# 2.3 phot_variable_time_series_gfov_statistical_parameters

Statistical parameters of field-of-view time series, only including observations with rejected_by_variability_processing=false as found in table phot_variable_time_series_gfov.
Note that only sources are included that have phot_variable_flag = "VARIABLE" in the gaia_source table.

Columns description:

solution_id : Solution Identifier (long)

The data in the MDB will be described by means of a ”Solution identifier” parameter. This will be a numeric field attached to each table row that can be used to unequivocally identify the version of all the subsystems that where used in the generation of the data as well as the input data used. Each DPC generating the data will have the freedom to choose the Solution identifier number, but they must ensure that given the Solution identifier they can provide detailed information about the ”conditions” used to generate the data: versions of the software, version of the data used…

source_id : Unique source identifier (long)

A unique single numerical identifier of the source obtained from gaia_source (for a detailed description see gaia_source.source_id)

num_observations_processed : Number of processed G-band observations for variability analyses (int, Dimensionless[see description])

The number of processed observations for variability analyses of this source, only including observations with rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

mean_obs_time : Mean observation time (with respect to T0) of G-band time series (double, Time[day])

Name: The mean observation time

Input: Gaia barycentric light-travel time corrected FOV observation times with respect to reference time $T_{0}$ = 2455197.5 days (=J2010.0 = 2010-01-01T00:00:00), only including observations with rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

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

 $\overline{t}=\frac{1}{N}\sum_{i=1}^{N}\ t_{i}.{}$ (2.1)

minimum : Minimum magnitude of the G-band time series (double, Magnitude[mag])

Minimum magnitude of the G-band time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

maximum : Maximum magnitude of the G-band time series (double, Magnitude[mag])

Maximum magnitude of the G-band time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

mean : Mean magnitude of the G-band time series (double, Magnitude[mag])

Name: The mean magnitude of the G-band time series

Input: Gaia time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

Control parameters: None

Output: Let $y_{i}$ be a time series of size N. The mean $\overline{y}$ is defined as

 $\overline{y}=\frac{1}{N}\sum_{i=1}^{N}\ y_{i}.{}$ (2.2)

median : Median magnitude of the G-band time series (double, Magnitude[mag])

Name: Median magnitude of the G-band time series

Input: Gaia time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

Control parameters: None

Output: The (unweighted) 50th percentile value.

Let $y_{i}$ be a time series of size $N$ ordered such as $y_{(1)}\leq y_{(2)}\leq\cdots\leq y_{(N)}$. The $m$-th (per cent) percentile $P_{m}$ is defined for $0 as follows:

 $P_{m}=\left\{\begin{array}[]{ll}y_{(1)}&\mbox{if }0

where $p_{i}=100\,i/(N+1)$.

range : Difference between the highest and lowest magnitude of the G-band time series (double, Magnitude[mag])

Name: Magnitude range of the G-band time series

Input: Gaia time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

Control parameters: None

Output: Let $y_{i}$ be a time series, $y_{\rm max}$ its largest element, and $y_{\rm min}$ its smallest element, then the range is defined as

 $R=y_{\rm max}-y_{\rm min}$ (2.3)

std_dev : Square root of the unweighted variance of the G-band time series values (double, Magnitude[mag])

Name: The square root of the unbiased unweighted variance.

Input: Gaia time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

Output: Let $y_{i}$ be a time series of size $N$. The unweighted standard deviation $\hat{\sigma}$ is defined as the square root of the sample-size unbiased unweighted variance:

 $\hat{\sigma}=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(y_{i}-\overline{y})^{2}}.$

skewness : Standardized unweighted skewness of the G-band time series values (double, Dimensionless[see description])

Name: The standardised unbiased unweighted skewness.

Input: Gaia time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

Output: Let $y_{i}$ be a time series of size $N$. The sample-size unbiased unweighted skewness moment $\mathcal{E}$ is defined as:

 $\mathcal{E}=\frac{N}{(N-1)(N-2)}\sum_{i=1}^{N}(y_{i}-\overline{y})^{3}.$

The standardized unbiased skewness $E$ is defined as:

 $E=\frac{\mathcal{E}}{\hat{\sigma}^{3}}$

where $\hat{\sigma}$ 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 $N$.

kurtosis : Standardized unweighted kurtosis of the G-band time series values (double, Dimensionless[see description])

Name: The standardised unbiased unweighted kurtosis.

Input: Gaia time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

Output: Let $y_{i}$ be a time series of size $N$. The sample-size unbiased unweighted kurtosis cumulant $\mathcal{K}$ is defined as:

 $\mathcal{K}=\frac{N(N+1)}{(N-1)(N-2)(N-3)}\sum_{i=1}^{N}(y_{i}-\overline{y})^{% 4}-\frac{3}{(N-2)(N-3)}\,\left[\sum_{i=1}^{N}(y_{i}-\overline{y})^{2}\right]^{% 2}.$

The standardized unbiased kurtosis $K$ is defined as:

 $K=\frac{\mathcal{K}}{\hat{\sigma}^{4}}$

where $\hat{\sigma}^{2}$ 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 $N$.

time_duration : Time duration of the G-band time series (double, Time[day])

Duration of Gaia time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

median_absolute_deviation : Median Absolute Deviation (MAD) of the G-band time series values (double, Magnitude[mag])

Name: The Median Absolute Deviation (MAD)

Input: Gaia time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

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 $1/\Phi^{-1}(3/4)\approx 1.4826$ (where $\Phi^{-1}$ 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:

 $\mbox{MAD}=1.4826~{}\mbox{median}\{|y_{i}-\mbox{median}\{y_{j},\forall j\in(1,% N)\}|,\forall i\in(1,N)\}.$ (2.4)

abbe : Abbe value of the G-band time series values (double, Dimensionless[see description])

Name: The Abbe value

Input: Gaia time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

Control parameters: None

Output: Let $\{t_{i},y_{i}\}$ be a time-sorted time series of size $N$, such that $t_{i} for all $i. The Abbe value $\cal{A}$ is defined as

 ${\cal A}=\frac{N}{2(N-1)}\frac{\sum_{i=1}^{N-1}(y_{i+1}-y_{i})^{2}}{\sum_{i=1}% ^{N}(y_{i}-\bar{y})}$ (2.5)

where $\bar{y}$ is the unweighted mean.

iqr : Interquartile range of the time series values (double, Magnitude[mag])

Name: The Interquartile Range (IQR)

Input: Gaia time series, only including observations with
rejected_by_variability_processing=false, as found in table
phot_variable_time_series_gfov.

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 $y_{(1)}\leq y_{(2)}\leq\cdots\leq y_{(N)}$. The $m$-th (per cent) percentile $P_{m}$ is defined for $0 as follows:

 $P_{m}=\left\{\begin{array}[]{ll}y_{(1)}&\mbox{if }0

where $p_{i}=100\,i/(N+1)$.