20.13.1 vari_summary
Summary table that provides the information on where a source_id can be found in the different Variability tables and statistical parameters of time series, using only transits not rejected.
Note that NULL is reported when the statistical parameter value is missing or cannot be calculated.
Columns description:
All Gaia data processed by the Data Processing and Analysis Consortium comes tagged with a solution identifier. This is a numeric field attached to each table row that can be used to unequivocally identify the version of all the subsystems that were used in the generation of the data as well as the input data used. It is mainly for internal DPAC use but is included in the published data releases to enable end users to examine the provenance of processed data products. To decode a given solution ID visit https://gaia.esac.esa.int/decoder/solnDecoder.jsp
A unique single numerical identifier of the source obtained from gaia_source (for a detailed description see gaia_source.source_id).
The number of processed observations for variability analyses of this source, using only transits not rejected.
mean_obs_time_g_fov : Mean observation time for G FoV transits (double, Time[Barycentric JD in TCB $$ 2 455 197.5 (day)])

Name: The mean observation time

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$. The mean $\overline{t}$ is defined as
$$\overline{t}=\frac{1}{N}\sum _{i=1}^{N}{t}_{i}.$$ (20.1)

Name: The time duration of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$, with $i=1$ to $N$. The time duration of the time series is equal to ${t}_{N}{t}_{1}$.

Name: The minimum value of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$, with $i=1$ to $N$. The minimum value of the time series is defined as:
$$min({y}_{i})\forall i\in (1,N)$$ (20.2)

Name: The maximum value of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$, with $i=1$ to $N$. The maximum value of the time series is defined as:
$$max({y}_{i})\forall i\in (1,N)$$ (20.3)

Name: The mean of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$. The mean $\overline{y}$ is defined as
$$\overline{y}=\frac{1}{N}\sum _{i=1}^{N}{y}_{i}.$$ (20.4)

Name: The median of the time series

Control parameters: None

Output: The 50th percentile unweighted value.
Let ${y}_{i}$ be a time series of size $N$ ordered such as ${y}_{(1)}\le {y}_{(2)}\le \mathrm{\cdots}\le {y}_{(N)}$. The $m$th (per cent) percentile ${P}_{m}$ is defined for $$ as follows:
$$ (20.5)
range_mag_g_fov : Difference between the highest and lowest G FoV magnitudes (float, Magnitude[mag])

Name: The range of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series, ${y}_{\mathrm{max}}$ its largest element, and ${y}_{\mathrm{min}}$ its smallest element, then the range is defined as
$$R={y}_{\mathrm{max}}{y}_{\mathrm{min}}$$ (20.6)
trimmed_range_mag_g_fov : Trimmed difference between the highest and lowest G FoV magnitudes (float, Magnitude[mag])

Name: The trimmed range

Control parameters: trim level $\alpha \in ]0,50[$

Output: The trimmed unweighted range.
Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. Let ${Y}_{\alpha}$ be the $\alpha $th percentile, and ${Y}_{100\alpha}$ be the $(100\alpha )$th percentile. Let ${y}_{i}^{\prime}$ be the trimmed time series of size N’ which is derived by iteratively removing the smallest or the largest values of the ${y}_{i}$ time series until $$ for all $i\in \{1,\mathrm{\cdots},{N}^{\prime}\}$. The trimmed range R’ is defined as
$${R}^{\prime}={y}_{\mathrm{max}}^{\prime}{y}_{\mathrm{min}}^{\prime}$$ (20.7) For DR3, the trim level $\alpha $ was set to 5.0.

Name: The square root of the unbiased unweighted variance.

Output: Let ${y}_{i}$ be a time series of size $N$. The unweighted standard deviation $\widehat{\sigma}$ is defined as the square root of the samplesize unbiased unweighted variance:
$$\widehat{\sigma}=\sqrt{\frac{1}{N1}\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{2}}.$$

Name: The standardised unbiased unweighted skewness.

Output: Let ${y}_{i}$ be a time series of size $N>2$. The samplesize unbiased unweighted skewness moment $\mathcal{E}$ is defined as:
$$\mathcal{E}=\frac{N}{(N1)(N2)}\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{3}.$$ The standardized unbiased skewness $E$ is defined as:
$$E=\frac{\mathcal{E}}{{\widehat{\sigma}}^{3}}$$ where $\widehat{\sigma}$ is the square root of the unbiased unweighted variance around the unweighted mean. While $\mathcal{E}$ is an unbiased estimate of the population value, $E$ becomes unbiased in the limit of large $N$.

Name: The standardised unbiased unweighted kurtosis.

Output: Let ${y}_{i}$ be a time series of size $N>3$. The samplesize unbiased unweighted kurtosis cumulant $\mathcal{K}$ is defined as:
$$\mathcal{K}=\frac{N(N+1)}{(N1)(N2)(N3)}\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{4}\frac{3}{(N2)(N3)}{\left[\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{2}\right]}^{2}.$$ The standardized unbiased kurtosis $K$ is defined as:
$$K=\frac{\mathcal{K}}{{\widehat{\sigma}}^{4}}$$ where ${\widehat{\sigma}}^{2}$ is the unbiased unweighted variance around the unweighted mean. While $\mathcal{K}$ is an unbiased estimate of the population value, $K$ becomes unbiased in the limit of large $N$.

Name: The Median Absolute Deviation (MAD)

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$. The MAD is defined as the median of the absolute deviations from the median of the data, scaled by a factor of $1/{\mathrm{\Phi}}^{1}(3/4)\approx 1.4826$ (where ${\mathrm{\Phi}}^{1}$ is the inverse of the cumulative distribution function for the standard normal distribution), so that the expectation of the scaled MAD at large $N$ equals the standard deviation of a normal distribution:
$$\text{MAD}=1.4826\text{median}\{{y}_{i}\text{median}\{{y}_{j},\forall j\in (1,N)\},\forall i\in (1,N)\}.$$ (20.8)

Name: The Abbe value

Control parameters: None

Output: Let $\{{t}_{i},{y}_{i}\}$ be a timesorted time series of size $N$, such that $$ for all $$. The Abbe value $\mathcal{A}$ is defined as
$$\mathcal{A}=\frac{{\sum}_{i=1}^{N1}{({y}_{i+1}{y}_{i})}^{2}}{2{\sum}_{i=1}^{N}{({y}_{i}\overline{y})}^{2}}$$ (20.9) where $\overline{y}$ is the unweighted mean.

Name: The Interquartile Range (IQR)

Control parameters: None

Output: The difference between the (unweighted) 75th and 25th percentile values: IQR$={P}_{75}{P}_{25}$.
Let ${y}_{i}$ be a time series of size $N$ ordered such as ${y}_{(1)}\le {y}_{(2)}\le \mathrm{\cdots}\le {y}_{(N)}$. The $m$th (per cent) percentile ${P}_{m}$ is defined for $$ as follows:
$$ (20.10)

Name: The singleband Stetson variability index

Control parameters: The pairing time interval

Output: Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. The Stetson value (Stetson 1996) is based on paired observations closely separated in time (within a chosen time interval). The time difference between the components of a pair should be small compared to the shortest timescale of variation being sought. If this condition is not satisfied, a transformation is applied to the unpaired measurement, in order to include its information together with paired observations. The maximum number $n$ of pairs may exceed $N/2$, since multiple pairs can be formed within a given time interval (as long as the latter is much smaller than the minimum variability timescale sought). The Stetson $J$ index is defined as
$$J=\frac{1}{n}\sum _{k=1}^{n}\text{sgn}({P}_{k})\sqrt{{P}_{k}},$$ where the index $k$ refers to the $k$th pair, composed of measurements $i$ and $j$, such that
$${P}_{k}=\{\begin{array}{cc}{\delta}_{i(k)}{\delta}_{j(k)},\text{if}i(k)\ne j(k)\hfill & \\ {\delta}_{i(k)}^{2}1,\text{if}i(k)=j(k)\hfill & \end{array}$$ with the normalized deviation of the $i$th measurement from the timeseries mean $\overline{y}$ expressed by:
$${\delta}_{i(k)}=\sqrt{\frac{N}{N1}}\frac{{y}_{i(k)}\overline{y}}{\mathrm{\Delta}{y}_{i(k)}}.$$

Name: The signaltonoise estimate

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. The signaltonoise ratio (S/N) is estimated as
$$\text{S/N}=\sqrt{\frac{{\sum}_{i=1}^{N}{({y}_{i}\overline{y})}^{2}}{{\sum}_{i=1}^{N}\mathrm{\Delta}{y}_{i}^{2}}}$$ (20.11) where $\overline{y}$ is the mean of the timeseries values.
outlier_median_g_fov : Greatest absolute deviation from the G FoV median normalized by the error (float)

Name: The most outlying measurement with respect to the median

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. The most outlying measurement is defined as the greatest absolute deviation from the median normalized by the error:
$$\text{outlierMedian}=\mathrm{max}\{\frac{{y}_{i}\text{median}\{{y}_{j},\forall j\in (1,N)\}}{\mathrm{\Delta}{y}_{i}},\forall i\in (1,N)\}.$$ (20.12)
The number of processed observations for variability analyses of this source, using only transits not rejected.
mean_obs_time_bp : Mean observation time for BP observations (double, Time[Barycentric JD in TCB $$ 2 455 197.5 (day)])

Name: The mean observation time

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$. The mean $\overline{t}$ is defined as
$$\overline{t}=\frac{1}{N}\sum _{i=1}^{N}{t}_{i}.$$ (20.13)

Name: The time duration of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$, with $i=1$ to $N$. The time duration of the time series is equal to ${t}_{N}{t}_{1}$.

Name: The minimum value of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$, with $i=1$ to $N$. The minimum value of the time series is defined as:
$$min({y}_{i})\forall i\in (1,N)$$ (20.14)

Name: The maximum value of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$, with $i=1$ to $N$. The maximum value of the time series is defined as:
$$max({y}_{i})\forall i\in (1,N)$$ (20.15)

Name: The mean of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$. The mean $\overline{y}$ is defined as
$$\overline{y}=\frac{1}{N}\sum _{i=1}^{N}{y}_{i}.$$ (20.16)

Name: The median of the time series

Control parameters: None

Output: The 50th unweighted percentile value.
Let ${y}_{i}$ be a time series of size $N$ ordered such as ${y}_{(1)}\le {y}_{(2)}\le \mathrm{\cdots}\le {y}_{(N)}$. The $m$th (per cent) percentile ${P}_{m}$ is defined for $$ as follows:
$$ (20.17)

Name: The range of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series, ${y}_{\mathrm{max}}$ its largest element, and ${y}_{\mathrm{min}}$ its smallest element, then the range is defined as
$$R={y}_{\mathrm{max}}{y}_{\mathrm{min}}$$ (20.18)
trimmed_range_mag_bp : Trimmed difference between the highest and lowest BP magnitudes (float, Magnitude[mag])

Name: The trimmed range

Control parameters: trim level $\alpha \in ]0,50[$

Output: The trimmed unweighted range.
Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. Let ${Y}_{\alpha}$ be the $\alpha $th percentile, and ${Y}_{100\alpha}$ be the $(100\alpha )$th percentile. Let ${y}_{i}^{\prime}$ be the trimmed time series of size N’ which is derived by iteratively removing the smallest or the largest values of the ${y}_{i}$ time series until $$ for all $i\in \{1,\mathrm{\cdots},{N}^{\prime}\}$. The trimmed range R’ is defined as
$${R}^{\prime}={y}_{\mathrm{max}}^{\prime}{y}_{\mathrm{min}}^{\prime}$$ (20.19) For DR3, the trim level $\alpha $ was set to 5.0.

Name: The square root of the unbiased unweighted variance.

Output: Let ${y}_{i}$ be a time series of size $N$. The unweighted standard deviation $\widehat{\sigma}$ is defined as the square root of the samplesize unbiased unweighted variance:
$$\widehat{\sigma}=\sqrt{\frac{1}{N1}\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{2}}.$$

Name: The standardised unbiased unweighted skewness.

Output: Let ${y}_{i}$ be a time series of size $N>2$. The samplesize unbiased unweighted skewness moment $\mathcal{E}$ is defined as:
$$\mathcal{E}=\frac{N}{(N1)(N2)}\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{3}.$$ The standardized unbiased skewness $E$ is defined as:
$$E=\frac{\mathcal{E}}{{\widehat{\sigma}}^{3}}$$ where $\widehat{\sigma}$ is the square root of the unbiased unweighted variance around the unweighted mean. While $\mathcal{E}$ is an unbiased estimate of the population value, $E$ becomes unbiased in the limit of large $N$.

Name: The standardised unbiased unweighted kurtosis.

Output: Let ${y}_{i}$ be a time series of size $N>3$. The samplesize unbiased unweighted kurtosis cumulant $\mathcal{K}$ is defined as:
$$\mathcal{K}=\frac{N(N+1)}{(N1)(N2)(N3)}\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{4}\frac{3}{(N2)(N3)}{\left[\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{2}\right]}^{2}.$$ The standardized unbiased kurtosis $K$ is defined as:
$$K=\frac{\mathcal{K}}{{\widehat{\sigma}}^{4}}$$ where ${\widehat{\sigma}}^{2}$ is the unbiased unweighted variance around the unweighted mean. While $\mathcal{K}$ is an unbiased estimate of the population value, $K$ becomes unbiased in the limit of large $N$.

Name: The Median Absolute Deviation (MAD)

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$. The MAD is defined as the median of the absolute deviations from the median of the data, scaled by a factor of $1/{\mathrm{\Phi}}^{1}(3/4)\approx 1.4826$ (where ${\mathrm{\Phi}}^{1}$ is the inverse of the cumulative distribution function for the standard normal distribution), so that the expectation of the scaled MAD at large $N$ equals the standard deviation of a normal distribution:
$$\text{MAD}=1.4826\text{median}\{{y}_{i}\text{median}\{{y}_{j},\forall j\in (1,N)\},\forall i\in (1,N)\}.$$ (20.20)

Name: The Abbe value

Control parameters: None

Output: Let $\{{t}_{i},{y}_{i}\}$ be a timesorted time series of size $N$, such that $$ for all $$. The Abbe value $\mathcal{A}$ is defined as
$$\mathcal{A}=\frac{{\sum}_{i=1}^{N1}{({y}_{i+1}{y}_{i})}^{2}}{2{\sum}_{i=1}^{N}{({y}_{i}\overline{y})}^{2}}$$ (20.21) where $\overline{y}$ is the unweighted mean.

Name: The Interquartile Range (IQR)

Control parameters: None

Output: The difference between the (unweighted) 75th and 25th percentile values: IQR$={P}_{75}{P}_{25}$.
Let ${y}_{i}$ be a time series of size $N$ ordered such as ${y}_{(1)}\le {y}_{(2)}\le \mathrm{\cdots}\le {y}_{(N)}$. The $m$th (per cent) percentile ${P}_{m}$ is defined for $$ as follows:
$$ (20.22)

Name: The singleband Stetson variability index

Control parameters: The pairing time interval

Output: Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. The Stetson value (Stetson 1996) is based on paired observations closely separated in time (within a chosen time interval). The time difference between the components of a pair should be small compared to the shortest timescale of variation being sought. If this condition is not satisfied, a transformation is applied to the unpaired measurement, in order to include its information together with paired observations. The maximum number $n$ of pairs may exceed $N/2$, since multiple pairs can be formed within a given time interval (as long as the latter is much smaller than the minimum variability timescale sought).
$$J=\frac{1}{n}\sum _{k=1}^{n}\text{sgn}({P}_{k})\sqrt{{P}_{k}},$$ where the index $k$ refers to the $k$th pair, composed of measurements $i$ and $j$, such that
$${P}_{k}=\{\begin{array}{cc}{\delta}_{i(k)}{\delta}_{j(k)},\text{if}i(k)\ne j(k)\hfill & \\ {\delta}_{i(k)}^{2}1,\text{if}i(k)=j(k)\hfill & \end{array}$$ with the normalized deviation of the $i$th measurement from the timeseries mean $\overline{y}$ expressed by:
$${\delta}_{i(k)}=\sqrt{\frac{N}{N1}}\frac{{y}_{i(k)}\overline{y}}{\mathrm{\Delta}{y}_{i(k)}}.$$

Name: The signaltonoise estimate

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. The signaltonoise ratio (S/N) is estimated as
$$\text{S/N}=\sqrt{\frac{{\sum}_{i=1}^{N}{({y}_{i}\overline{y})}^{2}}{{\sum}_{i=1}^{N}\mathrm{\Delta}{y}_{i}^{2}}}$$ (20.23) where $\overline{y}$ is the mean of the timeseries values.

Name: The most outlying measurement with respect to the median

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. The most outlying measurement is defined as the greatest absolute deviation from the median normalized by the error:
$$\text{outlierMedian}=\mathrm{max}\{\frac{{y}_{i}\text{median}\{{y}_{j},\forall j\in (1,N)\}}{\mathrm{\Delta}{y}_{i}},\forall i\in (1,N)\}.$$ (20.24)
The number of processed observations for variability analyses of this source, using only transits not rejected.
mean_obs_time_rp : Mean observation time for RP observations (double, Time[Barycentric JD in TCB $$ 2 455 197.5 (day)])

Name: The mean observation time

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$. The mean $\overline{t}$ is defined as
$$\overline{t}=\frac{1}{N}\sum _{i=1}^{N}{t}_{i}.$$ (20.25)

Name: The time duration of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$, with $i=1$ to $N$. The time duration of the time series is equal to ${t}_{N}{t}_{1}$.

Name: The minimum value of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$, with $i=1$ to $N$. The minimum value of the time series is defined as:
$$min({y}_{i})\forall i\in (1,N)$$ (20.26)

Name: The maximum value of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$ at times ${t}_{i}$, with $i=1$ to $N$. The maximum value of the time series is defined as:
$$max({y}_{i})\forall i\in (1,N)$$ (20.27)

Name: The mean of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$. The mean $\overline{y}$ is defined as
$$\overline{y}=\frac{1}{N}\sum _{i=1}^{N}{y}_{i}.$$ (20.28)

Name: The median of the time series

Control parameters: None

Output: The 50th unweighted percentile value.
Let ${y}_{i}$ be a time series of size $N$ ordered such as ${y}_{(1)}\le {y}_{(2)}\le \mathrm{\cdots}\le {y}_{(N)}$. The $m$th (per cent) percentile ${P}_{m}$ is defined for $$ as follows:
$$ (20.29)

Name: The range of the time series

Control parameters: None

Output: Let ${y}_{i}$ be a time series, ${y}_{\mathrm{max}}$ its largest element, and ${y}_{\mathrm{min}}$ its smallest element, then the range is defined as
$$R={y}_{\mathrm{max}}{y}_{\mathrm{min}}$$ (20.30)
trimmed_range_mag_rp : Trimmed difference between the highest and lowest RP magnitudes (float, Magnitude[mag])

Name: The trimmed range

Control parameters: trim level $\alpha \in ]0,50[$

Output: The trimmed unweighted range.
Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. Let ${Y}_{\alpha}$ be the $\alpha $th percentile, and ${Y}_{100\alpha}$ be the $(100\alpha )$th percentile. Let ${y}_{i}^{\prime}$ be the trimmed time series of size N’ which is derived by iteratively removing the smallest or the largest values of the ${y}_{i}$ time series until $$ for all $i\in \{1,\mathrm{\cdots},{N}^{\prime}\}$. The trimmed range R’ is defined as
$${R}^{\prime}={y}_{\mathrm{max}}^{\prime}{y}_{\mathrm{min}}^{\prime}$$ (20.31) For DR3, the trim level $\alpha $ was set to 5.0.

Name: The square root of the unbiased unweighted variance.

Output: Let ${y}_{i}$ be a time series of size $N$. The unweighted standard deviation $\widehat{\sigma}$ is defined as the square root of the samplesize unbiased unweighted variance:
$$\widehat{\sigma}=\sqrt{\frac{1}{N1}\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{2}}.$$

Name: The standardised unbiased unweighted skewness.

Output: Let ${y}_{i}$ be a time series of size $N>2$. The samplesize unbiased unweighted skewness moment $\mathcal{E}$ is defined as:
$$\mathcal{E}=\frac{N}{(N1)(N2)}\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{3}.$$ The standardized unbiased skewness $E$ is defined as:
$$E=\frac{\mathcal{E}}{{\widehat{\sigma}}^{3}}$$ where $\widehat{\sigma}$ is the square root of the unbiased unweighted variance around the unweighted mean. While $\mathcal{E}$ is an unbiased estimate of the population value, $E$ becomes unbiased in the limit of large $N$.

Name: The standardised unbiased unweighted kurtosis.

Output: Let ${y}_{i}$ be a time series of size $N>3$. The samplesize unbiased unweighted kurtosis cumulant $\mathcal{K}$ is defined as:
$$\mathcal{K}=\frac{N(N+1)}{(N1)(N2)(N3)}\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{4}\frac{3}{(N2)(N3)}{\left[\sum _{i=1}^{N}{({y}_{i}\overline{y})}^{2}\right]}^{2}.$$ The standardized unbiased kurtosis $K$ is defined as:
$$K=\frac{\mathcal{K}}{{\widehat{\sigma}}^{4}}$$ where ${\widehat{\sigma}}^{2}$ is the unbiased unweighted variance around the unweighted mean. While $\mathcal{K}$ is an unbiased estimate of the population value, $K$ becomes unbiased in the limit of large $N$.

Name: The Median Absolute Deviation (MAD)

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size $N$. The MAD is defined as the median of the absolute deviations from the median of the data, scaled by a factor of $1/{\mathrm{\Phi}}^{1}(3/4)\approx 1.4826$ (where ${\mathrm{\Phi}}^{1}$ is the inverse of the cumulative distribution function for the standard normal distribution), so that the expectation of the scaled MAD at large $N$ equals the standard deviation of a normal distribution:
$$\text{MAD}=1.4826\text{median}\{{y}_{i}\text{median}\{{y}_{j},\forall j\in (1,N)\},\forall i\in (1,N)\}.$$ (20.32)

Name: The Abbe value

Control parameters: None

Output: Let $\{{t}_{i},{y}_{i}\}$ be a timesorted time series of size $N$, such that $$ for all $$. The Abbe value $\mathcal{A}$ is defined as
$$\mathcal{A}=\frac{{\sum}_{i=1}^{N1}{({y}_{i+1}{y}_{i})}^{2}}{2{\sum}_{i=1}^{N}{({y}_{i}\overline{y})}^{2}}$$ (20.33) where $\overline{y}$ is the unweighted mean.

Name: The Interquartile Range (IQR)

Control parameters: None

Output: The difference between the (unweighted) 75th and 25th percentile values: IQR$={P}_{75}{P}_{25}$.
Let ${y}_{i}$ be a time series of size $N$ ordered such as ${y}_{(1)}\le {y}_{(2)}\le \mathrm{\cdots}\le {y}_{(N)}$. The $m$th (per cent) percentile ${P}_{m}$ is defined for $$ as follows:
$$ (20.34)

Name: The singleband Stetson variability index

Control parameters: The pairing time interval

Output: Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. The Stetson value (Stetson 1996) is based on paired observations closely separated in time (within a chosen time interval). The time difference between the components of a pair should be small compared to the shortest timescale of variation being sought. If this condition is not satisfied, a transformation is applied to the unpaired measurement, in order to include its information together with paired observations. The maximum number $n$ of pairs may exceed $N/2$, since multiple pairs can be formed within a given time interval (as long as the latter is much smaller than the minimum variability timescale sought). The Stetson $J$ index is defined as
$$J=\frac{1}{n}\sum _{k=1}^{n}\text{sgn}({P}_{k})\sqrt{{P}_{k}},$$ where the index $k$ refers to the $k$th pair, composed of measurements $i$ and $j$, such that
$${P}_{k}=\{\begin{array}{cc}{\delta}_{i(k)}{\delta}_{j(k)},\text{if}i(k)\ne j(k)\hfill & \\ {\delta}_{i(k)}^{2}1,\text{if}i(k)=j(k)\hfill & \end{array}$$ with the normalized deviation of the $i$th measurement from the timeseries mean $\overline{y}$ expressed by:
$${\delta}_{i(k)}=\sqrt{\frac{N}{N1}}\frac{{y}_{i(k)}\overline{y}}{\mathrm{\Delta}{y}_{i(k)}}.$$

Name: The signaltonoise estimate

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. The signaltonoise ratio (S/N) is estimated as
$$\text{S/N}=\sqrt{\frac{{\sum}_{i=1}^{N}{({y}_{i}\overline{y})}^{2}}{{\sum}_{i=1}^{N}\mathrm{\Delta}{y}_{i}^{2}}}$$ (20.35) where $\overline{y}$ is the mean of the timeseries values.

Name: The most outlying measurement with respect to the median

Control parameters: None

Output: Let ${y}_{i}$ be a time series of size N, with error bars $\mathrm{\Delta}{y}_{i}$. The most outlying measurement is defined as the greatest absolute deviation from the median normalized by the error:
$$\text{outlierMedian}=\mathrm{max}\{\frac{{y}_{i}\text{median}\{{y}_{j},\forall j\in (1,N)\}}{\mathrm{\Delta}{y}_{i}},\forall i\in (1,N)\}.$$ (20.36)
Source is present in vari_classifier_result (and not VariClassificationResult as the column name suggests).
Source is present in vari_rrlyrae
Source is present in vari_cepheid
Source is present in vari_planetary_transit
Source is present in vari_short_timescale
Source is present in vari_long_period_variable
Source is present in vari_eclipsing_binary
Source is present in vari_rotation_modulation
Source is present in vari_ms_oscillator
Source is present in vari_agn
Source is present in vari_microlensing
Source is present in vari_compact_companion