# 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 the result of multiple, uniformly sampled time bins, each with different rates, as opposed to the result of a single, uniform rate time bin. This probability is based upon the odd 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 hypothesis rejection test
described in the hardness
ratios and variability memo. Given the
\(N\) individual Bayesian
probability distribution of the aperture fluxes for
the same source in \(N\)
different observations (their means and standard
deviations), we estimate for each band the maximum
likelihood
\(\mathcal{L}_{1}^{\mathrm{max}}\)
and the corresponding maximizing arguments
\(F_{\left\langle band
\right\rangle}^{i,\mathrm{max}}\), of the
observed fluxes assuming a different flux for each
observation, as well as the maximum likelihood
\(\mathcal{L}_{2}^{\mathrm{max}}\)
and the corresponding maximizing argument
\(F_{\left\langle band
\right\rangle}^{\mathrm{max}}\) of the
observed fluxes assuming a single flux (the latter
is the null hypothesis of no variability). As per
Wilks'
theorem, the quantity:

follows \(\chi^{2}\) distribution with
\(N-1\) degrees of freedom,
under the null hypothesis. Therefore, the null
hypothesis (non-variability) is rejected with a
probability proportional to the cumulative
distribution of the \(\chi^{2}\) statistic for values
smaller than the estimated
\(D\). The quantity
*var_inter_prob* represents this cumulative
probability, and therefore gives the probability
that the source is variable.

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., *var_inter_prob* > 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
\([0,8]\) that is derived
according to the estimated value of the quantity
\(D/(N-1)\) defined
above. It is used 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. Below we tabulate the association
between the value of
\(D/(N-1)\) and
inter-observation variability index.

Variability Index | \(\frac{D}{N-1}\) | |||
---|---|---|---|---|

2 observations | >2 observations | |||

0 |
— | < 0.4 | — | < 0.8 |

3 |
≥ 0.4 | < 0.7 | ≥ 0.8 | < 1.0 |

4 |
≥ 0.7 | < 1.0 | ≥ 1.0 | < 1.15 |

5 |
≥ 1.0 | < 2.7 | ≥ 1.15 | < 2.1 |

6 |
≥ 2.7 | < 7.0 | ≥ 2.1 | < 3.8 |

7 |
≥ 7.0 | < 12.0 | ≥ 3.8 | < 5.5 |

8 |
≥ 12.0 | — | ≥ 5.5 | — |

#### 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 of the source region photon flux
density PDF when a single flux is assumed
\(\left( F_{\left\langle band
\right\rangle}^{\mathrm{max}} \right)\),
and the mean of the source region photon flux
density PDF for the individual observation that
maximizes the absolute value of the difference
\(\left( F_{\left\langle band
\right\rangle}^{i,\mathrm{max}}
\right)\):

Of all the contributing observations, the
observation that yields the highest value for this
equation, is used in computing this value, which is
recorded in *var_inter_sigma*. Intuitively,
this quantity can be interpreted as the variance of
the individual observation fluxes.

#### Inter-Observation Spectral Variability Probability

*var_inter_hard_prob*

The inter-observation
spectral variability probability
(*var_inter_hard_prob*) is a value that
records the probability that the source region
hardness ratios varied between the contributing
observations, based on the hypothesis rejection
test described in the hardness
ratios and variability memo. The
definition of this probability is identical to
that of the inter-observation source variability
(*var_inter_prob*), and also utilizes the
same hypothesis rejection test, but based on the
probability distributions (PDFs) for the
hardness ratios, rather than the probability
distributions for the fluxes. The definition of
the hardness ratio PDFs can be found in the
memo, and also in the hardness ratios columns
page. High values of
*var_inter_hard_prob* indicate that the
source is spectrally variable in the
corresponding combination of bands.

*var_inter_hard_sigma*

Similarly to *var_inter_sigma*, the
inter-observation hardness ratio variability
parameter (*var_inter_hard_sigma*) is the
absolute value of the difference between the
error weighted mean of the source region
hardness ratio PDF when a single hardness ratio
is assumed, and the mean of the source region
hardness ratio PDF for the individual
observation that maximizes the absolute value of
the difference:

Of all contributing observations, the
observation that yields the highest value for
this equation, is used in computing this value,
which is recorded in
*var_inter_hard_sigma*. Intuitively, this
quantity can be interpreted as the variance of
the individual observation hardness ratios.

*var_inter_hard_flag*

A Boolean set to FALSE if
*var_inter_hard_prob* is below 0.3 for
all three hardness ratios, and set to
TRUE otherwise.