Cyclic Weighted k-means Method with Application to Time-of-Day Interval Partition
Abstract
:1. Introduction
2. Problem Description
3. Methodology
3.1. Cyclic Distance
3.2. Cyclic Weighted k-means Method
Algorithm 1. The cyclic weighted k-means algorithm | |
Require: iterations , number of sampling units , cluster number . | |
Ensure: Determine the centroids and breakpoints in i-th iteration. | |
1: | Initialize the iterative number and centroids (see Algorithm 2). |
2: | for to do |
3: | for to do |
4: | Class label of t-th timescale in i-th iteration where , i.e., assign a centroid to timescale in i-th iteration. |
5: | End for |
6: | for to do |
7: | Update the centroids: . |
8: | End for |
9: | Ifthen |
10: | Output centroids and class labels . |
11: | Obtain subscripts which satisfy (let , ). |
12: | Obtain the corresponding breakpoints . |
13: | Ranking from small to large, the final breakpoints are obtained. |
14: | Break |
15: | End if |
16: | End for |
3.3. Initialization of Centroids
Algorithm 2. Initialization of the centroids under given | |
Require: Number of sampling units , cluster number , traffic flow series . | |
Ensure: Obtain the initialized centroids . | |
1: | Initialize the set of centroids to empty set, i.e., |
2: | Find the maximum from , whose corresponding timescale is . |
3: | Add into , i.e., . |
4: | for to do |
5: | Find -th timescale outside cyclic neighborhood of all elements in , satisfying , where . |
6: | Add into , i.e., . |
7: | End for |
8: | Sort the elements of set in ascending order and obtain the initialized centroids . |
3.4. Determination of K
3.5. Adjustment of Breakpoints
4. Case Study
4.1. Empirical Data
4.2. Results of Clustering
4.3. Evaluation of Methods
5. Conclusions
- i.
- The cyclic distance is the key for the cyclic weighted k-means algorithm, which makes it possible that the end point of the previous cycle and the start point of the current cycle are connected in the clustering result, and a complete cycle of data has been considered rather than separation from tail to head.
- ii.
- Some attached algorithms, i.e., centroid initialization, cluster number selection, and breakpoint adjustment, are helpful for further improvement of the cyclic weighted k-means algorithm to solve the TIP problem.
- iii.
- The feasibility of the proposed method is confirmed by empirical study. It is noted that the practical evaluation criteria (such as the average vehicle delay in benefits of traffic signal control) should serve the practice. From the perspective of application, the proposed method can also be applied to other scenes. For example, it can be applied to the inventory adjustment of e-commerce according to the daily sales, and the seat optimization of a call center according to the volume of calls.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
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Wang, G.; Qin, W.; Wang, Y. Cyclic Weighted k-means Method with Application to Time-of-Day Interval Partition. Sustainability 2021, 13, 4796. https://doi.org/10.3390/su13094796
Wang G, Qin W, Wang Y. Cyclic Weighted k-means Method with Application to Time-of-Day Interval Partition. Sustainability. 2021; 13(9):4796. https://doi.org/10.3390/su13094796
Chicago/Turabian StyleWang, Gaizhen, Wei Qin, and Yunhao Wang. 2021. "Cyclic Weighted k-means Method with Application to Time-of-Day Interval Partition" Sustainability 13, no. 9: 4796. https://doi.org/10.3390/su13094796