## Overview

The weird package contains functions and data used in the book *That’s Weird: Anomaly Detection Using R* by Rob J Hyndman. It also loads several packages needed to do the analysis described in the book.

## Installation

You can install the development version of weird from GitHub with:

```
# install.packages("devtools")
devtools::install_github("robjhyndman/weird-package")
```

## Usage

`library(weird)`

will load the following packages:

- dplyr, for data manipulation.
- ggplot2, for data visualisation.
- ks, for fitting models and producing forecasts.

You also get a condensed summary of conflicts with other packages you have loaded:

```
library(weird)
#> ── Attaching packages ────────────────────────────────────── weird 0.0.0.9000 ──
#> ✔ dplyr 1.1.4 ✔ ks 1.14.1
#> ✔ ggplot2 3.4.4
#> ── Conflicts ──────────────────────────────────────────────── weird_conflicts ──
#> ✖ dplyr::filter() masks stats::filter()
#> ✖ dplyr::lag() masks stats::lag()
```

## Example: Old Faithful Geyser data

The `oldfaithful`

data set contains eruption data from the Old Faithful Geyser in Yellowstone National Park, Wyoming, USA, from 1 January 2015 to 1 October 2021. The data were obtained from the geysertimes.org website. Recordings are incomplete, especially during the winter months when observers may not be present. There also appear to be some recording errors. The data set contains 2261 observations of 3 variables: `time`

giving the time at which each eruption began, `duration`

giving the length of the eruption in seconds, and `waiting`

giving the time to the next eruption in seconds. In the analysis below, we omit the eruption with `duration`

greater than 1 hour as this is likely to be a recording error. Some of the long `waiting`

values are probably due to omitted eruptions, and so we also omit eruptions with `waiting`

greater than 2 hours.

```
oldfaithful
#> # A tibble: 2,261 × 3
#> time duration waiting
#> <dttm> <dbl> <dbl>
#> 1 2015-01-02 14:53:00 271 5040
#> 2 2015-01-09 23:55:00 247 6060
#> 3 2015-02-07 00:49:00 203 5460
#> 4 2015-02-14 01:09:00 195 5221
#> 5 2015-02-21 01:12:00 210 5401
#> 6 2015-02-28 01:11:00 185 5520
#> 7 2015-03-07 00:50:00 160 5281
#> 8 2015-03-13 21:57:00 226 6000
#> 9 2015-03-13 23:37:00 190 5341
#> 10 2015-03-20 22:26:00 102 3961
#> # ℹ 2,251 more rows
```

## Kernel density estimates

The package provides the `kde_bandwidth()`

function for estimating the bandwidth of a kernel density estimate, and an `autoplot()`

method for plotting the resulting density. The figure below shows the kernel density estimate of the `duration`

variable obtained using these functions. The rug plot shows the actual data values.

```
of <- oldfaithful |>
filter(duration < 3600, waiting < 7200)
of_density <- kde(of$duration, h=kde_bandwidth(of$duration))
of_density |>
autoplot() +
geom_rug(aes(x=duration), of) +
labs(x = "Duration (seconds)")
```

The `kde_bandwidth()`

function can also be used to estimate the bandwidth for a bivariate kernel density estimate. The figure below shows the kernel density estimate of the `duration`

and `waiting`

variables using the bandwidth selected by the `kde_bandwidth()`

function. The rug plot shows the actual data values.

```
of_density <- of |>
select(duration, waiting) |>
kde(H = kde_bandwidth(of[,c("duration","waiting")]))
of_density |>
autoplot() +
geom_point(aes(duration, waiting), data = of, alpha=0.15) +
labs(x = "Duration (seconds)", y = "Waiting time (seconds)")
```

## Statistical tests

Some old methods of anomaly detection used statistical tests. While these are not recommended, they are still widely used, and are provided in the package for comparison purposes.

```
of |> filter(peirce_anomalies(duration))
#> # A tibble: 1 × 3
#> time duration waiting
#> <dttm> <dbl> <dbl>
#> 1 2018-04-25 19:08:00 1 5700
of |> filter(chauvenet_anomalies(duration))
#> # A tibble: 1 × 3
#> time duration waiting
#> <dttm> <dbl> <dbl>
#> 1 2018-04-25 19:08:00 1 5700
of |> filter(grubbs_anomalies(duration))
#> # A tibble: 1 × 3
#> time duration waiting
#> <dttm> <dbl> <dbl>
#> 1 2018-04-25 19:08:00 1 5700
of |> filter(dixon_anomalies(duration))
#> # A tibble: 1 × 3
#> time duration waiting
#> <dttm> <dbl> <dbl>
#> 1 2018-04-25 19:08:00 1 5700
```

In this example, they only detect the tiny 1-second duration, which is almost certainly a recording error. An explanation of these tests is provided in Chapter 4 of the book

## Boxplots

Boxplots are widely used for anomaly detection. Here are three variations of boxplots applied to the `duration`

variable.

```
of |>
ggplot(aes(x = duration)) +
geom_boxplot() +
scale_y_discrete() +
labs(y = "", x = "Duration (seconds)")
```

```
of |> gg_hdrboxplot(duration) +
labs(x = "Duration (seconds)")
```

```
of |> gg_hdrboxplot(duration, scatterplot = TRUE) +
labs(x = "Duration (seconds)")
```

The latter two plots are HDR boxplots, which allow the bimodality of the data to be seen. The dark shaded region contains 50% of the observations, while the lighter shaded region contains 99% of the observations. The plots use vertical jittering to reduce overplotting, and highlight potential outliers in red using the lookout algorithm (described in Chapter 6 of the book). An explanation of these plots is provided in Chapter 5 of the book.

It is also possible to produce bivariate boxplots. Several variations are provided in the package. Here are two types of bagplot.

```
of |>
gg_bagplot(duration, waiting) +
labs(x = "Duration (seconds)", y = "Waiting time (seconds)")
```

```
of |>
gg_bagplot(duration, waiting, scatterplot = TRUE) +
labs(x = "Duration (seconds)", y = "Waiting time (seconds)")
```

And here are two types of HDR boxplot

```
of |>
gg_hdrboxplot(duration, waiting) +
labs(x = "Duration (seconds)", y = "Waiting time (seconds)")
```

```
of |>
gg_hdrboxplot(duration, waiting, scatterplot = TRUE) +
labs(x = "Duration (seconds)", y = "Waiting time (seconds)")
```

The latter two plots show likely outliers in red, using the lookout algorithm.

## Scoring functions

Several functions are provided for providing anomaly scores for all observations.

- The
`density_scores()`

function uses either a fitted statistical model, or a kernel density estimate, to compute density scores. - The
`stray_scores()`

function uses the stray algorithm to compute anomaly scores. - The
`lof_scores()`

function uses local outlier factors to compute anomaly scores. - The
`glosh_scores()`

function uses the Global-Local Outlier Score from Hierarchies algorithm to compute anomaly scores. - The
`lookout()`

function uses the lookout algorithm to compute anomaly probabilities

Here are the top 0.02% most anomalous observations identified by each of the first four methods, along with the observations having lookout probability less than 0.05.

```
of |>
mutate(
denscore = density_scores(cbind(duration, waiting)),
strayscore = stray_scores(cbind(duration, waiting)),
lofscore = lof_scores(cbind(duration, waiting), k = 150),
gloshscore = glosh_scores(cbind(duration, waiting)),
lookout = lookout(cbind(duration, waiting))
) |>
filter(
denscore > quantile(denscore, prob=0.998) |
strayscore > quantile(strayscore, prob=0.998) |
lofscore > quantile(lofscore, prob=0.998) |
gloshscore > quantile(gloshscore, prob=0.998) |
lookout < 0.05
) |>
arrange(lookout)
#> # A tibble: 11 × 8
#> time duration waiting denscore strayscore lofscore gloshscore
#> <dttm> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2018-04-25 19:08:00 1 5700 17.5 0.380 3.78 1
#> 2 2020-06-01 21:04:00 120 6060 17.5 0.132 1.88 1
#> 3 2021-01-22 18:35:00 170 3600 16.8 0.0606 1.09 0.860
#> 4 2020-08-31 09:56:00 170 3840 16.7 0.0606 1.01 0.816
#> 5 2015-11-21 20:27:00 150 3420 16.7 0.0772 1.27 1
#> 6 2017-05-03 06:19:00 90 4740 16.4 0.0495 1.68 1
#> 7 2020-10-15 17:11:00 220 7080 15.7 0.0429 2.42 1
#> 8 2017-09-22 18:51:00 281 7140 15.5 0.0333 2.64 1
#> 9 2017-08-12 13:14:00 120 4920 15.2 0.0690 1.53 1
#> 10 2020-05-18 21:21:00 272 7080 14.9 0.0333 2.42 1
#> 11 2018-09-22 16:37:00 253 7140 14.7 0.0200 2.63 1
#> # ℹ 1 more variable: lookout <dbl>
```

## Robust multivariate scaling

Some anomaly detection methods require the data to be scaled first, so all observations are on the same scale. However, many scaling methods are not robust to anomalies. The `mvscale()`

function provides a multivariate robust scaling method, that optionally takes account of the relationships betwen variables, and uses robust estimates of center, scale and covariance by default. The centers are removed using medians, the scale function is the IQR, and the covariance matrix is estimated using a robust OGK estimate. The data are scaled using the Cholesky decomposition of the inverse covariance. Then the scaled data are returned. The scaled variables are rotated to be orthogonal, so are renamed as `z1`

, `z2`

, etc. Non-rotated scaling is possible by setting `cov = NULL`

.

```
mvscale(of)
#> Warning in mvscale(of): Ignoring non-numeric columns: time
#> # A tibble: 2,197 × 3
#> time z1 z2
#> <dttm> <dbl> <dbl>
#> 1 2015-01-02 14:53:00 2.06 -1.47
#> 2 2015-01-09 23:55:00 0.130 0.801
#> 3 2015-02-07 00:49:00 -1.78 -0.534
#> 4 2015-02-14 01:09:00 -2.04 -1.07
#> 5 2015-02-21 01:12:00 -1.38 -0.665
#> 6 2015-02-28 01:11:00 -2.76 -0.401
#> 7 2015-03-07 00:50:00 -3.92 -0.932
#> 8 2015-03-13 21:57:00 -0.932 0.668
#> 9 2015-03-13 23:37:00 -2.38 -0.799
#> 10 2015-03-20 22:26:00 -6.09 -3.87
#> # ℹ 2,187 more rows
mvscale(of, cov = NULL)
#> Warning in mvscale(of, cov = NULL): Ignoring non-numeric columns: time
#> # A tibble: 2,197 × 3
#> time duration waiting
#> <dttm> <dbl> <dbl>
#> 1 2015-01-02 14:53:00 1.61 -1.48
#> 2 2015-01-09 23:55:00 0.363 0.809
#> 3 2015-02-07 00:49:00 -1.92 -0.540
#> 4 2015-02-14 01:09:00 -2.33 -1.08
#> 5 2015-02-21 01:12:00 -1.56 -0.672
#> 6 2015-02-28 01:11:00 -2.85 -0.405
#> 7 2015-03-07 00:50:00 -4.15 -0.942
#> 8 2015-03-13 21:57:00 -0.726 0.674
#> 9 2015-03-13 23:37:00 -2.59 -0.807
#> 10 2015-03-20 22:26:00 -7.16 -3.91
#> # ℹ 2,187 more rows
```