DeCovarT MLE of cellular proportions for one bulk sample

Estimates $\hat{\boldsymbol{p}}=\arg\max_{\boldsymbol{p}} \ell_{\boldsymbol{y}\,|\,\boldsymbol{\zeta}}(\boldsymbol{p})$ under the Gaussian convolution model $$ \boldsymbol{y}\,|\,(\boldsymbol{\zeta},\boldsymbol{p}) \sim\mathcal{N}_{G}\!\bigl(\boldsymbol{\mu}\boldsymbol{p},\, \boldsymbol{\Sigma}(\boldsymbol{p})\bigr), \qquad \boldsymbol{\Sigma}(\boldsymbol{p})=\sum_{j=1}^{J}p_j^{2}\boldsymbol{\Sigma}_j, $$ subject to the simplex constraint $\mathbf{1}^{\mathsf{T}}\boldsymbol{p}=1$, $\boldsymbol{p}\ge\mathbf{0}$. Optimisation is performed in unconstrained coordinates $\boldsymbol{\rho}\in\mathbb{R}^{J-1}$ via $\boldsymbol{p}=\boldsymbol{\psi}(\boldsymbol{\rho})$ (Marquardt–Levenberg default; see other methods below and vignette("softmax-alr-derivatives", package = "DeCovarT")).

Usage

deconvolute_ratios_cibersort(y, mean_signature_matrix)

deconvolute_ratios_lsfit(y, mean_signature_matrix)

deconvolute_ratios_rlm(y, mean_signature_matrix)

deconvolute_ratios_nnls(y, mean_signature_matrix)

deconvolute_ratios_deconrnaseq(y, mean_signature_matrix)

deconvolute_ratios_Marquardt_Levenberg(
  y,
  mean_signature_matrix,
  Sigma,
  epsilon = 10^-4,
  itmax = 200
)

deconvolute_ratios_simulated_annealing(
  y,
  mean_signature_matrix,
  Sigma,
  epsilon = 10^-4,
  itmax = 200
)

deconvolute_ratios_L_BFGS_B(
  y,
  mean_signature_matrix,
  Sigma,
  epsilon = 10^-4,
  itmax = 200
)

deconvolute_ratios_Newton_Raphson(
  y,
  mean_signature_matrix,
  Sigma,
  epsilon = 10^-4,
  itmax = 200
)

deconvolute_ratios_gradient_descent(
  y,
  mean_signature_matrix,
  Sigma,
  epsilon = 10^-4,
  itmax = 200
)

Arguments

y: Bulk expression vector $\boldsymbol{y}\in\mathbb{R}^{G}$ (one heterogeneous sample).
mean_signature_matrix: Mean signature $\boldsymbol{\mu}\in\mathcal{M}_{G\times J}$ (columns = cell types; plug-in for latent profiles).
Sigma: Array $(\boldsymbol{\Sigma}_j)_{j=1}^{J}\in\mathcal{M}_{G\times G\times J}$ of cell-type covariances.
epsilon, itmax: Absolute convergence tolerance and maximum number of iterations for the optimiser.

Value

Named numeric vector $\hat{\boldsymbol{p}}$ on the simplex. Benchmark metrics are computed by deconvolute_ratios().

Details

Plug-in signature. Argument mean_signature_matrix is the mean $\boldsymbol{\mu}$, used as a proxy for the unobserved cell-type profiles $\boldsymbol{x}_{\cdot j}$. This is the frequentist plug-in; recovering sample-specific latents is a Bayesian / MAP problem (.map_gaussian_convolution()).

Functions

deconvolute_ratios_cibersort(): Linear baseline $\hat{\boldsymbol{y}}=\boldsymbol{\mu}\hat{\boldsymbol{p}}$ via nu-SVR (CIBERSORT-style); no covariance prior is used.
deconvolute_ratios_lsfit(): Ordinary least squares for $\boldsymbol{y}\approx\boldsymbol{\mu}\boldsymbol{p}$ (stats::lsfit()), following Abbas et al. (2009) ; estimates are projected back onto the simplex.
deconvolute_ratios_rlm(): Robust linear model $\boldsymbol{y}\approx\boldsymbol{\mu}\boldsymbol{p}$ (MASS::rlm()), as in Monaco et al. (2019) .
deconvolute_ratios_nnls(): Non-negative least squares for $\boldsymbol{y}\approx\boldsymbol{\mu}\boldsymbol{p}$ (nnls::nnls()), then simplex projection.
deconvolute_ratios_deconrnaseq(): Equality- and inequality-constrained least squares on the simplex (limSolve::lsei()), in the spirit of deconRNASeq.
deconvolute_ratios_simulated_annealing(): Simulated annealing on $\boldsymbol{\rho}$ (stats::optim() with method = "SANN").
deconvolute_ratios_L_BFGS_B(): Box-constrained L-BFGS-B directly in $\boldsymbol{p}$ (stats::optim() method = "L-BFGS-B").
deconvolute_ratios_Newton_Raphson(): Newton–Raphson / nlminb on $\boldsymbol{\rho}$ using analytic gradient and Hessian (stats::nlminb()).
deconvolute_ratios_gradient_descent(): BFGS quasi-Newton ascent on $\boldsymbol{\rho}$ (stats::optim() method = "BFGS").

References

Abbas AR, Wolslegel K, Seshasayee D, Modrusan Z, Clark HF (2009). “Deconvolution of Blood Microarray Data Identifies Cellular Activation Patterns in Systemic Lupus Erythematosus.” PloS One, 4. doi:10.1371/journal.pone.0006098 .

Monaco G, Lee B, Xu W, Mustafah S, Hwang YY, Carré C, Burdin N, Visan L, Ceccarelli M, Poidinger M, Zippelius A, Pedro de Magalhães J, Larbi A (2019). “RNA-Seq Signatures Normalized by mRNA Abundance Allow Absolute Deconvolution of Human Immune Cell Types.” Cell Reports, 26. doi:10.1016/j.celrep.2019.01.041 .

Examples

set.seed(1)
genes <- paste0("g", 1:2)
cts <- paste0("ct", 1:2)
mu <- matrix(c(20, 22, 22, 20), nrow = 2, dimnames = list(genes, cts))
Sigma <- array(
  c(1, 0, 0, 1, 1, 0, 0, 1),
  dim = c(2, 2, 2),
  dimnames = list(genes, genes, cts)
)
y <- drop(mu %*% c(0.6, 0.4) + rnorm(2, sd = 0.1))
deconvolute_ratios_Marquardt_Levenberg(y, mu, Sigma, itmax = 50)
#>       ct1       ct2 
#> 0.5605665 0.4394335