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 (2πNσ^{2}), where σ^{2} 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=time–_{N}time_{1}. |
---|---|

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(time / (2×_{N}`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