1544 lines
33 KiB
HTML
1544 lines
33 KiB
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obikmer
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<div class="md-header__topic" data-md-component="header-topic">
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<span class="md-ellipsis">
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Entropy filter
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obikmer
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Installation
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Theory
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Theory
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<ul class="md-nav__list" data-md-scrollfix>
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Kmers and super-kmers
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</span>
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</a>
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</li>
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<span class="md-ellipsis">
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DNA encoding
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</span>
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</a>
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</li>
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<span class="md-ellipsis">
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Entropy filter
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<span class="md-ellipsis">
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Entropy filter
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</span>
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
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Table of contents
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="#sub-word-frequencies" class="md-nav__link">
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<span class="md-ellipsis">
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Sub-word frequencies
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#corrected-shannon-entropy" class="md-nav__link">
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<span class="md-ellipsis">
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||
|
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Corrected Shannon entropy
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|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#maximum-entropy-correction-for-small-samples" class="md-nav__link">
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<span class="md-ellipsis">
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|
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Maximum entropy correction for small samples
|
||
|
||
</span>
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||
</a>
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</li>
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<li class="md-nav__item">
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<a href="#normalized-entropy" class="md-nav__link">
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||
<span class="md-ellipsis">
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||
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Normalized entropy
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||
|
||
</span>
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</a>
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||
|
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</li>
|
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|
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<li class="md-nav__item">
|
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<a href="#final-score" class="md-nav__link">
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<span class="md-ellipsis">
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Final score
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#why-no-equivalence-classes" class="md-nav__link">
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<span class="md-ellipsis">
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Why no equivalence classes
|
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</span>
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</a>
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</li>
|
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<li class="md-nav__item">
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<a href="#interpretation-as-an-effective-number-of-classes" class="md-nav__link">
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<span class="md-ellipsis">
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||
|
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Interpretation as an effective number of classes
|
||
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</span>
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||
</a>
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</li>
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<li class="md-nav__item">
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<a href="#properties" class="md-nav__link">
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<span class="md-ellipsis">
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Properties
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item">
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<a href="../minimizer/" class="md-nav__link">
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<span class="md-ellipsis">
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Minimizer selection
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||
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../indexing/" class="md-nav__link">
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<span class="md-ellipsis">
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Partitioning architecture
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||
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../evolutionary_distances/" class="md-nav__link">
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<span class="md-ellipsis">
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Central-position SNP distance (discussion)
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||
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||
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||
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||
</span>
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||
|
||
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
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<span class="md-ellipsis">
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|
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Implementation
|
||
|
||
|
||
|
||
</span>
|
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
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Implementation
|
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
|
||
<a href="../../implementation/superkmer/" class="md-nav__link">
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||
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||
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<span class="md-ellipsis">
|
||
|
||
|
||
SuperKmer
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
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||
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<li class="md-nav__item">
|
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<a href="../../implementation/kmer/" class="md-nav__link">
|
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||
<span class="md-ellipsis">
|
||
|
||
|
||
Kmer
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
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|
||
|
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|
||
<li class="md-nav__item">
|
||
<a href="../../implementation/chunkreader/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
Chunk reader
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
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<li class="md-nav__item">
|
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<a href="../../implementation/pipeline/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
Construction pipeline
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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<li class="md-nav__item">
|
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<a href="../../implementation/obipipeline/" class="md-nav__link">
|
||
|
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|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
obipipeline library
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
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|
||
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|
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|
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|
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|
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|
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|
||
On-disk storage
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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|
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MPHF selection
|
||
|
||
|
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|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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|
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Unitig evidence encoding
|
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</span>
|
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|
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|
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|
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|
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|
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|
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Evidence elimination (discussion)
|
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|
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|
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|
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|
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|
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|
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obilayeredmap crate
|
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|
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|
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</span>
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|
||
|
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|
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</a>
|
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<span class="md-ellipsis">
|
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|
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|
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PersistentCompactIntVec
|
||
|
||
|
||
|
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</span>
|
||
|
||
|
||
|
||
</a>
|
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|
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|
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|
||
|
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PersistentBitVec
|
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|
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|
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|
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</span>
|
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|
||
|
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|
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</a>
|
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|
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Merge command
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|
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|
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Merge parallelism & memory
|
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</span>
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|
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|
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|
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Kmer filtering
|
||
|
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|
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|
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|
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|
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Select command
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|
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obitaxonomy crate
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|
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Why no equivalence classes
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Interpretation as an effective number of classes
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<article class="md-content__inner md-typeset">
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|
||
|
||
<h1 id="kmer-entropy-filter">Kmer entropy filter</h1>
|
||
<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 one source of bias: the small number of observations within a single kmer relative to the number of possible sub-words.</p>
|
||
<h2 id="sub-word-frequencies">Sub-word frequencies</h2>
|
||
<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>
|
||
<div class="arithmatex">\[w_i = \text{kmer}[i \mathinner{..} i+ws-1], \quad i = 0, \ldots, n_{\text{words}}-1\]</div>
|
||
<p>Each sub-word is tallied under its own raw 2-bit-packed value — <strong>no canonicalization</strong>. Let <span class="arithmatex">\(f_j\)</span> be the count of raw word <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>), over the <span class="arithmatex">\(4^{ws}\)</span> possible raw words.</p>
|
||
<p>An earlier version of this filter first folded each sub-word into a circular+reverse-complement equivalence class, then "unfolded" the observed class frequency back onto its members to correct for unequal class sizes. That machinery bought nothing it was claimed for — see <em>Why no equivalence classes</em> below — while measurably weakening detection of the very sequences the filter exists to catch, so it was removed.</p>
|
||
<h2 id="corrected-shannon-entropy">Corrected Shannon entropy</h2>
|
||
<div class="arithmatex">\[H_{\text{corr}} = \log(n_{\text{words}}) - \frac{1}{n_{\text{words}}} \sum_j f_j \log f_j\]</div>
|
||
<p>This is a plain Shannon entropy over the observed raw-word frequencies.</p>
|
||
<h2 id="maximum-entropy-correction-for-small-samples">Maximum entropy correction for small samples</h2>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<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>
|
||
<h2 id="normalized-entropy">Normalized entropy</h2>
|
||
<div class="arithmatex">\[\hat{H}(ws) = \frac{H_{\text{corr}}}{H_{\max}} \in [0, 1]\]</div>
|
||
<h2 id="final-score">Final score</h2>
|
||
<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>
|
||
<div class="arithmatex">\[\text{entropy}(kmer) = \min_{ws=1}^{ws_{\max}} \hat{H}(ws)\]</div>
|
||
<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>
|
||
<h2 id="why-no-equivalence-classes">Why no equivalence classes</h2>
|
||
<p>A prior design folded each sub-word into the canonical form of its circular-rotation + reverse-complement equivalence class before tallying, on the reasoning that (a) it guarantees <span class="arithmatex">\(\text{entropy}(K) = \text{entropy}(\text{revcomp}(K))\)</span>, and (b) collapsing phase-shifted repeats (e.g. <code>ATG</code> ≡ <code>TGA</code> ≡ <code>GAT</code>) into one class better reflects that they are "the same" low-complexity pattern.</p>
|
||
<p>Both properties already hold for the raw, unfolded entropy above, without any class machinery:</p>
|
||
<ul>
|
||
<li><strong>Reverse complement</strong>: for any K of length n, window <span class="arithmatex">\(j\)</span> of <span class="arithmatex">\(\text{revcomp}(K)\)</span> equals <span class="arithmatex">\(\text{revcomp}\)</span> of window <span class="arithmatex">\((n{-}ws{-}j)\)</span> of K. This is a bijection between the window sets under which each window maps to its own revcomp — and revcomp is itself a bijection (involution) on the space of raw ws-mers. So the multiset of raw-word frequencies for <span class="arithmatex">\(\text{revcomp}(K)\)</span> is exactly a relabeling of the multiset for K, and Shannon entropy — a function of the frequency multiset alone — is exactly invariant. No folding required, for any K.</li>
|
||
<li><strong>Tandem repeats</strong>: a period-p repeat sampled by a stride-1 sliding window naturally cycles through its own rotations as raw tokens (e.g. <code>ATGATGATG…</code> yields the raw words <code>ATG</code>, <code>TGA</code>, <code>GAT</code> in rotation as the window slides). The low diversity this represents (few distinct raw words out of <span class="arithmatex">\(4^{ws}\)</span> possible) is already visible in the raw frequency distribution — no folding needed to detect it.</li>
|
||
</ul>
|
||
<p>What the fold-then-unfold step actually did was credit each observed class with the frequency of equivalence-class members that were <strong>never observed on the read strand</strong>, inflating <span class="arithmatex">\(H_{\text{corr}}\)</span> for genuine repeats. Worked example: k=31, ws=3, kmer = <code>ATG</code> repeated (<span class="arithmatex">\(n_{\text{words}}=29\)</span>, all 29 windows fall into one class of size 6 under the old scheme — 3 rotations × forward/revcomp):</p>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th></th>
|
||
<th><span class="arithmatex">\(H_{\text{corr}}\)</span></th>
|
||
<th>normalized</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td>old (folded, class size 6)</td>
|
||
<td><span class="arithmatex">\(\log 6 \approx 1.79\)</span></td>
|
||
<td><span class="arithmatex">\(\approx 0.53\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>current (raw, unfolded)</td>
|
||
<td><span class="arithmatex">\(\log 3 \approx 1.10\)</span></td>
|
||
<td><span class="arithmatex">\(\approx 0.33\)</span></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<p>The gap is not a rounding artifact: per sub-word order, the folded score for this same repeat swings from 0.53 (ws=3, aligned with the period) up to <strong>1.03</strong> (ws=5, misaligned with the period) — i.e. a period-3 repeat could score <em>above</em> the theoretical maximum for a random sequence, depending on which ws happens to divide the repeat's period. The raw formula stays flat at ≈0.33–0.40 across ws=2..6 regardless of alignment, which is the robustness the "minimum across ws" design was meant to provide in the first place.</p>
|
||
<h2 id="interpretation-as-an-effective-number-of-classes">Interpretation as an effective number of classes</h2>
|
||
<p><span class="arithmatex">\(H_{\text{corr}}\)</span> is a standard Shannon entropy over raw words, so the classical perplexity interpretation holds directly: <span class="arithmatex">\(N_{\text{eff}} = e^{H_{\text{corr}}}\)</span> is the number of equiprobable raw words that would yield the same entropy.</p>
|
||
<p>For the normalised score <span class="arithmatex">\(\hat{H}\)</span>, dividing by <span class="arithmatex">\(H_{\max}\)</span> changes the logarithm base:</p>
|
||
<div class="arithmatex">\[\hat{H} = \frac{\log N_{\text{eff}}}{\log N_{\max}} = \log_{N_{\max}} N_{\text{eff}} \quad \Longleftrightarrow \quad N_{\text{eff}} = N_{\max}^{\,\hat{H}}\]</div>
|
||
<p>The property is preserved: <span class="arithmatex">\(\hat{H}\)</span> is the logarithm (in base <span class="arithmatex">\(N_{\max}\)</span>) of the effective number of equi-represented raw words.</p>
|
||
<p>In the large-sample limit (<span class="arithmatex">\(n_{\text{words}} \gg 4^{ws}\)</span>), <span class="arithmatex">\(N_{\max} \approx 4^{ws}\)</span>, giving:</p>
|
||
<div class="arithmatex">\[N_{\text{eff}} \approx 4^{ws \cdot \hat{H}}\]</div>
|
||
<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 words out of 16 are occupied.</p>
|
||
<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">\(H_{\max} < \log(4^{ws})\)</span> due to the small-sample correction. The exact effective count is <span class="arithmatex">\(N_{\max}^{\hat{H}}\)</span>, not <span class="arithmatex">\(4^{ws \cdot \hat{H}}\)</span>.</p>
|
||
<h2 id="properties">Properties</h2>
|
||
<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>
|
||
<ul>
|
||
<li><strong>Orientation invariance</strong>: <span class="arithmatex">\(\text{entropy}(K) = \text{entropy}(\text{revcomp}(K))\)</span> — see <em>Why no equivalence classes</em> above for why this holds without any explicit strand-folding step.</li>
|
||
<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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