# R/corrDim.R In nonlinearTseries: Nonlinear Time Series Analysis

#### Documented in corrDimcorrMatrixcorrMatrix.corrDimembeddingDims.corrDimestimate.corrDimnlOrder.corrDimplot.corrDimplotLocalScalingExp.corrDimradius.corrDim

#' Correlation sum, correlation dimension and generalized correlation dimension
#' (order q >1).
#' @description
#' Functions for estimating the correlation sum and the correlation dimension
#' of a dynamical system from 1-dimensional time series using Takens' vectors.
#' @details
#' The correlation dimension is the most common measure of the fractal
#' dimensionality of a geometrical object embedded in a phase space. In
#' order to estimate the correlation dimension, the correlation sum is defined
#' over the points from the phase space:
#' \deqn{C(r) = \{(number\;of\;points\;(x_i,x_j)\;verifying\;that\;distance\;(x_i,x_j)<r\})/N^2}{C(r) = {number of points(xi,xj)  verifying distance(xi,xj)<r}/N^2}
#' However, this estimator is biased when the pairs in the sum are not
#' statistically independent. For example, Taken's vectors that are close in
#' time, are usually close in the phase space due to the non-zero
#' autocorrelation of the original time series. This is solved by using the
#' so-called Theiler window: two Takens' vectors must be separated by, at least,
#' the time steps specified with this window in order to be considered
#' neighbours. By using a Theiler window, we exclude temporally correlated
#' vectors from our estimations.
#'
#' The correlation dimension is estimated using the slope obtained by
#' performing a linear regression of
#' \eqn{\log10(C(r))\;Vs.\;\log10(r)}{log10(C(r)) Vs. log10(r)}. Since this
#' dimension is supposed to be an invariant of the system, it should not
#' depend on the dimension of the Taken's vectors used to estimate it. Thus,
#' the user should plot \eqn{\log10(C(r))\;Vs.\;\log10(r)}{log10(C(r)) Vs. log10(r)} for several embedding
#' dimensions when looking for the correlation dimension and, if for some range
#'  \eqn{\log10(C(r))}{log10(C(r))} shows a similar linear behaviour in
#'  different embedding dimensions (i.e. parallel slopes), these slopes are an
#'  estimate of the correlation dimension. The \emph{estimate} routine
#'  allows the user to get always an estimate of the correlation dimension, but
#'  the user must check that there is a linear region in the correlation sum
#'  over different dimensions. If such a region does not exist, the estimation
#' should be discarded.
#'
#' Note that the correlation sum  C(r) may be interpreted as:
#' \eqn{C(r) = <p(r)>,}
#' that is: the mean probability of finding a neighbour in a ball of radius r
#' surrounding a point in the phase space. Thus, it is possible to define a
#' generalization of the correlation dimension by writing:
#' \deqn{C_q(r) = <p(r)^{(q-1)}>}{Cq(r) = <p(r)^(q-1)>.}
#' Note that the correlation sum \deqn{C(r) = C_2(r)}{C(r) = C2(r).}
#'
#' It is possible to determine generalized dimensions Dq using the slope
#' obtained by performing a linear regression of
#' \eqn{log10(Cq(r))\;Vs.\;(q-1)log10(r)}. The case q=1 leads to the
#' information dimension, that is treated separately in this package
#' (\link{infDim}). The considerations discussed for the correlation dimension
#' estimate are also valid for these generalized dimensions.
#' @param time.series The original time series from which the correlation sum
#' will be estimated.
#' @param min.embedding.dim Integer denoting the minimum dimension in which we
#' shall embed the time.series (see \link{buildTakens}).
#' @param max.embedding.dim Integer denoting the maximum dimension in which we
#' shall embed the time.series (see \link{buildTakens}).Thus,
#' we shall estimate the correlation dimension between \emph{min.embedding.dim}
#' and \emph{max.embedding.dim}.
#' @param time.lag Integer denoting the number of time steps that will be use
#' to construct the Takens' vectors (see \link{buildTakens}).
#' @param min.radius Minimum distance used to compute the correlation sum C(r).
#' @param max.radius Maximum distance used to compute the correlation sum C(r).
#' @param corr.order Order of the generalized correlation Dimension q. It must
#' be greater than 1 (corr.order>1). Default, corr.order=2.
#' @param n.points.radius The number of different radius where we shall estimate.
#' C(r). Thus,  we will estimate C(r) in n.points.radius between min.radius and
#' @param theiler.window Integer denoting the Theiler window:  Two Takens'
#' vectors must be separated by more than theiler.window time steps in order to
#' be considered neighbours. By using a Theiler window, we exclude temporally
#' correlated vectors from our estimations.
#' @param do.plot Logical value. If TRUE (default value), a plot of the
#' correlation sum is shown.
#' @param number.boxes Number of boxes that will be used in the box assisted
#' algorithm (see \link{neighbourSearch}). If the user does not specify it, the
#' function uses a proper number of boxes.
#' @param ... Additional plotting parameters.
#' @return  A \emph{corrDim} object that consist of a list with four
#' components named \emph{radius}, \emph{embedding.dims}, \emph{order} and
#' \emph{corr.matrix}. \emph{radius} is a vector containing the different radius
#' where we have evaluated C(r). \emph{embedding.dims} is a vector containing
#' all the embedding dimensions in which we have estimated C(r). \emph{order}
#' stores the order of the generalized correlation dimension
#' that has been used. Finally, \emph{corr.matrix} stores all the correlation
#' sums that have been computed. Each row stores the correlation sum for a
#' concrete embedding dimension whereas each colum stores the correlation sum
#' for a specific radius.
#' @examples
#' \dontrun{
#' x=lorenz(sigma=10, rho = 28, beta =8/3, start = c(-10, -11, 47),
#' time =  seq(0, 70, by = 0.01), do.plot = FALSE)$x #' cd=corrDim(time.series=x,min.embedding.dim=3,max.embedding.dim=6, #' time.lag=10,min.radius=1e-3,max.radius=50, #' n.points.radius=100,theiler.window=100, #' number.boxes=100,do.plot=F) #' #' plot(cd,type="l") #' plotLocalScalingExp(cd,cex=0.5,xlim=c(1e-1,5)) #' cd.est = estimate(cd,regression.range=c(0.2,2)) #' cat("expected: 2.05 --- estimate: ",cd.est,"\n") #' } #' @references H. Kantz and T. Schreiber: Nonlinear Time series Analysis #' (Cambridge university press) #' @author Constantino A. Garcia #' @rdname corrDim #' @export corrDim #' @exportClass corrDim #' @useDynLib nonlinearTseries corrDim = function(time.series, min.embedding.dim = 2, max.embedding.dim = 5, time.lag = 1, min.radius, max.radius, corr.order = 2, n.points.radius = 5, theiler.window = 100, do.plot = TRUE, number.boxes = NULL, ...) { #estimate number of boxes for the box assisted algorithm if (is.null(number.boxes)) { number.boxes = estimateNumberBoxes(time.series, min.radius) } # Radius vector equally spaced points in the log10(radius) space log.radius = seq(log10(max.radius), log10(min.radius), len = n.points.radius) radius = 10 ^ log.radius corr.matrix = .Call("_nonlinearTseries_generalized_correlation_sum", PACKAGE = "nonlinearTseries", time.series, time.lag, theiler.window, radius, min.embedding.dim, max.embedding.dim, corr.order, number.boxes) dimnames(corr.matrix) = list(min.embedding.dim:max.embedding.dim, radius) #eliminate columns with at least one 0 wh = which(corr.matrix == 0, arr.ind = TRUE) wh = unique(wh[, 'col']) if (length(wh > 0)) { corr.matrix = corr.matrix[, -wh, drop = FALSE] #eliminate the corresponding radius values in the radius vector radius = -wh] } # create the corrDim object corr.dim = list(corr.matrix = corr.matrix, embedding.dims = min.embedding.dim:max.embedding.dim, radius = radius, corr.order = corr.order) class(corr.dim) = "corrDim" # add attributes id = deparse(substitute(time.series)) attr(corr.dim, "time.lag") = time.lag attr(corr.dim, "id") = id attr(corr.dim, "theiler.window") = theiler.window # plot if necessary if (do.plot) { tryCatch(plot(corr.dim,...), error = function(error){ warning("Error while trying to plot the correlation sum") }) } corr.dim } #' @return The \emph{nlOrder} function returns the order of the correlation sum. #' @rdname corrDim #' @export nlOrder.corrDim = function(x){ x$corr.order
}

#' Returns the correlation sums stored in the \emph{corrDim} object
#' @param x A \emph{corrDim} object.
#' @return The \emph{corrMatrix} function returns the correlations matrix
#' storing the correlation sums that have been computed for all the embedding
#' dimensions.
#' @export corrMatrix
corrMatrix = function(x){
UseMethod("corrMatrix")
}

#' @return The \emph{corrMatrix} function returns the correlations matrix
#' storing the correlation sums that have been computed for all the embedding
#' dimensions.
#' @rdname corrDim
#' @export
corrMatrix.corrDim = function(x){
x$corr.matrix } #' @return The \emph{radius} function returns the radius on which the #' correlation sum function has been evaluated. #' @rdname corrDim #' @export radius.corrDim = function(x){ radius.default(x) } #' @return The \emph{embeddingDims} function returns the embedding dimensions #' on which the correlation sum function has been evaluated. #' @rdname corrDim #' @export embeddingDims.corrDim = function(x){ embeddingDims.default(x) } #' @export #' @method print corrDim print.corrDim = function(x, ...){ print(x$corr.matrix)
}

#' @param main A title for the plot.
#' @param xlab A title for the x axis.
#' @param ylab A title for the y axis.
#' @param type Type of plot (see \code{\link[graphics]{plot}}).
#' @param log A character string which contains "x" if the x axis is to be
#' logarithmic, "y" if the y axis is to be logarithmic and "xy" or "yx" if both
#' axes are to be logarithmic.
#' @param ylim Numeric vector of length 2, giving the y coordinates range.
#' @param col Vector of colors for each of the dimensions of the plot.
#' @param pch Vector of symbols for each of the dimensions of the plot.
#' @param localScalingExp add a plot of the local scaling exponents of the
#' correlation sum.
#' @param add.legend add a legend to the plot?
#' @param cex.legend Magnification value for the legend.
#' @return The \emph{plot} function plots the correlation sum. It is possible to
#' plot the the correlation sum Vs the radius and also the local scaling
#' exponents of the correlation  sum Vs radius.
#' @rdname corrDim
#' @export
plot.corrDim = function(x, main = "Correlation Sum C(r)", xlab = NULL,
ylab = "C(r)", type = "b", log = "xy", ylim = NULL,
col = NULL, pch = NULL, localScalingExp = T,
add.legend = T, cex.legend = 1, ...) {
# set layout depending on options
if (add.legend || localScalingExp) {
current.par =  par(no.readonly = TRUE)
on.exit(par(current.par))
}
if (add.legend && localScalingExp) {
# 3 regions
layout(rbind(1, 2, 3), heights = c(4, 4, 2))
}else{
layout(rbind(1, 2), heights = c(8, 2))
}
if (localScalingExp) {
# add local slopes
layout(rbind(1, 2), heights = c(5, 5))
}
}
number.embeddings = nrow(x$corr.matrix) # plot options if (is.null(ylim)) { ylim = range(x$corr.matrix)
}
if (is.null(xlab)) {
xlab = ifelse(x$corr.order == 2, {"Radius r"}, { paste("Radius r^", x$corr.order - 1, "", sep = "")
})
}
col = vectorizePar(col,number.embeddings)
pch = vectorizePar(pch,number.embeddings)
plot(x$radius ^ (x$corr.order - 1), x$corr.matrix[1, ], type = type, log = log, col = col[[1]], pch = pch[[1]], ylim = ylim, xlab = xlab, ylab = ylab, main = main, ...) i = 2 while (i <= number.embeddings) { lines(x$radius ^ (x$corr.order - 1), x$corr.matrix[i, ], type = type,
col = col[[i]], pch = pch[[i]], ...)
i = i + 1
}
# Add local slopes if needed
if (localScalingExp) {
plotLocalScalingExp(x, xlab = xlab, type = type, col = col, pch = pch,
add.legend = F, ...)
}
par(mar = c(0, 0, 0, 0))
plot.new()
legend("center", "groups", ncol = ceiling(number.embeddings / 2),
bty = "n", col = col, lty = rep(1, number.embeddings), pch = pch,
lwd = rep(2.5, number.embeddings), cex = cex.legend,
legend = x$embedding.dims, title = "Embedding dimension") } } #' @return The \emph{plotLocalScalingExp} function plots the local scaling #' exponents of the correlation sum. #' @rdname corrDim #' @export plotLocalScalingExp.corrDim = function( x, main = "Correlation Dimension C(r)", xlab = NULL, ylab = "Local scaling exponents", type = "b", log = "x", ylim = NULL, col = NULL, pch = NULL, add.legend = T, ...) { # Check if it is possible to compute local slopes if ( ncol(x$corr.matrix) <= 1) {
stop("Cannot compute local scaling exponents (not enough points in the correlation matrix)")
}
number.embeddings = nrow(x$corr.matrix) if (add.legend) { current.par = par(no.readonly = TRUE) on.exit(par(current.par)) layout(rbind(1, 2), heights = c(8, 2)) } lcm = log10(x$corr.matrix)
dlcm = matrix(
t(apply(lcm, MARGIN = 1, differentiate,
h = (x$corr.order - 1) * (log10(x$radius[[2]]) - log10(x$radius[[1]])) )), nrow = number.embeddings) #dlcm=10^dlcm radius.axis = differentiateAxis(x$radius)
# obtain default parameters if not specified
if (is.null(ylim)) {
ylim = range(dlcm)
}
if (is.null(xlab)) {
xlab = ifelse(x$corr.order == 2, { "Radius r" }, { paste("Radius r^", x$corr.order - 1, "", sep = "")
})
}
col = vectorizePar(col,number.embeddings)
pch = vectorizePar(pch,number.embeddings)
# plot
plot(radius.axis ^ (x$corr.order - 1), dlcm[1, ], type = type, log = log, col = col[[1]], pch = pch[[1]], ylim = ylim, xlab = xlab, ylab = ylab, main = main, ...) i = 2 while (i <= number.embeddings) { lines(radius.axis ^ (x$corr.order - 1), dlcm[i, ],
type = type, col = col[[i]], pch = pch[[i]], ...)
i = i + 1
}
par(mar = c(0, 0, 0, 0))
plot.new()
legend("center", "groups", ncol = ceiling(number.embeddings / 2),
bty = "n", col = col, lty = rep(1, number.embeddings), pch = pch,
lwd = rep(2.5, number.embeddings),
legend = x$embedding.dims, title = "Embedding dimension") } } #' @return The \emph{estimate} function estimates the correlation dimension of #' the \emph{corr.dim} object by averaging the slopes of the embedding #' dimensions specified in the \emph{use.embeddings} parameter. The slopes #' are determined by performing a linear regression over the radius' range #' specified in \emph{regression.range}.If \emph{do.plot} is TRUE, a graphic of #' the regression over the data is shown. #' @param use.embeddings A numeric vector specifying which embedding dimensions #' should the \emph{estimate} function use to compute the correlation dimension. #' @param fit.col A vector of colors to plot the regression lines. #' @param fit.lty The type of line to plot the regression lines. #' @param fit.lwd The width of the line for the regression lines. #' @param lty The line type of the correlation sums. #' @param lwd The line width of the correlation sums. #' @param x A \emph{corrDim} object. #' @param regression.range Vector with 2 components denoting the range where #' the function will perform linear regression. #' @rdname corrDim #' @export #' estimate.corrDim = function(x, regression.range = NULL, do.plot = FALSE, use.embeddings = NULL, col = NULL, pch = NULL, fit.col = NULL, fit.lty = 2, fit.lwd = 2, add.legend = T, lty = 1, lwd = 1, ...) { corr.matrix = corrMatrix(x) if (!is.null(use.embeddings)) { corr.matrix = corr.matrix[as.character(use.embeddings),] } else { use.embeddings = as.numeric(rownames(corr.matrix)) } average = 0 #x axis q = nlOrder(x) radius = radius(x) number.embeddings = nrow(corr.matrix) log.radius = log10(radius) if (is.null(regression.range)) { r.min = min(radius) r.max = max(radius) } else { # transform the regression range in the corresponding radius r.min = (regression.range[[1]]) ^ (1 / (q - 1)) r.max = (regression.range[[2]]) ^ (1 / (q - 1)) } lcm = log10(corr.matrix) if (do.plot) { if (add.legend) { current.par = par(no.readonly = TRUE) on.exit(par(current.par)) layout(rbind(1, 2), heights = c(8, 2)) } # obtain vector of graphical parameters if not specified col = vectorizePar(col, number.embeddings) pch = vectorizePar(pch, number.embeddings) fit.col = vectorizePar(fit.col, number.embeddings, col) # eliminate thos dimensions that are innecesary for plotting reduced.x = x reduced.x$corr.matrix = NULL
reduced.x$corr.matrix = corr.matrix reduced.x$embedding.dims = NULL
reduced.x$embedding.dims = use.embeddings plot(reduced.x, col = col, pch = pch, lty = lty, lwd = lwd, add.legend = F, localScalingExp = F, ...) } #average over differents embedding dimensions for (i in 1:number.embeddings) { new.corr = eliminateDuplicates(corr.matrix[i,] , radius) indx = which(new.corr$radius >= r.min & new.corr$radius <= r.max) y.values = log10(new.corr$correlation[indx])
x.values = (q - 1) * log10(new.corr$radius[indx]) fit = lm(y.values ~ x.values) if (do.plot) { lines(new.corr$radius[indx] ^ (q - 1), 10 ^ fit$fitted.values, col = fit.col[[i]], lwd = fit.lwd, lty = fit.lty, ...) } average = average + fit$coefficients[[2]]
}
if (add.legend && do.plot ) {
par(mar = c(0, 0, 0, 0))
plot.new()
legend("center", "groups", ncol = ceiling(number.embeddings / 2),
bty = "n", col = col, lty = rep(1, number.embeddings), pch = pch,
lwd = rep(2.5, number.embeddings),
legend = use.embeddings, title = "Embedding dimension")
}
average / number.embeddings
}

# Private function
# Eliminate duplicate correlation.sums with different radius
eliminateDuplicates = function(correlation.sum, radius) {
len.correlation.sum  = length(correlation.sum)
unique.correlation.sum = unique(correlation.sum)
len.unique.correlation.sum = length(unique.correlation.sum)
if (len.unique.correlation.sum < len.correlation.sum) {
for (corr in unique.correlation.sum) {
index = which(correlation.sum == corr)