Common classification problems in Machine Learning (ML) are binary and multi-class (Sokolova and Lapalme, 2009). For binary classification, we have *accuracy*, *precision*, *recall*, and a combination of *precision* and *recall* which is so-called \(F_1\) score. However, the *precision* and *recall* from binary classification cannot be utilized literally to measure multi-class classification.

To measure the performance of multi-class classification, two important modifications on precision and recall of binary classification are introduced. Their names are **macro-average** and **micro-average**. Therefore, for example, the precision of multi-class classification shall become **macro-average** precision and **micro-average** precision.

Let’s begin with an example of multi-class classification with \(4\) classes (\(0\), \(1\), \(2\), and \(3\)). Suppose we have \(\text{our predictions}\) and the \(\text{true labels}\) for five data instances as follows:

\[\begin{align} \text{our predictions} &= [ 0, 0, 2, 2, 3 ], \\ \text{true labels} &= [ 0, 1, 3, 3, 3 ] \end{align}\]Our first prediction is $0$ and the true label is $0$. Next, our second prediction is $0$ and the true label is $1$. Our third prediction is $2$ while the true label is $3$ and so on. Let’s denote

\[\begin{align} tp_i &= \text{true positive for class }i \; (i = 0,1,2,3), \\ fp_i &= \text{false positive for class }i \; (i = 0,1,2,3). \end{align}\]After counting the true and false positives for each class, we obtain

\[\begin{align} tp_0 &= 1, \; tp_1 = 0, \; tp_2 = 0, \; tp_3 = 1, \text{ and} \\ fp_0 &= 1, \; fp_1 = 0, \; fp_2 = 2, \; fp_3 = 0. \end{align}\]As we’ve already known, $\text{precision}$ for class $i$ ($\text{precision}_i$) is defined as follows:

\[\begin{equation} \text{precision}_i = \frac{tp_i}{tp_i + fp_i}, \text{ for }i = 0,1,2,3. \tag{1}\label{eq:precision-formula} \end{equation}\]Therefore, employing Equation \eqref{eq:precision-formula}, we get

\[\begin{equation} \text{precision}_0 = 0.5, \; \text{precision}_1 = 0, \; \text{precision}_2 = 0, \; \text{precision}_3 = 1. \tag{2}\label{eq:precision-results} \end{equation}\]As explained in “Micro Average vs Macro average Performance in a Multiclass classification setting”, the **macro-average** precision ($\text{precision}_M$) for $4$ classes is defined as

However, the **micro-average** precision ($\text{precision}_\mu$) for $4$ classes is defined as

If there is a class imbalance problem, one of the options will be using **weighted macro-average** as performance metrics. The **weighted macro-average** precision ($\text{precision}_{WM}$) for $4$ classes is defined as

with $\text{weight}_i$ is the weight assigned to class $i$ as follows:

\[\begin{equation} \text{weight}_i = \frac{tp_i + fp_i}{\sum_{i=0}^{3}{(tp_i + fp_i)}} \tag{4}\label{eq:weight} \end{equation}\]Using Equation \eqref{eq:weighted-precision} and \eqref{eq:weight}, we compute the **weighted macro-average** precision as follows:

Next, we shall show that the **weighted macro-average** precision does equal to the **micro-average** precision.

Finally we have reached the end of this post. In brief, we have shown how to compute **macro-average**, **micro-average**, and **weighted macro-average**. Moreover, we have also shown that the **micro-average** equals to **weighted macro-average**.

#### References

Sokolova, M. and Lapalme, G. (2009). A systematic analysis of performance measures for classification tasks. *Information Processing & Management*, 45(4):427 - 437.