# MultiLabel Classification: objectives and metrics

## Objectives and metrics

### MultiLogloss

$\displaystyle\frac{-\sum\limits_{j=0}^{M-1} \sum\limits_{i=1}^{N} w_{i} (c_{ij} \log p_{ij} + (1-c_{ij}) \log (1 - p_{ij}) )}{M\sum\limits_{i=1}^{N}w_{i}} { ,}$

where $p_{ij} = \sigma(a_{ij}) = \frac{e^{a_{ij}}}{1 + e^{a_{ij}}}$ and $c_{ij} \in {0, 1}$

User-defined parameters:

use_weights

Use object/group weights to calculate metrics if the specified value is true and set all weights to 1 regardless of the input data if the specified value is false.

Default: true

### MultiCrossEntropy

$\displaystyle\frac{-\sum\limits_{j=0}^{M-1} \sum\limits_{i=1}^{N} w_{i} (t_{ij} \log p_{ij} + (1-t_{ij}) \log (1 - p_{ij}) )}{M\sum\limits_{i=1}^{N}w_{i}} { ,}$

where $p_{ij} = \sigma(a_{ij}) = \frac{e^{a_{ij}}}{1 + e^{a_{ij}}}$ and $t_{ij} \in [0, 1]$

User-defined parameters:

use_weights

Use object/group weights to calculate metrics if the specified value is true and set all weights to 1 regardless of the input data if the specified value is false.

Default: true

### Precision

This function is calculated separately for each class k numbered from 0 to M – 1.

$\frac{TP}{TP + FP}$

User-defined parameters:

use_weights

Use object/group weights to calculate metrics if the specified value is true and set all weights to 1 regardless of the input data if the specified value is false.

Default: true

### Recall

This function is calculated separately for each class k numbered from 0 to M – 1.

$\frac{TP}{TP+FN}$

User-defined parameters:

use_weights

Use object/group weights to calculate metrics if the specified value is true and set all weights to 1 regardless of the input data if the specified value is false.

Default: true

### F1

This function is calculated separately for each class k numbered from 0 to M – 1.

$2 \frac{Precision * Recall}{Precision + Recall}$

User-defined parameters:

use_weights

Use object/group weights to calculate metrics if the specified value is true and set all weights to 1 regardless of the input data if the specified value is false.

Default: true

### Accuracy

The formula depends on the value of the $type$ parameter:

#### Classic

$\displaystyle\frac{\sum\limits_{i=1}^{N}w_{i} \prod\limits_{j=0}^{M-1} [[p_{ij} > 0.5]==t_{ij}]}{\sum\limits_{i=1}^{N}w_{i}} { , }$

where $p_{ij} = \sigma(a_{ij}) = \frac{e^{a_{ij}}}{1 + e^{a_{ij}}}$

#### PerClass

This function is calculated separately for each class k numbered from 0 to M – 1.

$\frac{TP + TN}{\sum\limits_{i=1}^{N} w_{i}}$

User-defined parameters:

use_weights

Use object/group weights to calculate metrics if the specified value is true and set all weights to 1 regardless of the input data if the specified value is false.

Default: true

type

The type of calculated accuracy.

Possible values:

• Classic
• PerClass

Default: Classic

### HammingLoss

$\displaystyle\frac{\sum\limits_{j=0}^{M-1} \sum\limits_{i = 1}^{N} w_{i} [[p_{ij} > 0.5] == t_{ij}]]}{M \sum\limits_{i=1}^{N} w_{i}}$

User-defined parameters:

use_weights

Use object/group weights to calculate metrics if the specified value is true and set all weights to 1 regardless of the input data if the specified value is false.

Default: true

## Used for optimization

Name Optimization
MultiLogloss +
MultiCrossEntropy +
Precision -
Recall -
F1 -
Accuracy -
HammingLoss -