Stage-i scales for one single-cell bulk sample

Generalised method of moments for the two technical scales, at a composition that is either supplied or taken from the free-scale DeconRNASeq start \((c^{(0)},\boldsymbol{p}^{(0)}) =\arg\min_{c>0,\,p\in\Delta}\lVert y-c\tilde\mu p\rVert_2^2\). The start is floored into the open simplex. At that composition, \(\hat c =(\boldsymbol{m}^{\top}\boldsymbol{W}\boldsymbol{m})^{-1} \boldsymbol{m}^{\top}\boldsymbol{W}\boldsymbol{y}\) and \(\hat\phi=\boldsymbol{r}^{\top}\boldsymbol{W}\boldsymbol{r}/G\), with \(\boldsymbol{W}=\boldsymbol{S}_W(\boldsymbol{p})^{-1}\). Then \(\hat a=\hat\phi/\hat c\) and \(\hat n=\hat c^2/\hat\phi\). \(\hat n\) is a cell count only when the two platforms share their scale; it is not rounded.

\(\hat c\) is not the library-size ratio \(\mathbf{1}^{\top}y/\sum_j p_j S_j\).

Usage

single_cell_nuisance_scales(y, mean_signature_matrix, Sigma, p = NULL)

Arguments

y

Bulk vector of length \(G\).

mean_signature_matrix

Reference mean, \(G\times J\).

Sigma

Cell-level covariances, \(G\times G\times J\).

p

Optional composition. NULL runs the free-scale start.

Value

List with p, ols_scale (the least-squares \(c^{(0)}\), or NA when p was supplied), mean_scale, variance_scale, platform_scale and n_cells.

See also

deconvolute_single_cell(), deconvolute_ratios_deconrnaseq()

Examples

mu <- matrix(c(2, 0.2, 0.2, 2, 0.2, 3, 0.2, 3), nrow = 4)
sigma <- array(rep(diag(0.05, 4), 2), c(4, 4, 2))
set.seed(1)
y <- 30 * drop(mu %*% c(0.6, 0.4)) + rnorm(4, sd = 0.5)
single_cell_nuisance_scales(y, mu, sigma)
#> $p
#> [1] 0.596913 0.403087
#> 
#> $ols_scale
#> [1] 30.13053
#> 
#> $mean_scale
#> [1] 30.13053
#> 
#> $variance_scale
#> [1] 2.82171
#> 
#> $platform_scale
#> [1] 0.09364953
#> 
#> $n_cells
#> [1] 321.7371
#>