有根譜系樹中的無根夏普利值

Unrooted Shapley Value of Rooted Phylogenetic Trees

高毓翔(Yu-Hsiang Kao)

指導教授 ： 符麥克

國立交通大學/理學院/應用數學系暨研究所/碩士(2017年)

摘要

夏普利值 ( Shapley value)在1953年被Llody Shapley提出，在賽局理論 (Cooperative game theory)中是一個相當重要的參數，它在許多科學領域都被應用。近幾年來它被建議成一個優先的工具來估算譜系樹中的類群的重要性。
在此論文中，我們將會討論譜系樹中三種不同的夏普利值：有根夏普利值，變形的夏普利值和無根夏普利值。前面兩個已經被討論過在Jin and Fuchs (2014) 然後我們會複習一下一些已知結果。特別是他們已經證明變形的有根夏普利值跟fair proportion index有高度的相關性，fair proportion index 在演化生物學中是另一個經常使用的工具。在 Hartmann (2013)中，他們也提出了一些數據證明了他們的理論。
最近發現Hartmann的實驗數據中用的是無根夏普利值而不是變形的有根夏普利值。因此我們論文主要的目標是用數值分析還有理論的證明去討論無根夏普利值和fair proportion index的相關性。

關鍵字

譜系樹 ； 夏普利值

並列摘要

The Shapley value, proposed by Lloyd Shapley in 1953, is an important parameter in cooperative game theory and is widely used in several areas of science. Recently, it was suggested as a prioritization tool for taxa in phylogenetic trees.
In this thesis, we will discuss the three main notions of Shapley values in phylogenetic trees: the rooted Shapley value, the modified rooted Shapley value and the unrooted Shapley value. The first two were discussed in a recent work of Jin and Fuchs (2014) and we will review their results. In particular, they proved that the modified rooted Shapley value is strongly related to the fair proportion index, another commonly used prioritization tool in biodiversity. Moreover, they claimed that this gives a theoretical justification of data presented by Hartmann (2013).
Recently, it was pointed out that Hartmann actually used the unrooted Shapley value instead of the modified rooted Shapley value in his data. Thus, one of the main goal of this thesis is to study numerically and theoretically the relationship between unrooted Shapley value and fair proportion index. Our results give strong support to the common practice in biodiversity of replacing the unrooted Shapley value by the fair proportion index.

並列關鍵字

Phylogenetic trees ； Shapley value ； Fair proportion index

參考文獻

[1] Claus-Jochen Haake, Akemi Kashiwada, and Francis Edward Su. "The Shapley value of phylogenetic trees." Journal of Mathematical Biology 56.4 (2008): 479–497.

[2] David Aldous. "Probability distributions on cladograms." In: Random Discrete Structures, Springer New York, (1996): 118.

[3] Michael Fuchs and Emma Yu Jin. "Equality of Shapley value and fair proportion index in phylogenetic trees." Journal of Mathematical Biology 71.5 (2015): 1133–1147.

[4] Klaas Hartmann. "The equivalence of two phylogenetic biodiversity measures: the Shapley value and fair proportion index." Journal of Mathematical Biology 67.5 (2013): 1163–1170.

[5] Daniel P. Faith "Conservation evaluation and phylogenetic diversity." Biological Conservation 61.1 (1992): 1–10.