# Hexadecimal Aggregate Approximation Representation and Classification of Time Series Data
## 1 Introduction
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Representation Method** & **Year** & **Type** & **Complexity** & **References** \\ \hline Auto-regressive (AR) model & 1971 & T4 & & [36; 37] \\ \hline Discrete Fourier Transform(DFT) & 1993 & T1 & O(\(n(\log(n))\)) & [13; 38] \\ \hline Discrete Wavelet Transform (DWT) & 1999 & T1 & O(\(n\)) & [12; 39] \\ \hline Singular Value Decomposition (SVD) & 1997 & T2 & & [40] \\ \hline Discrete Cosine Transformation (DCT) & 1997 & T1 & — & [40] \\ \hline Piecewise Linear Approximation (PLA) & 1998 & T2 & O(\(n(\log(n))\)) & [17] \\ \hline Hidden Markov models (HMMs) & 1998 & T4 & — & [41] \\ \hline Piecewise Aggregate Approximation (PAA) or Segmented Means & 2000 & T1 & O(\(n\)) & [42] \\ \hline Piecewise Constant Approximation (PCA) & 2000 & T2 & — & [43] \\ \hline Adaptive Piecewise Constant Approximation (APCA) & 2002 & T2 & O(\(n\)) & [16] \\ \hline Perceptually important point (PIP) & 2001 & T1 & — & [44] \\ \hline Chebyshev Polynomials (CHEB) & 2004 & T1 & — & [45] \\ \hline Symbolic Aggregate Approximation (SAX) & 2003 & T2 & O(\(n\)) & [19; 22] \\ \hline HOT SAX & 2005 & T2 & & [46] \\ \hline Clipped Data & 2005 & T3 & — & [47] \\ \hline Group SAX & 2006 & T2 & & [48] \\ \hline Extended SAX & 2006 & T2 & & [49] \\ \hline Combining SAX and Piecewise Linear Approximation & 2007 & T2 & & [50] \\ \hline Indexable Piecewise Linear Approximation (IPLA) & 2007 & T1 & — & [51] \\ \hline
1d-SAX & 2013 & T2 & & [52] \\ \hline Move-Split-Merge (MSM) & 2013 & & & [53] \\ \hline SAX-VSM & 2013 & T2 & & [54] \\ \hline SAX-EFG & 2014 & T2 & & [55] \\ \hline Tree-based Representations & 2015 & & & [56] \\ \hline SC-DTW & 2015 & T1 & & [57] \\ \hline Representation based on Local Autopatterns & 2016 & & & [58] \\ \hline Grid Representation & 2019 & & & [59] \\ \hline SAX Navigator & 2019 & T2 & & [60] \\ \hline SAX-ARM & 2020 & T2 & & [61] \\ \hline SAX-BD & 2020 & T2 & & [62] \\ \hline Data-driven Kernel-based Probabilistic SAX & 2021 & T2 & & [63] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main representation methods for time series data (— no indicated by authors; \(n\) is the length of time series; T1 non-data adaptive; T2 data-adaptive; T3 data dictated; T4 model-based). feature. The mapping method between the new feature to SAX words is the same as the SAX method. Therefore, the length of the ESAX string is three times that in SAX. The SAX-TD method adds a trend distance for each segment; hence, the length of the symbol string is also two times the original string length. Therefore, both ESAX and SAX-TD methods require additional information to extend the original SAX string. The SAX-BD [62] method has a design integrating the SAX distance with a weighted boundary distance, resulting in it outperforming SAX-TD. Symbolic aggregate approximation methods have proven very effective in compacting the information content of time series. However, typical SAX-based techniques rely on a Gaussian assumption for the underlying data statistics, which often deteriorates their performance in practical scenarios [63]. To overcome this limitation, a novel method, the Hexadecimal Aggregate approXimation representation (HAX) of time series, is proposed in this paper. This method negates any assumption on the probability distribution of time series and represents each segment of a time series as a transformable interval object (TIO), using the transformation distance to measure the similarity between a pair of time series. Then, each TIO is mapped to a hexadecimal symbol by its location on a hexadecimal grid. Therefore, the HAX string is the same length as the SAX string with the same word size, the \(w\) parameter. We compare SAX, SAX-TD and SAX-BD methods with HAX methods. Our reason for choosing the SAX-TD and SAX-BD is that the outputs of the SAX-TD and SAX-BD still include the original SAX string, despite including some attachments; hence, there is comparability with the hex string of HAX. The experimental results show that HAX has higher accuracy than the SAX method.