In fxjollois/pad: Point Anomaly Detection in univariate time-serie

# For automatic figures and tables numbering and referencing
library(captioner)
table_nums <- captioner(prefix = "Table")
figure_nums <- captioner(prefix = "Figure")
captionfig <- function(fig) { 
  cap = figure_nums(fig)
  return (sub("(Figure[ ]+[0-9]+)", "**\\1**", cap))}
citefig <- function(fig) { return (figure_nums(fig, display = "cite"))}
figure_nums(name = "pointSample", 
            caption ="Univariate time serie, with an outlier (in red circle)")

Point Anomaly

The purpose of the pad package is to propose a large range of methods to detect point anomalies in a univariate time-serie. A point anomaly is a particular element, far enough to its neighborhood to be considered as an outlier, as shown in r citefig("pointSample").

x = runif(100)
x[50] = 3
opar <- par(mar = c(2, 2, 2, 0) + 0.1)
plot(x, type = "l", 
     xlab = "time", ylab = "value")
points(50, x[50], col = "red", cex = 2)
par(opar)

@gupta2014outlier [chap. 2.2.1] propose an interesting survey of methodologies to find outlier points for a time series. They define three different kind of methods:

Prediction Models: The outlier score for a point in the time series is computed as its deviation from the predicted value by a summary prediction model. The primary variation across models, is in terms of the particular prediction model used.

Profile Similarity-Based Approaches: These approaches maintain a normal profile and then compare a new time point against this profile to decide whether it is an outlier.

Deviants: Deviants are outlier points in time series from a minimum description length (MDL) point of view.

Notation

Given $t = (t_i)$ an univariate time serie, with $i = 1, \ldots, n$.

Detection methods

Standard Deviation

A point $i$ is considered as an anomaly if $t_i$ is outside the interval $[ \mu - \alpha * \sigma ; \mu + \alpha * sigma ]$.

Extreme Studentized Deviate

Also called ESD [@ESD], reference to put here -- in a simplified version

Prediction Models

Sliding window median (`SWMed`)

In @Basu2007, the authors want to suppress outliers in an univariate time serie. For that, they propose a simple method based on sliding window and median deviation. A point $i$ is considered as an anomaly if $$ | t_i - m_i | < \varepsilon $$ where $\varepsilon$ is a given threshold and $m_i$ is the median computed from the neighborhood of $i$, determined with two methods ($k$ is a given size):

two-sided: $m_i = median( t_{i - k}, \ldots, t_{i + k})$
one-sided: $m_i = median( t_{i - 2k}, \ldots, t_{i - 1})$