Difference between revisions of "User:Hussein Alasadi/Notebook/stephens/2013/10/13"
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− | We assume that <math> y = N(\mu, \Sigma) </math>. By properties of bi-variate normal distributions <math>y_2/y_1,M</math> ~ <math>N(\mu_2 + \rho \frac{\sigma_2}{\sigma_1}(y_1 - u_1), (1-\rho^2)\sigma_1^2)</math> where <math>\rho = \frac{E[y_1y_2]}{\sigma_1 \sigma_2}</math> | + | We assume that <math> y = N(\mu, \Sigma) </math>. By properties of bi-variate normal distributions <math>y_2/y_1,M</math> ~ <math>N(\mu_2 + \rho \frac{\sigma_2}{\sigma_1}(y_1 - u_1), (1-\rho^2)\sigma_1^2)</math> where <math>\rho = \frac{E[y_1y_2]}{\sigma_1 \sigma_2}</math>. The genius of Wen & Stephens lies in the idea that the distribution of <math>y_2</math> (assigned as the untyped SNP) is a function of both the panel data (<math>\mu_2</math>) and the typed SNPs <math>(y_1)</math>. |
== Li & Stephens in 2D == | == Li & Stephens in 2D == |
Revision as of 11:34, 17 October 2013
analyzing pooled sequenced data with selection | <html><img src="/images/9/94/Report.png" border="0" /></html> Main project page <html><img src="/images/c/c3/Resultset_previous.png" border="0" /></html>Previous entry<html> </html>Next entry<html><img src="/images/5/5c/Resultset_next.png" border="0" /></html> |
Intro to Wen & Stephens in 2DSuppose we have only summary-level data for haplotypes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_1, h_2, ..., h_{2n}} . Specifically let the summary-level data be denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle y = (y_1, y_2)' = \frac{1}{2n} \sum_i^{2n} h_i} . We assume in this two locus model, the first locus is typed and the second locus is untyped. We hope to predict what the allele frequency is at the untyped SNP using information from panel data (perhaps this can be interpreted as our prior). Formally, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle y_1} denote the allele frequency at the typed SNP and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle y_2} the allele frequency at the untyped SNP. We assume that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_1, h_2, ..., h_{2n}} are independent and identically distributed from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle P(M)} (our prior).
Li & Stephens in 2DWe describe the Li & Stephens haplotype copying model: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_1, h_2, ..., h_{k}} denote the k sampled haplotypes at 2 loci. Thus there are 4 possible haplotypes. The first haplotype is randomly chosen with equal probability from the four possible haplotypes. Consider now the conditional distribution of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_{k+1}} given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_1, h_2,...,h_k} . Recall the intuition is that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_{k+1}} is a mosaic of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_1, h_2,..,h_k} . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle X_j} denote which hapolotype Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_{k+1}} copies at site j (so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle X_j \in {1,2,..,k}} ). We model Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle X_j} as a markov chain on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle {1,..,k}} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle P(x_1 =x) = \frac{1}{k}} . The transition probabilities are: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle P(X_{j+1}=x'/X_j = x) = e^{-\frac{\rho_jd_j}{k}} + (1-e^{-\rho_jd_j})(1/k)} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x'=x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (1-e^{-\frac{\rho_jd_j}{k}})(1/k)} otherwise. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \rho_j} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle d_j} denote recombination and physical distances, respectively. Now in a hidden markov model, there is also the transmission process. To mimic the effects of mutation, the copying process may be imperfect. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle P(h_{k+1,j} =a / X_j = x, h_1,..,h_k) = \frac{k}{k+\theta} + \frac{\theta}{2(k+\theta)}} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h_{x,j} = a} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\theta}{2(k+\theta)}} otherwise. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \theta = (\sum_{m=1}^{n-1} \frac{1}{m})^{-1}} , where the motivation is the more haplotypes the less frequent mutation occurs. |