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Estimating a Series of Lower Bounds of Species Richness via a Generalized Cauchy-Schwarz Inequality
近年來一些生態調查資料的普及，因此文獻上的種類數估計量可利用這樣的資料來評估適用性，發現大多數的估計方法似乎在抽樣比例低時會有低估的情形，但是實務上低抽樣比例似乎是無可避免的，因為大多數的田野調查（field study）實際抽樣比例都不會超過1%，因此一直有新的估計方法被提出。最近Mao (2008) 利用廣義的柯西不等式 (a generalization of the Cauchy-Schwarz inequality) 推廣 Chao 1 與 Chao 2 這兩個估計式，Mao的估計式可以使用更多的頻率次數 （the frequencies of frequencies）來估計樣本中沒出現之物種數，而且利用矩陣來表示估計式仍然保有簡單的形式。但是當使用頻率次數較多時，矩陣也隨之變大，數值上的計算就會遇到問題，甚至無法計算。由於Mao (2008) 利用廣義的柯西不等式所提出的物種數估計擁有很好的理論基礎，針對上述該估計式所遇到的問題，本計畫書提出一個可行的方法來解決，即是考慮對tkk,,2,1 ,配適一統計模型，因此預期可以穩定tkk,,2,1 , 的估計表現，進而克服計算上的問題。本計畫利用電腦模擬比較現有的一些估計方法，以及新提出之方法。抽樣母體是三筆真實的森林數據，全部做過普查，因此相當珍貴，再者，它們是真實資料，因此比較能反映出母體的真實情況。一筆來自巴拿馬 (Panama) 的動態樣區, 另兩筆則來自馬來西亞(Malaysia)。由於是普查資料，因此可以任意切割抽樣區塊的大小，因此也可以研究區塊大小的選定如何影響估計方法的表現，進而提出一抽樣的準則。
Species richness (the number of species in an assemblage) estimation is recognized as a classic, basic and important issue with respect to biologists/ecologists in the literature. However, most existing methods result in considerable negative bias in applications, where field surveys typically represent only a small fraction of sampled communities. As a result, recently there are still several estimators put forward in the literature.Lately Mao (2008) proposed a series of lower bound estimators regarding species richness by means of a generalization of the Cauchy-Schwarz inequality. Two widely used estimators Chao 1 (Chao, 1987) and Chao 2 (Chao, 1987), which only use numbers of singletons and doubletons to estimate number of unseen species in the sample, can be viewed as a special case of Mao's estimators. While Mao's (2008) estimators take more species counts into account than Chao 1 and Chao 2 estimators, due to considerable statistical fluctuations in species counts some numerical difficulties may arise.In the proposal, we suggest fitting a statistical model on expected species counts (i 's) versus . Based on the fitted model, the induced surrogates iiˆ's will be more stable than observed species counts (the method of moment). It can be expected that substituting the surrogates iˆ's from the fitted model into the generalization of the Cauchy-Schwarz inequality will overcome numerical difficulties in Mao's (2008) formula.A simulation study would then be carried out to compare the performances of the conventional estimators as well as the developed one. In reality, real datasets should be more realistic than artificial ones. Thus three forest census datasets are employed to be sampling populations varying with quadart divisions. The data sets are briefly described as follows. The Barro Colorado Island (BCI) plot: 50 ha (1000500 m), locating in Barro Colorado Island, Panama. Six censuses have so far been surveyed. The second dataset is the Pasoh plot located in Pasoh Forest Reserve, Malaysia. The census carried out during 1985 to 1987 was used in this study. The other one is the Lambir plot with 52 ha (1040500 m) located in Lambir Hills National Park, Malaysia. This plot was initiated in 1991.
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