針對變量核心密度估計之帶寬設定進行最佳化之研究

A Study on Optimal Bandwidth Settings for Adaptive Kernel Density Estimation

劉柔均(Rou-Jun Liu)

指導教授 ： 歐陽彥正

國立臺灣大學/共同教育中心/統計碩士學位學程/碩士(2022年)

https://doi.org/10.6342/NTU202202644

摘要

核心密度估計方法(Kernel Density Estimation)是常見的無母數統計方法之一，相較於有母數統計，其不需要預先知道資料的分布假設，因此使用上具有較高的彈性。在核密度分析中，帶寬的選擇是影響結果的重要因素，因此如何挑選合適的帶寬成為重要的討論議題。
本研究改進 RVKDE 方法，使用最大概似估計法(Maximum Likelihood Estimation)來找出用來移動至經驗法則的最佳倍數，在多種人工合成資料集上實驗，將我們的方法與經驗法則、 Scott’s 法則、 Abramson提出的方法以及 ArcGIS 之方法進行比較，結果顯示雖然使用MLE 無法每一次都完全精準地找出真正的最優倍數，但距離正確的倍數亦不遠，且積分均方誤差(Mean Integrated Square Error)表現明顯優於其他常用的方法，估計結果的準確度大幅上升。

關鍵字

核密度估計 ； 帶寬選擇 ； 最大概似估計法

並列摘要

Kernel density estimation (KDE) is one of the most popular non-parametric methods to construct heatmap analysis. Due to the great influence on the result performance, the choice of bandwidth has become an important issue to discuss. This study improves Relaxed Variable Kernel Density Estimation (RVKDE) by implementing maximum likelihood estimation (MLE) to select the optimal multiple value of adjusted normal reference rule when shifting the median of bandwidth. We compare the performance of our method with the normal reference rule, Scott’s rule, the method proposed by Abramson, the method applied in ArcGIS, the original RVKDE method, and the improved ORAKDE method in 2-D and 3-D experiments. Although sometimes using MLE cannot precisely find out the exact optimal multiple number, it would not be far from the correct one and objectively provides an instructive suggestion of deciding the value. The measurement criteria Mean Integrated Square Error (MISE) of our method significantly outperforms than other methods, typically for the data whose patterns obviously differ from normal distribution.

並列關鍵字

Kernel Density Estimation ； Bandwidth Selection ； Maximum Likelihood Estimation

參考文獻

[1] Ross, S. M. (2014). Introduction to probability models. Academic press.

[2] Chainey, S., Tompson, L., Uhlig, S. (2008). The utility of hotspot mapping for predicting spatial patterns of crime. Security journal, 21(1), 4-28.

[3] Cleasby, I. R., Owen, E., Wilson, L., Wakefield, E. D., O'Connell, P., Bolton, M. (2020). Identifying important at-sea areas for seabirds using species distribution models and hotspot mapping. Biological Conservation, 241, 108375.

[4] Ngada, T., Bowers, K. (2018). Spatial and temporal analysis of crude oil theft in the Niger Delta. Security Journal, 31(2), 501-523.

[5] Guevara, C., Peñas, M. S. (2020). Surveillance Routing of COVID-19 Infection Spread Using an Intelligent Infectious Diseases Algorithm. IEEE Access, 8, 201925-201936.