## CHARACTERIZATION OF THE PARTIAL AUTOCORRELATION FUNCTION OF NONSTATIONARY TIME SERIES

Citations: | 9 - 5 self |

### BibTeX

@MISC{Dégerine_characterizationof,

author = {Serge Dégerine and Sophie Lambert-lacroix},

title = {CHARACTERIZATION OF THE PARTIAL AUTOCORRELATION FUNCTION OF NONSTATIONARY TIME SERIES},

year = {}

}

### OpenURL

### Abstract

The second order properties of a process are usually characterized by the autocovariance function. In the stationary case, the parameterization by the partial autocorrelation function is relatively recent. We extend this parameterization to the nonstationary case. The advantage of this function is that it is subject to very simple constraints in comparison with the auto-covariance function which must be nonnegative definite. As in the stationary case, this parameterization is well adapted to autoregressive models or to the identification of deterministic processes.

### Citations

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(Show Context)
Citation Context ...(PACF) is relatively recent. This result is established by Barndorff-Nielsen and Schou [1] for autoregressive processes and by Ramsey [18] for the general stationary case. It is also observed by Burg =-=[2]-=-, in the signal processing field, where the partial autocorrelation coefficients are called reflection coefficients. In fact, the one-to-one correspondence between autocorrelation function (ACF) and P... |

25 |
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(Show Context)
Citation Context ...years ago by Yule [19], the parameterization of a stationary time series by the partial autocorrelation function (PACF) is relatively recent. This result is established by Barndorff-Nielsen and Schou =-=[1]-=- for autoregressive processes and by Ramsey [18] for the general stationary case. It is also observed by Burg [2], in the signal processing field, where the partial autocorrelation coefficients are ca... |

15 |
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(Show Context)
Citation Context ... a stationary time series by the partial autocorrelation function (PACF) is relatively recent. This result is established by Barndorff-Nielsen and Schou [1] for autoregressive processes and by Ramsey =-=[18]-=- for the general stationary case. It is also observed by Burg [2], in the signal processing field, where the partial autocorrelation coefficients are called reflection coefficients. In fact, the one-t... |

14 | On the theory of correlation for any number of variables, treated by a new system of notation - Yule - 1907 |

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(Show Context)
Citation Context ...ath [16] propose to use the triangular square root. Dégerine [4] gives a general approach for the multivariate stationary case and introduces the canonical PACF. Estimation techniques are proposed in =-=[17]-=-, [10] and [6]. In the nonstationary case, the partial correlation coefficients appear in 2sthe generalization of Schur and Levinson-Durbin algorithms for non-locally deterministic processes by Lev-Ar... |

12 |
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(Show Context)
Citation Context ...�= 0 is satisfied. Moreover these last points establish clearly the sufficient condition of the theorem. ♦ A fundamental difference with the stationary case concerns the WoldCramér decomposition (see =-=[3]-=-) X(t)=U(t)+V (t), t ∈ ZZ , where U(·) is purely nondeterministic and V (·) is deterministic. When X(·) is a stationary AR(p) process, only one of these two components is possible 21saccording to σ 2 ... |

12 |
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(Show Context)
Citation Context ...the one-to-one correspondence between autocorrelation function (ACF) and PACF is a classical result in orthogonal polynomial theory when the spectral measure has an infinite set of growth points (see =-=[12]-=-). The simplicity of the constraints on the PACF gave birth to many autoregressive estimation methods ([2], [9], [5]). The extension of the PACF to the multivariate stationary case is a delicate point... |

6 |
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(Show Context)
Citation Context ...ynomial theory when the spectral measure has an infinite set of growth points (see [12]). The simplicity of the constraints on the PACF gave birth to many autoregressive estimation methods ([2], [9], =-=[5]-=-). The extension of the PACF to the multivariate stationary case is a delicate point. The difficulty to define a partial autocorrelation matrix lies in the square root choice in normalizing the partia... |

5 |
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(Show Context)
Citation Context ...y to define a partial autocorrelation matrix lies in the square root choice in normalizing the partial autocovariance. Morf, Viera and Kailath [16] propose to use the triangular square root. Dégerine =-=[4]-=- gives a general approach for the multivariate stationary case and introduces the canonical PACF. Estimation techniques are proposed in [17], [10] and [6]. In the nonstationary case, the partial corre... |

5 |
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(Show Context)
Citation Context ...l polynomial theory when the spectral measure has an infinite set of growth points (see [12]). The simplicity of the constraints on the PACF gave birth to many autoregressive estimation methods ([2], =-=[9]-=-, [5]). The extension of the PACF to the multivariate stationary case is a delicate point. The difficulty to define a partial autocorrelation matrix lies in the square root choice in normalizing the p... |

5 |
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(Show Context)
Citation Context ... PACF in order to estimate, in a nonparametric way, the second order characteristics of a process. Furthermore, this approach has lead to a new estimation method for periodic autoregressive processes =-=[14]-=-, which extends the one based on the empirical PACF in the stationary case [5]. Notice that a new time-dependent power spectrum is clearly related to the PACF (see [7]). The next section is devoted to... |

4 |
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(Show Context)
Citation Context .... In the nonstationary case, the partial correlation coefficients appear in 2sthe generalization of Schur and Levinson-Durbin algorithms for non-locally deterministic processes by Lev-Ari and Kailath =-=[15]-=-. In this note, we show that the PACF, like the ACF, can be used in order to parametrize any nonstationary time series. We precise its variation domain Dβ and present a new algorithm which constructs ... |

1 | Evolutive instantaneous spectrum associated with partial autocorrelation function
- Dégerine, Lambert-Lacroix
(Show Context)
Citation Context ...riodic autoregressive processes [14], which extends the one based on the empirical PACF in the stationary case [5]. Notice that a new time-dependent power spectrum is clearly related to the PACF (see =-=[7]-=-). The next section is devoted to the parametrization by the PACF and the 3slast one relates some interesting results coming from the use of this function. 2 Partial autocorrelation function 2.1 Notat... |

1 |
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(Show Context)
Citation Context ...(v; u) and ε b (u; v), s≤ u ≤ v ≤ t. This process, which allows to obtain a Cholesky decomposition of a n.n.d. hermitian matrix, is not common but follows the one proposed by Delsarte, Genin and Kamp =-=[8]-=- in the non-locally deterministic case. Furthermore, the extension of the Levinson-Durbin Algorithm to the locally deterministic case provides an easy way to check that a given matrix Rs,t is n.n.d. a... |

1 |
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(Show Context)
Citation Context ...6] propose to use the triangular square root. Dégerine [4] gives a general approach for the multivariate stationary case and introduces the canonical PACF. Estimation techniques are proposed in [17], =-=[10]-=- and [6]. In the nonstationary case, the partial correlation coefficients appear in 2sthe generalization of Schur and Levinson-Durbin algorithms for non-locally deterministic processes by Lev-Ari and ... |

1 |
Matrix theory Vol.I
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(Show Context)
Citation Context ...stationary, we obtain the formula |Rt−s| = |Rn| = β(0) n+1 n� j=1 � 1 −|β(j)| 2� n+1−j , where Rt−s = Rs,t,n= t − s, and β(j)=β(t, t − j),j ≥ 0. Rs,t is p.d. if and only if |Rs,s+j| > 0,j =0,...,t−s (=-=[11]-=-, Theorem 19 page 337). So the |Rs,t| expression and the GLD Algorithm give immediately the one-to-one correspondence between R(·, ·) and β(·, ·) in the non-locally deterministic case and then I(Dβ) c... |

1 |
Indéterminabilité pure et inversibilité des processus autorégressifs-moyenne mobile à coefficients dépendant du temps. Cahier du CERO
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- 1977
(Show Context)
Citation Context ...rself to purely nondeterministic processes. Another problem is to characterize the set of coefficients {at(k),k = 1,...,p} t∈ZZ for which a solution of (10) exists. A sufficient condition is given in =-=[13]-=-, using the theory of linear difference equations, when at(p) �=0 and σ 2 ε(t) = 1 for all t ∈ ZZ . Then this solution is purely nondeterministic and corresponds to U(·) in our example. Notice that no... |

1 |
Covariance characterization by autocorrelation matrices
- Morf, Vieira, et al.
- 1978
(Show Context)
Citation Context ...ariate stationary case is a delicate point. The difficulty to define a partial autocorrelation matrix lies in the square root choice in normalizing the partial autocovariance. Morf, Viera and Kailath =-=[16]-=- propose to use the triangular square root. Dégerine [4] gives a general approach for the multivariate stationary case and introduces the canonical PACF. Estimation techniques are proposed in [17], [1... |