

![0_image_0.png](0_image_0.png)

words, a good clustered dataset is the dataset that differentcluster are well separated, and the same-cluster data are well grouped together. The Silhouette index for each data i is defined by the following equation:

$$s(i)={\frac{b(i)-a(i)}{\operatorname*{max}\left\{b(i),a(i)\right\}}}$$
max {b(i), a(i)}(3)

![0_image_2.png](0_image_2.png)

$$b(i)=\operatorname*{min}_{c\in C\wedge c\neq l a b e l(i)}\left({\frac{1}{N_{D_{c}}}}\sum_{j\in D_{c}}d(i,j)\right)$$

![0_image_3.png](0_image_3.png)

$$\left(4\right)$$
$$a(i)={\frac{1}{N_{D_{I a b e l(i)}}}}\sum_{j\in D_{I a b e l(i)}\wedge i\neq j}d(i,j)$$

where s(i) is the Silhouette index of i th data, b(i) is the minimum average distance between the i th data and each of the different-cluster data, and a(i) is the average distance between the i th data and each of the same-cluster data. In Equations (4) and (5), C is a set of all possible clusters, Dc is a set of data in cluster c, NDcis the size of Dc, and d(**i, j**) is the distance measure function comparing between i th and j th data. Note that the s(i) ranges between -1 and 1. Having s(i) close to 1 means that data is well separated.

Global Silhouette index (GS) for a dataset is calculated as follows.

$$G S={\frac{1}{c}}{\sum_{j=1}^{c}}S$$
$$\mathbf{\tau}_{j}$$
Sj (6)
$$S_{c}={\frac{1}{M}}\sum_{i=1}^{M}s(i)$$

s(i) (7)

![0_image_1.png](0_image_1.png)

## 3. Methodology

In this section, we describe our approach, developed from the techniques described in Section 2, i.e., the DTW distance, the best warping window for Sakoe-Chiba band, multiple RK bands, and the Silhouette index. In brief, our approach consists of 5 major parts: 1) data preprocessing that reduces the length of time series data, 2) our proposed band boundary extraction algorithm, 3) finding the best warping window for Sakoe-Chiba band, 4) our proposed iterative R-K band learning, and 5) prediction for unlabeled data.

$\left(3\right)$. 
$$\left({\mathrm{5}}\right)$$
Our approach requires three input parameters, i.e., a set of training data T, a set of unlabeled data (test data) P, the maximum complexity that depends on time and computational resources, and the bound of a warping window size.

In data preprocessing step, we could reduce the computational complexity in case of very long time series data using interpolation function, both in training and test data. After that, we try to find the best R-K band by running the band boundary extraction algorithm. The best warping window is calculated and is used as an initial band of our proposed iterative learning. After learning have finished, two R-K bands are compared and the better one is selected. Finally, we calculate a training accuracy and make predictions for the test data using 1-NN with the DTW distance and the best band, enhanced with LB Keogh lower bound to speed up our classification approach. The prediction result A along with the training accuracy are returned as shown in Table 4.

$$({\mathfrak{G}})$$
$$\left(7\right)$$

## 3.1 Data Preprocessing

Since the classification prediction time may because a major constraint, a data preprocessing step is needed. In this step, we approximate the calculation complexity and try to reduce the complexity exceeding the threshold. Our approximated complexity is mainly based on the number of items in the training data, its length, and the number of heuristic function evaluations. Suppose we have n training data with