Scope
Material moved out of the Nature Methods Article appendix to keep the main manuscript within editorial length. Notation matches the article Methods section and MuSiC-style mean references (Wang et al. 2019).
Notation
| Symbol | Role |
|---|---|
| g=1,\ldots,G | genes |
| j=1,\ldots,J | cell types |
| i=1,\ldots,N | samples (subjects); independence across i |
| \boldsymbol{y}=(y_{gi})\in\mathbb{R}_{+}^{G\times N} | bulk; column \boldsymbol{y}_{\cdot i} |
| \boldsymbol{\mu}=(\mu_{gj})\in\mathbb{R}^{G\times J} | mean signature; column \boldsymbol{\mu}_{\cdot j} |
| \boldsymbol{p}=(p_{ji})\in\,]0,1[^{J\times N} | proportions; column \boldsymbol{p}_{\cdot i} |
Dot notation: \boldsymbol{\mu}_{\cdot j} is the j-th column of \boldsymbol{\mu}; \boldsymbol{y}_{g\cdot} would be the g-th row. Classical texts often write the design matrix as \boldsymbol{X}; here the fixed design is the mean signature \boldsymbol{\mu}, as in MuSiC cross-subject means.
Ideal noiseless model for sample i:
\boldsymbol{y}_{\cdot i}=\boldsymbol{\mu}\,\boldsymbol{p}_{\cdot i}, \qquad y_{gi}=\sum_{j=1}^{J}\mu_{gj}p_{ji}. \tag{1}
With G>J and \operatorname{rank}(\boldsymbol{\mu})=J, the system is overdetermined (Abbas et al. 2009); uniqueness in the square full-rank case follows from the Rouché–Capelli theorem (Shafarevich and Remizov 2013).
Ordinary least squares
With additive residual \epsilon_{gi}, OLS minimises squared error for each sample (independently):
\hat{\boldsymbol{p}}_{\cdot i}^{\mathrm{OLS}} \equiv \arg\min_{\boldsymbol{p}_{\cdot i}} \bigl\|\boldsymbol{y}_{\cdot i}-\boldsymbol{\mu}\,\boldsymbol{p}_{\cdot i}\bigr\|_{2}^{2} = \arg\min_{\boldsymbol{p}_{\cdot i}} \sum_{g=1}^{G} \Bigl(y_{gi}-\sum_{j=1}^{J}\mu_{gj}p_{ji}\Bigr)^{2}. \tag{2}
When \boldsymbol{\mu}^{\top}\boldsymbol{\mu} is invertible, the normal equations give
\hat{\boldsymbol{p}}_{\cdot i}^{\mathrm{OLS}} = \bigl(\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\bigr)^{-1} \boldsymbol{\mu}^{\top}\boldsymbol{y}_{\cdot i}. \tag{3}
Existence of this inverse requires full column rank J (no cell-type profile is an exact linear combination of the others; parent/child lineages that are collinear cannot both be estimated).
Homoscedastic Gaussian noise and MLE
Under the classical linear model (Article Fig.~DAG panel a),
y_{gi}=\sum_{j=1}^{J}\mu_{gj}p_{ji}+\epsilon_{gi}, \qquad \epsilon_{gi}\sim\mathcal{N}(0,\sigma_{i}^{2}), \tag{4}
i.e. y_{gi}\sim\mathcal{N}\bigl(\sum_{j}\mu_{gj}p_{ji},\,\sigma_{i}^{2}\bigr) with fixed (exogenous) \boldsymbol{\mu}. Then the MLE for \boldsymbol{p}_{\cdot i} coincides with Equation 3.
Gauss–Markov assumptions (sketch)
Under weak exogeneity of \boldsymbol{\mu}, homoscedasticity \operatorname{Var}(\epsilon_{gi})=\sigma_{i}^{2} for all g, zero mean \mathbb{E}(\epsilon_{gi})=0, and uncorrelated residuals across genes, the OLS estimator is the BLUE (best linear unbiased estimator). With i.i.d. Gaussian errors the sample log-likelihood for sample i is (up to constants)
\ell(\boldsymbol{p}_{\cdot i},\sigma_{i}\,|\,\boldsymbol{y}_{\cdot i},\boldsymbol{\mu}) = -G\log\sigma_{i} -\frac{1}{2\sigma_{i}^{2}} \sum_{g=1}^{G} \Bigl(y_{gi}-\sum_{j=1}^{J}\mu_{gj}p_{ji}\Bigr)^{2},
so maximising \ell in \boldsymbol{p}_{\cdot i} recovers Equation 2. The residual variance MLE is \hat\sigma_{i}^{2}=G^{-1}\sum_{g}\hat\epsilon_{gi}^{2}. Independence across samples i=1,\ldots,N is the modelling assumption used throughout DeCovarT; microarray literature sometimes questions gene-wise independence (Efron 2009).
What DeCovarT changes
DeCovarT replaces gene-wise scalar noise with a multivariate convolution: latent \boldsymbol{x}_{\cdot j}\sim\mathcal{N}_{G}(\boldsymbol{\mu}_{\cdot j}, \boldsymbol{\Sigma}_{j}), and \boldsymbol{y}_{\cdot i}\,|\,\boldsymbol{p}_{\cdot i} \sim\mathcal{N}_{G}\bigl(\boldsymbol{\mu}\,\boldsymbol{p}_{\cdot i}, \sum_{j}p_{ji}^{2}\boldsymbol{\Sigma}_{j}\bigr), with precision \boldsymbol{\Theta}_{j}=\boldsymbol{\Sigma}_{j}^{-1} (typically from
gLasso). See the article Methods and softmax / ALR derivatives.