  An approach is proposed for inferring Granger causality between jointly
stationary, Gaussian signals from quantized data. First, a necessary and
sufficient rank criterion for the equality of two conditional Gaussian
distributions is proved. Assuming a partial finite-order Markov property, a
characterization of Granger causality in terms of the rank of a matrix
involving the covariances is presented. We call this the causality matrix. The
smallest singular value of the causality matrix gives a lower bound on the
distance between the two conditional Gaussian distributions appearing in the
definition of Granger causality and yields a new measure of causality. Then,
conditions are derived under which Granger causality between jointly Gaussian
processes can be reliably inferred from the second order moments of quantized
measurements. A necessary and sufficient condition is proposed for Granger
causality inference under binary quantization. Furthermore, sufficient
conditions are introduced to infer Granger causality between jointly Gaussian
signals through measurements quantized via non-uniform, uniform or high
resolution quantizers. Apart from the assumed partial Markov order and joint
Gaussianity, this approach does not require the parameters of a system model to
be identified. No assumptions are made on the identifiability of the jointly
Gaussian random processes through the quantized observations. The effectiveness
of the proposed method is illustrated by simulation results.
