R-ymse: Ymse (Various)

#' Maurice Stevenson Bartlett's car data
#' 
#' @description This is an example data set Bartlett used for a lecture course on
#' stochastic processes, Statistics Department, University College, London. 
#' The data represents the times, in seconds, when cars passed an observation point
#' by a road. \cr
#' 
#' Bartlett attributes the data to a Dr A. J. Miller who supplied them as a class 
#' example. According to Adery C. A. Hope the data was recorded on a rural Swedish
#' road.
#' 
#' @section M. S. Bartlett's notes:
#' 
#' Analyse the above data with a view to examining:
#' \describe{
#'     \item{i}{whether the times of passing constitute a Poisson process;}
#'     \item{ii}{if not, whether some form of "bunching" or "clustering" seems to be
#'     present.}
#' }
#' Possible analyses include:
#' \describe{
#'     \item{a}{testing the homogeneity of the consecutive random time-intervals,
#' by means
#'     of a partitioning of the degrees of freedom for the total (approximate)
#' \eqn{\chi^2};}
#'     \item{b}{testing the homogeneity of counts in consecutive fixed time-intervals,
#' choosing
#'     an appropriate interval, and partitioning the degrees of freedom corresponding
#'     to the total dispersion by means of an analysis of variance;}
#'     \item{c}{testing the correlation between the consecutive random 
#' time-intervals;}
#'     \item{d}{examining the overall distribution of counts in fixed time-intervals;}
#'     \item{e}{examining the overall distribution of the consecutive random 
#' time-intervals}
#' }
#' You should undertake at least sufficient of these to answer the questions asked.
#' 
#' @format A numeric vector representing time points in seconds
#' 
#' @source The Spectral Analysis of Point Processes (p. 280), M. S. Bartlett, 1963
#' 
#' Also mentioned in: \cr
#' Statistical Estimation of Density Functions (p. 252), M. S. Bartlett, 1963 \cr
#' A Simplified Monte Carlo Significance Test Procedure (p. 583), 
#' Adery C. A. Hope, 1968
#' 
#' @examples
#' cpgram(diff(bartlett))
#' 
#' bartlett2 <- bartlett - bartlett[1]
#' 
#' x <- rep(0, tail(bartlett2, 1)*10)
#' x[bartlett2*10] <- 1
#' 
#' par(mfrow=c(2, 1), mar=c(2, 3, 1, 1))
#' plot(x, type="l", ann=FALSE)
#' lines(cumsum(x)/sum(x), col="red", lwd=2)
#' 
#' sp <- spectrum(x, main="", xlim=c(0, 0.1), ylim=c(1e-3, 0.04))
#' spec <- predict(loess(sp$spec[1:3000] ~ sp$freq[1:3000], span=0.15), se=TRUE)
#' lines(sp$freq[1:3000], spec$fit, col="red", lwd=2)
#' lines(sp$freq[1:3000], spec$fit - qt((0.99 + 1)/2, spec$df)*spec$se, 
#'   lty=1, col="lightblue")
#' lines(sp$freq[1:3000], spec$fit + qt((0.99 + 1)/2, spec$df)*spec$se,
#'   lty=1, col="lightblue")

"bartlett"


#' 2018 MarbleLympics speed skating times
#' 
#' @description Intermediate and total times for all 16 runs, arranged by lane and
#' heat number.
#' 
#' @format A list containing two data.frames, one for each lane. Columns are heat 
#' and rows are time checks in seconds.
#' 
#' @source https://www.youtube.com/watch?v=fA-O6f_jArk
#' 
#' @examples
#' tt <- t(do.call(cbind, speedskate))
#' pairs(tt)
#' cor(tt)
#' outer(
#'   colnames(tt), 
#'   colnames(tt), 
#'   Vectorize(function(i,j) cor.test(tt[,i],tt[,j])$p.value)
#' )

"speedskate"

#' Mathematical constants
#' 
#' Various mathemathical constants available as global variables
#' 
#' \describe{
#'   \item{\code{e}}{Euler's number}
#'   \item{\code{pi}}{Archimedes' number, the circle constant}
#'   \item{\code{phi}}{Golden ratio}
#'   \item{\code{feig1}}{Feigenbaum's first constant, \eqn{\delta}; 
#' bifurcation velocity}
#'   \item{\code{feig2}}{Feigenbaum's second constant, \eqn{\alpha}; 
#' reduction parameter}
#'   \item{\code{eu.ma}}{Euler–Mascheroni constant}
#'   \item{\code{khin}}{Khintchine's constant}
#'   \item{\code{glai.kin}}{Glaisher-Kinkelin constant}
#' }
#' 
#' @name math_constants
#' 
#' @usage NULL
#' 
#' @aliases 
#' e pi phi feig1 feig2 eu.ma khin glai.kin

"e"

#' High precision mathematical constants
#'
#' Character strings representing various mathemathical constants to ~100 
#' decimal points
#' 
#' \describe{
#'   \item{\code{e.char}}{Euler's number}
#'   \item{\code{pi.char}}{Archimedes' number, the circle constant}
#'   \item{\code{phi.char}}{Golden ratio}
#'   \item{\code{feig1.char}}{Feigenbaum's first constant, \eqn{\delta}; 
#' bifurcation velocity}
#'   \item{\code{feig2.char}}{Feigenbaum's second constant, \eqn{\alpha}; 
#' reduction parameter}
#'   \item{\code{eu.ma.char}}{Euler–Mascheroni constant}
#'   \item{\code{khin.char}}{Khintchine's constant}
#'   \item{\code{glai.kin.char}}{Glaisher-Kinkelin constant}
#' }
#' 
#' @name math_constants_char
#' 
#' @usage NULL
#' 
#' @aliases 
#' e.char pi.char phi.char feig1.char feig2.char eu.ma.char khin.char glai.kin.char

"e.char"