---
title: "Accuracy indices"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Accuracy indices}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

{lvmisc} contains a group of useful functions to compute basic indices of accuracy. These functions can be divided in those which compute element-wise values and those which compute average values:

* **Element-wise:**
    * `error()`
    * `error_abs()`
    * `error_pct()`
    * `error_abs_pct()`
    * `error_sqr()`
    
* **Average:**
    * `mean_error()`
    * `mean_error_abs()`
    * `mean_error_pct()`
    * `mean_error_abs_pct()`
    * `mean_error_sqr()`
    * `mean_error_sqr_root()`
    * `bias()`
    * `loa()`
    
You may notice that the majority of these functions have common prefixes (`error_` and `mean_error_`), intended to facilitate the use, as most text editors have an auto-complete feature. Also all of the accuracy indices functions take `actual` and `predicted` as arguments, and the functions that return average values have `na.rm = TRUE` in addition.

Let's now see how each function computes its results

## Element-wise

#### Error: `error()`

It simply subtracts the `predicted` from the `actual` values.

**Formula:** $$a_i - p_i$$

#### Absolute error: `error_abs()`

It returns the absolute values of the `error()` function.

**Formula:** $$|a_i - p_i|$$

#### Percent error: `error_pct()`

Divides the error by the `actual` values.

**Formula:** $$\frac{a_i - p_i}{a_i}\cdot100$$

#### Absolute percent error: `error_abs_pct()`

Returns the absolute values of the `error_pct()` function.

**Formula:** $$\frac{|a_i - p_i|}{|a_i|}\cdot100$$

#### Squared error: `error_sqr()`

It squares the values of the `error()` function.

**Formula:** $$(a_i - p_i)^2$$

## Average

#### Mean error: `mean_error()`

It is the average of the error.

**Formula:** $$\frac{1}{N}\sum_{i = 1}^{N}(a_i - p_i)$$

#### Mean absolute error: `mean_error_abs()`

Computes the average of the absolute error.

**Formula:** $$\frac{1}{N}\sum_{i = 1}^{N}|a_i - p_i|$$

#### Mean percent error: `mean_error_pct()`

The average of the percent error.

**Formula:** $$\frac{1}{N}\sum_{i = 1}^{N}\frac{a_i - p_i}{a_i}\cdot100$$

#### Mean absolute percent error: `mean_error_abs_pct()`

It is the average of the absolute percent error.

**Formula:** $$\frac{1}{N}\sum_{i = 1}^{N}\frac{|a_i - p_i|}{|a_i|}\cdot100$$

#### Mean squared error: `mean_error_sqr()`

Averages the mean squared error.

**Formula:** $$\frac{1}{N}\sum_{i = 1}^{N}(a_i - p_i)^2$$

#### Root mean squared error: `mean_error_sqr_root()`

It takes the square root of the mean squared error.

**Formula:** $$\sqrt{\frac{1}{N}\sum_{i = 1}^{N}(a_i - p_i)^2}$$

#### Bias: `bias()`

Alias to `mean_error()`.

#### Limits of agreement: `loa()`

**Formula:** $$bias \pm 1.96\sigma$$

Where $\sigma$ is the standard deviation.