Variables used in formulas
The following common variables are used in formulas of the described metrics:

$t_{i}$ is the label value for the ith object (from the input data for training).

$a_{i}$ is the result of applying the model to the ith object.

$p_{i}$ is the predicted success probability $\left(p_{i} = \frac{1}{1 + e^{a_{i}}}\right)$

$N$ is the total number of objects.

$M$ is the number of classes.

$c_{i}$ is the class of the object for binary classification.
$\begin{cases} c_{i} = 0{ , } & t_{i} \leqslant border \\ c_{i} = 1{ , } & t_{i} > border \end{cases}$

$w_{i}$ is the weight of the ith object. It is set in the dataset description in columns with the
Weight
type (if otherwise is not stated) or in thesample_weight
parameter of the Python package. The default is 1 for all objects. 
$P$, $TP$, $TN$, $FP$, $FN$ are abbreviations for Positive, True Positive, True Negative, False Positive and False Negative.
By default, $P$, $TP$, $TN$, $FP$, $FN$ use weights. For example, $TP = \sum\limits_{i=1}^{N} w_{i} [p_{i} > 0.5] c_{i}$

$Pairs$ is the array of pairs specified in the Pairs description or in the
pairs
parameter of the Python package. 
$N_{Pairs}$ is the number of pairs for the Pairwise metrics.

$a_{p}$ is the value calculated using the resulting model for the winner object for the Pairwise metrics.

$a_{n}$ is the value calculated using the resulting model for the loser object for the Pairwise metrics.

$w_{pn}$ is the weight of the ($p$; $n$) pair for the Pairwise metrics.

$Group$ is the array of object identifiers from the input dataset with a common
GroupId
. It is used to calculate the Groupwise metrics. 
$Groups$ is the set of all arrays of identifiers from the input dataset with a common
GroupId
. It is used to calculate the Groupwise metrics.