# 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.