# A Boundary Distance-Based Symbolic Aggregate Approximation Method for Time Series Data
## 1 Introduction  
Time series data are being generated every day in a wide range of application domains [1], such as bioinformatics, finance, engineering, etc. [2]. The parallel explosions of interest in streaming data and data mining of time series [3; 4; 5; 6; 7; 8; 9] have had little intersection. Time series classification methods can be divided into three main categories [10]: feature based, model based and distance based. There are many methods for feature extraction, for example: (1) spectral analysis such as discrete Fourier transform (DFT) [11], (2) discrete wavelet transform (DWT) [12], where features of the frequency domain are considered, and (3) singular value decomposition (SVD) [13], where eigenvalue analysis is carried out in order to find an optimal set of features. The model-based classification methods include auto-regressive models [14; 15] or hidden Markov models [16], among others. In distance-based methods, 1-NN [1] has been a widely used method due to its simplicity and good performance. Until now, almost all the research in distance-based classification has been oriented to defining different types of distance measures and then exploiting them within the 1-NN classifiers. The 1-NN classifier is probably the simplest classifier among all classifiers, while its performance is also good. Dynamic Time Warping(DTW) [17] as a distance method used for 1-NN classifier makes the classification accuracy reach the maximum at that time. However, due to the high dimensions, high volume, high feature correlation, and multiple noises, it has brought great challenges to the classification of time series, and even makes the DTW unusable. In fact, all non-trivial data mining and indexing algorithms decrease exponentially with dimensions. For example, above 16-20 dimensions, the index structure will be degraded to sequential scanning [18]. In order to reduce the time series dimensions and havea low bound to the Euclidean distance. The Piecewise Aggregate Approximation(PAA) [19] and Symbolic Aggregate Approximate(SAX) [20] were brought up. The distance in the SAX representation has a lower bound to the Euclidean distance. Therefore, the SAX representation speeds up the data mining process of time series data while maintaining the quality of the data mining results. SAX has been widely used in mobile data management [21], financial investment [22], feature extraction [23]. In recent years, with the popularity of deep learning, applying deep learning methods to multivariate time series classification has also received attention [24]. SAX allows a time series of arbitrary length \(n\) to be reduced to a string of arbitrary length \(w\)[20] (\(w<n\), typically \(w<<n\)). The alphabet size \(\alpha\) is also an arbitrary integer. The SAX representation has a major limitation. In the SAX representation, symbols are mapped from the average values of segments, and some important features may loss. For example, in Figure 1, if \(w=6\) and \(\alpha=6\), time series a represented as 'decfdb'. However, it can be clearly seen from the Figure 1 that the time series changes drastically. Therefore, a compromise is needed to reduce the dimension of time series while improving the accuracy. ESAX representation can express the characteristics of time series in more detail [25]. It chooses a maximum, a minimum and the average value in each time series segment as the new feature, then map the new feature to strings according to the SAX method. For the same time series, in Figure 1 time series a can be represented as 'adreeffdaabc'. SAX-TD (trend distance) method improves the accuracy of ESAX and reduce the complexity of symbol representation [26]. It uses fewer values than ESAX due to the strategy that one segment only needs one trend distance. In the Figure 1, the time series a is represented as'\({}_{-1.4}\)d\({}_{0.13}\)e\({}_{0.75}\)e\({}_{0.13}\)f\({}_{1.25}\)d\({}_{-0.25}\)b\({}_{-0.25}\)'. In this paper we propose a new method SAX-BD, in which BD means the boundary distance.