Introduction

We define here the evaluation measures used in TSB-UAD. We first introduce formal notations. Then, we review in detail previously proposed evaluation measures for time-series anomaly detection methods. We review notations for the time series and anomaly score sequence.

Time Series Notation

A time series \(T \in \mathbb{R}^n\) is a sequence of real-valued numbers \(T_i\in\mathbb{R}, [T_1,T_2,...,T_n]\), where \(n=|T|\) is the length of \(T\), and \(T_i\) is the \(i^{th}\) point of \(T\). We are typically interested in local regions of the time series, known as subsequences. A subsequence \(T_{i,\ell} \in \mathbb{R}^\ell\) of a time series \(T\) is a continuous subset of the values of \(T\) of length \(\ell\) starting at position \(i\). Formally, \(T_{i,\ell} = [T_i, T_{i+1},...,T_{i+\ell-1}]\).

Anomaly Score Sequence

For a time series \(T \in \mathbb{R}^n\), an AD method \(A\) returns an anomaly score sequence \(S_T\). For point-based approaches (i.e., methods that return a score for each point of \(T\)), we have \(S_T \in \mathbb{R}^n\). For range-based approaches (i.e., methods that return a score for each subsequence of a given length \(\ell\)), we have \(S_T \in \mathbb{R}^{n-\ell}\). Overall, for range-based (or subsequence-based) approaches, we define \(S_T = [S_{T,1},S_{T,2},...,S_{T,n-\ell}]\) with \(S_{T,i} \in [0,1]\).