### Abstract

We consider the problem of using either grouped observations of a counting process over a period or prescribed constant mean counts to create an intensity function for a Non-Homogeneous Poisson Process (NHPP) that estimates the observed process, and satisfies several constraints. First, we require that the estimator's mean value over an observation interval be equal to the mean number of observations in that interval; second, we require that the intensity function be continuous; third, we require that the function be piecewise linear. Optionally, we may also require that the intensity value at the end of the period be identical to the intensity value at the beginning of the period, for application in contexts in which the period of interest is inherently cyclic, e.g., a day, or a week. An objective of the estimator is that it should be 'smooth,' which will be defined subsequently. Our approach is to define a class of continuous piecewise-linear intensity functions and formulate the problem as a constrained quadratic programming problem, approachable through the solution of a simultaneous set of linear equations. We describe the method, identify conditions under which feasible solutions are assured to exist, and study the behavior of the solutions on an example problem.

Original language | English (US) |
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Title of host publication | Proceedings of the 2014 Winter Simulation Conference, WSC 2014 |

Editors | Andreas Tolk, Levent Yilmaz, Saikou Y. Diallo, Ilya O. Ryzhov |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 498-509 |

Number of pages | 12 |

ISBN (Electronic) | 9781479974863 |

DOIs | |

State | Published - Jan 23 2015 |

Event | 2014 Winter Simulation Conference, WSC 2014 - Savannah, United States Duration: Dec 7 2014 → Dec 10 2014 |

### Publication series

Name | Proceedings - Winter Simulation Conference |
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Volume | 2015-January |

ISSN (Print) | 0891-7736 |

### Other

Other | 2014 Winter Simulation Conference, WSC 2014 |
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Country | United States |

City | Savannah |

Period | 12/7/14 → 12/10/14 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Modeling and Simulation
- Computer Science Applications

### Cite this

*Proceedings of the 2014 Winter Simulation Conference, WSC 2014*(pp. 498-509). [7019915] (Proceedings - Winter Simulation Conference; Vol. 2015-January). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/WSC.2014.7019915

**A continuous piecewise-linear NHPP intensity function estimator.** / Nicol, David Malcolm; Leemis, Lawrence M.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 2014 Winter Simulation Conference, WSC 2014.*, 7019915, Proceedings - Winter Simulation Conference, vol. 2015-January, Institute of Electrical and Electronics Engineers Inc., pp. 498-509, 2014 Winter Simulation Conference, WSC 2014, Savannah, United States, 12/7/14. https://doi.org/10.1109/WSC.2014.7019915

}

TY - GEN

T1 - A continuous piecewise-linear NHPP intensity function estimator

AU - Nicol, David Malcolm

AU - Leemis, Lawrence M.

PY - 2015/1/23

Y1 - 2015/1/23

N2 - We consider the problem of using either grouped observations of a counting process over a period or prescribed constant mean counts to create an intensity function for a Non-Homogeneous Poisson Process (NHPP) that estimates the observed process, and satisfies several constraints. First, we require that the estimator's mean value over an observation interval be equal to the mean number of observations in that interval; second, we require that the intensity function be continuous; third, we require that the function be piecewise linear. Optionally, we may also require that the intensity value at the end of the period be identical to the intensity value at the beginning of the period, for application in contexts in which the period of interest is inherently cyclic, e.g., a day, or a week. An objective of the estimator is that it should be 'smooth,' which will be defined subsequently. Our approach is to define a class of continuous piecewise-linear intensity functions and formulate the problem as a constrained quadratic programming problem, approachable through the solution of a simultaneous set of linear equations. We describe the method, identify conditions under which feasible solutions are assured to exist, and study the behavior of the solutions on an example problem.

AB - We consider the problem of using either grouped observations of a counting process over a period or prescribed constant mean counts to create an intensity function for a Non-Homogeneous Poisson Process (NHPP) that estimates the observed process, and satisfies several constraints. First, we require that the estimator's mean value over an observation interval be equal to the mean number of observations in that interval; second, we require that the intensity function be continuous; third, we require that the function be piecewise linear. Optionally, we may also require that the intensity value at the end of the period be identical to the intensity value at the beginning of the period, for application in contexts in which the period of interest is inherently cyclic, e.g., a day, or a week. An objective of the estimator is that it should be 'smooth,' which will be defined subsequently. Our approach is to define a class of continuous piecewise-linear intensity functions and formulate the problem as a constrained quadratic programming problem, approachable through the solution of a simultaneous set of linear equations. We describe the method, identify conditions under which feasible solutions are assured to exist, and study the behavior of the solutions on an example problem.

UR - http://www.scopus.com/inward/record.url?scp=84940496852&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84940496852&partnerID=8YFLogxK

U2 - 10.1109/WSC.2014.7019915

DO - 10.1109/WSC.2014.7019915

M3 - Conference contribution

AN - SCOPUS:84940496852

T3 - Proceedings - Winter Simulation Conference

SP - 498

EP - 509

BT - Proceedings of the 2014 Winter Simulation Conference, WSC 2014

A2 - Tolk, Andreas

A2 - Yilmaz, Levent

A2 - Diallo, Saikou Y.

A2 - Ryzhov, Ilya O.

PB - Institute of Electrical and Electronics Engineers Inc.

ER -