In ctlab/ClusDec: Clustering approach to solve complete Deconvolution problem

Mixing process: linear model

Suppose that we want to mix some cell types. Let $W$ be an $n \times c$ matrix where $n$ is the number of genes and $c$ is the number of cell types. Matrix $W$ describes how every gene is expressed in concrete cell type. Let $H$ be an $c \times m$ matrix where $m$ is the number of samples we want to mix. Matrix $H$ describes in what proportions (frequencies) we want to mix our cell types, hence the sum of every column in $H$ has to be equal to one.

Mixing process can be modeled using linear model: $$ X = W \times H $$ where resulting $n \times m$ matrix describes how each gene is expressed in the resuling heterogenious mix.

For example if we wanted to mix brain and liver in three samples: pure brain, 50% / 50% and pure liver, our matrix H would look like

H = matrix(c(1.0, 0.0, 0.5, 0.5, 0.0, 1.0), nrow=2, ncol=3)
colnames(H) <- c("Sample 1", "Sample 2", "Sample 3")
rownames(H) <- c("Brain", "Liver")
H

if we also knew how some genes are expressed in brain and liver, our matrix W would look like

library(clusdec)
data("datasetLiverBrainLung")
W = 2^datasetLiverBrainLung[1:6, c(4, 1)]
colnames(W) <- c("Brain", "Liver")
W

then mixing processing would be modeled as simple matrix multiplication:

X <- W %*% H
X

Deconvolution

Deconvolution is opposite process.

Observed gene expression matrix $X$ is given and we want to find such matrices $W$ and $H$ that

$$ X \approx W \times H $$

where $X$, $W$ and $H$ are nonnegative matricies, where sum of every column of $H$ is close to one. There are a lot of methods that allow us to do such factorization, but we want to find biologically meaningful matrices $W$ and $H$.

If we know more information other than $X$ (e.g. we might know matrices $W$ or $H$, or we might know some signature genes for cell types that are in the mix) we will call it partial deconvolution problem. There are a lot of methods developed for solving partial deconvolution problem: they use different approaches and require different additional information.

If only X is known we will call it complete deconvolution problem. And that's what we are going to solve.

Signature genes

In terms of deconvolution we are going to call gene signature if it is higly expressed in only one cell type in given experiment. Knowledge of such genes allows to perform pretty sensitive deconvolution using DSA algorithm.

ClusDec

Main idea of ClusDec is to find such genes that may be signatures for cell types in the mix.

Accuracy

To decide which combination of clusters is better as putative sigantures we have to introduce a way to estimate accuracy of deconvolution. Lets assume we have already found such matrices $W$ and $H$ that $$ X \approx W \times H $$

and we want to estimate how good this factorization is. Let $\tilde{X} = W \times H$, so the most natural way is comparison of the matrices $X$ and $\tilde{X}$. Here we introduce the accuracy of deconvolution:

$$ accuracy(X, \tilde{X}) = || log(X) - log(\tilde{X}) ||_F $$

ctlab/ClusDec documentation built on May 14, 2019, 12:29 p.m.