# 14.3.9 vari_time_series_statistics

Statistical parameters of time series, using only transits not rejected, see rejected_by_variability column in the VO Table epoch_photometry_url in gaia_source.
Note that NaN will be reported when the parameter value is missing or cannot be calculated.

Columns description:

solution_id : Solution Identifier (long)

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 where 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

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_selected_g_fov : Total number of G FOV transits selected for variability analysis (int, Dimensionless[see description])

The number of processed observations for variability analyses of this source, using only transits not rejected, see rejected_by_variability column in the VO Table epoch_photometry_url in gaia_source.

mean_obs_time_g_fov : Mean observation time for G FoV transits (double, Time[Barycentric JD in TCB - 2455197.5 (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

 $\overline{t}=\frac{1}{N}\sum_{i=1}^{N}\ t_{i}.$ (14.6)

time_duration_g_fov : Time duration of the time series for G FoV transits (double, Time[day])

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 $t_{N}-t_{1}$.

min_mag_g_fov : Minimum G FoV magnitude (double, Magnitude[mag])

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:

 $min(y_{i})\ \forall i\in(1,N)$ (14.7)

max_mag_g_fov : Maximum G FoV magnitude (double, Magnitude[mag])

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:

 $max(y_{i})\ \forall i\in(1,N)$ (14.8)

mean_mag_g_fov : Mean G FoV magnitude (double, Magnitude[mag])

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

 $\overline{y}=\frac{1}{N}\sum_{i=1}^{N}\ y_{i}.$ (14.9)

median_mag_g_fov : Median G FoV magnitude (double, Magnitude[mag])

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 $y_{(1)}\leq y_{(2)}\leq\cdots\leq y_{(N)}$. 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_mag_g_fov : Difference between the highest and lowest G FoV magnitudes (double, Magnitude[mag])

Name: The range of the time series

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

 $R=y_{\rm max}-y_{\rm min}$ (14.10)

std_dev_mag_g_fov : Square root of the unweighted G FoV magnitude variance (double, Magnitude[mag])

Name: The square root of the unbiased unweighted variance.

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_mag_g_fov : Standardized unweighted G FoV magnitude skewness (double, Dimensionless[see description])

Name: The standardised unbiased unweighted skewness.

Output: Let $y_{i}$ be a time series of size $N>2$. 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_mag_g_fov : Standardized unweighted G FoV magnitude kurtosis (double, Dimensionless[see description])

Name: The standardised unbiased unweighted kurtosis.

Output: Let $y_{i}$ be a time series of size $N>3$. 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 $N$.

mad_mag_g_fov : Median Absolute Deviation (MAD) for G FoV transits (double, Magnitude[mag])

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:

 $\mbox{MAD}=\mbox{median}\{|y_{i}-\mbox{median}\{y_{j},\forall j\in(1,N)\}|,% \forall i\in(1,N)\}.$ (14.11)

abbe_mag_g_fov : Abbe value for G FoV transits (double, Dimensionless[see description])

Name: The Abbe value

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

 ${\cal A}=\frac{N}{2(N-1)}\frac{\sum_{i=1}^{N-1}(y_{i+1}-y_{i})^{2}}{\sum_{i=1}% ^{N}(y_{i}-\bar{y})}$ (14.12)

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

iqr_mag_g_fov : Interquartile range for G FoV transits (double, Magnitude[mag])

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 $y_{(1)}\leq y_{(2)}\leq\cdots\leq y_{(N)}$. 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 $p_{i}=100\,i/(N+1)$.

num_selected_bp : Total number of BP observations selected for variability analysis (int, Dimensionless[see description])

The number of processed observations for variability analyses of this source, using only transits not rejected, see rejected_by_variability column in the VO Table epoch_photometry_url in gaia_source.

mean_obs_time_bp : Mean observation time for BP observations (double, Time[Barycentric JD in TCB - 2455197.5 (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

 $\overline{t}=\frac{1}{N}\sum_{i=1}^{N}\ t_{i}.$ (14.13)

time_duration_bp : Time duration of the BP time series (double, Time[day])

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 $t_{N}-t_{1}$.

min_mag_bp : Minimum BP magnitude (double, Magnitude[mag])

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:

 $min(y_{i})\ \forall i\in(1,N)$ (14.14)

max_mag_bp : Maximum BP magnitude (double, Magnitude[mag])

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:

 $max(y_{i})\ \forall i\in(1,N)$ (14.15)

mean_mag_bp : Mean BP magnitude (double, Magnitude[mag])

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

 $\overline{y}=\frac{1}{N}\sum_{i=1}^{N}\ y_{i}.$ (14.16)

median_mag_bp : Median BP magnitude (double, Magnitude[mag])

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 $y_{(1)}\leq y_{(2)}\leq\cdots\leq y_{(N)}$. 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_mag_bp : Difference between the highest and lowest BP magnitudes (double, Magnitude[mag])

Name: The range of the time series

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

 $R=y_{\rm max}-y_{\rm min}$ (14.17)

std_dev_mag_bp : Square root of the unweighted BP magnitude variance (double, Magnitude[mag])

Name: The square root of the unbiased unweighted variance.

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_mag_bp : Standardized unweighted BP magnitude skewness (double, Dimensionless[see description])

Name: The standardised unbiased unweighted skewness.

Output: Let $y_{i}$ be a time series of size $N>2$. 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_mag_bp : Standardized unweighted BP magnitude kurtosis (double, Dimensionless[see description])

Name: The standardised unbiased unweighted kurtosis.

Output: Let $y_{i}$ be a time series of size $N>3$. 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 $N$.

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:

 $\mbox{MAD}=\mbox{median}\{|y_{i}-\mbox{median}\{y_{j},\forall j\in(1,N)\}|,% \forall i\in(1,N)\}.$ (14.18)

abbe_mag_bp : Abbe value for BP observations (double, Dimensionless[see description])

Name: The Abbe value

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

 ${\cal A}=\frac{N}{2(N-1)}\frac{\sum_{i=1}^{N-1}(y_{i+1}-y_{i})^{2}}{\sum_{i=1}% ^{N}(y_{i}-\bar{y})}$ (14.19)

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

iqr_mag_bp : Interquartile BP magnitude range (double, Magnitude[mag])

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 $y_{(1)}\leq y_{(2)}\leq\cdots\leq y_{(N)}$. 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 $p_{i}=100\,i/(N+1)$.

num_selected_rp : Total number of RP observations selected for variability analysis (int, Dimensionless[see description])

The number of processed observations for variability analyses of this source, using only transits not rejected, see rejected_by_variability column in the VO Table epoch_photometry_url in gaia_source.

mean_obs_time_rp : Mean observation time for RP observations (double, Time[Barycentric JD in TCB - 2455197.5 (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

 $\overline{t}=\frac{1}{N}\sum_{i=1}^{N}\ t_{i}.$ (14.20)

time_duration_rp : Time duration of the RP time series (double, Time[day])

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 $t_{N}-t_{1}$.

min_mag_rp : Minimum RP magnitude (double, Magnitude[mag])

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:

 $min(y_{i})\ \forall i\in(1,N)$ (14.21)

max_mag_rp : Maximum RP magnitude (double, Magnitude[mag])

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:

 $max(y_{i})\ \forall i\in(1,N)$ (14.22)

mean_mag_rp : Mean RP magnitude (double, Magnitude[mag])

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

 $\overline{y}=\frac{1}{N}\sum_{i=1}^{N}\ y_{i}.$ (14.23)

median_mag_rp : Median RP magnitude (double, Magnitude[mag])

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 $y_{(1)}\leq y_{(2)}\leq\cdots\leq y_{(N)}$. 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_mag_rp : Difference between the highest and lowest RP magnitudes (double, Magnitude[mag])

Name: The range of the time series

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

 $R=y_{\rm max}-y_{\rm min}$ (14.24)

std_dev_mag_rp : Square root of the unweighted RP magnitude variance (double, Magnitude[mag])

Name: The square root of the unbiased unweighted variance.

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_mag_rp : Standardized unweighted RP magnitude skewness (double, Dimensionless[see description])

Name: The standardised unbiased unweighted skewness.

Output: Let $y_{i}$ be a time series of size $N>2$. 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_mag_rp : Standardized unweighted RP magnitude kurtosis (double, Dimensionless[see description])

Name: The standardised unbiased unweighted kurtosis.

Output: Let $y_{i}$ be a time series of size $N>3$. 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 $N$.

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:

 $\mbox{MAD}=\mbox{median}\{|y_{i}-\mbox{median}\{y_{j},\forall j\in(1,N)\}|,% \forall i\in(1,N)\}.$ (14.25)

abbe_mag_rp : Abbe value for RP observations (double, Dimensionless[see description])

Name: The Abbe value

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

 ${\cal A}=\frac{N}{2(N-1)}\frac{\sum_{i=1}^{N-1}(y_{i+1}-y_{i})^{2}}{\sum_{i=1}% ^{N}(y_{i}-\bar{y})}$ (14.26)

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

iqr_mag_rp : Interquartile RP magnitude range (double, Magnitude[mag])

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 $y_{(1)}\leq y_{(2)}\leq\cdots\leq y_{(N)}$. 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 $p_{i}=100\,i/(N+1)$.