# Solved – How to compute precision/recall for multiclass-multilabel classification

classificationmachine learningmulti-classprecision-recall

I'm wondering how to calculate precision and recall measures for multiclass multilabel classification, i.e. classification where there are more than two labels, and where each instance can have multiple labels?

For multi-label classification you have two ways to go First consider the following.

• $n$ is the number of examples.
• $Y_i$ is the ground truth label assignment of the $i^{th}$ example..
• $x_i$ is the $i^{th}$ example.
• $h(x_i)$ is the predicted labels for the $i^{th}$ example.

# Example based

The metrics are computed in a per datapoint manner. For each predicted label its only its score is computed, and then these scores are aggregated over all the datapoints.

• Precision = $\frac{1}{n}\sum_{i=1}^{n}\frac{|Y_{i}\cap h(x_{i})|}{|h(x_{i})|}$ , The ratio of how much of the predicted is correct. The numerator finds how many labels in the predicted vector has common with the ground truth, and the ratio computes, how many of the predicted true labels are actually in the ground truth.
• Recall = $\frac{1}{n}\sum_{i=1}^{n}\frac{|Y_{i}\cap h(x_{i})|}{|Y_{i}|}$ , The ratio of how many of the actual labels were predicted. The numerator finds how many labels in the predicted vector has common with the ground truth (as above), then finds the ratio to the number of actual labels, therefore getting what fraction of the actual labels were predicted.

There are other metrics as well.

# Label based

Here the things are done labels-wise. For each label the metrics (eg. precision, recall) are computed and then these label-wise metrics are aggregated. Hence, in this case you end up computing the precision/recall for each label over the entire dataset, as you do for a binary classification (as each label has a binary assignment), then aggregate it.

The easy way is to present the general form.

This is just an extension of the standard multi-class equivalent.

• Macro averaged $\frac{1}{q}\sum_{j=1}^{q}B(TP_{j},FP_{j},TN_{j},FN_{j})$

• Micro averaged $B(\sum_{j=1}^{q}TP_{j},\sum_{j=1}^{q}FP_{j},\sum_{j=1}^{q}TN_{j},\sum_{j=1}^{q}FN_{j})$

Here the $TP_{j},FP_{j},TN_{j},FN_{j}$ are the true positive, false positive, true negative and false negative counts respectively for only the $j^{th}$ label.

Here $B$ stands for any of the confusion-matrix based metric. In your case you would plug in the standard precision and recall formulas. For macro average you pass in the per label count and then sum, for micro average you average the counts first, then apply your metric function.

You might be interested to have a look into the code for the mult-label metrics here , which a part of the package mldr in R. Also you might be interested to look into the Java multi-label library MULAN.

This is a nice paper to get into the different metrics: A Review on Multi-Label Learning Algorithms