Input Data
- A numeric matrix with two samples and five species.
- A phylogenetic tree for those five species.
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The different UniFrac algorithms are listed below, along with examples for calculating them.
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The branch indices (green circles) are used for ordering the \(L\), \(A\), and \(B\) arrays. Values for \(L\) are drawn from the input phylogenetic tree. Values for \(A\) and \(B\) are the total number of species observations descending from that branch; \(A\) for Sample_A, and \(B\) for Sample_B.
| \(n = 8\) | Number of branches |
| \(A = \{9, 0, 0, 0, 9, 6, 3, 3\}\) | Branch weights for Sample_A. |
| \(B = \{7, 5, 1, 4, 2, 8, 8, 0\}\) | Branch weights for Sample_B. |
| \(A_T = 15\) | Total observations for Sample_A. |
| \(B_T = 15\) | Total observations for Sample_B. |
| \(L = \{0.2, 0.4, 0.8, 0.5, 0.9, 0.6, 0.7, 0.3\}\) | The branch lengths. |
unweighted_unifrac()unweighted_unifrac()UniFrac(weighted=FALSE)qiime diversity beta-phylogenetic --p-metric unweighted_unifracunifrac.unweighted()First, transform A and B into presence (1) and absence (0) indicators.
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\[\begin{align*} A &= \{9, 0, 0, 0, 9, 6, 3, 3\} \\ A' &= \{1, 0, 0, 0, 1, 1, 1, 1\} \end{align*}\] |
\[\begin{align*} B &= \{7, 5, 1, 4, 2, 8, 8, 0\} \\ B' &= \{1, 1, 1, 1, 1, 1, 1, 0\} \end{align*}\] |
Then apply the formula:
\[\begin{align*} U &= \displaystyle \frac{\sum_{i = 1}^{n} L_i(|A'_i - B'_i|)}{\sum_{i = 1}^{n} L_i(max(A'_i,B'_i))} \\ \\ U &= \displaystyle \frac{L_1(|A'_1-B'_1|) + L_2(|A'_2-B'_2|) + \cdots + L_n(|A'_n-B'_n|)}{L_1(max(A'_1,B'_1)) + L_2(max(A'_2,B'_2)) + \cdots + L_n(max(A'_n,B'_n))} \\ \\ U &= \displaystyle \frac{0.2(|1-1|) + 0.4(|0-1|) + \cdots + 0.3(|1-0|)}{0.2(max(1,1)) + 0.4(max(0,1)) + \cdots + 0.3(max(1,0))} \\ \\ U &= \displaystyle \frac{0.2(0) + 0.4(1) + 0.8(1) + 0.5(1) + 0.9(0) + 0.6(0) + 0.7(0) + 0.3(1)}{0.2(1) + 0.4(1) + 0.8(1) + 0.5(1) + 0.9(1) + 0.6(1) + 0.7(1) + 0.3(1)} \\ \\ U &= \displaystyle \frac{0.4 + 0.8 + 0.5 + 0.3}{0.2 + 0.4 + 0.8 + 0.5 + 0.9 + 0.6 + 0.7 + 0.3} \\ \\ U &= \displaystyle \frac{2}{4.4} \\ \\ U &= 0.4545455 \end{align*}\]
weighted_unifrac()weighted_unifrac()UniFrac(weighted=TRUE, normalized=FALSE)qiime diversity beta-phylogenetic --p-metric weighted_unifrac\[\begin{align*} W &= \sum_{i = 1}^{n} L_i|\frac{A_i}{A_T} - \frac{B_i}{B_T}| \\ \\ W &= L_1|\frac{A_1}{A_T} - \frac{B_1}{B_T}| + L_2|\frac{A_2}{A_T} - \frac{B_2}{B_T}| + \cdots + L_n|\frac{A_n}{A_T} - \frac{B_n}{B_T}| \\ \\ W &= 0.2|\frac{9}{15} - \frac{7}{15}| + 0.4|\frac{0}{15} - \frac{5}{15}| + \cdots + 0.3|\frac{3}{15} - \frac{0}{15}| \\ \\ W &= 0.02\overline{6} + 0.1\overline{3} + 0.05\overline{3} + 0.1\overline{3} + 0.42 + 0.08 + 0.2\overline{3} + 0.06 \\ \\ W &= 1.14 \end{align*}\]
weighted_normalized_unifrac()weighted_normalized_unifrac()UniFrac(weighted=TRUE, normalized=TRUE)qiime diversity beta-phylogenetic --p-metric weighted_normalized_unifracunifrac.weighted()\[\begin{align*} N &= \displaystyle \frac {\sum_{i = 1}^{n} L_i|\frac{A_i}{A_T} - \frac{B_i}{B_T}|} {\sum_{i = 1}^{n} L_i(\frac{A_i}{A_T} + \frac{B_i}{B_T})} \\ \\ N &= \displaystyle \frac {L_1|\frac{A_1}{A_T} - \frac{B_1}{B_T}| + L_2|\frac{A_2}{A_T} - \frac{B_2}{B_T}| + \cdots + L_n|\frac{A_n}{A_T} - \frac{B_n}{B_T}|} {L_1(\frac{A_1}{A_T} + \frac{B_1}{B_T}) + L_2(\frac{A_2}{A_T} + \frac{B_2}{B_T}) + \cdots + L_n(\frac{A_n}{A_T} + \frac{B_n}{B_T})} \\ \\ N &= \displaystyle \frac {0.2|\frac{9}{15} - \frac{7}{15}| + 0.4|\frac{0}{15} - \frac{5}{15}| + \cdots + 0.3|\frac{3}{15} - \frac{0}{15}|} {0.2(\frac{9}{15} + \frac{7}{15}) + 0.4(\frac{0}{15} + \frac{5}{15}) + \cdots + 0.3(\frac{3}{15} + \frac{0}{15})} \\ \\ N &= \displaystyle \frac {0.02\overline{6} + 0.1\overline{3} + 0.05\overline{3} + 0.1\overline{3} + 0.42 + 0.08 + 0.2\overline{3} + 0.06} {0.21\overline{3} + 0.1\overline{3} + 0.05\overline{3} + 0.1\overline{3} + 0.66 + 0.56 + 0.51\overline{3} + 0.06} \\ \\ N &= \displaystyle \frac{1.14}{2.326667} \\ \\ N &= 0.4899713 \end{align*}\]
generalized_unifrac(alpha = 0.5)generalized_unifrac(alpha = 0.5)GUniFrac(alpha = 0.5)qiime diversity beta-phylogenetic --p-metric generalized_unifrac -a 0.5\[\begin{align*} G &= \displaystyle \frac {\sum_{i = 1}^{n} L_i(\frac{A_i}{A_T} + \frac{B_i}{B_T})^{\alpha} |\displaystyle \frac {\frac{A_i}{A_T} - \frac{B_i}{B_T}} {\frac{A_i}{A_T} + \frac{B_i}{B_T}} |} {\sum_{i = 1}^{n} L_i(\frac{A_i}{A_T} + \frac{B_i}{B_T})^{\alpha}} \\ \\ G &= \displaystyle \frac { L_1(\frac{A_1}{A_T} + \frac{B_1}{B_T})^{0.5} |\displaystyle \frac {\frac{A_1}{A_T} - \frac{B_1}{B_T}} {\frac{A_1}{A_T} + \frac{B_1}{B_T}}| + \cdots + L_n(\frac{A_n}{A_T} + \frac{B_n}{B_T})^{0.5} |\displaystyle \frac {\frac{A_n}{A_T} - \frac{B_n}{B_T}} {\frac{A_n}{A_T} + \frac{B_n}{B_T}}| }{ L_1(\frac{A_1}{A_T} + \frac{B_1}{B_T})^{0.5} + \cdots + L_n(\frac{A_n}{A_T} + \frac{B_n}{B_T})^{0.5} } \\ \\ G &= \displaystyle \frac { 0.2(\frac{9}{15} + \frac{7}{15})^{0.5} |\displaystyle \frac {\frac{9}{15} - \frac{7}{15}} {\frac{9}{15} + \frac{7}{15}}| + \cdots + 0.3(\frac{3}{15} + \frac{0}{15})^{0.5} |\displaystyle \frac {\frac{3}{15} - \frac{0}{15}} {\frac{3}{15} + \frac{0}{15}}| }{ 0.2(\frac{9}{15} + \frac{7}{15})^{0.5} + \cdots + 0.3(\frac{3}{15} + \frac{0}{15})^{0.5} } \\ \\ G &\approx \displaystyle \frac {0.03 + 0.23 + 0.21 + 0.26 + 0.49 + 0.08 + 0.27 + 0.13} {0.21 + 0.23 + 0.21 + 0.26 + 0.77 + 0.58 + 0.60 + 0.13} \\ \\ G &= \displaystyle \frac{1.701419}{2.986235} \\ \\ G &= 0.569754 \end{align*}\]
variance_adjusted_unifrac()variance_adjusted_unifrac()qiime diversity beta-phylogenetic --p-metric weighted_normalized_unifrac --p-variance-adjusted\[\begin{align*} V &= \displaystyle \frac {\sum_{i = 1}^{n} L_i\displaystyle \frac {|\frac{A_i}{A_T} - \frac{B_i}{B_T}|} {\sqrt{(A_i + B_i)(A_T + B_T - A_i - B_i)}} } {\sum_{i = 1}^{n} L_i\displaystyle \frac {\frac{A_i}{A_T} + \frac{B_i}{B_T}} {\sqrt{(A_i + B_i)(A_T + B_T - A_i - B_i)}} } \\ \\ V &= \displaystyle \frac { L_1\displaystyle \frac {|\frac{A_1}{A_T} - \frac{B_1}{B_T}|} {\sqrt{(A_1 + B_1)(A_T + B_T - A_1 - B_1)}} + \cdots + L_n\displaystyle \frac {|\frac{A_n}{A_T} - \frac{B_n}{B_T}|} {\sqrt{(A_n + B_n)(A_T + B_T - A_n - B_n)}} }{ L_1\displaystyle \frac {\frac{A_1}{A_T} + \frac{B_1}{B_T}} {\sqrt{(A_1 + B_1)(A_T + B_T - A_1 - B_1)}} + \cdots + L_n\displaystyle \frac {\frac{A_n}{A_T} + \frac{B_n}{B_T}} {\sqrt{(A_n + B_n)(A_T + B_T - A_n - B_n)}} } \\ \\ V &= \displaystyle \frac { 0.2\displaystyle \frac {|\frac{9}{15} - \frac{7}{15}|} {\sqrt{(9 + 7)(15 + 15 - 9 - 7)}} + \cdots + 0.3\displaystyle \frac {|\frac{3}{15} - \frac{0}{15}|} {\sqrt{(3 + 0)(15 + 15 - 3 - 0)}} }{ 0.2\displaystyle \frac {\frac{9}{15} + \frac{7}{15}} {\sqrt{(9 + 7)(15 + 15 - 9 - 7)}} + \cdots + 0.3\displaystyle \frac {\frac{3}{15} + \frac{0}{15}} {\sqrt{(3 + 0)(15 + 15 - 3 - 0)}} } \\ \\ V &\approx \displaystyle \frac {0.002 + 0.012 + 0.010 + 0.013 + 0.029 + 0.005 + 0.016 + 0.007} {0.014 + 0.012 + 0.010 + 0.013 + 0.046 + 0.037 + 0.036 + 0.007} \\ \\ V &= \displaystyle \frac{4.09389}{4.174402} \\ \\ V &= 0.9807128 \end{align*}\]