PTDESCRIBE procedure
Gives summary and second order statistics for a point process (R.P. Littlejohn & R.C.
Butler).
Options
PRINT = string
Whether to print (statistics); default stat
SELECTION = strings
What to print (interval, trend, poisson,
icorrelation, ispectrum, cspectrum, cintensity, vtcurve, all); default
inte
REPRESENTATION = string
How the point process is represented in the DATA
variate (time, interval, zeroone); default time
GRAPHICS = string
Style of graphical output, or GRAPHICS=* to avoid any
graphs (lineprinter, highresolution); default high
Parameters
DATA = variates
Variate containing point process to be analysed
START = scalars
Initial time (if REPRESENTATION=time); default 0
LENGTH = scalars
Length of time over which process is observed; default takes
the time of the last event
CITAU = scalars
Window width for calculating count intensity; default 0.5 ×
mean interval length
VTTAU = scalars
Window width for calculating variance-time curve; default 0.5
× mean interval length
SAVE = pointers
Pointer to save calculated values
Description
A point process, or series of events, is characterized both by the times at which events
occur, and the intervals between events. The Poisson process is the most basic point
process, with Poisson counts in any interval, and independent exponentially distributed
intervals between events.
A comprehensive account of methods for analysing point processes is given by Cox &
Lewis (1966). PTDESCRIBE implements many of the test and summary statistics they give
and should be used in conjunction with the text for a full discussion of the motivation and
context of their use. All equations referred to below are from Cox & Lewis (1966).
The DATA variate may contain either the times at which events occur, the intervals
between events, or a sequence of 0's and 1's, with 1's indicating the times of events on
an integer time scale. The option REPRESENTATION specifies which of these is used. If
REPRESENTATION=time and the process is measured from some time other than zero, the
initial time should be given in the parameter START. Otherwise the START time is assumed
to be zero. The first interval is taken to lie between the START time and the first event. If
the process is observed beyond the last event, the total duration of the process should be
given in the parameter LENGTH. Checks are carried out on START, LENGTH and the length
of each interval, and the procedure terminates if these are inconsistent. If
REPRESENTATION=time, the DATA variate may be restricted, facilitating the analysis of
truncated or thinned point processes.
If SAVE is set, time and interval are saved, together with summary interval or second
order statistics specified by SELECTION as detailed below. SAVE sets up a pointer, with
each element labeled by the name of the relevant statistics saved. For example, if
SAVE=clstats, then the intervals between the events will be saved in
clstats['interval'].
The option SELECTION can be used to obtain any combination of eight available
analyses, with the PRINT and GRAPHICS options controlling the output. The default setting
is SELECTION=interval, while SELECTION=all gives all eight analyses. In what follows,
the number of events is denoted by N and the variate carrying the times of events by time.
The rate of a point process is calculated as the reciprocal of the average interval length.
interval - plots data and summarises the interval distribution
print:
summary statistics for the interval process.
graph: times of events; histogram of the intervals between
events; histogram of the intervals with bins appropriate
for the exponential distribution.
save:
summary summary statistics.
trend - tests for trend in the process
print:
an N(0,1) test statistic (Ch 3.3 (11)), which is optimal against certain
specifications of trend; Bartlett's test for the homogeneity of variance of groups of
3, 8 and 20 contiguous intervals.
poisson - tests whether the point process is Poisson
print:
Kolmogorov-Smirnov tests for the empirical distribution function of
times of events (Ch 6.2 (27-29, 38)) and for Durbin's order statistic transformation
of the intervals (Ch 6.2 (43)); Moran's test against a gamma renewal process for the
empirical distribution function (Ch 6.2 (43)); N(0,1) test for trend (see trend above)
is applied to Durbin's transformed process.
graph:
log survivor function of the interval distribution, compared to the
Poisson case (a straight line through the origin with slope = -rate); plots of the
empirical distribution function of times of events and Durbin's order statistics with
Kolmogorov-Smirnov bounds.
icorrelation - autocorrelations for the interval sequence.
print:
the first (N/2-1) end-adjusted autocorrelations (Ch 5.2 (17, 18)) for the
interval sequence and their standardization; the end-adjustments are derived using
the autocorrelations from CORRELATE.
graph:
plot of the autocorrelations of the interval sequence and 95%
confidence bounds.
save:
order the order of the autocorrelations, icorrelation the
autocorrelations of the interval sequence.
ispectrum - periodogram for the interval process
print:
the periodogram for the interval process (Ch 5.3 (6, 8)) obtained from
FOURIER divided by (2πNσ2), where σ2 is the variance of the interval
lengths; since for the Poisson process the ordinates of the periodogram are iid
exponentially distributed r.v.s, the ordinates are also tested as the intervals of a
Poisson process as provided for by the SELECTION settings trend and poisson
above.
graph:
the periodogram and Poisson level (π/2) plotted against frequency;
plot of the scaled cumulative periodogram with Kolmogorov-Smirnov bounds.
save:
ifrequency frequencies at which periodogram is calculated,
ispectrum interval periodogram.
cspectrum - periodogram for the count process
print:
periodogram for the count process (Ch 5.5 (16)) calculated at
frequencies 2πω = 2πn/T, for n=1...2N, T=timeN-time1.
graph:
count periodogram and Poisson level (=2) graphed against
frequency.
save:
cfrequency frequencies at which periodogram is calculated,
cspectrum interval periodogram.
cintensity - intensity function for the counting process
print:
intensity function for the counting process (Ch 5.4(v) (20)) calculated
for times CITAU × (j-0.5), j=1...integer-part(timeN / (2×CITAU)); if CITAU is not set,
PTDESCRIBE sets it to 0.5 times the average interval length; a preliminary screening
precludes an inappropriate setting of CITAU.
graph:
intensity function with asymptotic 95% confidence intervals for the
Poisson level, the intensity for which = rate, plotted against time.
save:
citime times for which intensity is calculated, cintensity intensity
function.
vtcurve variance-time curve V(t) and index of dispersion I(t)
print:
V(t) scaled by 1-time/LENGTH (Ch 5.4(iii) (12) and following), and I(t)
(Ch 4.5(3)) calculated for times VTTAU × j, j=1...integer-part(T/(2×VTTAU)); the
setting of VTTAU is screened to preclude inappropriate values, and if unset is
assigned the value 0.5 times the average interval length.
graph:
V(t) and I(t) against time.
save:
vtime times at which V(t) and I(t) are calculated, vtcurve V(t),
dispersion I(t).
Options: PRINT, SELECTION, REPRESENTATION, GRAPHICS.
Parameters: DATA, START, LENGTH, CITAU, VTTAU, SAVE.
Method
The procedure tests of whether a point process is a Poisson process and calculates
summary statistics in the time and frequency domains for a point process following Cox
& Lewis (1966). Most statistics are obtained using CALCULATE, with
FOURIER being used for ispectrum and CORRELATE for the pre-adjusted autocorrelations.
Action with
RESTRICT
DATA may be restricted only if REPRESENTATION=time, in which case only the units not
excluded by the restriction are involved in the analysis.
Reference
Cox, D.R. & Lewis, P.A.W. (1966). The Statistical Analysis of Series of Events. Methuen,
London.