---
title: "Multiclass averaging"
author: "Davis Vaughan"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Multiclass averaging}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

## Introduction

Classification metrics in `yardstick` where both the `truth` and `estimate` 
columns are factors are implemented for the binary and the
multiclass case. The multiclass implementations use `micro`, `macro`,
and `macro_weighted` averaging where applicable, and some metrics have their
own specialized multiclass implementations. 

## Macro averaging

Macro averaging reduces your multiclass predictions down to multiple sets of
binary predictions, calculates the corresponding metric for each of the binary
cases, and then averages the results together. As an example, consider 
`precision` for the binary case.

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

In the multiclass case, if there were levels `A`, `B`, `C` and `D`, macro averaging
reduces the problem to multiple one-vs-all comparisons. The `truth` and
`estimate` columns are recoded such that the only two levels are `A` and `other`,
and then precision is calculated based on those recoded columns, with `A` being
the "relevant" column. This process is repeated for the other 3 levels to get
a total of 4 precision values. The results are then averaged together.

The formula representation looks like this. For `k` classes:

$$
Pr_{macro} = \frac{Pr_1 + Pr_2 + \ldots + Pr_k}{k} = Pr_1 \frac{1}{k} + Pr_2 \frac{1}{k} + \ldots + Pr_k \frac{1}{k}
$$

where $PR_1$ is the precision calculated from recoding the multiclass predictions
down to just `class 1` and `other`.

Note that in macro averaging, all classes get equal weight when contributing
their portion of the precision value to the total (here `1/4`). This might not
be a realistic calculation when you have a large amount of class imbalance. In
that case, a _weighted macro average_ might make more sense, where the weights
are calculated by the frequency of that class in the `truth` column.

$$
Pr_{weighted-macro} = Pr_1 \frac{\#Obs_1}{N} + Pr_2 \frac{\#Obs_2}{N} + \ldots + Pr_k \frac{\#Obs_k}{N}
$$

## Micro averaging

Micro averaging treats the entire set of data as an aggregate result, and 
calculates 1 metric rather than `k` metrics that get averaged together.

For precision, this works by calculating all of the true positive results for
each class and using that as the numerator, and then calculating all of the
true positive and false positive results for each class, and using that
as the denominator.

$$
Pr_{micro} = \frac{TP_1 + TP_2 + \ldots + TP_k}{(TP_1 + TP_2 + \ldots + TP_k) + (FP_1 + FP_2 + \ldots + FP_k)}
$$
In this case, rather than each _class_ having equal weight, each _observation_
gets equal weight. This gives the classes with the most observations more 
power.


## Specialized multiclass implementations

Some metrics have known analytical multiclass extensions, and do not need to
use averaging to get an estimate of multiclass performance. 

Accuracy and Kappa use the same definitions as their binary counterpart,
with accuracy counting up the number of correctly predicted true values out of the total 
number of true values, and kappa being a linear combination of two accuracy 
values.

Matthews correlation coefficient (MCC) has a known multiclass generalization
as well, sometimes called the $R_K$ statistic. Refer to [this page](https://en.wikipedia.org/wiki/Matthews_correlation_coefficient#Multiclass_case) 
for more details.

ROC AUC is an interesting metric in that it intuitively makes sense to perform 
macro averaging, which computes a multiclass AUC as the average of the 
area under multiple binary ROC curves. However, this loses an important 
property of the ROC AUC statistic in that its binary case is insensitive to 
class distribution. To combat this, a multiclass metric was created that retains
insensitivity to class distribution, but does not have an easy visual interpretation
like macro averaging. This is implemented as the `"hand_till"` method, and is
the default for this metric.
