From 013bcbf01ed797a197f23977b361023fb0eabdcb Mon Sep 17 00:00:00 2001 From: Anton Goloborodko Date: Mon, 25 Mar 2024 13:00:39 +0100 Subject: [PATCH] docs: update math in stats --- doc/stats.rst | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/doc/stats.rst b/doc/stats.rst index 5ac5455..6c55714 100644 --- a/doc/stats.rst +++ b/doc/stats.rst @@ -61,7 +61,7 @@ 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. @@ -69,13 +69,13 @@ 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.