本研究是針對出現與否地圖去檢驗物種的相關性，首先透過核估計法去估計物種在各個格子中的發生機率，再代入肯德爾等級相關係數檢定來檢驗物種的相關性。此外，我們還比較了二項分配方法、Veech 方法、t 檢定方法，前兩種方法可以直接應用於出現與否地圖，而後一種方法也需先透過核估計來估計物種在格子的發生率，再進行檢定。在模擬研究中，我們比較了這四種方法在不同條件下的表現。當其中一物種的發生率為 10%，且兩物種的發生率差異超過 40% 時，我們方法的左尾檢定之型一誤差會較接近的顯著水準，並且比其他方法有更高檢定力。最後，本研究的實例是使用宜蘭福山植物園的資料，將原資料轉為出現與否資料後去檢測物種共現性，而結果顯示在不同的樹種之間是存在顯著的正相關或負相關。

In this research, we present a novel approach for examining species co-occurrence
in presence-absence maps. Our methodology involves estimating the probability of
species presence in each grid through kernel estimation, followed by evaluating species co-occurrence using Kendall’s rank correlation coefficient. Furthermore, we conduct comparative analyses with established methods including the binomial distribution method, Veech's method, and t-test method. While the former two can be directly applied to presence-absence data, the latter requires prior estimation of species presence rates using kernel estimation. In our simulation study, we assess the performance of these four methods across various conditions. Overall, our approach exhibits superior performance in left-tailed testing, especially when a species’ presence rate is below 10% and the other species’ presence rate exceeds it by 40%. In such scenarios, our method yields a Type I error rate closer to the significance level, accompanied by higher statistical power compared to alternative methods. Finally, we apply our methodology to real forestry data from Fushan Botanical Garden. The findings reveal significant positive or negative associations among different tree species.

目錄
圖目錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II
表目錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
第一章 緒論 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
第二章 研究方法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
第一節 機率方法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
第一小節 二項分配方法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
第二小節 Veech 方法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
第二節 假設檢定方法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
第一小節 皮爾森相關係數 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
第二小節 肯德爾等級相關係數 . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
第三章 模擬研究. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
第一小節 隨機相關 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
第二小節 右尾檢定 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
第三小節 左尾檢定 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
第四章 實例分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
第五章 結論 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
參考文獻 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
圖目錄
2.1 A 與 B 物種出現與否資料示意圖 . . . . . . . . . . . . . . . . . . . . . 5
2.2 和諧與不和諧配對圖 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 在同一強度函數 λ(u1,u2) ∝ 1 下所生成的出現與否地圖 . . . . . . . . 16
3.2 在同一強度函數 λ(u1,u2) ∝ u1 下所生成的出現與否地圖 . . . . . . . 16
3.3 在同一強度函數 λ(u1,u2) ∝ u1 + u2 下所生成的出現與否地圖 . . . . 17
3.4 在同一強度函數 λ(u1,u2) ∝ 21 − u1 下所生成的出現與否地圖 . . . . 17
3.5 在同一強度函數 λ(u1,u2) ∝ 41 − u1 − u2 下所生成的出現與否地圖 . 17
4.1 宜蘭福山植物園的等高線圖 (藍色的線條代表溪流) . . . . . . . . . . . 29
4.2 福山植物園中八種樹種之點分布位置 . . . . . . . . . . . . . . . . . . . 30
4.3 福山植物園中八種樹種之出現與否地圖 . . . . . . . . . . . . . . . . . . 30
表目錄
2.1 兩物種出現與否所對應之格點個數觀察值 . . . . . . . . . . . . . . . . 4
2.2 兩物種出現與否所對應之格點個數觀察值與期望值 . . . . . . . . . . . 5
2.3 肯德爾數值運算例子 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 雙尾檢定之型一誤差 (粗體為最接近顯著水準 0.05) . . . . . . . . . . . 19
3.2 相關性檢驗 (雙尾檢定) . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 右尾檢定之型一誤差 (粗體為最接近顯著水準 0.05) . . . . . . . . . . . 21
3.4 右尾檢定之檢定力 (粗體為最高的檢定力) . . . . . . . . . . . . . . . . 22
3.5 正相關檢定 (右尾檢定) . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6 左尾檢定之型一誤差 (粗體為最接近顯著水準 0.05) . . . . . . . . . . . 24
3.7 左尾檢定之檢定力 (粗體為最高的檢定力) . . . . . . . . . . . . . . . . 25
3.8 負相關檢定 (左尾檢定) . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1 福山植物園中物種實際個數、存在格數和物種出現率 . . . . . . . . . . 28
4.2 八種物種間雙尾檢定 P 值 (括號內的數值代表發生率) . . . . . . . . . 31
4.3 八種物種間單尾檢定之 P 值 . . . . . . . . . . . . . . . . . . . . . . . . 33

[1] Agresti, A. (1984). Analysis of ordinal categorical data. Wiley series in probability and mathematical statistics Show all parts in this series.
[2] Arita, H. T., Christen, A., Rodríguez, P., and Soberón, J. (2012). The presence–
absence matrix reloaded: the use and interpretation of range–diversity plots. Global
Ecology and Biogeography, 21(2):282–292.
[3] Blanchet, F. G., Cazelles, K., and Gravel, D. (2020). Co-occurrence is not evidence of ecological interactions. Ecology Letters, 23(7):1050–1063.
[4] Carstensen, D. W. and Olesen, J. M. (2009). Wallacea and its nectarivorous birds:
nestedness and modules. Journal of Biogeography, 36(8):1540–1550.
[5] Cazelles, K., Araújo, M. B., Mouquet, N., and Gravel, D. (2016). A theory for
species co-occurrence in interaction networks. Theoretical Ecology, 9:39–48.
[6] Chang, Y., Rakshit, S., Huang, C., and Wu, W. (2023). Probabilistic approaches
for investigating species co-occurrence from presence-absence maps. Peerj, 11.
[7] Dormann, C. F., Bobrowski, M., Dehling, D. M., Harris, D. J., Hartig, F., Lischke, H., Moretti, M. D., Pagel, J., Pinkert, S., Schleuning, M., et al. (2018). Biotic interactions in species distribution modelling: 10 questions to guide interpretation and avoid false conclusions. Global Ecology and Biogeography, 27(9):1004–1016.
[8] Gotelli, N. J. (2000). Null model analysis of species co-occurrence patterns. Ecology, 81(9):2606–2621.
[9] Gotelli, N. J. and Ulrich, W. (2010). The empirical bayes approach as a tool to
identify non-random species associations. Oecologia, 162:463–477.
[10] Griffith, D. M., Veech, J. A., and Marsh, C. J. (2016). cooccur: Probabilistic
species co-occurrence analysis in Ｒ. Journal of Statistical Software, 69(c02).
[11] Kendall, M. G. (1938). A new measure of rank correlation. Biometrika, 30(1/2):81–93.
[12] Patterson, B. D. and Atmar, W. (1986). Nested subsets and the structure of insular mammalian faunas and archipelagos. Biological Journal of the Linnean Society, 28(1-2):65–82.
[13] Pearson, K. (1895). Vii. note on regression and inheritance in the case of two
parents. Proceedings of the Royal Society of London, 58(347-352):240–242.
[14] Pitta, E., Giokas, S., and Sfenthourakis, S. (2012). Significant pairwise co-occurrence patterns are not the rule in the majority of biotic communities. Diversity, 4(2):179–193.
[15] Sanderson, J. G. (2000). Testing ecological patterns. American Scientist, 88(4):332.
[16] Sanderson, J. G., Diamond, J. M., and Pimm, S. L. (2009). Pairwise co-existence
of bismarck and solomon landbird species. Evolutionary Ecology Research, 11(5):771–
786.
[17] Sfenthourakis, S., Giokas, S., and Tzanatos, E. (2004). From sampling stations to archipelagos: investigating aspects of the assemblage of insular biota. Global Ecology and Biogeography, 13(1):23–35.
[18] Sfenthourakis, S., Tzanatos, E., and Giokas, S. (2006). Species co-occurrence: the case of congeneric species and a causal approach to patterns of species association. Global Ecology and Biogeography, 15(1):39–49.
[19] Su, S.-H. (2009). The complete datasets of the fushan forest dynamics plot: the
first and second tree censuses and the topographic survey.
[20] Su, S.-H., Chang-Yang, C., Lu, C., Tsui, C., Lin, T., Lin, C., Chiou, W., Kuan, L., Chen, Z., and Hsieh, C. (2007). Fushan subtropical forest dynamics plot: tree species
characteristics and distribution patterns. Taiwan Forestry Research Institude.
[21] Ulrich, W., Almeida-Neto, M., and Gotelli, N. J. (2009). A consumer’s guide to
nestedness analysis. Oikos, 118(1):3–17.
[22] Ulrich, W. and Gotelli, N. J. (2013). Pattern detection in null model analysis.
Oikos, 122(1):2–18.
[23] Veech, J. A. (2006). A probability-based analysis of temporal and spatial co-occurrence in grassland birds. Journal of Biogeography, 33(12):2145–2153.
[24] Veech, J. A. (2013). A probabilistic model for analysing species co-occurrence.
Global Ecology and Biogeography, 22(2):252–260.
[25] Veech, J. A. (2014). The pairwise approach to analysing species co-occurrence.
Journal of Biogeography, 6(41):1029–1035.
[26] Wright, D. H. and Reeves, J. H. (1992). On the meaning and measurement of nestedness of species assemblages. Oecologia, 92:416–428.