BUGS_ · SEE_ALSO
vartst - non-parametric statistical tests for source variability
vartst source_file var_file
VARTST computes test statistics for the Kolmogorov-Smirnov (KS) and Cramer-von Mises (CvM) one-sample goodness-of-fit tests, in order to test the null hypothesis of constant source intensity. These non-parametric statistical tests compare the observed cumulative distribution function (cdf) of photon arrival times with a model cdf expected for a constant source. In each test, the hypothesis of constant source intensity may be rejected at the confidence level C if the test statistic exceeds the (1-C) quantile for the appropriate distribution. Arrival times are automatically corrected for data gaps, and energy channels and time intervals to be tested may be specified on the command line.
The KS test used is the "two-sided" test, so called because deviations of the observed cdf both above and below the model cdf are considered. The test statistic is the supremum of the absolute value of the difference between the observed and model cdf's. Large N approximations for the quantiles of the KS test statistic are used. These approximations are generally considered valid for N > 40 events, but comparisons of approximate and tabulated quantiles for 90%, 95%, and 99% confidence levels indicate that the approximate values differ from the tabulated ones by no more than 0.5% for N as small as 10 events. Confidence bands for the unknown true cdf are also computed. These bands form an "envelope" about the observed cdf and may be used to test non-constant source models. Any model cdf which lies wholly within the region defined by a band may be considered consistent with the data at the confidence level appropriate to that band.
The CvM test provides an alternative to the KS test. Whereas the latter considers only the maximum difference between the observed and model cdf's, the CvM test uses the sum of the squares of the differences between the two distributions (plus a small number of order 1/N) as its test statistic. Again, large N approximations for quantiles of the CvM test statistic are used, and comparisons of approximate and tabulated values for N=10 indicate differences of less than 2% for 90% and 95% confidence level quantiles, and 5% for the 99% quantile.
The output consists of the values of the KS and CvM test statistics plus the 90%, 95%, and 99% quantiles for each statistic. An STSDAS table file containing more detailed information (see below) is also output. Results of the KS test may be plotted with the task KSPLOT.
- source_file = prompt = Input Source Timing File
Required input in time ordered QPOE format. The file is created by running TIMSORT on a specified source region. The file may be specified by supplying the rootname and a "_sti.qp" extension will be appended by the task before looking for the file. For example input "i6757" is interpreted as "i6757_sti.qp".
- var_file = prompt = root name for output files
Required Output filename root. Three files are output. The statistic results are written in table format, and a "_var.tab" extension will be appended.. For example input "i6757" becomes "i6757_ltc.tab". Each row in the table represents the statistics for each photon of data. The column titles and definitions are as follows:
time ----------- degapped photon arrival time dist ----------- vertical distance between the model and data model ---------- cumulative distribution function of model cdf ------------ cumulative distribution function of data cdfplus -------- upper confidence band cdfminus ------- lower confidence bandA table file header is also written. It consists of parameters copied from the input QPOE file and parameters recorded from running this task.
Also 2 plot command files are written. The ascii files consist of igi plot commnds. root_ig1.cmd = plot commands for the integral and max diff plot, and root_ig2.cmd consists of the commands to overlay the confidence bands.
- (bandwidth = 95.0) prompt = cdf confidence band width
The bandwidth for the confidence band computation. Choices are 90.0, 95.0, 99.0. 95.0 is the default.
- (display = 1) [int]
The display level. Input 0 for no output and 5 for the maximum level.
- (clobber = no) [boolean]
Boolean flag specifying whether or not the table file can be overwritten if it already exists.
- (get_gintvs = yes) [boolean]
Boolean flag specifying whether to filter the input events through the good time intervals stored in the QPOE file. It is useful to turn this parameter off when analyzing lab data where good intervals are not applicable or in a file without the intervals, but otherwise the value should remain set to yes.
Type `help timfilter` and `help timsort` for information concerning input files.1. Test whether the source is variable over the entire observation. Write the cumulative distribution of arrival times, with 95% confidence bands to rp110590_var.tab. The statistics displayed below are written to the table header. Write the igi plot command files to rp110590_ig1.cmd, and rp110590_ig2.cmd.
xt> vartst Input Source Timing File (root_sti.qp): rp110590_sti.qp Root name for output files [root(_var.tab,_ig1,_ig2.cmd)]: . Number of Events = 1749 Ks-test Thresh. : 90% = 0.02906, 95% = 0.03240, 99% = 0.03883 Max diff = 0.02017, No Variability detected CvM Thresh. : 90% = 0.34700, 95% = 0.46100, 99% = 0.74300 CvM = 0.07542, No Variability detected Creating igi Ksplot cmd file: ./rp110590_ig1.cmd Creating igi Conf band cmd file: ./rp110590_ig2.cmd Creating Var file: ./rp110590_var.tab (The data can be plotted with task 'ksplot')
No Corrections are being applied for Fractional Spatial Exposure.
Since these tests deal with individual events, they are unable to distinguish between source and background. However, since these tests are only useful in testing the simple null hypothesis of constant intensity, this is not an issue unless the background itself is variable. In such cases the tests may indicate variability when there is none. The ability to deal with background variability will be added in a later release.
Documentation on region filtering (help regions ) for a description of the spatial filter user interface.
Documentation on QPOE filtering (help qpoe ) for a description of the QPOE filter user interface.
Documentation on file extensions (help extensions ) for a description of PROS file extensions.
Documentation on coordinates (help coords ) for a description of PROS coordinate conventions.
For an in-depth discussion of the KS and CvM tests, the user is referred to W. J. Conover Practical Nonparametric Statistics, 1971 (Wiley), and W. W. Daniel Applied Nonparametric Statistics, 2nd edition, 1990 (PWS-Kent). For tabulations of test statistic quantiles for finite sample sizes, the user is referred to L. H. Miller, J. Amer. Statist. Assoc., 1956, v. 51, pp. 111-121 for the KS test statistic, and M. A. Stevens and U. R. Maag, Biometrika, 1968, v. 55, pp. 428-430 for the CvM test statistic.
We are grateful to Prof. Eric D. Feigelson of Penn State for his help in developing this task.