } else{

    if(is.null(Drift)){
      if(!is.null(Phi)){
        CTMp <- CTMparam(DeltaT, Phi)
        if(is.null(CTMp$ErrorMessage)){
          Drift <- CTMp$Drift  # Drift <- logm(Phi)/DeltaT  # Phi <- expm(Drift * DeltaT)
        }else{
          ErrorMessage <- CTMp$ErrorMessage
          stop(ErrorMessage)
        }
      }else{ # is.null(Phi)
        cat("Either the drift matrix Drift or the autoregressive matrix Phi should be input to the function.")
        #("Note that Phi(DeltaT) = expm(Drift*DeltaT).")
        stop(ErrorMessage)
      }
    }
    #
    # Check on B
    if(length(Drift) > 1){
      Check_B_or_Phi(B=-Drift)
      if(all(Re(eigen(Drift)$val) > 0)){
        cat("All (the real parts of) the eigenvalues of the drift matrix Drift are positive. Therefore, it is assumed the input for Drift was B = -A instead of A (or -Phi instead of Phi). Drift = -B = A will be used.")
        cat("Note that Phi(DeltaT) = expm(-B*DeltaT) = expm(A*DeltaT) = expm(Drift*DeltaT).")
        Drift = -Drift
      }
      if(any(Re(eigen(Drift)$val) > 0)){
        #ErrorMessage <- ("The function stopped, since some of (the real parts of) the eigenvalues of the drift matrix Drift are positive.")
        #return(ErrorMessage)
        #stop(ErrorMessage)
        cat("If the function stopped, this is because some of (the real parts of) the eigenvalues of the drift matrix Drift are positive.")
      }
    }
  }
  #
  if(length(Drift) == 1){
    q <- 1
  }else{
    q <- dim(Drift)[1]
  }
  #
  #
  if(Stand == 1){
    # Check on SigmaVAR, Sigma, and Gamma
    if(any(class(Phi) == "varest")){
      SigmaVAR <- cov(resid(Phi))
      Phi <- Phi_VARest
      Gamma <- Gamma.fromVAR(Phi, SigmaVAR)
    }else if(any(class(Phi) == "ctsemFit")){
      Sigma <- summary(Phi)$DIFFUSION
      Gamma <- Gamma.fromCTM(Drift, Sigma)
    }else if(is.null(SigmaVAR) & is.null(Gamma) & is.null(Sigma)){ # All three unknown
      ErrorMessage <- (paste0("One of the arguments SigmaVAR, Sigma, or Gamma is not found: it should be part of the input (when Stand = 1). In case of the first matrix, specify 'SigmaVAR = SigmaVAR'."))
      stop(ErrorMessage)
    }else if(is.null(Gamma)){ # Gamma unknown, calculate Gamma from Phi & SigmaVAR or Drift & Sigma

      if(!is.null(SigmaVAR)){ # SigmaVAR known, calculate Gamma from Phi & SigmaVAR

        # Check on SigmaVAR
        Check_SigmaVAR(SigmaVAR, q) # TO DO lijkt niet te runnen.... "could not find function "Check_SigmaVAR""

        # Calculate Gamma
        if(is.null(Phi)){
          if(q == 1){
            Phi <- exp(-B*DeltaT)
          }else{
            Phi <- expm(-B*DeltaT)
          }
        }
        Gamma <- Gamma.fromVAR(Phi, SigmaVAR)


      }else if(!is.null(Sigma)){ # Sigma known, calculate Gamma from Drift & Sigma

        # Check on Sigma
        Check_Sigma(Sigma, q)

        # Calculate Gamma
        if(is.null(Drift)){
          CTMp <- CTMparam(DeltaT, Phi)
          if(is.null(CTMp$ErrorMessage)){
            Drift <- CTMp$Drift  # Drift <- logm(Phi)/DeltaT  # Phi <- expm(Drift * DeltaT)
          }else{
            ErrorMessage <- CTMp$ErrorMessage
            stop(ErrorMessage)
          }
        }
        Gamma <- Gamma.fromCTM(Drift, Sigma)

      }

    }else if(!is.null(Gamma)){ # Gamma known, only check on Gamma needed

      # Checks on Gamma
      Check_Gamma(Gamma, q)

    }

    # Standardize Drift and Gamma
    Sxy <- sqrt(diag(diag(Gamma)))
    Gamma <- solve(Sxy) %*% Gamma %*% solve(Sxy)
    Drift <- solve(Sxy) %*% Drift %*% Sxy
    #Sigma_s <- solve(Sxy) %*% Sigma %*% solve(Sxy)
  }
  #
  #
  if(!is.null(WhichElements)){
    # Check on WhichElements
    Check_WhichElts(WhichElements, q)
    nrLines <- sum(WhichElements)
  } else{
    WhichElements <- matrix(1, ncol = q, nrow = q)
    nrLines <- q*q #<- sum(WhichElements)
  }
  #
  if(!is.null(Labels)){
    if(length(Labels) != nrLines){
      ErrorMessage <- (paste0("The argument Labels should contain ", nrLines, " elements, that is, q*q or the number of 1s in WhichElements (or WhichElements is incorrectly specified); not ", length(Labels)))
      stop(ErrorMessage)
    }
    #if(any(!is.character(Labels))){ # Note: This does not suffice, since it could also be an expression
    #  ErrorMessage <- (paste0("The argument Labels should consist of solely characters."))
    #  return(ErrorMessage)
    #  stop(ErrorMessage)
    #}
  }
  if(!is.null(Col)){
    if(length(Col) != nrLines){
      ErrorMessage <- (paste0("The argument Col should contain ", nrLines, " elements, that is, q*q or the number of 1s in WhichElements (or WhichElements is incorrectly specified); not ", length(Col)))
      stop(ErrorMessage)
    }
    if(any(Col %% 1 != 0)){
      ErrorMessage <- (paste0("The argument Col should consist solely of integers."))
      stop(ErrorMessage)
    }
  }
  if(!is.null(Lty)){
    if(length(Lty) != nrLines){
      ErrorMessage <- (paste0("The argument Lty should contain ", nrLines, " elements, that is, q*q or the number of 1s in WhichElements (or WhichElements is incorrectly specified); not ", length(Lty)))
      stop(ErrorMessage)
    }
    if(any(Lty %% 1 != 0)){
      ErrorMessage <- (paste0("The argument Lty should consist solely of integers."))
      stop(ErrorMessage)
    }
  }
  if(!is.null(Title)){
    if(length(Title) != 1 & !is.list(Title)){
      ErrorMessage <- (paste0("The argument Title should be a character or a list (containing maximum 3 items)."))
      stop(ErrorMessage)
    }
    if(length(Title) > 3){
      ErrorMessage <- (paste0("The list Title should contain maximum 3 items. In the given input, it consists of ", length(Title), " items."))
      stop(ErrorMessage)
    }
    # Option: Also check whether each element in list either a "call" or a 'character' is...
  }


  #def.par <- par(no.readonly = TRUE) # save default, for resetting...
  #par(def.par)  #- reset to default

  if(is.null(Labels)){
    subscripts = NULL
    for(i in 1:q){
      subscripts = c(subscripts, paste0(i, 1:q, sep=""))
    }
    legendT = NULL
    for(i in 1:(q*q)){
      e <- bquote(expression(Phi(Delta[t])[.(subscripts[i])]))
      legendT <- c(legendT, eval(e))
    }
  } else{
    legendT <- as.vector(Labels)
  }

  if(is.null(Col)){
    Col <- matrix(NA, ncol = q, nrow = q)
    for(i in 1:q){
      Col[i, 1:q] <- i
    }
    Col <- as.vector(t(Col))
  }

  if(is.null(Lty)){
    Lty <- matrix(NA, ncol = q, nrow = q)
    diag(Lty) <- 1
    Lty[upper.tri(Lty, diag = FALSE)] <- 2:(1+length(Lty[upper.tri(Lty, diag = FALSE)]))
    Lty[lower.tri(Lty, diag = FALSE)] <- Lty[upper.tri(Lty, diag = FALSE)]
    Lty <- as.vector(t(Lty))
  }

  if(is.null(Title)){
    Title_1 <- expression(Phi(Delta[t])~plot)
    Title_2 <- "How do the lagged parameters vary"
    #Title_2 <- "How do the overall lagged parameters"
    Title_3 <- "as a function of the time interval?"
  }else{
    Title_1 <- NULL
    Title_2 <- NULL
    if(length(Title) == 1){
      if(is.list(Title)){
        Title_3 <- Title[[1]]
      }else{
        Title_3 <- Title
      }
    }else if(length(Title) == 2){
      Title_2 <- Title[[1]]
      Title_3 <- Title[[2]]
    }else if(length(Title) == 3){
      Title_1 <- Title[[1]]
      Title_2 <- Title[[2]]
      Title_3 <- Title[[3]]
    }
  }

  ##########################################################################################

  if(any(is.complex(eigen(Drift)$val))){
    while (!is.null(dev.list()))  dev.off()  # to reset the graphics pars to defaults
    # Multiple solutions, then 2x2 plots
    op <- par(mfrow=c(2,2))
    complex <- TRUE
    nf <- layout(matrix(c(1,2,5,3,4,6),2,3,byrow = TRUE), c(3,3,1), c(2,2,1), TRUE)
    #layout.show(nf)
  } else{
    op <- par(mfrow=c(1,1))
    complex <- FALSE
    #
    while (!is.null(dev.list()))  dev.off()  # to reset the graphics pars to defaults
    par(mar=c(par('mar')[1:3], 0)) # optional, removes extraneous right inner margin space
    plot.new() # TO DO Alex, here I create an empty plot such that I can determine the location of the legend (later on I also do this for the case of complex eigenvalues)
    # Alex, can you look into how to do this in another way?
    l <- legend(0, 0,
                legend = legendT, #cex=CEX,
                bty = "n",
                lty=Lty, # gives the legend appropriate symbols (lines)
                lwd=rep(2, q*q),
                col=Col # gives the legend lines the correct color and width
    )
    # calculate right margin width in ndc
    w <- 1.5 *( grconvertX(l$rect$w, to='ndc') - grconvertX(0, to='ndc') )
    par(omd=c(0, 1-w, 0, 1))
    #
  }


  DeltaTs<-seq(Min,Max,by=Step)

  PhiDeltaTs<-array(data=NA,dim=c(q,q,length(DeltaTs)))
  if(length(Drift) == 1){
    for(i in 1:length(DeltaTs)){
      PhiDeltaTs[,,i]<-exp(Drift*DeltaTs[i])
    }
  }else{
    for(i in 1:length(DeltaTs)){
      PhiDeltaTs[,,i]<-expm(Drift*DeltaTs[i])
    }
  }

  Xlab <- expression(Time~interval (Delta[t]))
  Ylab <- expression(Phi(Delta[t])~values)
  #
  #wd <- getwd()
  #dev.copy(png, filename = paste0(wd, "/www/PhiPlot.png"))
  teller <- 1
  plot(y=rep(0, length(DeltaTs)), x=DeltaTs, type="l", ylim=c(min(PhiDeltaTs), max(PhiDeltaTs)),
       #ylab = expression(Overall~Phi(Delta[t])~values),
       ylab = Ylab,
       xlab = Xlab,
       col=1000, lwd=2, lty=1,
       main = mtext(side=3, line=2, adj=0, as.expression(Title_1)),
       sub = mtext(side=3, line=c(1,0), adj=0, c(as.expression(Title_2), as.expression(Title_3)))
  )
  #
  teller <- 0
  for(j in 1:q){
    for(i in 1:q){
      if(WhichElements[j,i] == 1){
        teller <- teller + 1
        lines(y=PhiDeltaTs[j,i,], x=DeltaTs, col=Col[teller], lwd=2, lty=Lty[teller])
      }
    }
  }

  #Add lines for max or min of Phi
  if(MaxMinPhi == TRUE){
    if(is.null(Phi)){
      Phi <- expm(Drift*DeltaT)
    }
    MaxD <- MaxDeltaT(Phi = Phi)
    if(is.null(MaxD$ErrorMessage)){
      Max_DeltaT <- MaxD$DeltaT_MinOrMaxPhi
      Phi_MinMax <- MaxD$MinOrMaxPhi
    }else{
      ErrorMessage <- MaxD$ErrorMessage
      stop(ErrorMessage)
    }
    #
    teller <- 0
    for(j in 1:q){
      for(i in 1:q){
        if(WhichElements[j,i] == 1){
          teller <- teller + 1
          #abline(v=Max_DeltaT[j,i], col=Col[teller], lwd=1, lty=Lty[teller])
          #abline(v=Max_DeltaT[j,i], col="white", lwd=0.5, lty=Lty[teller])
          segments(x0=Max_DeltaT[j,i], y0=0, x1=Max_DeltaT[j,i], y1=Phi_MinMax[j,i], col=Col[teller], lwd=0.5, lty=Lty[teller])
        }
      }
    }
  }


  if(complex == FALSE){
    if(q<4){CEX = 1}else{CEX = (1.4-q/10)} # Check for optimal values!
    #
    #legend("topright",
    legend(par('usr')[2], par('usr')[4], xpd=NA,
           legend = legendT, cex=CEX,
           bty = "n",
           lty=Lty, # gives the legend appropriate symbols (lines)
           lwd=rep(2, q*q),
           col=Col # gives the legend lines the correct color and width
    )
  }




  if(complex){
    # Multiple solutions and add 3 plots (2 for 2 different solutions and one scatter plot)
    Title_2_N <- "using an 'aliasing' matrix"
    Title_3_N <- "(i.e., another solution for A)"
    Title_1_N2 <- expression(Phi(Delta[t])~scatter~plot~'for'~multiples~of~Delta[t])
    Title_2_N2 <- expression(Note~that~'for'~multiples~of~Delta[t])
    Title_3_N2 <- expression(Phi(Delta[t])~is~unique)

    EigenDrift <- eigen(Drift)
    V <- EigenDrift$vector

    for(N in 1:2){ # Note: last plot is scatter plot
      im <- complex(real = 0, imaginary = 1)
      diagN <- matrix(0, ncol = q, nrow = q)
      # Note: ordering eigenvalues is based on Mod(eigenvalues): so, if find one complex then next is its conjugate.
      W_complex <- which(Im(EigenDrift$val) != 0)
      NrComplexPairs <- length(W_complex)/2
      tellerComplex = -1
      for(i in 1:NrComplexPairs){
        tellerComplex = tellerComplex + 2
        index <- W_complex[tellerComplex]
        diagN[index,index] <- 1
        diagN[index+1,index+1] <- -diagN[index,index]
        # Note if nr of complex pairs > 1: 'diagN' should always be x and -x within a conjugate pair, but over the complex pairs x does not have to be the same...
      }
      diagN <- N * diagN
      A_N = Drift + (2 * base::pi * im / DeltaT) * V %*% diagN %*% solve(V)
      #A_N
      #print(A_N)
      Drift_N <- Re(A_N)
      PhiDeltaTs_N<-array(data=NA,dim=c(q,q,length(DeltaTs)))
      for(i in 1:length(DeltaTs)){
        PhiDeltaTs_N[,,i] <- expm(Drift_N*DeltaTs[i])
      }
      #
      plot(y=rep(0, length(DeltaTs)), x=DeltaTs, type="l", ylim=c(min(PhiDeltaTs_N), max(PhiDeltaTs_N)),
           ylab = Ylab, xlab = Xlab,
           col=1000, lwd=2, lty=1,
           main = mtext(side=3, line=2, adj=0, as.expression(Title_1)),
           sub = mtext(side=3, line=c(1,0), adj=0, c(as.expression(Title_2_N), as.expression(Title_3_N)))
      )
      #
      teller <- 0
      for(j in 1:q){
        for(i in 1:q){
          if(WhichElements[j,i] == 1){
            teller <- teller + 1
            lines(y=PhiDeltaTs_N[j,i,], x=DeltaTs, col=Col[teller], lwd=2, lty=Lty[teller])
          }
        }
      }
    }
    # In case last plot is scatter plot
    # In last plot a scatter plot, for multiples of DeltaT, from Min to Max.
    Min_ <- Min + Min%%DeltaT # last part is remainder after integer division
    Max_ <- Max - Max%%DeltaT # last part is remainder after integer division
    DeltaTs <- seq(Min_, Max_, by=DeltaT)
    PhiDeltaTs_N<-array(data=NA,dim=c(q,q,length(DeltaTs)))
    for(i in 1:length(DeltaTs)){
      PhiDeltaTs_N[,,i]<-expm(Drift_N*DeltaTs[i])
    }
    #
    plot(y=rep(0, length(DeltaTs)), x=DeltaTs, type="l", ylim=c(min(PhiDeltaTs_N), max(PhiDeltaTs_N)),
         ylab = Ylab, xlab = Xlab,
         col=1000, lwd=2, lty=1,
         main = mtext(side=3, line=2, adj=0, as.expression(Title_1_N2)),
         sub = mtext(side=3, line=c(1,0), adj=0, c(as.expression(Title_2_N2), as.expression(Title_3_N2)))
    )
    #
    teller <- 0
    for(j in 1:q){
      for(i in 1:q){
        if(WhichElements[j,i] == 1){
          teller <- teller + 1
          points(y=PhiDeltaTs_N[j,i,], x=DeltaTs, col=Col[teller], lwd=2, pch=Lty[teller])
        }
      }
    }
  } # end if complex


  if(complex == TRUE){
    if(q<4){CEX = 1}else{CEX = (1.4-q/10)} # Check for optimal values!
    par(mar = c(0,0,0,0))
    plot.new()
    legend("center",
           legend = legendT, cex=CEX,
           bty = "n",
           lty=Lty, # gives the legend appropriate symbols (lines)
           lwd=rep(2, q*q),
           col=Col # gives the legend lines the correct color and width
    )
    plot.new()
    legend("center",
           legend = legendT, cex=CEX,
           bty = "n",
           pch=Lty, # gives the legend appropriate symbols (lines)
           #lwd=rep(2, q*q),
           col=Col # gives the legend lines the correct color and width
    )
  }

  #dev.off()

  par(op)





  ############################################################################################################

  #final <- list(.. = ...)
  #return(final)

}