Page 257 - Special Topic Session (STS) - Volume 1

STS426 Guillaume B.
We present a general transient detection method based on the likelihood
function applicable to a wide range of problems. We present two new
1
periodogram statistics: the generalized (kth order) modified Rayleigh statistic,
2
2
2
labeled ℛ ; and the modified Z statistic, labeled Ƶ , that benefit from all the

features of their predecessors but do not suffer from the artifacts caused by
unaccounted for correlations in the trigonometric moments. Finally, we
2
present two fit statistics, the K and B statistics, optimal for χ and exponential
variables respectively, and thus relevant to the model-testing in frequency
space.

2. Methodology
A transient can only be identified as such in relation to the underlying
background, which can either be constant or variable. In most applications
where transient sources are searched for, the background is constant or nearly
so. Intensity ratios and variability timescales must be worked with as
parameters in order to establish optimal detection algorithms and thresholds.
The first measurement gives the first estimate of the reference value: the value
we expect to measure under usual conditions when there is no transient. The
second measurement gives a second estimate of that reference value, but it
can also be evaluated for its potential of being a transient. The third
measurement gives a third estimate of the reference, the likelihood of
measuring such a value is evaluated by the ratio of the single-measurement
likelihood function centered on the maximum likelihood reference value given
by the previously calculated joint likelihood.
With each subsequent measurement we: (1) compute the likelihood of the
newly measured value based on the single-measurement function defined by
the current reference value; (2) if the likelihood is less than the defined
threshold, issue a transient event trigger. Do not update the estimate of the
reference value; (3) if the likelihood is more than the defined threshold (within
the likelihood interval), recalculate the joint likelihood function including the
new measurement and update the reference. The thresholds must be
optimized for the application.
The detection of transients in images, time series, energy spectra, and
power spectra follows the same methodology, applying different analytical
probability functions depending on the input data. The illustration uses the
periodogram for which the ℛ statistic is used for greatest sensitivity to weak
2

signals. Unlike images, time series, and energy spectra, the values of power
2
estimates in frequency channels are related to χ and exponential distributions.

1 The mathematical statistics of likelihood are from the work of Fisher (1912, 1922); the
philosophical basis is primarily from Royall (1997); and other technical details of data analysis
and statistics are mostly from Cowan (1997).
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