# Variables used in formulas

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

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

• $a_{i}$ is the result of applying the model to the i-th 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 i-th object. It is set in the dataset description in columns with the Weighttype (if otherwise is not stated) or in the sample_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 theGroupwise 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.