# Transitional SAX Representation for Knowledge Discovery for Time Series
## 5 Conclusions  
In this work, we described the popularity and universality of SAX, which is a symbolic aggregate approximation in the field of dimensionality reduction for time series data. The original SAX barely captures trend information from the perspective of time-series shape. Therefore, we proposed a symbolic aggregate approximation with transitional information, which can represent trend information by appending transition information to basic SAX. In a given time window, a SAX word is created, and we can trace how data points travel from the current quantile region to the next location. We call this moving behavior from the current location to the next location a transition. When in a current location, data points in a window can choose from three movements--upward transition, downward transition, and sojourn transition. These movements are saved in the data format of a matrix. First, we conducted experiments to verify the effectiveness of SAX-TM compared with other state-of-the-art methods such as SAX-TD and SAX-SD. The experimental results show SAX-TM has the lowest 1 NN classification error among the algorithms. Next, we identified intrinsic statistical properties of SAX-TM. From [20], we selected information loss, KL-divergence, and information embedding cost as important measurements. Overall, the information loss of SAX is lower than that of SAX-TM. However, the number of datasets with lower KL-divergence for SAX-TM is slightly larger than that for SAX. This tendency is also preserved in terms of the IEC score. Nonetheless, SAX-TM substantially reduces classification error compared with SAX. SAX-TM shows explicit increases in accuracy even while appending transition information to SAX. In spite of the aforementioned advantages, the proposed algorithm has several limitations. Basically, SAX compresses raw data for smoothing. However, SAX-TM increases the complexity of SAX representation by appending a transition matrix. We plan to investigate the minimal effective information to add to SAX and compare it with well-known non-SAX methods. In addition, future research directions include the theoretical aspects of the transition information in several time-series models. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multirow{2}{*}{**N**} & \multicolumn{2}{c}{**Information Loss**} & \multicolumn{2}{c}{**KL-Divergence**} & \multicolumn{2}{c}{**IEC Score**} & \multicolumn{2}{c}{**Classification Error**} \\ \cline{3-10}  & & **SAX** & **SAX-TM** & **SAX** & **SAX-TM** & **SAX** & **SAX-TM** & **Raw Data** & **SAX** & **SAX-TM** \\ \hline
50 words & 270 & 0.056 & 0.06 & 0.323 & 0.355 & 0.304 & 0.332 & 0.361 & 0.338 & 0.286 \\ Adiac & 176 & 0.009 & 0.189 & 0.07 & 0.103 & 0.07 & 0.087 & 0.407 & 0.383 & 0.491 \\ Beef & 470 & 0.038 & 0.03 & 0.346 & 0.493 & 0.332 & 0.477 & 0.4 & 0.466 & 0.3 \\ Coffee & 286 & 0.014 & 0.016 & 0.123 & 0.088 & 0.122 & 0.086 & 0.25 & 0.107 & 0 \\ ECG & 96 & 0.053 & 0.084 & 0.504 & 0.44 & 0.474 & 0.405 & 0.11 & 0.22 & 0.07 \\ Face(four) & 350 & 0.069 & 0.104 & 0.597 & 0.702 & 0.556 & 0.631 & 0.276 & 0.171 & 0.158 \\ Gun-point & 150 & 0.025 & 0.027 & 0.357 & 0.35 & 0.346 & 0.338 & 0.13 & 0.17 & 0.04 \\ Lighting2 & 637 & 0.112 & 0.387 & 0.969 & 0.934 & 0.862 & 0.666 & 0.197 & 0.229 & 0.164 \\ Lighting7 & 319 & 0.123 & 0.232 & 0.912 & 0.898 & 0.8 & 0.716 & 0.37 & 0.397 & 0.274 \\ Oliveoil & 570 & 0.064 & 0.121 & 0.261 & 0.342 & 0.246 & 0.305 & 0.233 & 0.166 & 0.067 \\ SwedishLeaf & 128 & 0.025 & 0.015 & 0.174 & 0.142 & 0.169 & 0.14 & 0.201 & 0.441 & 0.171 \\ Synthetic control & 60 & 0.092 & 0.073 & 0.2 & 0.158 & 0.182 & 0.148 & 0.12 & 0.02 & 0.02 \\ \hline Average & & 0.057 & 0.112 & 0.403 & 0.417 & 0.372 & 0.361 & 0.255 & 0.259 & 0.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Information loss and efficiency of information embedding in SAX and SAX-TM. **Author Contributions:** Conceptualization, K.S. and K.L. ; Investigation, K.S. ; Methodology, K.S. and K.L. ; Software, K.S. ; resources, K.S. and M.R. ; data curation, K.S.