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Efficient Multiple Temporal Network Kernel Density Estimation

Yu Shao, Peng Cheng, Xiang Lian, Lei Chen, Wangze Ni, Xuemin Lin, Chen Zhang, Liping Wang

TL;DR

This work tackles efficient temporal-network kernel density estimation on road graphs by introducing Range Forest Solution (RFS) and its streaming extension Dynamic Range Forest Solution (DRFS), together with Lixel Sharing to exploit redundancy across neighboring lixels. The approach uses per-edge range forests to maintain spatiotemporal aggregations and supports non-polynomial kernels via a Q–A decomposition, enabling exact KDE values with reduced memory and fast queries. Empirical results show RFS yielding up to 6× speedups over ADA and 88.9× over SPS, while DRFS achieves over 99.9% accuracy with tunable depth and quantization for significant time/memory savings. The framework offers flexible kernel choices (including Exponential and Cosine) and strong applicability to real-world traffic, policing, and mobility heatmaps, providing a scalable solution for online and batch TN-KDE tasks.

Abstract

Kernel density estimation (KDE) has become a popular method for visual analysis in various fields, such as financial risk forecasting, crime clustering, and traffic monitoring. KDE can identify high-density areas from discrete datasets. However, most existing works only consider planar distance and spatial data. In this paper, we introduce a new model, called TN-KDE, that applies KDE-based techniques to road networks with temporal data. Specifically, we introduce a novel solution, Range Forest Solution (RFS), which can efficiently compute KDE values on spatiotemporal road networks. To support the insertion operation, we present a dynamic version, called Dynamic Range Forest Solution (DRFS). We also propose an optimization called Lixel Sharing (LS) to share similar KDE values between two adjacent lixels. Furthermore, our solutions support many non-polynomial kernel functions and still report exact values. Experimental results show that our solutions achieve up to 6 times faster than the state-of-the-art method.

Efficient Multiple Temporal Network Kernel Density Estimation

TL;DR

This work tackles efficient temporal-network kernel density estimation on road graphs by introducing Range Forest Solution (RFS) and its streaming extension Dynamic Range Forest Solution (DRFS), together with Lixel Sharing to exploit redundancy across neighboring lixels. The approach uses per-edge range forests to maintain spatiotemporal aggregations and supports non-polynomial kernels via a Q–A decomposition, enabling exact KDE values with reduced memory and fast queries. Empirical results show RFS yielding up to 6× speedups over ADA and 88.9× over SPS, while DRFS achieves over 99.9% accuracy with tunable depth and quantization for significant time/memory savings. The framework offers flexible kernel choices (including Exponential and Cosine) and strong applicability to real-world traffic, policing, and mobility heatmaps, providing a scalable solution for online and batch TN-KDE tasks.

Abstract

Kernel density estimation (KDE) has become a popular method for visual analysis in various fields, such as financial risk forecasting, crime clustering, and traffic monitoring. KDE can identify high-density areas from discrete datasets. However, most existing works only consider planar distance and spatial data. In this paper, we introduce a new model, called TN-KDE, that applies KDE-based techniques to road networks with temporal data. Specifically, we introduce a novel solution, Range Forest Solution (RFS), which can efficiently compute KDE values on spatiotemporal road networks. To support the insertion operation, we present a dynamic version, called Dynamic Range Forest Solution (DRFS). We also propose an optimization called Lixel Sharing (LS) to share similar KDE values between two adjacent lixels. Furthermore, our solutions support many non-polynomial kernel functions and still report exact values. Experimental results show that our solutions achieve up to 6 times faster than the state-of-the-art method.
Paper Structure (28 sections, 7 theorems, 22 equations, 22 figures, 3 tables, 5 algorithms)

This paper contains 28 sections, 7 theorems, 22 equations, 22 figures, 3 tables, 5 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Related Work
  3. Preliminaries
  4. Problem Definition
  5. The State-of-the-Art Solutions
  6. Framework
  7. Efficient Aggregation Method
  8. Range Forest Solution
  9. Query on the Range Forest
  10. Construction of the Range Forest
  11. Streaming Update
  12. Dynamic Range Forest Solution
  13. Quantization
  14. Lixels Sharing
  15. Domination Determination
  16. ...and 13 more sections

Key Result

lemma 1

The time complexity of the ADA method is $O(\vert E \vert \cdot T_{sp} + L \cdot \vert E \vert \cdot \log(\frac{N}{\vert E \vert}))$chan_fast_2021.

Figures (22)

  • Figure 1: Mobility Heatmaps in San Francisco Area
  • Figure 2: An example of over-estimation bandwidth.
  • Figure 3: An example of a query $q$ (denoted by black square) and an event $o_i$ (denoted by black dot).
  • Figure 4: The illustration of the ADA method.
  • Figure 5: A range forest with four event points. The insertion order is $\{o_1, o_3, o_4, o_2\}$, forming four range trees $T_1,T_2,T_3,T_4$.
  • ...and 17 more figures

Theorems & Definitions (11)

  • definition 1: Road Network
  • definition 2: Lixel
  • definition 3: Event
  • definition 4
  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • lemma 6
  • ...and 1 more