Skip to content

Update admixslug.tex #13

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
arevsumer wants to merge 1 commit into
base: master
Choose a base branch
from
Open

Update admixslug.tex #13

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
36 changes: 11 additions & 25 deletions docs/admixslug.tex
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -6,7 +6,7 @@
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\author{Benjamin Peter, Arev}
\author{Benjamin Peter, Arev Sümer}

\newcommand{\BE}[1]{\mathbb{E}\left[#1\right]}
\newcommand{\BFZ}{\mathbf{Z}}
Expand All @@ -27,7 +27,7 @@ \section*{Model Overview}



We are primarily interested in estimating the latent states $\BFZ$ and $\BFG$, but we also estimate the transition matrix $A$ between states (which, in turn, is informative about admixture proportion and times), the contamination and error rate for each read group, the substructure in each source $\tau_k$, and the average drift since admixture from each source $F_k$.
We are primarily interested in estimating the latent states $\BFZ$ and $\BFG$, but we also estimate the contamination rate ($c_r$), error rate ($e), and the reference bias ($b$) for each read group, the substructure in each source $\tau_k$, and the average drift since admixture from each source $F_k$.
\section*{Notation overview}
To summarize, the notation is as follows:
\begin{itemize}
Expand Down Expand Up @@ -117,27 +117,13 @@ \subsection*{Contamination model}
\end{align}
independent of the locus.

\subsection*{Genotype likelihoods}
The genotype likelihood for locus $l$ can be written as $P(O_l | G_l) = \prod_{rj} P(O_{lrj} | G_l)$, where the product is over all reads aligning to this locus (double indexing because we multiply over all read-groups (indexed by $r$) and all reads per read-group (indicated by $j$).


The backwards probabilities
\begin{equation}
P(O_{lrj} | G_l) = P(O | C_{lrj}=1)c_{lrj} + P(O | G_l, C_{lrj}=0)(1 - c_{lrj})
\end{equation}
where
\begin{equation*}
P(O | C_{lrj}=1) = \begin{cases}
\psi_l &\text{ if } O=1\\ (1-\psi_l) &\text{ if } O=0
\end{cases}
\end{equation*}

\subsection*{Genotype model}
We estimate the genotype given the conditional-SFS entry $Z_l=k$, $F_k$ is the probability that both alleles are IBD, and $\tau_k$ is the probability that the individual has a derived allele at position $k$
Thus
\begin{align}
P(G_l = 0| Z_l=k, \tau_k, F_k) &= F_k (1-\tau_k) + (1-F_k) (1-\tau_k)^2\nonumber\\
P(G_l = 1| Z_l=k, \tau_k, F_k) &= 2(1-F_k) \tau(1-\tau_k)\nonumber\\
P(G_l = 1| Z_l=k, \tau_k, F_k) &= 2(1-F_k) \tau_k(1-\tau_k)\nonumber\\
P(G_l = 2| Z_l=k, \tau_k, F_k) &= F_k \tau_k + (1-F_k) \tau_k^2\label{eq:prg}
\end{align}

Expand Down Expand Up @@ -166,7 +152,7 @@ \subsection*{Genotype model}


\subsection*{Likelihood}
We observe the data $\mathbf{O}$, and we know the parameters $\theta = (\tau_k, F_k, c_r, e,b)$, the
We observe the data $\mathbf{O}$, we provide initial values (and later estimate values using an EM algorithm) for the parameters $\theta = (\tau_k, F_k, c_r, e,b)$, and we know the
contamination panel $\psi$ and the conditional SFS $\mathbf{Z}$. The variables
$C_r, X_{lri}$ and $G_l$ are latent variables we need to sum over.
\begin{align}
Expand All @@ -177,16 +163,16 @@ \subsection*{Likelihood}
\subsection*{Forward Probabilities}
\paragraph{Read probabilities}
\begin{align*}
P(X_{lrj} | G_l, C_r, \psi_l) &= P(X_{lrj} | C=0) Pr( C=0) + \sum P(X_{lrj} | C=1) Pr(C=1) \\
P(X_{lrj} | C=0) &= \sum_{g=0}^2 P(X_{lrj} | G_{lrj}=g)P(G_l=g | Z_l) \frac{ P(O_l | G_l=g)} {P(O_{lrj} | G_l=g) }
P(X_{lrj} | G_l, C_r, \psi_l) &= P(X_{lrj} | C=0) Pr( C=0) + P(X_{lrj} | C=1) Pr(C=1) \\
P(X_{lrj} | C=0) &= \sum_{g=0}^2 P(X_{lrj} | G_{lrj}=g)P(G_l=g | Z_l)
\end{align*}
the ratio in the last equation is the probability of all other observations given the genotype
where $g$ is all possible genotypes.

\subsection*{Backward Probabilities}
Calculate the probability of all observations given a genotype (interpreted as function of the genotype $G_l = 0,1,2$
\begin{align*}
P(O_{l} | G_l) &= \prod_{rj} P(O_{lrj} | G_l)\\
P(O_{lrj} | G_l) &= \sum_a P(O_{lrj}|X_{lrj}=a) P(X_{lrj}=a | G_l, C_r, \psi_l)\\
P(O_{lrj} | G_l) &= \sum_x P(O_{lrj}|X_{lrj}=x) P(X_{lrj}=x | G_l, C_r, \psi_l)\\
P(X_{lrj} | G_l, C_r, \psi_l) &= P(X_{lrj} | C_{lrj}=0)P(C_{lrj}=0) + P(X_{lrj} | \psi_l, C_{lrj}=1)P(C_{lrj}=1)]
\end{align*}

Expand All @@ -200,8 +186,8 @@ \subsection*{Posterior}
$$P(X_{lrj} | O) \propto P(X_{lrj} | C, G, Z, \psi) P(O_{lrj} | X_{lrj})$$
\paragraph{Posterior Contamination}
Calculate the posterior probability that read $rij$ is contamination
$$P(C_{rij}) = \frac{\sum_a P(X=a|C_r=1) P(O|X=a) P(C_r=1)}{\sum_i \big[ P(X=a|C=1) P(O|X=a)P(C_r=1) + P(X=a|C=0) P(O|X=ia)P(C_r=0)\big]}$$
where $a=0,1$
$$P(C_{rij}) = \frac{\sum_x P(X=a|C_r=1) P(O|X=x) P(C_r=1)}{\sum_i \big[ P(X=x|C=1) P(O|X=x)P(C_r=1) + P(X=x|C=0) P(O|X=ix)P(C_r=0)\big]}$$
where $x=0,1$, the states X can have.

\subsection*{Parameter estimation}
We estimate parameters using the complete-data log-likelihood using an EM-algorithm.
Expand All @@ -223,7 +209,7 @@ \subsection*{Parameter estimation}


\subsubsection*{Estimating $e$ and $b$}
Let $n_{a,b,c}$ be the number of reads where $O_{lri}=0, X_{lri}=b, W_l=c$
Let $n_{a,b,c}$ be the number of reads where $O_{lri}=a, X_{lri}=b, W_l=c$
\begin{align*}
\hat{e} & = \frac{n_{1,0,0} + n_{1,1,1}}{n_{1,0,0} + n_{1,1,1} + n_{0,0,0} +n_{0, 1, 1}}\\
\hat{b} & = \frac{n_{0,1,0} + n_{0, 0,1}}{n_{0, 1, 0} + n_{0, 0, 1} + n_{1, 1, 0} +n_{1, 0, 1}}
Expand Down