Abstract
Extending the notion of an information loss rate to general processes is not trivial. It is even more difficult than generalizing the concept of information loss from discrete to continuous RVs. In this section we propose one possible generalization, making similar restrictions as in Chap. 3. Specifically, we focus on piecewise bijective functions (PBFs) and continuous-valued, one-dimensional, discrete-time stationary stochastic processes.
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Notes
- 1.
If the upper and lower bound coincide, then \(\mathbf {Y}\) is Markov—this suggests a continuous-valued equivalent of lumpability and a corresponding information-theoretic characterization (cf. [Gei14, Proposition 3.14] and [GT14]).
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Geiger, B.C., Kubin, G. (2018). Piecewise Bijective Functions and Continuous Inputs. In: Information Loss in Deterministic Signal Processing Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-59533-7_7
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DOI: https://doi.org/10.1007/978-3-319-59533-7_7
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