Gives summary and second order statistics for a point process (R.P. Littlejohn & R.C. Butler).
Options
PRINT = string token |
Whether to print (statistics); default stat |
|---|---|
SELECTION = string tokens |
What to print (interval, trend, poisson, icorrelation, ispectrum, cspectrum, cintensity, vtcurve, all); default inte |
REPRESENTATION = string token |
How the point process is represented in the DATA variate (time, interval, zeroone); default time |
GRAPHICS = string token |
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.
See also
Procedures: CDESCRIBE, DESCRIBE.
Commands for: Spatial statistics.
Example
CAPTION 'PTDESCRIBE example',\
!t('Data from Vere-Jones & Deng (1988),',\
'A point process analysis of historical earthquakes from',\
'North China, Earthquake Research in China, 2(2), 165-181.',\
'Dates of earthquakes 1480-1980, Richter magnitude > 6.0.');\
STYLE=meta,plain
VARIATE [VALUES=\
1484.1,1487.6,1501.1,1502.8,1505.8,1536.8,1548.7,1556.1,\
1568.3,1568.4,1587.3,1597.8,1614.8,1618.4,1618.9,1622.2,\
1624.1,1624.3,1626.5,1642.5,1652.2,1658.1,1665.3,1668.6,\
1679.7,1683.9,1695.4,1720.5,1730.7,1739.0,1764.5,1815.8,\
1820.6,1829.9,1830.4,1831.7,1846.6,1853.0,1861.5,1882.9,\
1888.5,1910.0,1917.1,1921.0,1921.9,1922.7,1927.1,1929.0,\
1932.3,1932.6,1934.1,1937.6,1945.0,1945.7,1948.4,1966.2,\
1967.2,1969.5,1975.1,1976.3,1976.6,1976.7,1979.5,1979.6] date
CALCULATE intv=MVREPLACE(DIFF(date); date$[1]-1480)
PTDESCRIBE [SELECTION=all; REPRESENTATION=time; GRAPHICS=*] date;\
START=1480; LENGTH=500; CITAU=2; VTTAU=0.5; SAVE=clstats
PTDESCRIBE [PRINT=*; SELECTION=icorrelation,ispectrum;\
REPRESENTATION=interval; GRAPHICS=high] intv; LENGTH=500