# Transitional SAX Representation for Knowledge Discovery for Time Series
## 4 Experiments
### Information Analysis  
Next, we evaluate the performance of SAX-TM from the viewpoint of information. SAX has been regarded as a de facto standard to reduce the dimensionality of time series data. Despite its popularity and universality, the structural properties of SAX from the information viewpoint have been rarely researched to the best of our knowledge. Among the statistical facets, [PERSON] et al. proposed in the investigation of time-series dimensionality reduction [20] that we focus on information loss and efficiency of information embedding. Both minimizing the loss of useful information and preserving useful information in a raw time series are practical goals. Thus, we adopted procedures to discover intrinsic properties of the proposed method from the perspective of information loss and information embedding. Figure 2: Comparison of error rates in 1 NN classification between the existing methods and the proposed algorithm. For this purpose, we calculated the information loss, denoted by \(L_{info}\), by mean squared error (MSE) between a raw signal and reconstructed symbolic words:  
\[L_{info}(\hat{T},T)=\frac{\sum(\tilde{t}_{i}-t_{i})^{2}}{n-1},\;\tilde{t}_{i}\in \hat{T},\;t_{i}\in T, \tag{8}\]  
in which \(T\) is raw signal and \(\hat{T}\) is a reconstructed one. To conduct the comparison, we scaled raw time series and SAX words to \([0,1]\). We also calculated the Kullback-Leibler (KL) divergence, which is a non-symmetric similarity measure between two different probability distributions. For distributions \(P\) and \(Q\) with k points, the KL divergence is defined as follows:  
Figure 3: Parameters among SAX, SAX-SD, SAX-TD, and SAX-TM. \[KL(P\parallel Q)=\sum_{i=1}^{k}p_{i}\log\frac{p_{i}}{q_{i}},\ p_{i}\in P,\ q_{i}\in Q \tag{9}\]  
In our experiments, we take \(P\) as the distribution of the original signal \(T\) and \(Q\) as that of the reconstructed signal \(\hat{T}\) by a histogram with \(\alpha\) as the number of bins. Information loss measures the amount of information abandoned when converting the original time series to a symbolic representation. KL-divergence represents the closeness between the distribution of a raw signal and that of a reconstructed signal. To combine the two measures, [PERSON] et al. defined information embedding cost (IEC) as a ratio of KL-divergence and information loss as follows:  
\[IEC_{T}(P,Q)=\frac{KL_{T}(P\parallel Q)}{1+L_{info}(\hat{T},T)}. \tag{10}\]  
Given a time series \(T\) in distribution \(P\) and reconstructed signal \(\hat{T}\) with distribution \(Q\), the IEC score describes the number of extra bits needed to transform the output \(\hat{T}\) when information loss incurs by one unit, revealing how much useful information is abandoned when transforming a raw \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **ED** & **SAX** & **SAX** & **SAX** & **SAX** & **SAX** & **SAX** & **SAX** & **SAX** & **SAX** & **SAX** \\  & **Error** & **Error** & **\(n\)** & **\(\alpha\)** & **-TD** & **-TD** & **-TD** & **-SD** & **-SD** & **-TD** & **-TM** & **-TM** & **-TM** \\  & & & & & & **Error** & **\(n\)** & **\(\alpha\)** & **Error** & **\(n\)** & **\(\alpha\)** & **Error** & **\(n\)** & **\(\alpha\)** \\ \hline  
133 & 128 & 3 & **0.0670** & 128 & 5 \\ \hline average & 0.234 & 0.340 & & & 0.172 & & 0.154 & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: 1 NN classification error rates of ED (Euclidean distance); 1 NN best classification error rates, length \(n\), and dimensionality reduction ratio \(\alpha\) of the SAX, SAX-TD, SAX-SD, and SAX-TM on 20 datasets. The lowest error rates are highlighted in bold.