# Similarity Measurement and Classification of Temporal Data Based on Double Mean Representation
## 1 Introduction
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Representation** & **Publication Time** & **Type** & \begin{tabular}{c} **Algorithm** \\ **Complexity** \\ \end{tabular} & \
\begin{tabular}{c} **Method** \\ **Source** \\ \end{tabular} \\ \hline Indexable Piecewise Linear & 2007 & T1 & N & [8] \\ Approximation (IPLA) & 2013 & T1 & N & [9] \\ Move Split Merge (MSM) & 2013 & T1 & N & [9] \\ Graphical-Content-Based DTW (SC) & 2015 & T1 & N & [10] \\ Singular Value Decomposition & 1997 & T2 & O(Mn2) & [4] \\ Piecewise Linear Approximation (PLA) & 1998 & T2 & O(n(log(n))) & [11] \\ Piecewise Constant Approximation (PCA) & 2000 & T2 & N & [12] \\ Adaptive Partitioning Constant & 2002 & T2 & O(n) & [13] \\ Approximation (APCA) & 2003 & T2 & O(n) & [14,15] \\ Symbolized Aggregate & 2014 & T2 & N & [16] \\ Approximation (SAX) & 2020 & T2 & N & [17] \\
\begin{tabular}{c} Hexadecimal Aggregate \\ Approximation (HAX) \\ (PAX) \\ (PX) \\ \end{tabular} & 2021 & T2 & N & [18] \\ Symbol Aggregation & 2021 & T2 & N & [18] \\ Approximation Based on Distance & 2022 & T2 & N & [19] \\ and Momentum & & & & \\ Convergence Trend Symbol & 2023 & T2 & N & [20] \\ Aggregation Approximation (SAX-TI) & 2005 & T3 & N & [21] \\ Clipped Data & 2015 & T3 & N & [22] \\ Tree-Based Representation Method & 2015 & T3 & N & [22] \\ Hidden Markov Models (HMMs) & 1998 & T4 & N & [23] \\ Automatic Regression Model & 2012 & T4 & N & [24] \\ Representation Method Based on & 2016 & T4 & N & [25] \\ Local Automatic Mode & 2019 & T4 & N & [26] \\ Grid-Based Representation Method & 2019 & T4 & N & [26] \\ Based on Random Forest and & & & & \\ Ranking Importance & 2022 & T4 & N & [27] \\ \hline \hline \end{tabular}  
* N: author not listed. T1: non-data-adaptive representation methods. T2: data-adaptive representation methods. T3: data-indicative representation methods. T4: model-based representation methods.  
\end{table} 
Table 1: \(Cont.\)(2) In non-data-adaptive representation methods, the non-data-adaptive representation methods based on frequency domain transformation convert time series data from the time domain to the frequency domain and represent the time series data using the spectral information in the frequency domain. Commonly used methods include Discrete Fourier Transform (DFT) [2], Discrete Wavelet Transform (DWT) [3], Discrete Cosine Transform (DCT) [4], and others. These methods approximate the original time series data by taking the first k coefficients after a certain transformation, achieving compression and reducing the dimensionality of the time series data. Although parameter determination for domain-transformation-based representation methods is challenging and requires extensive parameter tuning experiments, non-data-adaptive methods based on piecewise approximation have also emerged. Examples include Piecewise Aggregate Approximation (PAA) [5], Indexable Piecewise Linear Aggregation (IPLA) [8], and other methods. Non-data-adaptive representation methods use equally sized segments to approximate the time series data. While the approximation quality may not be as good as that of data-adaptive representation methods, using such methods for similarity measurement is relatively straightforward and simple. Non-data-adaptive representation methods use equally sized segments to approximate the time series data. Although the approximation quality may not be as good as that of data-adaptive representation methods, using these methods for similarity measurement is more direct and simpler. (3) Model-based representation methods assume that time series data are stochastic and use models such as AutoRegressive (AR) [24], AutoRegressive Moving Average (ARMA) [25], and Hidden Markov Models (HMMs) [26] for fitting. This approach requires the data to conform to certain assumptions and mathematical deduction theories; otherwise, distortion may occur. Additionally, time series data are complex in structure and contain a significant amount of noise, which poses significant challenges in constructing accurate models.