Table 2 List of notation
From: A Dirichlet-multinomial mixed model for determining differential abundance of mutational signatures
Matrix factorisation | ||
|---|---|---|
Data | N | Number of samples or observations (input of NMF) |
K | Number of signatures | |
F | Number of mutational categories or features | |
\(\textbf{V}\) | (\(F \times N\)) Count matrix of categorised mutations | |
Parameters | \(\textbf{Y}^{*}\) | (\(K \times N\)) Matrix of exposures. For the sections below, \(\textbf{Y}=\textbf{Y}^{*\top }\) |
\(\textbf{S}\) | (\(F \times K\)) matrix of signature definitions | |
Indices | f | Index for features (mutational types such as ACA\(\rightarrow\)ATA, for instance) |
Models of differential abundance | ||
|---|---|---|
Data | N | Number of observations with \(N\ge N_s\) |
\(N_s\) | Number of patients, equivalent to the number of samples | |
K | Number of categories/signatures | |
P | Number of fixed effect covariates | |
Q | Number of precision covariates | |
\(\textbf{X}\) | (\(N \times P\)) covariate matrix with P=2 in our case, without loss of generality | |
\(\textbf{Z}\) | (\(N \times N_s\)) boolean matrix for random intercepts | |
\(\textbf{D}\) | (\(N \times Q\)) precision covariate matrix with \(\textbf{D}=\textbf{X}\) in our case | |
\(\textbf{Y}\) | (\(N \times K\)) response matrix of exposures | |
\(T_j\) | Mutational toll of observation j. \(T_j=\sum _{k'}Y_{jk'}\) | |
\(M_i^{(1)}\), \(M_i^{(2)}\) | (\(T_i^{(1)} \times 1\)) sequence of mutations of each observation of patient i | |
Parameters | \(\varvec{\beta }\) | (\(P \times (K-1)\)) Matrix of coefficients for fixed effects |
\(\mathbf {\Sigma _{\varvec{\beta }_p}}\) | (\((K-1) \times (K-1)\)) covariance matrix of the pth row of \(\varvec{\beta }\) | |
\(\textbf{U}\) | (\(N_s \times (K-1)\)) matrix of coefficients for random effects | |
\(\mathbf {\Sigma }\) | (\((K-1) \times (K-1)\)) random effect covariance matrix | |
\(\varvec{\lambda }\) | (\(Q \times 1\)) vector of precision parameters | |
Indices | i | Index for patients |
j | Index for observations (patient and group combinations) | |
p | Index for covariates | |
\(k'\) | Index for signatures/categories | |
k | Index for signatures/categories (log-ratio, or the first \(K-1\)) | |
l | Index for mutations in an observation | |
Random variables | \(\varvec{\alpha }\) | Parameter for the Dirichlet or Dirichlet-multinomial |
\(\bar{\varvec{\alpha }}\) | Parameter for the Dirichlet or Dirichlet-multinomial (compositional) | |
Other notation | ||
|---|---|---|
| Â | \(\mathbf {0_K}\) | Vector of zeros of length K |
| Â | \(\textbf{Y}, \mathbf {y_j}, {y_{jk}}\) | Notation for matrices, vectors and scalars. |
| Â | P, p | Dimensions, indices |
Simulations | \(\pi\) | For simulations C1-3, mixing proportion. Lower values indicate no mixing (no differential abundance). |