## Abstract

The maximum entropy (ME) principle, as it is often invoked in the context of time-series analysis, suggests the selection of a power spectrum which is consistent with autocorrelation data and corresponds to a random process least predictable from past observations. We introduce and compare a class of spectra with the property that the underlying random process is least predictable at any given point from the complete set of past and future observations. In this context, randomness is quantified by the size of the corresponding smoothing error and deterministic processes are characterized by integrability of the inverse of their power spectral densities - as opposed to the log-integrability in the classical setting. The power spectrum which is consistent with a partial autocorrelation sequence and corresponds to the most random (MR) process in this new sense, is no longer rational but generated by finitely many fractional-poles.

Original language | English (US) |
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Pages (from-to) | 2841-2851 |

Number of pages | 11 |

Journal | IEEE Transactions on Information Theory |

Volume | 53 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2007 |

### Bibliographical note

Funding Information:Manuscript received January 26, 2006; revised January 17, 2007. This work was supported by the National Science Foundation and the Air Force Office of naval Research. The material in this paper was presented in part at the 45th IEEE Conference on Decision and Control, San Diego, CA, December 2006. The author is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: tryphon@ece. umn.edu). Communicated by V. A. Vaishampayan, Associate Editor for At Large. Color versions of Figures 1–3 are available online at http://ieeexplore.ieee. org. Digital Object Identifier 10.1109/TIT.2007.901149

## Keywords

- Entropy rate
- Predictability
- Randomness
- Smoothing
- Time-arrow