### Abstract

Consider a time series T and the set of polynomial models of T. We discuss two types of linearalities of T. The first type is measured by the maximal number of linear members of a polynomial model may have, denoted LN(T). An upper bound for LN(T) is given. The other is measured by the number of linear elements in a Gröbner Basis G of the ideal vanishing at all points of T, denoted LIN(G). Note that for each selected term order on the monomials of F[x_{1}, ..., x_{n}], there is a unique generating set, called the reduced Gröbner Basis, for the vanishing ideal mentioned above. We give a method to find linear members in G with respect to any term order. When selecting a graded term order (total degree prefered), we give a formula for the cardinality of LIN(G). Sample models are illustrated to support the theorems and propositions and they are constructed using the Buchberger Möller Algorithm.

Original language | English |
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Title of host publication | Proceedings of the Fifth IASTED International Conference on Modelling, Simulation, and Optimization |

Pages | 125-128 |

Number of pages | 4 |

State | Published - 1 Dec 2005 |

Event | 5th IASTED International Conference on Modelling, Simulation, and Optimization - Oranjestad, Aruba Duration: 29 Aug 2005 → 31 Aug 2005 |

### Publication series

Name | Proceedings of the IASTED International Conference on Modelling, Simulation, and Optimization |
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Volume | 2005 |

### Other

Other | 5th IASTED International Conference on Modelling, Simulation, and Optimization |
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Country | Aruba |

City | Oranjestad |

Period | 29/08/05 → 31/08/05 |

### Fingerprint

### Keywords

- Gröbner basis
- Polynomial models
- Time series

### Cite this

*Proceedings of the Fifth IASTED International Conference on Modelling, Simulation, and Optimization*(pp. 125-128). (Proceedings of the IASTED International Conference on Modelling, Simulation, and Optimization; Vol. 2005).

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*Proceedings of the Fifth IASTED International Conference on Modelling, Simulation, and Optimization.*Proceedings of the IASTED International Conference on Modelling, Simulation, and Optimization, vol. 2005, pp. 125-128, 5th IASTED International Conference on Modelling, Simulation, and Optimization, Oranjestad, Aruba, 29/08/05.

**Linearality of polynomial models of discrete time series.** / Li, Aihua; Saydam, Serpil.

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

TY - GEN

T1 - Linearality of polynomial models of discrete time series

AU - Li, Aihua

AU - Saydam, Serpil

PY - 2005/12/1

Y1 - 2005/12/1

N2 - Consider a time series T and the set of polynomial models of T. We discuss two types of linearalities of T. The first type is measured by the maximal number of linear members of a polynomial model may have, denoted LN(T). An upper bound for LN(T) is given. The other is measured by the number of linear elements in a Gröbner Basis G of the ideal vanishing at all points of T, denoted LIN(G). Note that for each selected term order on the monomials of F[x1, ..., xn], there is a unique generating set, called the reduced Gröbner Basis, for the vanishing ideal mentioned above. We give a method to find linear members in G with respect to any term order. When selecting a graded term order (total degree prefered), we give a formula for the cardinality of LIN(G). Sample models are illustrated to support the theorems and propositions and they are constructed using the Buchberger Möller Algorithm.

AB - Consider a time series T and the set of polynomial models of T. We discuss two types of linearalities of T. The first type is measured by the maximal number of linear members of a polynomial model may have, denoted LN(T). An upper bound for LN(T) is given. The other is measured by the number of linear elements in a Gröbner Basis G of the ideal vanishing at all points of T, denoted LIN(G). Note that for each selected term order on the monomials of F[x1, ..., xn], there is a unique generating set, called the reduced Gröbner Basis, for the vanishing ideal mentioned above. We give a method to find linear members in G with respect to any term order. When selecting a graded term order (total degree prefered), we give a formula for the cardinality of LIN(G). Sample models are illustrated to support the theorems and propositions and they are constructed using the Buchberger Möller Algorithm.

KW - Gröbner basis

KW - Polynomial models

KW - Time series

UR - http://www.scopus.com/inward/record.url?scp=33751212784&partnerID=8YFLogxK

M3 - Conference contribution

SN - 0889865264

SN - 9780889865266

T3 - Proceedings of the IASTED International Conference on Modelling, Simulation, and Optimization

SP - 125

EP - 128

BT - Proceedings of the Fifth IASTED International Conference on Modelling, Simulation, and Optimization

ER -