The paper presents a dynamic data-driven symbolic approach to construct generative models of causal cross-dependence among different sources of (possibly heterogeneous) measurements. The main objective here is to identify the input-output relationships in the underlying dynamical system using sensory data only. Synchronized pairs of input and output time series are first independently symbolized via partitioning the individual data sets in their respective range spaces. A generative model is then obtained to capture cross-dependency in the symbolic input-output dynamics as a variable-memory cross D-Markov (also called xD-Markov) machine, which is different from the standard PFSA. The proposed input-output model has been validated on charging-discharging data sets of a lead-acid battery. The cross-dependency features of current-voltage patterns during charging-discharging cycles have been used to estimate and predict the parameters of battery performance (e.g., State-of-Charge (SOC)) and health (e.g., State-of-Health (SOH)).