PERIOD A Time-Series Analysis Package
Contents
1. SUN 167 6 History DATE 1993 Mar 1993 Mar 1993 Mar 1993 Mar 1993 Mar 1993 Mar 16 17 17 17 17 LT 1993 Jul 5 1993 Aug 21 1993 Aug 2 Author VSD VSD VSD VSD VSD VSD VSD VSD VSD 18 Comments Bug in FOLD option found by Martin Still Sussex Traced to PROMPT_ZEROPT parameter not setting ZEROPT parameter correctly Modified PERIOD_FIT and PERIOD_PHASE subroutines accordingly by removing the PROMPT_ZEROPT parameter completely Included an on line help option A new subroutine PERIOD_HELP reads a set of help files contained in the new PERIOD subdirectory PERIOD_HLP The help files are named according to the convention PERIOD_ COMMAND HLP PERIOD_MAIN and PERIOD_PERIOD have been modified accordingly as have the various command procedures and documents LSN 1 3 and LUN 2 3 Minor changes made to PERIOD_INPUT to reduce references made to RUBY TYPE replaced by WRITE in PERIOD_INPUT and PERIOD_STATUS to conform to standard F77 Bug in PERIOD_LOG which prevented the opening of an old log file fixed VERSION 3 0 of PERIOD released to incorporate the above changes Installed on ING Vax and copies sent to STARLINK RLVAD STAR Sussex SUSTAR SYSTEM and Amsterdam PSI IBERPAC DWINGELOO UVAAO1 ERIK This was the first version to be released on STARLINK VERSION 3 0 x Included line number as diagnostic of corrupted x axis data in2. The variance of each of these bins or samples is then calculated This gives a measure of the scatter around the mean light curve defined by the means of the data in each sample The PDM statistic can then be calculated by dividing the overall variance of allthe samples by the variance of the original unbinned dataset This process is then repeated for the next trial frequency Note that windowed data cannot be passed to this option since its variance is zero If the trial period is not a true period then the overall sample variance will be approximately equal to the variance of the original dataset ie the PDM statistic will be approximately equal to 1 If the trial period is a correct period the PDM statistic will reach a local minimum compared with neighbouring periods hopefully near zero SCARGLE By redefining the classical periodogram ie the discrete Fourier periodogram in such a manner as to make it invariant to a shift of the origin of time Lomb 1976 and Scargle 1982 developed a novel type of periodogram analysis quite powerful for finding and testing the significance of weak periodic signals in otherwise random unevenly sampled data Horne and Baliunas 1986 have elaborated on the method and Press and Rybicki 1989 present a fast implementation of the algorithm a modified version of which is used here This implementation uses FFTs to increase the speed of computation although it is in no way equivalent to conventional
3. FFT periodogram analysis Note that windowed data cannot be passed to this option since it needs to calculate the variance which is zero to normalize the power of the periodogram STRING The string length method is an intuitively simple method described in detail by Dworetsky 1983 and Friend et al 1990 The data is folded on a series of trial periods and at each period the sum of the lengths of line segments joining successive points the string length is calculated Minima in a plot of string length versus trial frequency indicate possible periods The string length method is especially useful in the limit of a very small number about 20 or more of randomly spaced observations of periodic phenomena Note that windowed data cannot be passed to this option due to the y data scaling process see Dworetsky 1983 PEAKS This option should be run once a periodogram has been obtained It finds the highest peak in the periodogram or lowest trough if it is a string length PDM or reduced x plot between user specified frequencies The resulting period is calculated along with an error Errors on period detections are notoriously difficult to estimate The estimate used in the previous version of PERIOD v3 0 employed a formula derived by Kovacs 1981 The derivation assumed a single signal Gaussian noise and even data spacing This is clearly not the case with most astronomical datasets and the formula is hence of little use see Horne
4. PERIOD_SCARGLE modified to remove calculation of false alarm probability from subroutine PEAKS in PERIOD_PERIOD modified to include a more rigorous calculation of the SCARGLE false alarm probability using an analytical expression for the number of independent frequencies NWK parameter specifying sizes of the workspace arrays in PERIOD_PERIOD and PERIOD_SCARGLE reduced from MXROW 100 to MXROW 16 in order to cut down on memory usage PERIOD_MOMENT modified to remove redundant skew and kurtosis calculations PERIOD_DOC TEX split into two documents a user document PERIOD_USER TEX LUN 2 1 and a system document PERIOD_SYSTEM TEX LSN 1 1 PERIOD_INTRO modified to give my address as RGO La Palma and my e mail account as CAVAD VSD PERIOD_USER TEX largely rewritten in order to make document clearer and more comprehensive Sampling interval in MEM option of PERIOD_PERIOD now output in PEAKS instead of in MEM 17 DATE 1992 1992 1992 1993 1993 1993 1993 1993 1993 1993 1993 Aug 17 Aug 17 Sept 24 Mar 3 Mar 3 Mar 3 Mar 4 Mar 4 Mar 4 Mar 4 Mar 4 Author VSD VSD VSD VSD VSD VSD VSD VSD VSD VSD VSD SUN 167 6 History Comments Dimension of NWK in PERIOD_PERIOD changed from 16 to 27 This is because the maximum possible workspace usage assuming 10000 data points and 10000 frequency steps in PERIOD_SCARGLE is 262144 VERSION 1 2 o
5. STARLINK RLVAD STAR and Amsterdam ERIK astro uva nl VERSION 4 0 SUN 167 6 History gt gt DATE Author Comments 1995 June 15 GJP PERIOD Version 4 0 ported to UNIX Functionality remains the same as version 4 0 Now requires an environmental variable to find the help files SUN167 and SSN25 re written dropping references to VMS 1995 Sept 26 GJP Added the OGIP reading option and modified the documents accordingly SSN167 becomes SSN25 1995 Oct 12 GJP Modified SUN167 SSN25 and PERIOD_HISTORY to be compatible with star2html so that WWW versions can be created by latex2html Also removed conversion of input files names to upper case Submitted to STARLINK 1995 Dec 15 BLY Removed dependencies on XANADU versions on PLT and FITSIO by a providing xanlib a and plt dhf for supported systems b using Starlink version of FITSIO Makefile and documents modified accordingly Hypertext versions of documents created Starlink Release x VERSION 4 1 1997 Mar 10 GJP PERIOD 4 1 ported to LINUX QDP PLT routine replaced by PGPLOT with some small loss of interactive plotting capability Slightly clearer error messages in some routines modified help files and a minor VAXism in period_output was fixed A number of routines have been changed to ensure that the variables used are preinitialised to zero 1997 May O7 BLY Modified period_poly f to use integer powers not double precision powers when raising X to higher po
6. periodogram Removed the estimates from PEAKS which were all incorrect added an extra option SIG and modified each of the 6 remaining PERIOD finding options in order to perform a Fisher randomisation test Also included new INFO parameter in PERIOD_FT PERIOD_SCARGLE and PERIOD_CLEAN Modified PERIOD_INTRO to reflect new version date and removed request for acknowledgments Modified PERIOD_DETREND and PERIOD_POLY by setting maximum polynomial order from 20 to 10 since this was causing underflow errors Changed variable type of SEED in PERIOD_NOISE from REAL to INTEGER since this caused a crash Changed check of minimum number of points in PERIOD_SINFIT from LT 3 to LE 3 since this caused a crash Updated all the HLP files to reflect the above changes and included a new file for the SIG option 21 DATE 1993 1993 1993 1994 1994 1994 1994 1994 Nov 18 Nov 18 Nov 18 Jan 15 Jan 24 Jan 25 Jan 26 Feb 1 Author VSD VSD VSD VSD VSD VSD VSD VSD SUN 167 6 History Comments Restructured PERIOD subdirectories Now only 4 subdirectories FOR HLP DOC and SYS The first three directories contain the fortran source code help files and user documents as before The SYS directory contains the executable image and the command procedures for initialising and making the PERIOD package Restructured the command procedures There are now only two PERIOD
7. permutations is set to 100 You are then prompted for a seed for the random number generator this number determines the starting point in a number series of infinite period Therefore entering the same seed on two calls to SIG will result in the same sequence of random numbers If SIG is already enabled one can disable the significance calculation by typing SIG again With the significance calculation enabled every time a period finding option is run CHISQ FT SCARGLE CLEAN STRING PDM a Fisher randomization test is performed see for example Nemec and Nemec 1985 This consists of calculating the periodogram as usual and loading the specified output slot The y axis data is then shuffled to form a new randomized time series The periodogram of this dataset is then calculated but not stored in the output slot which will always contain the periodogram of the real time series This randomization and periodogram calculation loop is then performed for the number of permutations specified by the user This can take a considerable amount of time depending on the number of data points in the time series the frequency search parameters and the number of permutations Once the loop is complete you should enter the PEAKS option to view the resulting significances Two significance estimates are given in PEAKS The first denoted FAP1 represents the proportion of permutations ie shuffled time series that contained a trough lower than in
8. 261 125 Lomb N R 1976 Astrophys Space Sci 39 447 Nemec A F L amp Nemec J M 1985 Astron J 90 2317 Press W H amp Rybicki G B 1989 Astrophys J 338 277 Roberts D H Leh r J amp Dreher J W 1987 Astron J 93 968 Scargle J D 1982 Astrophys J 263 835 Scargle J D 1990a Astrophys J 359 469 Scargle J D 1990b In Errors Bias amp Uncertainties in Astronomy Eds Jaschek C amp Murtagh F Cambridge University Press Cambridge Schwarzenberg Czerny A 1989 Mon Not R astr Soc 241 153 Schwarzenberg Czerny A 1991 Mon Not R astr Soc 253 198 Stellingwerf R F 1978 Astrophys J 224 953 15 SUN 167 6 History This appendix lists all of the changes made to the PERIOD package and the release dates of the various versions DATE Author Comments 1992 July 10 VSD VERSION 1 0 released to Robert Smith Sussex Mark 0O Dell Sussex and Erik Kuulkers Amsterdam VERSION 1 0 1992 July 21 VSD This history file added to the package 1992 July 21 VSD PERIOD_SCARGLE modified to output EFFM instead of PROB and JMAX This enables PEAKS to calculate te false alarm probability of any peak in the periodogram 1992 July 21 VSD PERIOD_PERIOD modified to cope with changes in PERIOD_SCARGLE and calculate the false alarm probability of any peak selected in PEAKS Also included a trap for invalid values of the loop gain in CLEAN greater t
9. PERIOD you will be confronted with the following menu options which are described in greater detail below FEROS OBR PERIOD gt A time series analysis package Version gt 5 0 for UNIX Date gt 12 December 2001 Ritter tt ttt ttt tet ett tt eter r tt etree Options INPUT Input ASCII file data OGIP Input OGIP FITS table data FAKE Create fake data NOISE Add noise to data DETREND Detrend the data WINDOW Set data points to unity OPEN Open a log file CLOSE Close the log file PERIOD Find periodicities FIT Fit sine curve to folded data FOLD Fold data on given period SINE or sine curves PLT Call PLT STATUS Information on stored data OUTPUT Output data HELP On line help QUIT Quit PERIOD PERIOD gt Any one of these commands can be entered by typing anything from the shortest unambiguous string up to the full command name Therefore P would be ambiguous but PE would not INPUT As described in section 1 this option allows you to input ASCII data into PERIOD The routine determines the number of columns in the input files and then prompts the user for which columns refer to the x axis y axis and y axis errors if desired see section For example if the user is inputting radial velocity data the x axis would most probably be HJD s the y axis the heliocentric radial velocities and there would most likely b
10. PERIOD_INPUT in response to request from Margaret Penston RGO Bug in PERIOD_PHASE found by Martin Still Sussex When folding data using binning on a number of files the folded data is all placed into a single incorrect slot Problem traced to the use of the COUNTER variable for two different tasks slot number and data array elements Changed the data array variable to BINCOUNT Checked all PERIOD subroutines for a similar error and nothing found Bug in PERIOD_CLEAN found and corrected by Christian Knigge Oxford When processing a number of files using CLEAN the first periodogram is calculated correctly but subsequent files are 19 DATE 1993 1993 1993 1993 1993 1993 1993 1993 Aug Aug Aug Aug Aug Aug Aug Aug 22 22 22 22 22 22 2T 27 Author VSD VSD VSD VSD VSD VSD VSD VSD SUN 167 6 History Comments wrongly processed as are files processed if CLEAN is run twice without exiting PERIOD Problem traced to the C array which is now initialised on each call to PERIOD_CLEAN Checked all PERIOD_PERIOD subroutines for a similar error and nothing found PERIOD_CLEAN made more robust by checking the HWHM calculated by PERIOD_HWHM is not zero before calling PERIOD_FILLB this used to cause a crash PEAKS in PERIOD_PERIOD now checks that different numbers are specified for the periodogram and time series Also now quits if a O i
11. SUN 167 6 Starlink Project Starlink User Note 167 6 V S Dhillon G J Privett K P Duffey 12th December 2001 PERIOD A Time Series Analysis Package Version 5 0 User s Manual SUN 167 6 Contents Contents 1 Introduction 2 Initializing and Running 3 Data Format and Storage 3 1 Reading ASCII files as inpu 3 2 Reading OGIP FITS files as input er Using this Document Menu Options A Simple Recipe Future Improvements Acknowledgments References A History ii 12 13 13 14 15 1 SUN 167 6 Data Format and Storage 1 Introduction This document describes how to use PERIOD version 5 0 a software package designed to search for periodicities in data Version 5 0 is the successor to 4 2 of which it is a Double Precision implementation also re configured to utilise dynamic array allocation deallocation for the majority of its input output and work arrays This document is intended for use with the UNIX software version 5 0 If you have previously used a VMS version you should be aware that the UNIX is case sensitive with regard to directory structure and file names For this reason the instructions below are in lower case and may appear odd to those familiar with VMS 2 Initializing and Running To run PERIOD under UNIX you need only type the word period at the shell prompt 3 Data Format and Storage 3 1 Reading ASCII files as input The simplest form of input for PERIOD is a
12. _START COM and PERIOD_MAKE COM which calls PERIOD_MAKE OPT This is to bring it more into line with STARLINK standard PERIOD_START COM now defines the new directories and defines a new logical PERIOD_DISK PERIOD_MAKE COM compiles creates and fills the object library links and then deletes the object files and object library To bring into line with STARLINK the PERIOD initialisation command is now PERIODSTART and not PERIOD_START Included a new subroutine PERIOD_CASE which converts strings from lower to upper case Replaces calls to the VAX VMS specific STR UPCASE Modified PERIOD_STRING and PERIOD_SINFIT so as not to crash when presented with windowed data All PERIOD_PERIOD options can now either cope with windowed data or else report an error Changes made to PERIOD_DOC files This file converted to LaTeX and renamed PERIOD_HISTORY TEX PERIOD_SYSTEM TEX renamed to SSN167 TEX and rewritten to reflect above changes SUN167 TEX also rewritten to reflect above changes PERIOD_USER TEX the old Sussex LUN deleted Changes made to PERIOD_INPUT so that if it finds x axis data which is not in ascending order it prompts users for whether they would like to sort it or not If they do it sorts the data using PERIOD_SHELLSORT If not it aborts as before Follows suggestions by Ed Zuiderwijk RGO Following extensive testing VERSION 4 0 of PERIOD released to incorporate the above changes Installed on ING Vax and copies sent to
13. a files may contain any number of rows There is also no restricition on the number of FITS file extensions within the file to be read Again you must specify which columns of the table refers to the x axis which column refers to the y axis and which column refers to the y axis errors The y axis errors are optional and if included are used or handled by all operations in the main PERIOD menu see section 5 After you have entered the name of the file to be read the file is opened and the keywords relating to telescope type etc are read These are displayed so that you can make sure you are looking at the right file The output will look something like this PERIOD gt ogip Enter name of OGIP FITS file lt CR gt to quit test fits File details are Telescope ROSAT Instrument PSPCC Detector Unknown Object XRT PSPC PSF AR LA Total number of extensions found 4 Number of OGIP extensions found 4 Extension al A binary table Size 3 by 2 EXTNNAME is STDGTI HDUCLAS1 is GTI HDUCLAS2 is STANDARD 3 SUN 167 6 Using this Document Extension 2 A binary table Size 8 by 29380 EXTNNAME is STDEVT HDUCLAS1 is EVENTS HDUCLAS2 is ACCEPTED Extension 3 A binary table Size 8 by 1246 EXTNNAME is REJEVT HDUCLAS1 is EVENTS HDUCLAS2 is REJECTED You are then asked to select the extension to be examined and are shown the header descriptions for each table column From this information you can select which column
14. and Baliunas 1986 Schwarzenberg Czerny 1991 presents a detailed account of the accuracy of period determinations and advises a post mortem analysis by measuring the width and heights of peaks in a periodogram Although virtually impossible to automate it is possible to do this manually from within PERIOD using the fitting routines of QDP PLT see above Therefore for the sake of generality and to avoid uncertainties version 4 0 of PERIOD now only outputs an error derived by calculating the half size of a single frequency bin centred on the peak or trough in a periodogram and SUN 167 6 Menu Options then converting to period units This error gives an indication of the accuracy to which a peak can be located in a periodogram due to the frequency sampling Clearly with a larger frequency search interval it is more difficult to locate a peak precisely and this is reflected in the error estimate However this error estimate does not take into account the fact that the peak or trough may not represent the true period which can be shifted due to anumber of effects and it should therefore be regarded as a minimum error and not a formal error If the significance calculation is enabled with the SIG command see below two false alarm probabilities are quoted alongside the period The first FAP1 is the probability that given the frequency search parameters there is no periodic component present in the data with this period The seco
15. ation as the original Selecting the 0 1d dataset allows you to apply noise to data create errorbars on the data points and or add noise to the data sampling so that for instance an evenly sampled dataset becomes unevenly sampled This routine is useful not only in creating realistic artificial datasets in conjunction with FAKE but also in investigating the effects of noise on a period detection DETREND This option removes the D C bias from data which if not removed gives rise to significant power at 0 Hz There are two options If the data show no long term trends it is best to simply subtract the mean and divide by the standard deviation the M option This gives a dataset with a mean of zero and a standard deviation of one Otherwise it is best to subtract a low order polynomial fit to the data the P option since if these are not removed a Fourier transform will inject a significant amount of power at the frequency of the long term variations SUN 167 6 Menu Options WINDOW One of the main problems with the classical periodogram see Scargle 1982 for a definition is spectral leakage of which there are several forms Leakage to nearby frequencies sidelobes is due to the finite total interval over which the data is sampled Leakage to distant frequencies is due to the finite size of the interval between samples The WINDOW option sets all the y axis data points to unity A discrete Fourier transform of the resulti
16. data on a given period and zero point Hence this option transforms the data onto a phase scale where one phase unit is equal to one period and phase zero is defined by the zero point If the zero point is not known the data can be folded by taking the first data point as the zero point This option is useful for checking whether derived periods actually give sensible results when applied to the data In addition to normal folding it is also possible to phase bin the data which folds the data and then averages all the data points falling into each bin SINE Adds subtracts multiplies or divides a sine curve from data The sine curve has the form Y GAMMA AMPLITUDE SIN 2 0 PI PERIOD X ZEROPT This option is useful for removing or adding known periods from to data thus enabling or testing the detection of other periods PLT This routine calls PGPLOT routines to display the graphs of the slots requested The layout of the displays is fixed but output file types such as landscape postscript files can be created This represents slightly less functionality than the original XANADU based QDP PLT routine but no ODP PLT routine is currently available for LINUX In order to receive on line help simply type HELP at the PERIOD PLT prompt To exit PERIOD PLT and return to the PERIOD menu type EXIT STATUS Returns information on the data slots or on the stored fits in the log file This command is useful in order to check wh
17. e errors associated with each radial 5 SUN 167 6 Menu Options velocity value Note that the x axis values must be in ascending order otherwise INPUT will report a warning and either sort the data if requested to do so or abort Note also that the y axis errors are used by all options in the main PERIOD menu but by only the CHISQ periodicity finding option in the period_period sub menu OGIP As described in section 3 2 this option allows you to input data from an OGIP FITS table into PERIOD The routine displays some information about the file requested and allows you to choose which of its available tables is to be examined You then select which of the columns in the file refers to the x axis y axis and y axis errors if desired see section 3 FAKE Allows you to create fake data with which to test or experiment with PERIOD Two options are catered for periodic data or chaotic data The periodic data are created by summing a user specified number of sine curves of the form Y GAMMA AMPLITUDE SIN 2 0 PI PERIOD X ZEROPT The chaotic data are created using a simple logistic equation of the form Xn 1 LAMBDA Xn 1 Xn see for example Scargle 1990ab NOISE Using this option it is possible to add noise to data or randomize data The latter operation is carried out by specifying the N ew dataset option which will construct an artificial dataset of the same mean value and the same standard devi
18. f PERIOD released to incorporate the above changes Installed on SUSTAR as local STARLINK software Copy sent to Erik Kuulkers Amsterdam VERSION 1 2 Annoying information written to screen when running PLT removed from PERIOD_PLT Now at ING not Sussex PERIOD_PDM and PERIOD_FTEST subroutines added for the new Phase Dispersion Minimization PDM option PERIOD_PERIOD extensively modified to cope with the new PDM option The two major changes are the inclusion of the code to call PERIOD_PDM which calculates the PDM statistic and PEAKS which now does an F test to determine the significance of a PDM minimum by calling PERIOD_FTEST Error check on zero period added to PERIOD_FAKE PERIOD_FIT PERIOD_PHASE and PERIOD_SINE Reduced the size of the INFILEARRAY string in PERIOD_PERIOD by shortening Frequency to Freq Adapted PERIOD_SYSTEM TEX LSN 1 2 and PERIOD_USER TEX LUN 2 2 to incorporate new PDM option and other changes to bring the documentation into line with the STARLINK release New e mail address version number and date added to PERIOD_INTRO PERIOD_COMPILE PERIOD_LIBRARY PERIOD_MAIL all modified to include the new PERIOD_PDM and PERIOD_FTEST subroutines VERSION 2 0 of PERIOD released to incorporate the above changes Installed on ING Vax and copies sent to STARLINK RLVAD STAR Sussex SUSTAR SYSTEM and Amsterdam PSI IBERPAC DWINGELOO UVAAO1 ERIK VERSION 2 0
19. ford SUN 167 6 History DATE 1993 Aug 1993 Nov 1993 Nov 1993 Nov 1993 Nov 1993 Nov 1993 Nov 1993 Nov 1993 Nov 1993 Nov 1993 Nov 27 13 13 13 13 14 14 14 14 14 18 Author VSD VSD VSD VSD VSD VSD VSD VSD VSD VSD VSD 20 Comments Removed chi squared value of polynomial fit from PERIOD_DETREND and replaced with simple RMS calculation VERSION 3 1 never released VERSION 3 1 Removed CHAOS option from PERIOD_PERIOD since it is of no use in its present form Modified PERIOD_HELP and various COM files and deleted the HLP file to reflect this change Removed MEM option from PERIOD_PERIOD since it is of no use in its present form Modified PERIOD_HELP and various COM files and deleted the HLP file to reflect this change Removed the now redundant subroutines PERIOD_EVLMEM PERIOD_MEMCOF and PERIOD_MDIAN1 from the PERIOD_FOR directory and placed them in the PERIOD_DEV directory Minor bug in PERIOD_NOISE discovered Removed the option which allows one to add Poissonian noise to the data since it incorrectly just added the SQRT of the data value Removed PERIOD_FTEST from the PERIOD_FOR directory and placed in PERIOD_DEV directory This routine is now redundant since significances are calculated in PERIOD_PERIOD using a Fisher test see below Major changes to PERIOD_PERIOD in order to include significance estimate for each
20. grams using PLT Check to see the validity of the highest peak selected 17 Check the results in the log file using STATUS In particular look closely at the false alarm probabilities 18 Fold the original data on the most likely period using FOLD 19 Plot the folded data using PLT If this looks sensible the period may well be correct Make a postscript file by typing epsf_1 To see the other options type instead 20 Output the periodograms and folded data to an ASCII file on disk using OUTPUT 21 Exit PERIOD by typing QUIT The above description is intended only to be a very brief guide Clearly a great deal more experimentation is required before it can definitely be said that a period has been detected For example you should investigate other large peaks in the periodogram try smaller or larger frequency ranges or try one of the other periodicity finding options a useful comparison of a number of different techniques is given by Carbonell Oliver and Ballester 1992 and Heck Manfroid and Mersch 1985 Other analysis techniques might also be attempted such as subtracting a sine curve from the data in order to investigate its effects on the harmonics and enable the detection of less dominant periods 7 Future Improvements Include more periodicity finding options such as AoV Schwarzenberg Czerny 1989 maximum entropy method Fahlman and Ulrych 1982 an FFT and chaos analysis Canizzo and Gooding
21. han 0 and less than 2 1992 July 21 VSD PERIOD_INPUT modified to prevent crashes if the input file contains two contiguous x axis points with the same or decreasing values 1992 July 21 VSD PERIOD_DOC extended by adding a recipe SUN 167 6 History DATE 1992 1992 1992 1992 1992 1992 1992 1992 1992 1992 1992 1992 1992 July July July July July July July July July Aug 6 Aug 7 Aug 1 Aug 1 21 21 21 21 21 29 29 30 30 7 7 Author VSD VSD VSD VSD VSD VSD VSD VSD VSD VSD VSD VSD VSD 16 Comments PERIOD_MAIN _CLEAN _DETREND _FIT _FOLD _FT _INPUT _NOISE _PERIOD _PHASE _POLYFIT and _STATUS modified in order to handle 10000 data or frequency points and 40 data files containing 20 columns This ensures a page file quota of no more than 34000 is required to run PERIOD PERIOD_LOG _READFREE _STATUS modified by setting large loop variables in data or parameter declarations Included a MAXCOL parameter in PERIOD_MAIN and PERIOD_INPUT in order to cut down array sizes Also set size of MAXANT to MXROW in order to ensure that only the maximum number of permitted rows are loaded PERIOD_COM PERIOD_MAIL COM added in order to make e mailing new versions of PERIOD easier VERSION 1 1 of PERIOD released to incorporate the above changes Copy given to Mark 0 Dell Sussex VERSION 1 1
22. hough strictly speaking the power spectrum is a theoretical quantity defined as an integral over continuous time of which the periodogram is merely an estimate based on a finite amount of discrete data Scargle 1982 SUN 167 6 Menu Options FREQ Sets the frequency search parameters The minimum frequency maximum fre quency and frequency interval can be selected by you Generally there is no restriction on the number of frequencies to be stepped though in the processing Alternatively by entering 0 s default values can be accepted Note that the default values are set on entering the PERIOD package and thus the FREQ option need not be run if default frequencies are required The default values are calculated as follows minimum frequency 0 ie infinite period maximum frequency 1 2 x Smallest Data Interval ie Nyquist frequency interval 1 4 x Total Time Interval CHISQ This is a straight forward technique where the input data is folded on a series of trial periods At each trial period the data is fitted with a sine curve The resulting reduced x values are plotted as a function of trial frequency and the minima in the plot suggest the most likely periods See Horne Wade and Szkody 1986 for an example of the use of this method which is ideally suited to the study of radial velocity data or any other sinusoidal variations Note that windowed data cannot be processed by this option since no sine fit is possib
23. ich slots contain which datasets and also as a means of obtaining some elementary statistics on the stored data You can also use this option to check the fits from the SINE and PEAKS options stored in the log file without having to exit the package and read the log file OUTPUT Writes any selected slot to an ASCII file on disk This is the only way of saving data created by PERIOD it does not write to FITS files and should therefore be run before QUITing in order to store Say a power spectrum SUN 167 6 A Simple Recipe 12 HELP This command provides on line help for PERIOD Detailed information about individual com mands can be obtained by typing HELP COMMAND eg HELP PERIOD QUIT or EXIT This option quits a PERIOD session However it does provide a last chance to stay in the package This is essential to prevent accidental exit since any data files created using PERIOD will be lost on exit from the package unless one OUTPUTs the data first 6 A Simple Recipe A simple guide designed to introduce inexperienced users to the steps involved in detecting periodicities is outlined below Detailed descriptions of the individual PERIOD commands can be found in section 5 and in the on line help facility which also gives help on the individual prompts one is confronted with 1 Create an ASCII data file containing the time series 2 Read ASCII data into PERIOD using INPUT 3 or read OGIP FITS files into PERIOD
24. le CLEAN The CLEAN algorithm was originally developed for use in aperture synthesis and was later applied to one dimensional data by Roberts Leh r and Dreher 1987 An adapted version of Leh r s code is used here and is particularly useful for unequally spaced data The algorithm basically deconvolves the spectral window from the discrete Fourier power spectrum or dirty spectrum This produces a CLEAN spectrum which is largely free of the many effects of spectral leakage In order to prevent small errors from destabilizing the CLEAN procedure the user is prompted for two parameters the loop gain and the number of iterations Briefly with each iteration some fraction governed by the loop gain of the window function is removed from the dirty spectrum For convergence the loop gain must lie between 0 and 2 typical values being between 0 1 and 1 Values at the bottom of this range require more iterations but should provide more stability Hence the number of iterations should be large if the loop gain is small typical values lying between 1 and 100 Note that an increase in the number of cleans produces a less noisy spectrum but in general the amplitude of the peaks is decreased sometimes by a substantial amount See Roberts Leh r and Dreher 1987 for further details on choosing these parameters FT This option performs a classical discrete Fourier transform on the data and sums the mean square amplitudes of the result to f
25. lots can be overwritten Typing 0 0 will return the you to the menu Most PERIOD commands prompt not only for an input slot but also for an output slot SUN 167 6 Data Format and Storage 2 Enter first and last slots for input 0 0 to quit Enter first and last slots for output 0 0 to quit The input should contain the dataset or datasets and the output will contain the result of the operation on the dataset or datasets For example given a set of 5 time series which have previously been loaded into slots 1 to 5 using INPUT which need to be fitted with a sine curve using the command FIT see section 5 you would type 1 5 in reply to the first prompt and 6 10 in reply to the second PERIOD will then fit sine curves to the data files loaded in slots 1 to 5 and put the fits in slots 6 to 10 Clearly the number of output slots must be equal to the number of input slots In addition if any selected input slot is empty PERIOD will abort the operation and return you to the main menu It should be noted that in order to save on storage space you could have typed 1 5 in reply to the second prompt and the original data files would have been overwritten by the resulting sine curves 3 2 Reading OGIP FITS files as input Because of the increasing use of OGIP standard FITS files within the X ray community PERIOD has been extended to allow FITS file table extensions where the FITS keyword HDUCLASS has the value OGIP As with ASCII data dat
26. n ASCII file The files may contain any number of rows and columns It is possible to specify which column refers to the x axis which column refers to the y axis and which column refers to the y axis errors The y axis errors are optional and if included are used or handled by all operations in the main PERIOD menu see section 5 However the only periodicity finding option in the period_period sub menu which actually uses the errors on data points is CHISQ see section 5 The remaining five techniques CLEAN FT PDM SCARGLE STRING ignore errorbars if they are present Note also that the input file must contain x axis values which are in ascending order otherwise the program will report a warning and either sort the data if requested to do so or abort the input Data is stored and processed within PERIOD using a slot system A data slot is simply an array holding one dataset The maximum number of data slots which can be handled at any one time in PERIOD is 40 hence PERIOD has the capability of analysing a large amount of data simultaneously The first command that is usually run in PERIOD is INPUT see section bh which loads datasets into data slots Enter first and last slots for input 0 0 to quit In order to load the first slot with a single dataset you should reply 1 1 to the above prompt Similarly if you want to load slots 4 through to 9 with 6 datasets one should reply 4 9 to the above prompt It is important to note that s
27. nd FAP2 is the probability that the period is not actually equal to the quoted value but is equal to some other value Note that FAP1 is only output if the whole frequency range is specified to be analysed in PEAKS see below One sigma errors on both significance values are also given If the significance values are zero these errors are displayed as 1 implying that the false alarm probabilities lie between 0 00 and 0 01 with 95 confidence Clearly the lower a significance value and its error the more likely the quoted period is a correct one If both the significances and errors are displayed as 1 this means that the input periodogram has not been subjected to a significance calculation ie the significance calculation has been disabled Note that the results can be written to a log file if one is open For more information on the SIG option see below For useful discussions on errors and significances of period determinations see Schwarzenberg Czerny 1991 and Nemec and Nemec 1985 SIG This option works as a switch either turning on or turning off the significance calculation The default on entering PERIOD is for the significance calculation to be disabled This means that no significance values are calculated or attached to period determinations By typing SIG the significance calculation is enabled You are first prompted for the number of permutations in the sample To ensure reliable significance values the minimum number of
28. ng data using for example the FT option see below yields the window function or spectrum which shows the effects of spectral leakage OPEN It is possible to store the fits calculated by SINE and PEAKS in a log file This option opens a new log file if it does not already exist or else re opens an old log file and skips over the existing entries CLOSE This option closes the currently open log file PERIOD This is where all the work is done You will be confronted by the following sub menu Options SELECT FREQ CHISQ CLEAN FT PDM SCARGLE STRING PEAKS SIG HELP QUIT Select data slots Set show frequency search limits Chi squared of sine fit vs frequency CLEANed power spectrum Discrete Fourier power spectrum Phase dispersion minimization Lomb Scargle normalized periodogram String length vs frequency Calculate period from periodogram Enable disable significance calc On line help Quit PERIOD_PERIOD PERIOD_PERIOD gt e SELECT Selects input and output slots for processing as described in section 3 The input slots should contain the time series the output slots will contain for example the power spectra SELECT must be run every time a periodicity finding option is about to be executed although tedious this prevents one from accidentally overwriting slots Throughout the PERIOD package and this document the terms power spectrum and periodogram are used inter changeably alt
29. on evenly spaced functions and it is possible to obtain values below this even for pure noise data with certain data spacings The well known SCARGLE false alarm probabilities are also incorrect since the Horne and Baliunas 1986 equation for the number of independent frequencies has been shown to be incorrect Christian Knigge Oxford private communication Even if correct the Horne and Baliunas formula would be incorrect to apply in a general way since itisa poor approximation to small datasets The only reliable method of estimating significances from such non parametric tests is by some sort of Monte Carlo or randomization method As described above one such method Fisher randomization has been implemented in this version of PERIOD v4 0 following the prescription described by Nemec and Nemec 1985 HELP This command provides on line help for PERIOD Detailed information about indi vidual commands can be obtained by typing HELP COMMAND eg HELP PEAKS QUIT or EXIT This quits the PERIOD_PERIOD sub menu and returns the user to the main PERIOD menu 11 SUN 167 6 Menu Options Returning to the main PERIOD menu FIT Folds the data on a given period and zero point and then fits the data with a sine curve The sine curve has the form Y GAMMA AMPLITUDE SIN 2 0 PI PERIOD X ZEROPT Outputs the fit parameters which can be written to a log file and the resulting sine curve FOLD Folds the
30. orm a power spectrum see for example Deeming 1975 This discrete Fourier transform is defined for arbitrary data spacing and is equal to the convolution of the true Fourier transform with a spectral window Hence the effects of data spacing such as aliasing are all contained in the spectral window which can be generated using the WINDOW option see above This spectral window should be analysed in conjunction with the discrete Fourier transform generated here in order to estimate the effects of aliasing PDM The phase dispersion minimization PDM technique is simply an automated version of the classical method of distinguishing between possible periods in which the period producing the least observational scatter about the mean light curve or for example radial velocity curve is chosen This technique which is described in detail by Stellingwerf 1978 is well suited to cases in which only a few observations are available over a limited period of time especially if the light curve is highly non sinusoidal The data is first folded on a SUN 167 6 Menu Options 8 series of trial frequencies For each trial frequency the full phase interval 0 1 is divided into a user specified number of bins The width of each bin is specified by the user such that a point need not be picked if a bin width narrower than the bin spacing is selected or a point can belong to more than one bin if a bin width wider than the bin spacing is selected
31. s 1988 Scargle 1990ab and Lehto Czerny and McHardy 1993 8 Acknowledgments The author gratefully acknowledges the use of subroutines written by Joseph Leh r Cambridge Keith Horne STScI and Tom Marsh Oxford as well as those written by Press and Rybicki 1989 I also thank the bug finding guinea pigs Martin Still Sussex Mark O Dell Sussex Erik Kuulkers Amsterdam and Thomas Augusteijn Amsterdam for their many useful comments and suggestions Finally I am indebted to Christian Knigge Oxford for his help with bug fixes and his many suggestions regarding significance estimates SUN 167 6 References 14 9 References Canizzo J amp Goodings D 1986 Astrophys J Lett 334 L31 Carbonell M Oliver R amp Ballester J L 1992 Astron Astrophys 264 350 Deeming T J 1975 Astrophys Space Sci 36 137 Dworetsky M M 1983 Mon Not R astr Soc 203 917 Fahlman G G amp Ulrych T J 1982 Mon Not R astr Soc 199 53 Friend M T Martin J S Smith R C amp Jones D H P 1990 Mon Not R astr Soc 246 637 Heck A Manfroid J amp Mersch G 1985 Astron Astrophys Suppl Ser 59 63 Horne J H amp Baliunas S L 1986 Astrophys J 302 757 Horne K Wade R A amp Szkody P 1986 Mon Not R astr Soc 219 791 Kovacs G 1981 Astrophys Space Sci 78 175 Lehto H J Czerny B amp McHardy I M 1993 Mon Not R astr Soc
32. s entered for the slot number this did not work properly before Bug in CHAOS option of PERIOD_FAKE did not reset the initial value parameter if processing more than one slot Problem remedied by introducing a new parameter INITVAL PERIOD_DETREND made more robust by removing the PERIOD_POLYFIT subroutine and calling PERIOD_LSQUAR directly from PERIOD_DETREND Program now outputs the chi squared value of the polynomial fit Also rewrote PERIOD_POLY to handle polynomials of order up to 20 the previous limit was 12 and removed the redundant subroutine PERIOD_DPOLY Updated the command procedures in PERIOD_COM by removing any reference to PERIOD_POLYFIT which has now been deleted from the PERIOD_FOR subdirectory Bug found in DETREND the polynomial fit was divided and not subtracted from the time series Have modified PERIOD_DETREND accordingly Added my INTERNET e mail address to PERIOD_INTRO and incremented the date and version number 3 1 Removed string length significance limits from PEAKS in PERIOD_PERIOD since misleading the theoretical minimum only applied to an evenly spaced sinusoid Followed comment by Christian Knigge Oxford Corrected reduced chi squared values in the CHISQ option of PERIOD_PERIOD and PERIOD_FIT by dividing by the number of points minus the number of free parameters 3 in PERIOD_SINFIT Previously the chi squared value was only divided by the number of points Bug spotted by Christian Knigge Ox
33. should be in x axis y axis or the y axis errors part of your chosen data slot Enter the extension number to be considered 0 to quit 3 Column Contents 1 X Y PHA PI TIME DETX DETY 8 STATUS Enter the TTYPE column number to be read as X 0 to quit 1 Enter the TTYPE column number to be read as Y 0 to quit 2 Enter the TTYPE column number to be read as Y errors 0 to ignore 0 NO oP WN Enter the slot number into which the data should be read 0 to quit 1 Reading data into slot 1 Read 1246 out of 1246 OK Filled Slot 1 After this you are given the option to reread the same document and fill a different slot While slightly more complicated to read than normal ASCII the program is still very easy to use 4 Using this Document There are two main sections in this document Section 5 should be consulted by experienced users who require detailed information on individual PERIOD commands while Section 6 is intended for inexperienced users who require an introductory recipe detailing the steps that should follow when searching for periodicities in data SUN 167 6 Menu Options 4 Please note that PERIOD also has an extensive on line help facility This provides information at a level of detail similar to that found in section However it also provides a detailed description of the individual prompts something which cannot be found in this document 5 Menu Options PERIOD is a menu driven package On entering
34. the case of the CHISQ STRING and PDM options or a peak higher than in the SUN 167 6 Menu Options 10 case ofthe FT SCARGLE and CLEAN options that of the periodogram of the unrandomized dataset at any frequency This therefore represents the probability that given the frequency search parameters no periodic component is present in the data with this period and it is only output in PEAKS if the whole frequency range is specified to be analysed The second significance denoted FAP2 represents the proportion of permutations that at the frequency given by the period output by PEAKS contained troughs lower than or peaks higher than the peak or trough in the periodogram of the real dataset This therefore represents the probability that the period is not actually equal to the quoted value but is equal to some other value and is quoted for any frequency range specified in PEAKS Standard errors on both of these false alarm probabilities are also given see Nemec and Nemec 1985 It is perhaps worth mentioning here that significance estimates of period detections are notoriously unreliable The methods used in the previous version of PERIOD v3 0 suffered from a number of problems For example the F test used with the PDM method Stellingwerf 1978 has been proved to be incorrect see for example Heck Manfroid and Mersch 1985 Similarly the theoretical minimum string lengths quoted by Dworetsky 1983 are misleading since they are based
35. using OGIP Doo wae wa 4 Detrend the data using the M option in DETREND if the data show long term variations use the P option instead 5 Open a log file for the fits using OPEN 6 Enter the PERIOD_PERIOD menu by typing PERIOD 7 Select the slots which contain the time series data and specify the output slots for the periodograms using SELECT 8 Set the frequency search limits using FREQ If you have no idea what the period is accept the default values by typing 0 or alternatively by not typing FREQ in the first place 9 Enable the significance calculation by typing SIG and specifying say 200 permutations 10 Calculate the Lomb Scargle periodogram by typing SCARGLE 11 Now run PEAKS on the resulting periodogram specifying the frequency range which contains the peak you wish to measure you may enter 0 0 if you wish to process the entire range Write the results to the log file 12 Now reselect the time series slots and different output slots for a new periodogram using SELECT 13 SUN 167 6 Acknowledgments 13 Type CLEAN with 5 iterations and a loop gain of 0 2 for example Before doing this you may wish to disable the significance calculation by typing SIG again since the CLEAN algorithm can take a considerable amount of processing time 14 Run PEAKS on the resulting periodogram and store the results 15 Now quit the PERIOD_PERIOD sub menu by typing QUIT 16 Plot the periodo
36. wers since X can be negative Old method caused core dump on Digital Unix k VERSION 4 2 k VERSION 5 0 2001 Dec 12 KPD PERIOD implemented in Double Precision Dynamic array allocation deallocation introduced for the majority of the input output and work arrays
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