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pi.lyx
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pi.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass article
\use_default_options true
\maintain_unincluded_children false
\language english
\language_package default
\inputencoding auto
\fontencoding global
\font_roman "default" "default"
\font_sans "default" "default"
\font_typewriter "default" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype false
\use_dash_ligatures true
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\use_hyperref false
\papersize default
\use_geometry false
\use_package amsmath 1
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine basic
\cite_engine_type default
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 1
\use_minted 0
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\is_math_indent 0
\math_numbering_side default
\quotes_style english
\dynamic_quotes 0
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header
\begin_body
\begin_layout Standard
\begin_inset Formula
\[
\pi_{A}=\frac{1}{N(N-1)/2}\sum_{j>i}\frac{n_{\textrm{diff}}(i,j)}{n_{\textrm{sites}}},
\]
\end_inset
where
\begin_inset Formula $N$
\end_inset
is number of sequences,
\begin_inset Formula $n_{\textrm{diff}}(i,j)$
\end_inset
is number of sites that differ between sequences
\begin_inset Formula $i$
\end_inset
and
\begin_inset Formula $j$
\end_inset
and
\begin_inset Formula $n_{\textrm{sites}}$
\end_inset
is the total number of sites (padded).
\begin_inset Formula
\[
\theta_{A}=\frac{n_{\textrm{seg}}}{\sum_{1}^{N-1}\frac{1}{i}},
\]
\end_inset
where
\begin_inset Formula $n_{\textrm{seg}}$
\end_inset
is the number of sites that are not fixed in the population.
\end_layout
\begin_layout Standard
Note that Tajima's D is calculated as
\begin_inset Formula
\[
D=\frac{\pi_{A}n_{\textrm{sites}}-\theta_{A}}{f(N,n_{\textrm{seg}})}.
\]
\end_inset
\end_layout
\begin_layout Standard
Msums and libsequence are essentially the same with
\begin_inset Formula $\theta_{\pi}:=\pi_{A}n_{\textrm{sites}}$
\end_inset
and
\begin_inset Formula $\theta_{W}:=\theta_{A}$
\end_inset
.
\end_layout
\end_body
\end_document