R/pvar-package.R

In pvar: Calculation and Application of p-Variation

#' p-variation calculation and application
#' 
#' This package deals with p-variation for the sample (i.e. the sequence of data values). 
#' It gives opportunity to calculate the p-variation for the sample -- this is the main purpose of this package.
#' Nonetheless, it could be used to calculate p-variation for arbitrary
#' piecewise monotonic function as well. 
#' Moreover, the package includes one example of practical application of the p-variation.
#' 
#' \tabular{ll}{
#' Package: \tab pvar\cr
#' Type: \tab Package\cr
#' Version: \tab 2.2.5\cr
#' Date: \tab 2016-05-17\cr
#' License: \tab GPL-2\cr
#' Institution: \tab Vilnius University Faculty of Mathematics and Informatics \cr
#' }
#' 
#' This package is about p-variation. It deals with p-variation of a finite sample data values.
#' To be precise, lets star with the definitions. Originally p-variation is defined for a functions.
#' 
#' For a function \eqn{f:[0,1] \rightarrow R}{f:[0,1] -> R} and \eqn{0 < p < \infty}{0 < p <  \infty} 
#' p-variation is defined as
#' 
#' \deqn{
#'   v_p(f) = \sup \left\{ \sum_{i=1}^m |f(t_i) - f(t_{i-1})|^p : 0=t_0<t_1<\dots<t_m=1, m \geq 1 \right\}
#' }{
#'   v_p(f) = sup { \sum |f(t_i) - f(t_{i-1})|^p : 0=t_0<t_1<\dots<t_m=1, m>=1}
#' }
#' 
#' Analogically, for a sequences of values \eqn{X_0, X_1,..., X_n}, the p-variation is defined as
#' \deqn{
#'   v_p(\{X_i\}_{i=0}^n) = \max\left\{ \sum_{i=1}^k |X_{j_i}-X_{j_{i-1}}|^p: 0=j_0<j_1<\dots<j_k=n, \; k=1,2,...,n \right\} 
#' }{
#'   v_p(\{X_i\}_{i=0}^n) = max { \sum |X_{j_i}-X_{j_{i-1}}|^p :0=j_0<j_1<\dots<j_k=n, \; k=1,2,\dots,n }
#' }
#' 
#' The points \eqn{0=t_0<t_1<\dots<t_m=1} (or \eqn{0=j_0<j_1<\dots<j_k=n}) that achieves the maximums is called a supreme partition (or just a partition for short).
#' 
#' There are two main functions that this package is all about, namely it is \code{\link{pvar}} and \code{\link{PvarBreakTest}}.
#' The main function in this package is \code{\link{pvar}}. 
#' It calculates the p-variation and the partition.
#' And the function \code{\link{PvarBreakTest}} is one of the examples of p-variation applications.
#' It performs structural break test of vector \code{x} that exams whether there are multiple
#' shifts in mean inside vector \code{x}.
#' 
#' All other functions are loaded only for supporting and illustrating purposes.
#' @author Author and Maintainer: Vygantas Butkus <Vygantas.Butkus@@gmail.com>.
#' 
#' Special thanks to Rimas Norvaisa the supervisor of my studies.
#' 
#' @references
#' 
#' [1] V. Butkus, R. Norvaisa. Lith Math J (2018). https://doi.org/10.1007/s10986-018-9414-3
#' 
#' [2] R. M. Dudley, R. Norvaisa. An Introduction to
#' p-variation and Young Integrals, Cambridge, Mass., 1998.
#' 
#' [3] R. M. Dudley, R. Norvaisa. Differentiability of 
#' Six Operators on Nonsmooth Functions and p-Variation, Springer Berlin Heidelberg, 
#' Print ISBN 978-3-540-65975-4, Lecture Notes in Mathematics Vol. 1703, 1999.
#' 
#' [4] R. Norvaisa, A. Rackauskas. Convergence in law of partial sum processes in p-variation norm. 
#' Lth. Math. J., 2008., Vol. 48, No. 2, 212-227.   
#' 
#' [5] J. Qian. The p-variation of Partial Sum Processes and the Empirical Process. 
#' The Annals of Probability, 1998, Vol. 26, No. 3, 1370-1383.
#'
#' @seealso
#' The main function is \code{\link{pvar}} - it finds p-variation and the partition that maximizes \code{\link{Sum_p}} function.
#'
#' Other important functions is \code{\link{PvarBreakTest}} it performs structural break test of vector \code{x}
#' by calculating p-variations of \code{BridgeT(x)} (see \code{\link{BridgeT}}). 
#'
#' @encoding utf8 
#' @import Rcpp
#' @importFrom graphics plot points par
#' @importFrom stats filter rnorm time ts sd var
#' @docType package
#' @name pvar-package
#' @useDynLib pvar
#' @aliases pvar-package
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