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docs: update math in stats
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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) + \left(1 - \frac{U(N)}{C} \right),
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\left( - \frac{N}{C} \right),
which can be solved as

.. math::
C = \Re(W_{Lambert}( - { \exp( - {1 \over u} ) \over u} ) ) + {1 \over u}
C = \Re \left( W_{Lambert} \left( - \frac{ \exp\left( - \frac{1}{U} \right) } {U} \right) \right) + \frac{1}{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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