Eric Coissac
bb7adc1154
Expands MkDocs navigation and documentation for evidence elimination, the merge command, and kmer filtering. Refactors kmer representation to a generic `KmerOf<L>` type with a bitwise reverse complement algorithm. Unifies MPHF construction, introduces approximate fingerprint-based indexing, and updates the pipeline, chunkreader, and storage layouts. Adds code coverage reports and clarifies architectural invariants for improved maintainability.
1270 lines
28 KiB
HTML
1270 lines
28 KiB
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obikmer
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Entropy filter
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DNA encoding
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Entropy filter
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Entropy filter
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Sub-word frequencies
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Corrected Shannon entropy
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Maximum entropy correction for small samples
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Normalized entropy
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Final score
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Interpretation as an effective number of classes
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Properties
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Minimizer selection
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Partitioning architecture
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SuperKmer
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Kmer
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Chunk reader
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Construction pipeline
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obipipeline library
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On-disk storage
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MPHF selection
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<h1 id="kmer-entropy-filter">Kmer entropy filter</h1>
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<p>Low-complexity kmers (polyA, polyT, tandem repeats) are detected and excluded during phase 1. The filter computes a <strong>normalized Shannon entropy</strong> over sub-words of multiple sizes, corrected for two sources of bias: the small number of observations within a single kmer, and the unequal sizes of circular equivalence classes.</p>
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<h2 id="sub-word-frequencies">Sub-word frequencies</h2>
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<p>For a kmer of length k and a sub-word size ws (1 ≤ ws ≤ ws_max, typically ws_max = 6), extract the <span class="arithmatex">\(n_{\text{words}} = k - ws + 1\)</span> overlapping sub-words by sliding a window of length ws:</p>
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<div class="arithmatex">\[w_i = \text{kmer}[i \mathinner{..} i+ws-1], \quad i = 0, \ldots, n_{\text{words}}-1\]</div>
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<p>Each sub-word is mapped to its <strong>circular canonical form</strong>: the lexicographic minimum among all cyclic rotations of the word <strong>and all cyclic rotations of its reverse complement</strong>. This extended equivalence relation ensures that entropy(K) = entropy(revcomp(K)) — the filter is strand-symmetric. Let <span class="arithmatex">\(s_j\)</span> be the size of equivalence class <span class="arithmatex">\(j\)</span> (number of distinct raw words mapping to canonical form <span class="arithmatex">\(j\)</span>), and <span class="arithmatex">\(f_j\)</span> the count of canonical form <span class="arithmatex">\(j\)</span> among the <span class="arithmatex">\(n_{\text{words}}\)</span> sub-words (<span class="arithmatex">\(\sum_j f_j = n_{\text{words}}\)</span>).</p>
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<h2 id="corrected-shannon-entropy">Corrected Shannon entropy</h2>
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<p>The circular equivalence classes have unequal sizes: under a uniform distribution over all <span class="arithmatex">\(4^{ws}\)</span> raw words, class <span class="arithmatex">\(j\)</span> is visited with probability <span class="arithmatex">\(s_j / 4^{ws}\)</span>, not <span class="arithmatex">\(1/n_a\)</span>. Computing entropy directly over canonical classes therefore underestimates the entropy of a random sequence.</p>
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<p>The correction "unfolds" each canonical class back to its member raw words, redistributing each observation of class <span class="arithmatex">\(j\)</span> equally among its <span class="arithmatex">\(s_j\)</span> members:</p>
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<div class="arithmatex">\[H_{\text{corr}} = \log(n_{\text{words}}) - \frac{1}{n_{\text{words}}} \sum_j f_j \log f_j + \frac{1}{n_{\text{words}}} \sum_j f_j \log s_j\]</div>
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<p>The last term is the correction for unequal class sizes. For a uniformly random sequence (<span class="arithmatex">\(f_j \approx n_{\text{words}} \cdot s_j / 4^{ws}\)</span>), this gives <span class="arithmatex">\(H_{\text{corr}} \approx \log(4^{ws}) = 2 \cdot ws \cdot \log 2\)</span>, the maximum entropy over raw words.</p>
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<h2 id="maximum-entropy-correction-for-small-samples">Maximum entropy correction for small samples</h2>
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<p>With only <span class="arithmatex">\(n_{\text{words}}\)</span> observations over <span class="arithmatex">\(4^{ws}\)</span> possible raw words, the achievable maximum entropy is bounded by the most uniform integer distribution over <span class="arithmatex">\(4^{ws}\)</span> categories.</p>
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<p>Let <span class="arithmatex">\(c = \lfloor n_{\text{words}} / 4^{ws} \rfloor\)</span> and <span class="arithmatex">\(r = n_{\text{words}} \bmod 4^{ws}\)</span>. The most uniform integer distribution assigns frequency <span class="arithmatex">\(c+1\)</span> to <span class="arithmatex">\(r\)</span> categories and <span class="arithmatex">\(c\)</span> to the remaining <span class="arithmatex">\(4^{ws} - r\)</span>, with the convention <span class="arithmatex">\(0 \log 0 = 0\)</span>:</p>
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<div class="arithmatex">\[H_{\max} = -\left[(4^{ws} - r)\,\frac{c}{n_{\text{words}}}\log\frac{c}{n_{\text{words}}} + r\,\frac{c+1}{n_{\text{words}}}\log\frac{c+1}{n_{\text{words}}}\right]\]</div>
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<p>When <span class="arithmatex">\(n_{\text{words}} < 4^{ws}\)</span>: <span class="arithmatex">\(c=0\)</span>, <span class="arithmatex">\(r=n_{\text{words}}\)</span>, and the formula reduces to <span class="arithmatex">\(H_{\max} = \log(n_{\text{words}})\)</span> — a single unified expression covers both regimes. A truly random sequence achieves <span class="arithmatex">\(H_{\text{corr}} \approx H_{\max}\)</span>.</p>
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<h2 id="normalized-entropy">Normalized entropy</h2>
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<div class="arithmatex">\[\hat{H}(ws) = \frac{H_{\text{corr}}}{H_{\max}} \in [0, 1]\]</div>
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<h2 id="final-score">Final score</h2>
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<p>The filter computes <span class="arithmatex">\(\hat{H}(ws)\)</span> for each word size ws from 1 to ws_max and returns the <strong>minimum</strong>:</p>
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<div class="arithmatex">\[\text{entropy}(kmer) = \min_{ws=1}^{ws_{\max}} \hat{H}(ws)\]</div>
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<p>A value near 0 indicates low complexity (e.g. AAAA…); near 1 indicates high complexity. A kmer is rejected if <span class="arithmatex">\(\text{entropy}(kmer) < \theta\)</span>, where <span class="arithmatex">\(\theta\)</span> is a collection parameter (default 0.7). The minimum across word sizes ensures that any scale of repetition is detected independently: polyA is caught at ws=1, dinucleotide repeats at ws=2, etc.</p>
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<h2 id="interpretation-as-an-effective-number-of-classes">Interpretation as an effective number of classes</h2>
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<p><span class="arithmatex">\(H_{\text{corr}}\)</span> is a standard Shannon entropy over raw words (after unfolding the equivalence classes), so the classical perplexity interpretation holds directly: <span class="arithmatex">\(N_{\text{eff}} = e^{H_{\text{corr}}}\)</span> is the number of equiprobable classes that would yield the same entropy.</p>
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<p>For the normalised score <span class="arithmatex">\(\hat{H}\)</span>, dividing by <span class="arithmatex">\(H_{\text{max}}\)</span> changes the logarithm base:</p>
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<div class="arithmatex">\[\hat{H} = \frac{\log N_{\text{eff}}}{\log N_{\text{max}}} = \log_{N_{\text{max}}} N_{\text{eff}} \quad \Longleftrightarrow \quad N_{\text{eff}} = N_{\text{max}}^{\,\hat{H}}\]</div>
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<p>The property is preserved: <span class="arithmatex">\(\hat{H}\)</span> is the logarithm (in base <span class="arithmatex">\(N_{\text{max}}\)</span>) of the effective number of equi-represented classes.</p>
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<p>In the large-sample limit (<span class="arithmatex">\(n_{\text{words}} \gg 4^{ws}\)</span>), <span class="arithmatex">\(N_{\text{max}} \approx 4^{ws}\)</span>, giving:</p>
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<div class="arithmatex">\[N_{\text{eff}} \approx 4^{ws \cdot \hat{H}}\]</div>
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<p>This has a clean interpretation: <span class="arithmatex">\(ws \cdot \hat{H}\)</span> is the <strong>effective word length</strong> (in bases) of a perfectly uniform distribution that would produce the same entropy. At <span class="arithmatex">\(\hat{H} = 1\)</span> the full space of <span class="arithmatex">\(4^{ws}\)</span> words is used; at <span class="arithmatex">\(\hat{H} = 0.5\)</span> with ws=2, only <span class="arithmatex">\(4^1 = 4\)</span> effective classes out of 16 are occupied.</p>
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<p>In our actual regime, <span class="arithmatex">\(n_{\text{words}}\)</span> is small and <span class="arithmatex">\(4^{ws}\)</span> can exceed <span class="arithmatex">\(n_{\text{words}}\)</span>, so <span class="arithmatex">\(H_{\text{max}} < \log(4^{ws})\)</span> due to the small-sample correction. The exact effective count is <span class="arithmatex">\(N_{\text{max}}^{\hat{H}}\)</span>, not <span class="arithmatex">\(4^{ws \cdot \hat{H}}\)</span>.</p>
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<h2 id="properties">Properties</h2>
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<p>The entropy score is a function of the kmer sequence alone — it does not depend on the surrounding context or on the position within any genome. Two consequences:</p>
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<ul>
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<li><strong>Orientation invariance</strong>: <span class="arithmatex">\(\text{entropy}(K) = \text{entropy}(\text{revcomp}(K))\)</span>, guaranteed by the strand-symmetric canonical form.</li>
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<li><strong>Context independence</strong>: the same kmer is always rejected or always kept, regardless of which genome it occurs in, where in that genome it appears, or which strand is considered. The filter defines a fixed partition of the kmer space into low-complexity and valid kmers.</li>
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</ul>
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