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update docs
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golobor committed Mar 25, 2024
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6 changes: 3 additions & 3 deletions doc/stats.rst
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Expand Up @@ -61,21 +61,21 @@ increases according to a simple equation:

.. math::
U(N+1) = U(N) + (1 - {U(N) \\over C}),
U(N+1) = U(N) + (1 - {U(N) \over C}),
where :math:`N` is the number of sequenced molecules, :math:`U(N)` is the expected number
of observed unique molecules after sequencing :math:`N` molecules, and :math:`C` is the library complexity.
This differential equation yields [1, 2]:

.. math::
{U(N) \\over C} = 1 - exp( - {N \\over C}),
{U(N) \over C} = 1 - exp( - {N \over C}),
which can be solved as

.. math::
C = \Re(lambert W( - { \exp( - {1 \\over u} ) \\over u} ) ) + {1 \\over u}
C = \Re(W_{Lambert}( - { \exp( - {1 \over u} ) \over u} ) ) + {1 \over u}
Library complexity can guide in the choice of sequencing depth of the library
and provide an estimate of library quality.
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