Fig. 3. Extended framework for inferring a regulatory network from noisy data.

a A regulatory network with 1D regulation from A to B and 2D regulation from A and B to C. b The threshold for the regulation-detection score ($S^{\mathrm{thres}}=0.9-0.005\times \left(\text{noise level}\right)$, green dashed line) distinguishes true (A ⊣ B) and false regulation (A → C). c The fraction of data satisfying $S_{{\mathbf{X}}^{\sigma }}^Y>S^{\mathrm{thres}}$, total regulation score (${\mathrm{TRS}}_{{\mathbf{X}}^{\sigma }}^Y$), is used to infer the regulation. Specifically, ${\mathrm{TRS}}_{{\mathbf{X}}^{\sigma }}^Y>{\mathrm{TRS}}^{\mathrm{thres}}$ is used where ${\mathrm{TRS}}^{\mathrm{thres}}=0.9-0.01\times \text{(noise level)}$ (green dashed line). Here, data are presented as box plots (n = 100), in which the box bounds the IQR divided by the median, and whiskers extend to a maximum of 1.5 × IQR beyond the box. d In CFL, direct negative regulation from A to C exists. e On the other hand, in SFL, the regulatory chain A ⊣ B → C induces an indirect negative regulation from A to C. f, g ${\mathrm{\Delta }}_{B^{+}}^C\left(A\right)$ cannot distinguish between the direct and indirect regulations in the presence of noise because $S_{A^{-}{B}^{+}}^C>S_{A^{+}{B}^{+}}^C$ for both CFL and SFL, indicating the presence of regulation $\begin{array}{c}\hfill A⊣\hfill \\ \hfill B\to \hfill \end{array}C$. h, i $S_{A_{\mathrm{shuffled}}^{-}{B}^{+}}^C$ with the surrogate time series of A can be used to distinguish between the indirect and direct regulations. To disrupt the information of A, the time series of A is shuffled (h). In the presence of direct regulation (CFL), but not indirect regulation (SFL), $S_{A_{\mathrm{shuffled}}^{-}{B}^{+}}^C$ is significantly smaller than the original $S_{A^{-}{B}^{+}}^C$ (p-value < 0.001). j By including the surrogate test, our extended framework can successfully infer IFL, CFL, and SFL even from noisy time series. k F₂ score of our inference method when the level of noise increases from 0 to 20%. Here, the mean of the F₂ score for 10 data sets is calculated. Each data set consists of 100 time series, which are simulated with different initial conditions. Source data are provided as a Source Data file.