dorsal/arxiv
View SchemaOn Statistical Methods of Parameter Estimation for Deterministically Chaotic Time-Series
| Authors | V. F. Pisarenko, D. Sornette |
|---|---|
| Categories | |
| ArXiv ID | physics/0308059 |
| URL | https://arxiv.org/abs/physics/0308059 |
| DOI | 10.1103/PhysRevE.69.036122 |
| Journal | Phys. Rev. E 69, 036122 (2004) |
Abstract
We discuss the possibility of applying some standard statistical methods (the least square method, the maximum likelihood method, the method of statistical moments for estimation of parameters) to deterministically chaotic low-dimensional dynamic system (the logistic map) containing an observational noise. A ``pure'' Maximum Likelihood (ML) method is suggested to estimate the structural parameter of the logistic map along with the initial value $x_1$ considered as an additional unknown parameter. Comparisons with previously proposed techniques on simulated numerical examples give favorable results (at least, for the investigated combinations of sample size $N$ and noise level). Besides, unlike some suggested techniques, our method does not require the a priori knowledge of the noise variance. We also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed Bayesian approaches. We note the trade-off between the need of using a large number of data points in the ML analysis to decrease the bias (to guarantee consistency of the estimation) and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition. The method of statistical moments for the estimation of the parameter of the logistic map is discussed. This method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically (not only numerically).
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"abstract": "We discuss the possibility of applying some standard statistical methods (the\nleast square method, the maximum likelihood method, the method of statistical\nmoments for estimation of parameters) to deterministically chaotic\nlow-dimensional dynamic system (the logistic map) containing an observational\nnoise. A ``pure\u0027\u0027 Maximum Likelihood (ML) method is suggested to estimate the\nstructural parameter of the logistic map along with the initial value $x_1$\nconsidered as an additional unknown parameter. Comparisons with previously\nproposed techniques on simulated numerical examples give favorable results (at\nleast, for the investigated combinations of sample size $N$ and noise level).\nBesides, unlike some suggested techniques, our method does not require the a\npriori knowledge of the noise variance. We also clarify the nature of the\ninherent difficulties in the statistical analysis of deterministically chaotic\ntime series and the status of previously proposed Bayesian approaches. We note\nthe trade-off between the need of using a large number of data points in the ML\nanalysis to decrease the bias (to guarantee consistency of the estimation) and\nthe unstable nature of dynamical trajectories with exponentially fast loss of\nmemory of the initial condition. The method of statistical moments for the\nestimation of the parameter of the logistic map is discussed. This method seems\nto be the unique method whose consistency for deterministically chaotic time\nseries is proved so far theoretically (not only numerically).",
"arxiv_id": "physics/0308059",
"authors": [
"V. F. Pisarenko",
"D. Sornette"
],
"categories": [
"physics.data-an",
"physics.class-ph"
],
"doi": "10.1103/PhysRevE.69.036122",
"journal_ref": "Phys. Rev. E 69, 036122 (2004)",
"title": "On Statistical Methods of Parameter Estimation for Deterministically Chaotic Time-Series",
"url": "https://arxiv.org/abs/physics/0308059"
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