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machine-learningclassificationmultilabel-classificationprecision-recall
MaVe
MaVe

Reputation: 1725

Precision/recall for multiclass-multilabel classification

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?

Upvotes: 32

Views: 33666

Answers (5)

Cristian Garcia
Cristian Garcia

Reputation: 9859

In python using sklearn and numpy:

from sklearn.metrics import confusion_matrix
import numpy as np

labels = ...
predictions = ...

cm = confusion_matrix(labels, predictions)
recall = np.diag(cm) / np.sum(cm, axis = 1)
precision = np.diag(cm) / np.sum(cm, axis = 0)

Upvotes: 4

phoxis
phoxis

Reputation: 61910

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

Upvotes: 24

David M W Powers
David M W Powers

Reputation: 360

Simple averaging will do if the classes are balanced.

Otherwise, recall for each real class needs to be weighted by prevalence of the class, and precision for each predicted label needs to be weighted by the bias (probability) for each label. Either way you get Rand Accuracy.

A more direct way is to make a normalized contingency table (divide by N so table adds up to 1 for each combination of label and class) and add the diagonal to get Rand Accuracy.

But if classes aren't balanced, the bias remains and a chance corrected method such as kappa is more appropriate, or better still ROC analysis or a chance correct measure such as informedness (height above the chance line in ROC).

Upvotes: 1

dolly
dolly

Reputation: 196

  • Let us assume that we have a 3-class multi classification problem with labels A, B and C.
  • The first thing to do is to generate a confusion matrix. Note that the values in the diagonal are always the true positives (TP).

  • Now, to compute recall for label A you can read off the values from the confusion matrix and compute:

    = TP_A/(TP_A+FN_A)
= TP_A/(Total gold labels for A)

  • Now, let us compute precision for label A, you can read off the values from the confusion matrix and compute:

    = TP_A/(TP_A+FP_A)
= TP_A/(Total predicted as A)

  • You just need to do the same for the remaining labels B and C. This applies to any multi-class classification problem.

Here is the full article that talks about how to compute precision and recall for any multi-class classification problem, including examples.

Upvotes: 6

Mark E. Haase
Mark E. Haase

Reputation: 26881

The answer is that you have to compute precision and recall for each class, then average them together. E.g. if you classes A, B, and C, then your precision is:

(precision(A) + precision(B) + precision(C)) / 3

Same for recall.

I'm no expert, but this is what I have determined based on the following sources:

https://list.scms.waikato.ac.nz/pipermail/wekalist/2011-March/051575.html http://stats.stackexchange.com/questions/21551/how-to-compute-precision-recall-for-multiclass-multilabel-classification

Upvotes: 8

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