Measure observability by the generalized informational correlation

Informational correlation coefficient (ICC) can be used to measure the degree of observability for a system. In this paper, we define the generalized informational correlation coefficient (GICC), which is suitable for both discrete and continuous random variables. For the case in which the probability density functions (PDFs) are regularly supersummable, we obtain the exact value of GICC. Moreover, for the linear, stochastically autonomous system, we derive the explicit formula for the degree of observability, and prove the equivalence between the proposed measure and the traditional rank condition. Finally, a simple example is given to compare the discrete state case and the continuous state case.