The Living Thing / Notebooks :


labelling losses, fitting classifiers etc



Precision/Recall and f-scores all work for multi-label classification, although they have bad qualities in unbalanced classes.

Unbalanced class problems


Metric Zoo

One of the less abstruse summaries of these is the scikit-learn classifier loss page, which includes both formulae and verbal descriptions; this is surprisingly hard to find on, e.g. the documentation for deep learning toolkits, in keeping with the field’s general taste for magical black boxes.

Matthews correlation coefficient

Due to Matthews (Matt75). This is the first choice for seamlessly handling multi-label problems, since its behaviour is reasonable for 2 class or multi class, balanced or unbalanced, and it’s computationally very cheap. Unless you have a very different importance for your classes, this is a good default.

However, it is not differentiable with respect to classification certainties, so you can’t use it as, e.g., a target in neural nets; Therefore you use surrogate measures which are differentiable and use this to track your progress.

2-class case

Take your \(2 times 2\). confusion matrix of true positive, false positives etc.

\begin{equation*} {\text{MCC}}={\frac {TP\times TN-FP\times FN}{{\sqrt {(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}}} \end{equation*}
\begin{equation*} |{\text{MCC}}|={\sqrt {{\frac {\chi ^{2}}{n}}}} \end{equation*}

Multiclass case

Take your \(K times K\) confusion matrix \(C\), then

\begin{equation*} {\displaystyle {\text{MCC}}={\frac {\sum _{k}\sum _{l}\sum _{m}C_{kk}C_{lm}-C_{kl}C_{mk}}{{\sqrt {\sum _{k}(\sum _{l}C_{kl})(\sum _{k'|k'\neq k}\sum _{l'}C_{k'l'})}}{\sqrt {\sum _{k}(\sum _{l}C_{lk})(\sum _{k'|k'\neq k}\sum _{l'}C_{l'k'})}}}}} \end{equation*}


Receiver Operator Characteristic/Area Under Curve. Supposedly dates back to radar operators in the mid-century. HaMc83 talk about the AUC for radiology; Supposedly Spac89 introduced it to machine learning, but I haven’t read the article in question. Allows you to trade off importance of false positive/false negatives.

Cross entropy

I’d better write down form for this, since most ML toolkits are curiously shy about it.

Let \(x\) be the estimated probability and \(z\) be the supervised class label. Then the binary cross entropy loss is

\begin{equation*} \ell(x,z) = -z\log(x) - (1-z)\log(1-x) \end{equation*}

If \(y=\operatorname{logit}(x)\) is not a probability but a logit, then the numerically stable version is

\begin{equation*} \ell(y,z) = \max\{y,0\} - y + \log(1+\exp(-|x|)) \end{equation*}


Flach, P., Hernández-Orallo, J., & Ferri, C. (2011) A Coherent Interpretation of AUC as a Measure of Aggregated Classification Performance. In Proceedings of the 28th International Conference on Machine Learning (ICML-11) (pp. 657–664).
Gorodkin, J. (2004) Comparing two K-category assignments by a K-category correlation coefficient. Computational Biology and Chemistry, 28(5–6), 367–374. DOI.
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Matthews, B. W.(1975) Comparison of the predicted and observed secondary structure of T4 phage lysozyme. Biochimica et Biophysica Acta (BBA) - Protein Structure, 405(2), 442–451. DOI.
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