Model

We consider a stochastic dynamical system of \(N\) variables in which the state at time \(t+1\), \(\vec{\sigma} (t+1)\) is governed by the state at time \(t\), \(\vec{\sigma} (t)\), according to the kinetic Ising model

\[\label{eq:kIsingProb} P[\sigma_i(t+1)|\vec{\sigma}(t)] = \frac{\exp [ \sigma_i(t+1) H_i(t)]}{\mathcal{N}}\]

where \(H_i(t) = \sum_j W_{ij} \sigma_j(t)\) represents the local field acting on \(\sigma_i(t+1)\), \(W_{ij}\) represents the interaction from variable \(j\) to variable \(i\), and \(\mathcal{N} = \sum_{\sigma_i(t+1)} \exp[\sigma_i(t+1) H_i(t)]\) represents a normalizing factor. Intuitively, the state \(\sigma_i(t+1)\) tends to align with the local field \(H_i(t)\).

The inference methods in the literature usually extract the interactions \(W_{ij}\) from the time series \(\{ \vec{\sigma}(t) \}\) of entire variables. In realistic situations, however, the experimental data often contains only subset of variables. Our aim was to develop a data driven approach that can infer

(i) the interactions between variables (including observed-to-observed, hidden-to-observed, observed-to-hidden, and hidden-to-hidden interactions);

(ii) the configurations of hidden variables;

(iii) the number of hidden variables.