bartlett: Maurice Stevenson Bartlett's car data

In AkselA/R-ymse: Ymse (Various)

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.

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.

Usage

bartlett

Format

A numeric vector representing time points in seconds

M. S. Bartlett's notes

Analyse the above data with a view to examining:

i

whether the times of passing constitute a Poisson process;

ii

if not, whether some form of "bunching" or "clustering" seems to be present.

Possible analyses include:

a

testing the homogeneity of the consecutive random time-intervals, by means of a partitioning of the degrees of freedom for the total (approximate) χ^2;

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;

c

testing the correlation between the consecutive random time-intervals;

d

examining the overall distribution of counts in fixed time-intervals;

e

examining the overall distribution of the consecutive random time-intervals

You should undertake at least sufficient of these to answer the questions asked.

Source

The Spectral Analysis of Point Processes (p. 280), M. S. Bartlett, 1963

Also mentioned in:
Statistical Estimation of Density Functions (p. 252), M. S. Bartlett, 1963
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")

AkselA/R-ymse documentation built on March 21, 2020, 9:52 a.m.