Introduce k-mer based distances (Hamming and Mash).

Defines the Hamming distance based on k-mer presence/absence vectors and introduces the Mash distance, noting it is an evolutionary estimator rather than a metric.

theory-phylogeny-kmer_based-kmer_set_distances.md

@@ -28,7 +28,7 @@ $$D = \sum_i \mathbb{1}[a_i \ne b_i],$$
 
 where (a\_i,b\_i\\in{0,1}) indicate the presence of k-mer (i) in the two genomes. It counts the number of k-mers for which the two incidence vectors differ, without normalization by the size of the k-mer space. As an (L^1) distance, Hamming distance is a metric. Unlike abundance-based measures, it cannot be recovered from k-mer counts alone once the presence/absence representation has been discarded.
 
-**Mash** is not an independent set dissimilarity but a nonlinear transformation of the Jaccard similarity. If
+**Mash** is not an independent set dissimilarity but a nonlinear transformation of the Jaccard similarity ([Ondov et al. 2016](#ref-Ondov2016-dq)). If
 
 $$J = 1-D_{\mathrm{Jaccard}},$$
 
@@ -36,7 +36,7 @@ the Mash distance is defined as
 
 $$D = -\frac{1}{k}\ln\left(\frac{2J}{1+J}\right) = -\frac{1}{k}\ln\left(\frac{2(1-J_{\mathrm{Jaccard}})} {2-J_{\mathrm{Jaccard}}}\right),$$
 
-with the result clamped to (1.0) when the Jaccard similarity reaches zero. This transformation is monotone increasing with Jaccard distance and therefore preserves the ordering of pairwise dissimilarities. Monotonicity alone, however, does not guarantee preservation of the triangle inequality. Mash should consequently be regarded as an evolutionary distance estimator derived from k-mer similarity rather than assumed to be a metric.
+with the result clamped to $1.0$ when the Jaccard similarity reaches zero. This transformation is monotone increasing with Jaccard distance and therefore preserves the ordering of pairwise dissimilarities. Monotonicity alone, however, does not guarantee preservation of the triangle inequality. Mash should consequently be regarded as an evolutionary distance estimator derived from k-mer similarity rather than assumed to be a metric.
 
 ## Abundance-based distances
 
@@ -136,4 +136,10 @@ Legendre, P. & Gallagher, E.D. (2001). <a href="https://doi.org/10.1007/s0044201
 
 </div>
 
+<div id="ref-Ondov2016-dq" class="csl-entry">
+
+Ondov, B.D., Treangen, T.J., Melsted, P., Mallonee, A.B., Bergman, N.H., Koren, S., <span style="font-style: italic;">et al.</span> (2016). <a href="https://doi.org/10.1186/s13059-016-0997-x">Mash: fast genome and metagenome distance estimation using MinHash</a>. <span style="font-style: italic;">Genome biology</span>, 17, 132.
+
+</div>
+
 </div>