Causality in branching time series
Theoretical Systems Biology Retreat
(German Cancer Resarch Center, Heidelberg)
Ellwangen · 22 June 2016
Gene expression data for branching lineages
- branching pseudotime evolution of gene expression
Gene expression data for branching lineages
- branching pseudotime evolution of gene expression
Gene expression data for branching lineages
- branching pseudotime evolution of gene expression
- here: ordered by switching time
▷ Learn about gene regulation?
- Gene that switches early and does not branch candidate for regulator?
- Gene that branches in different fates candidate for regulated gene?
(Causal) time series analysis
Model fitting
- ODE based model
Oconce, Haghverdi, Müller & Theis, Bioinformatics (2015) - Boolean network
Moignard, Woodhoouse ... Göttgens, Nat. Biotechn. (2015)
Granger Causality/Transfer Entropy
- Granger Causality $\simeq$ Transfer Entropy
Barnett, Phys. Rev. Lett. (2009)
Causal Inference Methods
- PC algorithm
Spirtes et al. (2000) - Structural equation models
e.g. Peters, ..., Schölkopf, NIPS (2013)
Convergent Cross Mapping
- Taken's Theorem
Takens (1980)
Model fitting
- ODE based model
Oconce, Haghverdi, Müller & Theis, Bioinformatics (2015)
▷ edges using GENIE3, then optimize for $ u_g(\hat{\boldsymbol{x}})$
$$\frac{d}{dt} \hat x_g = \alpha u_g(\hat{\boldsymbol{x}}) - \lambda \hat x_g,~~ \textstyle p(\mathcal{D}|\theta) \propto \prod_{t=1}^T \exp\big(-\frac{(x_{tg} - \hat x_g(t,\theta))^2}{2 \sigma^2}\big) $$ - Discrete state space model
Moignard, Woodhouse ... Goettgens, Nat. Biotech. (2015) - generate discrete state graph of 1-gene transitions
- check consistency of trial Boolean networks
(Causal) time series analysis
Model fitting
- ODE based model
Oconce, Haghverdi, Müller & Theis, Bioinformatics (2015) - Boolean network
Moignard, Woodhoouse ... Göttgens, Nat. Biotechn. (2015)
Granger Causality/Transfer Entropy
- Granger Causality $\simeq$ Transfer Entropy
Barnett, Phys. Rev. Lett. (2009)
Causal Inference Methods
- PC algorithm
Spirtes et al. (2000) - Structural equation models
e.g. Peters, ..., Schölkopf, NIPS (2013)
Convergent Cross Mapping
- Taken's Theorem
Takens (1980)
Granger Causality / Causal Inference Methods
Conditional independence tests: $$ X_{ti} \perp\!\!\!\perp X_{(t-1)j} \;|\; X_{(t-1)k_1}, X_{(t-1)k_2}, ... $$
- Only works well if there is a high amount of dynamic information and no error in time variable. Both is not the case!
- Instead: time series with geometric constraints associated with branchings. Cannot be exploited by local independence tests.
(Causal) time series analysis
Model fitting
- ODE based model
Oconce, Haghverdi, Müller & Theis, Bioinformatics (2015) - Boolean network
Moignard, Woodhoouse ... Göttgens, Nat. Biotechn. (2015)
Granger Causality/Transfer Entropy
- Granger Causality $\simeq$ Transfer Entropy
Barnett, Phys. Rev. Lett. (2009)
Causal Inference Methods
- PC algorithm
Spirtes et al. (2000) - Structural equation models
e.g. Peters, ..., Schölkopf, NIPS (2013)
Convergent Cross Mapping
- Taken's Theorem
Takens (1980)
Convergent Cross Mapping
Idea: Reconstruct regulating gene from the history of the regulated gene.
Convergent Cross Mapping
Idea: Reconstruct regulating gene from the history of the regulated gene.
▷ Can exploit branching geometry!
▷ But: no statistical formulation!
New method: statistical test for coupling functions
Rationale
\begin{align} X_{tj} & = f_j(X_{(t-1)j}, X_{(t-1)i_1}, X_{(t-1)i_2}, ...) \\ \Leftrightarrow \quad & F_j(X_{tj},X_{(t-1)j}, X_{(t-1)i_1}, X_{(t-1)i_2}, ...) = 0 \end{align}Statistically test whether $X_{(t-1)i_1}$ contributes to $F_j$ by testing for the existence of a function $g$ $$ g(X_{tj},X_{(t-1)j}, X_{(t-1)i_2}, ...) = X_{(t-1)i_1}. $$
Implicit function theorem: if there is no $g$, $F_j$ is constant w.r.t $X_{(t-1)i_1}$.
Definition of test: bivariate case
Consider $X_{tj} = f_j(X_{(t-1)j},X_{(t-1)i})$. How to statistically test for the existence of mapping the $g : \mathcal{X}_j \rightarrow \mathcal{X}_i ?$
$$ z_{ji} = \frac{\overline{\Delta}_{ji} - \overline{\delta}_{i}}{\sqrt{(\hat\sigma_{ji}^{\Delta})^2/n + (\hat\sigma_{ji}^{\delta})^2/n}}$$ where $$ \Delta_{tji}^{(a)} = \frac{1}{|{\mathcal{N}}_{tj}^{(a)}|} \sum_{t' \in {\mathcal{N}}_{tj}^{(a)}} d_{tt'i} $$ is the variation across realizations for a fixed value in the domain $\mathcal{X}_j$ $$ {\delta}_{ti}^{(a)} = \frac{1}{2}({d}_{t(t-1)i}^{(a)} + {d}_{(t+1)ti}^{(a)}) $$ is the dynamic variation in the codomain.
Numerical experiment
Summary
- Goal: infer gene regulation from pseudotime order.
- Problem: very little dynamic information, traditional approaches fail, instead use branching geometry.
- Convergent Cross Mapping: can exploit branching geometry but serious flaws.
- New method: statistical test for coupling functions via implicit function theorem.