Last updated: 2019-10-08
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To develop a method for smoothed count matrix factorization. Suppose we observe a count matrix \(Y\) whose underlying rows(true row parameters) are smooth/spatially-structured and we are interested in finding a factorization/low-ranking approximation of the matrix.
Potential applications are to bulk data eg RNA-seq data from GTEX project. If data are from gene splicing, then we should take care of cuts that have overlaps on two exons.
It might not be a good application to single cell data as each sample is a cell.
We discussed 3 possible methods.
The first method is borrowing ideas from generalized linear model.
Suppose \[\begin{equation} Y\sim Poi(\Lambda),\Lambda = \exp(M), M=LF, \end{equation}\] where \(Y\) is a \(N\times p\) matrix, \(L\) is a \(N\times K\) matrix and \(F\) is a \(K\times p\) matrix.
Using Taylor series expansion, we have \[\begin{equation} \log(Y_{ij})\approx \log(\lambda_{ij})+\frac{Y_{ij}-\lambda_{ij}}{\lambda_{ij}} \end{equation}\]
Define \(Z_{ij} = \log(\lambda_{ij})+\frac{Y_{ij}-\lambda_{ij}}{\lambda_{ij}}\), then \(E(Z_{ij}|\lambda_{ij}) = \log(\lambda_{ij})\), \(Var(Z_{ij}|\lambda_{ij}) = \frac{1}{\lambda_{ij}}\). So we have \(Z = M+E = LF+E\), where \(E_{ij}\sim N(0,\frac{1}{\lambda_{ij}})\)
In my simulations, this method does not perform well. Possible reasons could be too many zeros in the matrix. So I’ll try to some ‘dense’ example first.
The multiscale approach utilizes the distribution of Poisson variable conditional on another Poisson distribution. Denote \(Y_+\) as the sum of two Poisson variables and \(Y_+^L, Y_+^R\) the left and right one. Then \[\begin{equation} Y^L_+|Y_+ \sim Binomial(Y_+,\frac{\Lambda_+^L}{\Lambda_+}). \end{equation}\]
Let \(p = \frac{\Lambda_+^L}{\Lambda_+}\). Define \[\begin{equation} \alpha = \log\frac{p}{1-p} = \log\frac{\Lambda_+^L}{\Lambda_+^R}, \end{equation}\] then \[\begin{equation} \hat{\alpha} = \log\frac{Y_+^L}{Y_+^R}. \end{equation}\]
Then we can use normal approximation to likelihood of \(\alpha\). One choice is due to Gart and Zweifel(1967).
The \(\hat{\alpha}\)s obtained can be seen as wavelet coefficients of Poisson data.
Further, we can assume \(A = \{\alpha\}\) has low rank structure which might be more plausible for certain data type.
Topic model assumes each row of \(Y\)
\[\begin{equation} Y_i|n_i\sim MN(n_i,\sum_{k=1}^K l_{ik}f_k) , \end{equation}\] where \(n_i\) is the sum of \(Y_i\) and \(f_k\) is a vector of length \(p\).
Zhenrong Xing described a method to smooth \(f_k\) in his PhD thesis, which utilizes EM algorithm. When updating \(f_k\), the method first updates using EM, then multiply the updates by \(c_k\) to make it be close to those of actual sequencing read counts(I’ll look into these counts see how they are derived), then apply smash to the ‘counts’.
The method uses smash which gives an approximate estimate of true smooth \(f_k\). But there is no convergence guarantee.
Suppose there is only one factor, \(Y=lf\), then \(Y_i = l_if\). Given \(l_i\), the log likelihood of \(f\) is
\[\begin{equation} (\sum_{i=1}^n Y_i)\log f - f\sum_{i=1}^n l_i, \end{equation}\]
which only require knowing the row sums of \(Y\). So we can use ‘exact’ method to estimate \(f\) in EM algorithm and obtain convergence guarantee. For example, the Poisson multiscale method by Kolaczyk and Nowak.
Alternative fomulation of topic model:
\[\begin{equation} Y \sim Poisson(\Lambda), \Lambda = LF, Y = \sum_k Z_k, Z_k\sim Poisson(l_kf_k'). \end{equation}\]
Then we can get \(E(Z_k|l,f,Y)\) and use EM update \(L,F\).