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題 名 | 平滑曲線逼近法修正Hellinger距離指標=Spline Interpolation to Modifying Hellinger Distance Index |
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作 者 | 潘宏裕; 林政瑋; | 書刊名 | 中國統計學報 |
卷 期 | 51:1 2013.03[民102.03] |
頁 次 | 頁74-97 |
分類號 | 360.13 |
關鍵詞 | 相似性指標; 距離指標; Hellinger距離指標; 平滑曲線逼近法; Similarity index; Distance index; Hellinger distance; Spline interpolation; |
語 文 | 中文(Chinese) |
中文摘要 | 在環境生態學中,生態學家常會利用相似性指標(S)來探討兩個群落中的物種分布結構之相似程度,而在機率統計方面也常用距離指標(D)來量化兩個機率分自己的相似程度。然而距離指標是用來衡量兩個機率分配的相異程度,其與相似性指標常呈某種關係,如D=1-S和D=√(1-S)等。機率論中,f-divergence是用來測量兩個機率分配差異的函數,其相關的距離指標有Kullback-Leibler (dKL)、Hellinger(dH)和χ^2-divergence等,而且f-divergence指標具有非負(non-negativity)、單調(monotonicity)和凸性(convexity)等好性質。本研究探討f-divergence指標中的Hellinger距離指標,其除擁有上述的好性質外,亦可以用歐幾里得範數(Euclidean norm)來表示;此外它與Bhattacharyya coefficient (dBC)的關係為dH=√2(1-dBC),也可表示為Kullback-Leibler的下界函數(dKL≥2dH ^2)。本研究利用動差法求出Hellinger距離指標的估計量,然而此估計量有高估的情形,因僅考慮樣本中出現的共同種,並末將樣本中未出現的共同種納入;利用泰勒展開式來修正其偏差,亦無法獲得好的改善。因此本研究使用平滑曲線逼近法進行修正,且利用矩陣條件數評估所採用逼近法的優劣。最後,以嘉義縣好美里海岸線水生生物的調查資料來說明本研究所提之指標的可行性。 |
英文摘要 | In environmental ecology, ecologists often use similarity indices to explore the similarity of species structures between two difference communities. In probability and statistics, distance indices are also used to quantify the dis-similarity between two probability distributions. Some relationships of distance index with similarity index are D=1-Sand D=√(1-S). In probability, f-divergence is a function that measures the difference between two probability distributions, for example Kullback-Leibler (dKL), Hellinger (dH) and χ^2-divergence etc. These indictors have some good properties, such as non-negativity, monotonicity, and convexity. In this study, we discuss the index of Hellinger distance, which is directly related to the Euclidean norm of the difference. Moreover, the relationship of Hellinger distance and Bhattacharyya coefficient (dBC) can be expressed as dH=√2(1-dBC), and Hellinger distance is also the lower bound function of Kullback-Leibler distance (dKL≥2d^2H). We discuss the index of Hellinger distance, and derived it's estimator by the method of moments (MME). The MME is overestimated due to only consider the observed shared species in the sample. Using the Taylor expansion to adjust the bias, but it does not work. So, we correct the MME by the spline interpolation and use condition number of matrix to judge the approach. Finally, we use the survey data of the marine organism in Hao-Mei Village, Chiayi County, Taiwan to illustrate the feasibility of the estimator. |
本系統中英文摘要資訊取自各篇刊載內容。