• Motivation

• The ZINB-WaVE model

• Comparison with existing approaches

• General and flexible a zero-inflated negative binomial (ZINB) model.

• Extract low-dimensional signal from the data.

• Accounting for zero inflation (dropouts), over-dispersion, and the count nature of the data.

• The model includes a sample-level intercept, which serves as a global-scaling normalization factor.

• Gives the user the ability to include both gene-level and sample-level covariates.

• Inclusion of observed and unobserved sample-level covariates enables normalization for complex, non-linear effects (batch effects)

• Gene-level covariates may be used to adjust for sequence composition effects, such as gene length and GC-content effects.

The mean of the NB distribution is \(\mu\) and its variance is: \[ \sigma^2 = \mu + \frac{\mu^2}{\theta}. \] In particular, the NB distribution boils down to a Poisson distribution when \(\theta = +\infty\).

For any \(\pi\in[0,1]\), the PMF of the zero-inflated negative binomial (ZINB) distribution is given by

\[ f(y;\mu,\theta, \pi) = \pi \delta_0(y) + (1 - \pi) f_N(y;\mu,\theta), \quad \forall y\in\mathbb{N}, \]

where \(\delta_0(\cdot)\) is the Dirac function.

Here, \(\pi\) can be interpreted as the probability that a \(0\) is observed instead of the actual count, resulting in an inflation of zeros compared to the NB distribution, hence the name ZINB.

Given \(n\) samples and \(J\) genes, let \(Y_{ij}\) denote the count of gene \(j\) (for \(j=1,\ldots,J\)) for sample \(i\) (for \(i=1,\ldots,n\)).

• ZINB-WaVE is a general and flexible method for low-dimensional signal extraction from noisy, zero-inflated count data.

• It works well for projecting single-cell data in two dimensions as a dimensionality reduction step prior to clustering or pseudotime inference.

• We are extending the model to provide normalized values for visualization (residuals).

• And to supervised differential expression problems (test \(\beta = 0\)).

Co-authors

• Fanny Perradeau
• Svetlana Gribkova
• Sandrine Dudoit
• Jean-Philippe Vert

John Ngai Lab

• John Ngai
• David Stafford
• Justin Choi

ZINB-Wave preprint: doi.org/10.1101/125112

zinbwave package: bioconductor.org/packages/zinbwave

These slides: rpubs.com/daviderisso

To estimate the parameters \((\alpha, \beta, \gamma, W, \theta)\), we follow a penalized maximum likelihood approach, to reduce overfitting and improve the numerical stability of the optimization problem in the setting of many parameters.

\[ \max_{\beta,\gamma,W,\alpha,\zeta} \left\{\ell(\beta,\gamma,W,\alpha,\zeta) - \text{Pen}(\beta,\gamma,W,\alpha,\zeta) \right\}\,, \]

where \(\zeta = \log \theta\) and

\[ \text{Pen}(\beta,\gamma,W,\alpha,\zeta) = \frac{\epsilon_{\beta}}{2} \|\beta^0\|^2 + \frac{\epsilon_{\gamma}}{2} \|\gamma^0\|^2 + \frac{\epsilon_{W}}{2}\|W\|^2 + \frac{\epsilon_{\alpha}}{2}\|\alpha\|^2 + \frac{\epsilon_{\zeta}}{2}\text{Var}(\zeta) \,. \]

• \(\beta^0\) and \(\gamma^0\) denote the matrices \(\beta\) and \(\gamma\) without the rows corresponding to the intercepts if an unpenalized intercept is included in the model.

• \(\|\cdot\|\) is the Frobenius matrix norm ( \(\|A\| = \sqrt{\text{tr}(A^*A)}\), where \(A^*\) denotes the conjugate transpose of \(A\)).

• The penalty shrinks the estimated parameters to \(0\), except for the cell and gene-specific intercepts which are not penalized and the dispersion parameters which are shrunk towards a constant value across genes (cf. edgeR).

• The penalty also ensures that at the optimum \(W\) and \(\alpha^T\) have orthogonal columns, which is useful for visualization or interpretation of latent factors.

The penalized likelihood is not concave, making its maximization computationally challenging.

We instead find a local maximum, starting from a smart initialization and iterating a numerical optimization scheme until local convergence.