### Abstract

Language | English |
---|---|

Pages | 1077-1092 |

Number of pages | 16 |

Journal | IEEE Journal on Selected Topics in Signal Processing |

Volume | 12 |

Issue number | 5 |

Early online date | 29 Jun 2018 |

DOIs | |

Publication status | Published - 31 Oct 2018 |

### Fingerprint

### Keywords

- multi-state processes
- hidden semi Markov models
- explicit random duration
- hypothesis testing
- error exponent
- large deviations principle
- threshold effect
- Lyapunov exponent

### Cite this

*IEEE Journal on Selected Topics in Signal Processing*,

*12*(5), 1077-1092. https://doi.org/10.1109/JSTSP.2018.2851506

}

*IEEE Journal on Selected Topics in Signal Processing*, vol. 12, no. 5, pp. 1077-1092. https://doi.org/10.1109/JSTSP.2018.2851506

**Optimal detection and error exponents for hidden semi-Markov models.** / Bajović, Dragana; He, Kanghang; Stanković, Lina; Vukobratović, Dejan; Stanković, Vladimir.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Optimal detection and error exponents for hidden semi-Markov models

AU - Bajović, Dragana

AU - He, Kanghang

AU - Stanković, Lina

AU - Vukobratović, Dejan

AU - Stanković, Vladimir

PY - 2018/10/31

Y1 - 2018/10/31

N2 - We study detection of random signals corrupted by noise that over time switch their values (states) between a finite set of possible values, where the switchings occur at unknown points in time. We model such signals as hidden semi-Markov signals (HSMS), which generalize classical Markov chains by introducing explicit (possibly non-geometric) distribution for the time spent in each state. Assuming two possible signal states and Gaussian noise, we derive optimal likelihood ratio test and show that it has a computationally tractable form of a matrix product, with the number of matrices involved in the product being the number of process observations. The product matrices are independent and identically distributed, constructed by a simple measurement modulation of the sparse semi-Markov model transition matrix that we define in the paper. Using this result, we show that the Neyman-Pearson error exponent is equal to the top Lyapunov exponent for the corresponding random matrices. Using theory of large deviations, we derive a lower bound on the error exponent. Finally, we show that this bound is tight by means of numerical simulations.

AB - We study detection of random signals corrupted by noise that over time switch their values (states) between a finite set of possible values, where the switchings occur at unknown points in time. We model such signals as hidden semi-Markov signals (HSMS), which generalize classical Markov chains by introducing explicit (possibly non-geometric) distribution for the time spent in each state. Assuming two possible signal states and Gaussian noise, we derive optimal likelihood ratio test and show that it has a computationally tractable form of a matrix product, with the number of matrices involved in the product being the number of process observations. The product matrices are independent and identically distributed, constructed by a simple measurement modulation of the sparse semi-Markov model transition matrix that we define in the paper. Using this result, we show that the Neyman-Pearson error exponent is equal to the top Lyapunov exponent for the corresponding random matrices. Using theory of large deviations, we derive a lower bound on the error exponent. Finally, we show that this bound is tight by means of numerical simulations.

KW - multi-state processes

KW - hidden semi Markov models

KW - explicit random duration

KW - hypothesis testing

KW - error exponent

KW - large deviations principle

KW - threshold effect

KW - Lyapunov exponent

U2 - 10.1109/JSTSP.2018.2851506

DO - 10.1109/JSTSP.2018.2851506

M3 - Article

VL - 12

SP - 1077

EP - 1092

JO - IEEE Journal on Selected Topics in Signal Processing

T2 - IEEE Journal on Selected Topics in Signal Processing

JF - IEEE Journal on Selected Topics in Signal Processing

SN - 1932-4553

IS - 5

ER -