# Source Variability

Source variability within an observation is assessed by three methods: (1) the Kolmogorov-Smirnov (K-S) test, (2) the Kuiper's test, and (3) computation of the Gregory-Loredo variability probability, all based on the source region counts. Intra-observation source variability within any contributing observations to a master source entry is assessed according to the highest level of variability seen within any single contributing observation. Inter-observation source variability between any contributing observations to a master source entry is assessed by application of a \(\chi^{2}\) hypothesis test applied to the source region photon fluxes observed in the contributing observations.

## Properties of Individual Per-Observation Detections

### Gregory-Loredo Variability Probability

*var_prob*

The probability that the source region count rate lightcurve is better described by multiple, uniformly sampled time bins with potentially different rates in each bin, as opposed to being described by a single, uniform rate time bin. This probability is based upon the odds ratios (for describing the lightcurve with two or more bins of potentially different rates) calculated from a Gregory-Loredo analysis of the arrival times of the events within the source region. Corrections to the event rate are applied accounting for good time intervals and for the source region dithering across regions of variable exposure (e.g., chip edges) during the observation. Probability values are calculated for each science energy band.

### Kolmogorov-Smirnov (K-S) Test Probability

*ks_prob*

The probability that the arrival times of the events within the source region are inconsistent with a constant source count rate throughout the observation. High values of this quantity imply that the source is not consistent with a constant rate, and that the source is likely variable. The probability is computed by means of a hypothesis rejection test from a one-sample K-S test applied to the unbinned event data, with corrections applied for good time intervals and for the source region dithering across regions of variable exposure (e.g., chip edges) during the observation. Probability values are calculated for each science energy band. Note that this variability diagnostic does not treat the source and background separately.

### Kuiper's Test Probability

*kp_prob*

The probability that the arrival times of the events within the source region are inconsistent with a constant source count rate throughout the observation. High values of this quantity imply that the source is not consistent with a constant rate, and that the source is likely variable. The probability is computed by means of a hypothesis rejection test from a one-sample Kuiper's test applied to the unbinned event data, with corrections applied for good time intervals and for the source region dithering across regions of variable exposure (e.g., chip edges) during the observation. Probability values are calculated for each science energy band. Note that this variability diagnostic does not treat the source and background separately.

### Variability Index

*var_index*

An index in the range [0,10] that combines (a) the Gregory-Loredo variability probability with (b) the fractions of the multi-resolution light curve output by the Gregory-Loredo analysis that are within 3σ and 5σ of the average count rate, to evaluate whether the source region flux is uniform throughout the observation. See the Gregory-Loredo Probability How and Why topic for a definition of this index value, which is calculated for each science energy band.

### Count Rate Variability

*var_mean*,

*var_sigma*,

*var_min*,

*var_max*

#### Mean Count Rate

The mean count rate (*var_mean*) is the
time-averaged source region count rate derived from
the multi-resolution light curve output by
the Gregory-Loredo
analysis. This value is calculated for
each science energy
band.

#### Count Rate Standard Deviation

The count rate standard deviation
(*var_sigma*) is the time-averaged
1σ statistical variability of the source region count
rate derived from the multi-resolution light
curve output by the Gregory-Loredo analysis. This value is
calculated for each science energy band.

#### Minimum Count Rate

The minimum count rate (*var_min*) is the
minimum value of the source
region count rate derived from the multi-resolution light
curve output by the Gregory-Loredo analysis. This value is
calculated for each science energy band.

#### Maximum Count Rate

The maximum count rate (*var_max*) is the
maximum value of the source region count rate
derived from the multi-resolution light curve output
by the Gregory-Loredo analysis. This value is
calculated for
each science energy
band.

### Dither Warning Flag

*dither_warning_flag*

The dither warning flag consists of a Boolean whose value is TRUE if the highest statistically significant peak in the power spectrum of the source region count rate, for the science energy band with the highest variability index, occurs either at the dither frequency of the observation or at a beat frequency of the dither frequency. Otherwise, the dither warning flag is FALSE. This value is calculated for each science energy band.

### Gregory-Loredo Light Curve File

*see Data Products page*

Each light curve file records the multi-resolution light curve output by the Gregory-Loredo analysis of the arrival times of the source events within the source region, per observation and science energy band. A background light curve with identical time-binning to the source light curve is derived from an analysis of the events within the background region. Note that the source lightcurve is not strictly a rate derived from binned counts. Instead, it is a probabilistic model of the lightcurve, derived from a probability weighted average of the lightcurve models calculated by the Gregory-Loredo algorithm at different uniform binnings.

## Master Source and Stacked Observation Detection Properties

### Intra-Observation:

#### Intra-Observation Gregory-Loredo, Kolmogorov-Smirnov, and Kuiper's Variability Probability

*var_intra_prob*,

*ks_intra_prob*,

*kp_intra_prob*

The Gregory-Loredo,
Kolmogorov-Smirnov (K-S) test, and
Kuiper's test
intra-observation
variability probabilities
represent the highest values of the variability probabilities
(*var_prob*, *ks_prob*, *kp_prob*)
calculated for each of the contributing
observations (i.e., the highest level of variability among
the observations contributing to the master source entry).

#### Intra-Observation Variability Index

*var_intra_index*

The intra-observation
variability index (*var_intra_index*)
represents the highest value of the variability
indices (*var_index*) calculated for each of
the contributing observations.

### Inter-observation:

#### Inter-Observation Variability Probability

*var_inter_prob*

The inter-observation
variability probability (*var_inter_prob*)
is a value that records the probability that the source
region photon flux varied between the contributing
observations, based on the \(\chi^{2}\) distribution of the
photon fluxes and the errors (standard deviation) of the individual
observations. In other words, (1 - *var_inter_prob*)
is the probability that the measured fluxes are consistent
with a non-varying source.

The reason for this careful definition is that the probabilities for intra-observation and inter-observation variability are, by necessity, of a different nature. Whereas one can say with reasonable certainty whether a source was variable during an observation covering a contiguous time interval, when comparing measured fluxes from different observations one knows nothing about the source's behavior during the intervening interval(s). Consequently, when the inter-observation variability probability is high (e.g., >0.7), one can confidently state that the source is variable on longer time scales, but when the probability is low, all one can say is that the observations are consistent with a constant flux.

#### Inter-Observation Variability Index

*var_inter_index*

The inter-observation
variability index (*var_inter_index*) is an
integer value in the range [0,10] that is derived
from the inter-observation variability
probability to evaluate whether the source
region photon flux is constant between the
observations. The degree of confidence in
variability expressed by this index is similar to
that of the intra-observation variability
index.

#### Inter-Observation Count Rate Variability

*var_inter_sigma*

The inter-observation
flux variability (*var_inter_sigma*) is the
absolute value of the difference between the error
weighted mean source region photon flux density of
all the contributing observations and the mean
source region photon flux density of the single
observation for which the absolute value of the
difference, divided by the standard deviation for
that observation, is maximized:

Here, \(\left\langle
F_{ew}\right\rangle\) represents
the *inter*-observation error weighted mean
source region photon flux
density; \(\left\langle
f_{i=x}\right\rangle\) is
the *intra*-observation mean source region
photon flux density for the single
observation \(x\);
and \(\sigma_{i=x}\) is the
standard deviation corresponding to the
observation \(x\)
[where \(x \in \left\{ k \in
\mathbb{Z}^{+} | k \leq N \right\}\)
for \(N\) contributing
observations]. Of all contributing observations,
observation \(x\) yields the
highest value for this equation, which is the value
recorded by *var_inter_sigma*.